GUIDING CHILDREN'S LEARNING OF MATHEMATICS

228 Computational fluency (special issue). (2003, February).
Teaching Children Mathematics 9(6).
For Further Reading
This focus issue on computational fluency contains
Baroody, A. (2006). Why children have difficulties mas- nine featured articles on the topic.
tering the basic number combinations and how to help
them. Teaching Children Mathematics 13(1), 22–31. Russell, S. (2001). Developing computational fluency
with whole numbers. Teaching Children Mathematics
Baroody compares a number sense view of learning 7(3), 154–158.
number combinations with what he calls conventional
wisdom. He suggests that problems with learning Russell provides strategies, rather than rote proce-
combinations are based on poor instructional proce- dures, that build number understanding and computa-
dures that concentrate on memorization rather than tion fluency.
on building understanding of number operations and
relationships. Whitenack, J., Knipping, N., Novinger, S., & Underwood,
G. (2001). Second graders circumvent addition and
Behrend, J. (2001). Are rules interfering with children’s subtraction difficulties. Teaching Children Mathematics
mathematical understanding? Teaching Children Math- 8(1), 228–233.
ematics 8(1), 36–40.
Second graders confront and solve addition and
Behrend uses a case study to explore misinterpre- subtraction difficulties.
tation of rules for addition and the problems chil-
dren experience as a result of learning rules without
understanding.

CHAPTER 12

Extending
Computational
Fluency with
Larger Numbers

n the primary grades students begin work with number
and number operations encountered in the context of
realistic situations and problems. They learn to count
objects and sets; they represent numerical situations

with materials, pictures, and numerals; and they establish
basic understandings of place value and properties of number
operations. These concepts and skills are the foundation for computational
fluency, and their use continues through intermediate and middle grades.
Computational fluency refers to the use of numbers with confidence and
ease in problem-solving situations. Students with computational fluency
know when and how to calculate to solve problems with the four basic op-
erations. Fluency is more than memorizing computational rules or learning
key words; it includes estimation, number sense, and incorporating higher-
order thinking skills, such as judgments about the reasonableness of com-
puted answers. In this chapter activities and examples extend basic concepts
and skills with number operations by introducing written algorithms, includ-
ing alternative approaches for all four operations, estimation, mental compu-
tation, and use of technology.

In this chapter you will read about:
1 Computational fluency with larger numbers and development of four

approaches to computation: paper-and-pencil algorithms, estimation,
calculators, and mental computation

229

230 Part 2 Mathematical Concepts, Skills, and Problem Solving

2 Activities for addition and subtraction of whole numbers to 100 using
base-10 blocks and the hundreds chart

3 Representing addition and subtraction of larger numbers with base-
10 models and with traditional and alternative algorithms

4 Estimation strategies for addition and subtraction of larger numbers
5 Activities and models for multiplication and division of larger

numbers
6 Traditional and alternative algorithms for multiplication and division

of larger numbers
7 Estimation strategies and mental computation for multiplication and

division

Number Operations In prekindergarten through grade 2 all students should:
with Larger Numbers • Develop a sense of whole numbers and represent and use

In primary grades students begin their work with them in flexible ways, including relating, composing, and
numbers up to 100 or 200 that allow them to explore decomposing numbers;
addition, subtraction, multiplication, and division in • Connect number words and numerals to the quanti-
problem situations. They learn properties of num- ties they represent, using various physical models and
ber operations and strategies for number facts to representations;
develop accuracy and speed. In intermediate and • Understand various meanings of addition and subtraction
middle grades students extend their understanding of whole numbers and the relationship between the two
of operations, strategies, and facts when working operations;
with two-digit and three-digit numbers. Operations • Understand the effects of adding and subtracting whole
with smaller numbers can be modeled with base-10 numbers;
blocks and represented in pictures, and these mod- • Understand situations that entail multiplication and di-
els are useful for understanding that the same prop- vision, such as equal groupings of objects and sharing
erties of operations also apply to larger numbers. equally;
Solving problems through concrete, pictorial, and • Develop and use strategies for whole-number computa-
symbolic representations builds the foundation for tions, with a focus on addition and subtraction;
computational fluency with larger numbers, which • Develop fluency with basic number combinations for ad-
are more difficult to represent physically. dition and subtraction;
• Use a variety of methods and tools to compute, includ-
The NCTM standards on extending number and ing objects, mental computation, estimation, paper and
number operations are the following (National pencil, and calculators.
Council of Teachers of Mathematics, 2000): In grades 3–5 all students should:
• Understand the place-value structure of the base-10
NCTM Standards on Extending number system and be able to represent and compare
Number and Operations whole numbers and decimals;
• Recognize equivalent representations for the same num-
Instructional programs from prekindergarten through grade ber and generate them by decomposing and composing
12 should enable all students to: numbers;
• Understand numbers, ways of representing numbers, re- • Explore numbers less than 0 by extending the number line
and through familiar applications;
lationships among numbers, and number systems; • Describe classes of numbers according to characteristics
• Understand meanings of operations and how they relate such as the nature of their factors;
• Understand various meanings of multiplication and
to one another; division;
• Compute fluently and make reasonable estimates. • Understand the effects of multiplying and dividing whole
numbers;
• Identify and use relationships between operations,
such as division as the inverse of multiplication, to solve
problems;

Chapter 12 Extending Computational Fluency with Larger Numbers 231

• Understand and use properties of operations, such as the quence used for introducing and teaching children
distributivity of multiplication over addition; basic addition. Problem situations are represented
with materials and pictures and finally recorded
• Develop fluency with basic number combinations for with numerals and symbols. For example:
multiplication and division and use these combinations to
mentally compute related problems, such as 30 ϫ 50; • Yesterday Jorje had 40 baseball cards. Today he
bought 30 more. How many cards does he have
• Develop fluency in adding, subtracting, multiplying, and (see Figure 12.1a)?
dividing whole numbers;
• Angela had 72 cents in dimes and pennies. She
• Develop and use strategies to estimate the results of spent 45 cents on a pencil. How much money
whole-number computations and to judge the reason- does she have now (see Figure 12.1b)?
ableness of such results;
Students model problems such as these with base-
• Select appropriate methods and tools for computing with ball cards or coins. They can represent the op-
whole numbers from among mental computation, esti- erations with index cards, plastic coins, or base-10
mation, calculators, and paper and pencil according to blocks showing tens and ones. They can also draw
the context and nature of the computation and use the pictures and write number sentences.
selected method or tools.
Modeling operations with materials shows stu-
In grades 6–8 all students should: dents how adding (combining) or subtracting (sep-
• Develop an understanding of large numbers and recog- arating) larger numbers follows the same rules as
basic facts. While working with materials and ex-
nize and appropriately use exponential, scientific, and cal- plaining their actions, students may “invent” the tra-
culator notation; ditional algorithm or alternative strategies (Activity
• Use factors, multiples, prime factorization, and relatively 12.1). Each operation has several valid algorithms,
prime numbers to solve problems; and some are the dominant ones in other cultures
• Develop meaning for integers and represent and compare and countries.
quantities with them;
• Use the associative and commutative properties of ad- Algorithms for Addition. The traditional, or
dition and multiplication and the distributive property conventional, algorithm for addition is commonly
of multiplication over addition to simplify computations referred to as carrying, but it is more accurately
with integers, fractions, and decimals; called regrouping and renaming or trading up.
• Understand and use the inverse relationships of addition Students who have played the exchange game in
and subtraction, multiplication and division, and squaring
and finding square roots to simplify computations and
solve problems;
• Develop and use strategies to estimate the results of
rational-number computations and judge the reasonable-
ness of the results.

Addition and Subtraction Strategies
for Larger Numbers

Introducing children to addition strategies with

larger numbers follows the same instructional se-

Figure 12.1 (a) Base-10 L IBER TY L IBER TY L IBER TY UN ITE D ME RIC AS TAT ES OF A
rods illustrating 40 ϩ 30
ϭ 70. (b) Six dimes and 10¢ 10¢ 10¢ O 10¢
12 pennies (with 4 dimes
and 5 pennies circled) IN GOD 1981 NE DIME
WE TRUST
40 + 30 = 70 INGOD WETRUST
(a) IN GOD 1981
WE TRUST 1¢

IN GOD 1981 LIBERTY
WE TRUST
1981
INGOD WETRUST UN ITE D ME RIC AS TAT ES OF A

1¢L I B E R T Y O10¢ L IBER TY

1981 NE DIME 10¢

IN GOD 1981
WE TRUST

U INT RICAEDSTATES OFAMEINGOD WETRUST INGOD WETRUST 1¢

O1¢ 1¢L I B E R T Y 1¢L I B E R T Y ONE CENT

RICANE CENT 1981 1981 INGOD WETRUST

OINGOD WETRUST EDSTATES OFAME EDSTATES OFAME 1¢L I B E R T Y

RICA1¢L I B E R T Y 1¢ 1¢ 1981
1981
O NE CENT NE CENT
U INT INGOD WETRUST INGOD WETRUST
U INT
1¢L I B E R T Y
1981 1¢L I B E R T Y

1981

(b)

232 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 12.1 Thinking Strategies for Two-Digit Addition (Problem Solving,
Reasoning, and Mathematical Representation)

Level: Grades 1–3 set of 29 objects, and others use place-value materials to
Setting: Whole class show 20 ϩ 8 ϩ 20 ϩ 9 ϭ 57. Some might draw a picture,
Objective: Students add two numbers between 9 and 100. or use a different sequence such as 30 ϩ 30 ϭ 60 and
Materials: Counters, base-10 materials then subtract 3.

• Tell a story based on a classroom situation. For example, • Ask students to discuss their thinking. Compare the an-
“There are 28 children in Ms. Quay’s class and 29 in our swers to see if the strategies work.
class. If everybody can go on the field trip to the zoo, how
many children are going?” • Record the thinking that students present in number sen-
tences and/or vertical notation.
• Ask students to model the situation with objects, place-
value materials, or pictures. Some count a set of 28 and a

Chapter 10 understand the rules for trading 10 ones Liane: Then, we regroup 10 beans and exchange
for 1 ten and 10 tens for 1 hundred. For example: them for a stick.

• Genevieve had 26 colored markers in her art kit. Johanna: Write down 3 for the number of beans,
Her uncle gave her 17 markers. How many did she and put the tens stick with the other tens.
have?
Teacher: How can we record that we have an
Bean sticks show the regrouping for Genevieve’s extra ten?
markers in Figure 12.2. After putting the three bean
sticks together, combine the loose beans (6 ϩ 7), Justin: Write it in the tens column because it is 1 ten.
and then trade 10 beans for a bean stick, leaving 3
loose beans. The new bean stick combines with the Huang: Now we have 1 ten ϩ 2 tens ϩ 1 ten, or 4
existing 3 for a total of 4 bean sticks, or 4 tens. tens. Write the 4 in the tens column below the line.

Using manipulatives as a model for operations 1
allows students to record what they see, to talk 26
through the process they have completed, and then ϩ17
to write the steps: 43

Ronnie: We put the sticks together. That gives us 3 Manipulative materials are a physical model to
sticks, or 30. demonstrate the operation, but they have served
their purpose when students understand the actions
Enrique: We’ll add the beans 6 ϩ 7 for 13 beans. with numbers up to 1,000. Some students need con-
crete models longer than other students and con-
؉‫؍‬ ‫؍‬ tinue to use them until they feel confident with the
algorithm. Adding and subtracting numbers larger
(a) (b) (c) than 1,000 becomes cumbersome to model with
Figure 12.2 Bean sticks showing 26 ϩ 17 ϭ 43 base-10 materials, but students have learned that
the same exchange processes take place with any
size number.

Work with the hundreds chart also illustrates ad-
dition (and subtraction) with tens and ones. Starting
at 26, Genevieve could count forward 17 places to
reach 43. After several examples, children usually
find the shortcut for adding 10 by moving down one
row rather than counting forward 10. They may also
notice that they can go down two rows to 46 and left
3 numbers to 43. Adding 20 and subtracting 3 has
the same result as adding 17 (Figure 12.3).

Chapter 12 Extending Computational Fluency with Larger Numbers 233

1 2 3 4 5 6 7 8 9 10 pencils leads to an expanded alternative algorithm
11 12 13 14 15 16 17 18 19 20 and a low-stress alternative algorithm. Note that the
low-stress algorithm is a form of expanded notation
21 22 23 24 25 26 27 28 29 30 addition.

31 32 33 34 35 36 37 38 39 40 Traditional Expanded Low Stress

41 42 43 44 45 46 47 48 49 50 1 20 ϩ 6 26
26 10 ϩ 7 ϩ17 Think:
51 52 53 54 55 56 57 58 59 60 ϩ17 30 ϩ 13 ϭ 43
43 30 20 ϩ 10
61 62 63 64 65 66 67 68 69 70 13 6 ϩ 7
43 30 ϩ 13
71 72 73 74 75 76 77 78 79 80
The usefulness of the low-stress algorithm is more
81 82 83 84 85 86 87 88 89 90 apparent as the numbers grow in size:

91 92 93 94 95 96 97 98 99 100 • Mr. Gilbert ordered bags of birdseed. He had 358
in the storeroom and received a new shipment of
Figure 12.3 Addition on the hundreds chart 267. How many bags of birdseed did he have?

While students are learning about regrouping, Figure 12.4 shows a student model for adding the
they are also learning that regrouping is not always ones, tens, and hundreds with materials. These mod-
necessary. If they survey the problem, they can de- els can be recorded in either expanded notation or
cide whether the sums in any column are greater the low-stress form. These models provide a bridge
than 10. By presenting regrouping and nonregroup- to the traditional algorithm as students see that re-
ing problems, teachers encourage students to think grouping is represented in three different formats.
about each problem before starting. Different prob-
lems can be solved in different ways. Because both the expanded notation and the
low-stress algorithm move students toward under-
Most people learned the traditional addition al- standing the traditional algorithm, many teachers
gorithm with regrouping, but regrouping is not the introduce them first, thus the names transitional
only, or even the best, algorithm in all situations. or teaching algorithms. Base-10 blocks can also
All the operations have alternative algorithms with be used to model the steps for the expanded nota-
steps that differ from the traditional algorithm, but tion or low-stress algorithm by putting the hundreds
the results are equally valid mathematical pro- together and writing that sum, then the tens, then
cesses. Alternative algorithms, also referred to as the ones:
low-stress, transitional, or teaching algorithms, have
advantages for many students. They often preserve Mr. Gilbert’s birdseed
the meaning of the number and place value better,
are easy to model with materials, connect to esti- Low Stress Traditional
mation and mental computation processes, and re-
quire less memorization of steps than the traditional 11 Think:
algorithms. Alternative algorithms also require stu-
dents to think about the procedure and recognize 358 358 8 ϩ 7 ϭ 15
that several solution strategies are possible. Instead
of learning only one right way, they begin to think, ϩ267 Think: ϩ267 Write 5 in ones place and
“Which procedure makes more sense to me?” and
“Which procedure is most efficient and effective in 500 300 ϩ 200 625 regroup 1 ten
this situation?” 110 50 ϩ 60 10 ϩ 50 ϩ 60 ϭ 120
15 8 ϩ 7 Write 2 in tens place and
In addition, a common alternative algorithm in- 625 500 ϩ 110 ϩ 15 regroup 1 hundred
volves adding numbers in the largest place value 100 ϩ 300 ϩ 200 ϭ 600
(left digits) first and recording that partial sum. In
turn, each place value to the right is then added, Write 6 in hundreds place
and partial sums are recorded. For example, the
model and discussion for adding 26 pencils and 17 The side-by-side comparison illustrates the simplic-
ity of the expanded and low-stress algorithms. Both
algorithms develop estimation and mental compu-
tation skills. Front-end estimation uses the left-most

234 Part 2 Mathematical Concepts, Skills, and Problem Solving

digit because its place value gives it the greatest column before they determine whether regrouping
value. is needed because they have not looked to see if
regrouping is needed or because they do not under-
Many students have problems with the traditional stand what is going on at all. The low-stress algo-
algorithm. They do not know why they put “flying rithm changes the old rule from “Always begin in
1’s” above the addends. Many also have alignment the ones place” to “Keep the hundreds together, the
problems writing the regrouped numbers above the tens together, and the ones together.” In algebra this
problem. Some even begin by putting 1’s over every rule will become “Combine the x values, then the y
values.”
358
ϩ267 Compensation is another alternative approach
that encourages mental computation. Students ad-
(a) Two sets of bean sticks model the problem. just the original addends to simplify the addition.
Then they correct, or compensate, in the final an-
358 swer. Students who have played the exchange game
ϩ267 from Chapter 10 are familiar with compensation. If
they have 7 ones on the trading board and roll 5 on
5 the dice, instead of counting out five more cubes,
students learn to pick up a rod for 10 and put 5 cubes
(b) 15 single beans are exchanged for 1 tens stick, back in the bank. The result is still 12 on the board,
with 5 loose beans remaining. but students have arrived at the result through com-
pensating—adding 10 and subtracting 5. Here is an-
358 other example:
ϩ267
Teacher: Ms. Gideon started driving to Salt Lake
25 City at 8:00 in the morning. At 1:00 in the afternoon,
her trip odometer read 287 miles, and she saw a
(c) 10 tens sticks are exchanged for a hundreds raft, sign “Salt Lake City 100 miles.” How many miles will
with 2 sticks remaining. she drive on her trip? Work the problem on your
place-value mat, and tell me what you did.
358
ϩ267 Keisha: We put 2 hundreds, 8 tens, and 7 ones on
our mat. Then we added 100 on the mat. The trip
625 was 387 miles long (Figure 12.5a).

(d) The hundreds rafts are combined. Teacher: What is the distance if the sign reads 99
Figure 12.4 Traditional algorithm with bean sticks illus- miles to Salt Lake City?
trating 358 ϩ 267 ϭ 625
Lindo: It would be 386 miles because it was 1 mile
shorter.

Teacher: Start with 287 on your board. How could
you show addition of 99? Is there more than one
way?

Huang: We put 9 tens and 9 ones on the mat and
regrouped to get 386.

Walter: We put 100 on the mat and took off 1 be-
cause it is the same as adding 99 (Figure 12.5b).

When students are comfortable with the process
of adding 100 and subtracting 1, they find ways to use
compensation to adjust a cumbersome regrouping
problem into an easier problem for mental computa-
tion and eliminate regrouping in many problems.

Chapter 12 Extending Computational Fluency with Larger Numbers 235

ϩ ing down by tens and using the hundreds chart to
(a) 287 ϩ 100 ϭ 387 model subtraction:

• A length of rope 70 feet long has 40 feet cut from
it. How many feet remain?

In Figure 12.6, 70 feet of rope is represented by 7 ten
rods; 4 rods (40 feet of rope) are removed and 3 rods
(30 feet of rope) remain.

ϩ remove 4 rods
(b) 287 ϩ 99 ϭ 386
Figure 12.5 Compensation algorithm for addition Figure 12.6 Rods representing 70 feet Ϫ 40 feet

Traditional Decomposition, the traditional subtraction al-
gorithm, is modeled in the exchange game from
11 Think: Chapter 10 when students start with 100 or 1,000 and
499 9 ϩ 9 ϭ 18 remove chips on each turn. Although commonly
ϩ789 Write 8 in ones place and regroup 1 ten called borrowing, decomposition is another exam-
1,288 10 ϩ 90 ϩ 80 ϭ 180 ple of regrouping and renaming, or trading down.
Decomposition involves the same steps as addition
Write 8 in tens place and regroup 1 hundred but involves trading down rather than trading up:
100 ϩ 400 ϩ 700 ϭ 1,200
Write 1 in thousands place and 2 in • Bandar saved $50 and spent $30 for clothes. How
much does he have?
hundreds place
• The sneakers cost $47. I have $29 now. How much
Compensation more do I need?

Think: • Ani’s family was shopping for a digital camera. The
500 Ϫ 1 499 ϭ 500 Ϫ 1 list price was $573. The same camera was for sale
ϩ789 on the Internet for $385. How much can they save?
1,289 Ϫ 1 ϭ 1,288
The stories can be acted out using play money, then
Decrease the answer by 1 with base-10 blocks. Students regroup and rename
hundreds to tens and tens to ones with materials in
E XERCISE each situation, or they draw simple diagrams. Stu-
dents can represent $573 with base-10 materials, as
Assess understanding of addition by interviewing shown in Figure 12.7a. In Figure 12.7b a ten rod has
three to five students. Present each of the follow- been exchanged for 13 one rods. Figure 12.7c shows
ing problems, and ask them to “think aloud” as that 100 is exchanged to make a total of 16 tens. Af-
they work. Do they use the traditional algorithm? ter regrouping, 5 ones, 8 tens, and 3 hundreds are
Does their thinking reflect knowledge of regroup- removed (Figure 12.7d). As in addition, decomposi-
ing? Do any of them use the low-stress algorithm or tion is related to expanded notation:
compensation?

123 ؉ 100 ‫؍‬ 123 ؉ 999 ‫؍‬ Expanded
345 ؉ 300 ‫؍‬ 345 ؉ 289 ‫؍‬
1,297 ؉ 500 ‫؍‬
1,297 ؉ 511 ‫••• ؍‬
500 ϩ 70 ϩ 3 400 ϩ 160 ϩ 13 Regroup

Algorithms for Subtraction. Procedures for sub- Ϫ300 Ϫ 80 Ϫ 5 Ϫ300 Ϫ 80 Ϫ5 Subtract
traction with larger numbers are related to addition
processes. Subtraction between 20 and 100 can be 100 ϩ 80 ϩ 8 ϭ 188
modeled with manipulatives and simple pictures or
on the hundreds chart. Subtraction of tens is eas- Decomposition 4 16 13 Regroup
ily understood when children have been count- Ϫ3 8 5 Subtract
573
Ϫ3 8 5 1 88

236 Part 2 Mathematical Concepts, Skills, and Problem Solving

573 process understood and retained skill in subtracting
(a) _Ϫ__3_8_5_ two-digit numbers better than those who watched
demonstrations but did not actually exchange the
6 13 materials. For example:

(b) 573 • If 306 plants were purchased for the garden and
_Ϫ__3_8_5_ 148 were already planted, how many still needed
to be planted?
4 16 13
Expanded Think:
(c) 573
_Ϫ__3_8_5_ 306 ϭ 30 tens ϩ 6 or

4 16 13 306 29 tens ϩ 16 200 ϩ 90 ϩ 16

(d) 573 Ϫ148 Ϫ14 tens Ϫ 8 Ϫ100 Ϫ 40 Ϫ 8
_Ϫ__3_8_5_
188 15 tens ϩ 8 ϭ 158 100 ϩ 50 ϩ 8 ϭ 158

Decomposition

Figure 12.7 Base-10 blocks show subtraction with re- 29 16 Regroup 30 tens to 29 tens and 10 ones
grouping: (a) 5 flats, 7 rods, 3 units; (b) regroup 1 rod; 306 Subtract 29 tens Ϫ 14 tens, 16 ones Ϫ 8 ones
(c) regroup 1 flat; (d) subtract by removing 385. Ϫ1 4 8
158

A cooperative activity in Activity 12.2 illustrates stu- Students who are comfortable with various rep-
dents exploring decomposition. resentations of 306 know that it can be called 30
tens and 6 ones. By regrouping 1 ten as 10 ones,
Decomposition is difficult for many children if they have 29 tens and 16 ones, or 2 hundreds, 9 tens,
they memorize steps because each problem has and 16 ones. Subtraction is completed by subtract-
a unique regrouping situation. Instead, they can ing 8 from 16, 4 tens from 9 tens, and 1 hundred
look at each place value and decide if regrouping from 2 hundreds, or 29 tens Ϫ 14 tens and 16 ones
is needed. Ϫ 8 ones.

Experience with physically trading from hun- Alternative Algorithms for Subtraction. Several
dreds to tens to ones is important. Frances Thomp- alternative algorithms are possible for subtraction
son (1991) found that students who actually ma- of larger numbers. The equal-addition algorithm
nipulated materials showing the decomposition for subtraction is well known in Europe and other
parts of the world but has been used infrequently
MISCONCEPTION in the United States. The mathematical concept
behind the algorithm is simple—many subtrac-
Zero in subtraction can create several problems for students. tion combinations have the same answer. For exam-
If students find 0 in the tens place, they may skip to the hun- ple, many subtraction problems have a difference
dreds place and regroup directly to the ones place without of 28:
first making 100 into 10 tens. The other problem that students
experience with 0 in the subtraction is to switch the subtrac- 57 58 59 60 61 62 164 168 781 780
tion order from, say, 0 Ϫ 8 to 8 Ϫ 0. Modeling problems with Ϫ29 Ϫ30 Ϫ31 Ϫ32 Ϫ33 Ϫ34 Ϫ136 Ϫ140 Ϫ763 Ϫ762
place-value materials and using expanded notation helps
students understand the correct procedure for both of these 28 28 28 28 28 28 28 28 28 28
problems. In this example the student demonstrates both
mistakes in decomposition: Some number combinations are easy to calculate
(58 Ϫ 30), whereas others (e.g., 164 Ϫ 136) are per-
Incorrect Correct Expanded ceived as more difficult.
regrouping regrouping notation
Let each student list their current age and the
14 9 14 400 ϩ 90 ϩ 14 age of another person who is older. Ask what the
504 4 0 4 404 Ϫ300 Ϫ 80 Ϫ 9 difference in age is now. Then ask what the differ-
Ϫ389 Ϫ3 8 9 Ϫ3 8 9
100 ϩ 10 ϩ 5
185 115

Chapter 12 Extending Computational Fluency with Larger Numbers 237

ACTIVITY 12.2 Decomposition Algorithm (Reasoning and Representation)

Level: Grades 3–5 • Tell each team to solve the problem, using any way they
choose. Ask each team to write the mathematical sen-
Setting: Cooperative learning tence for the situation with their answer: 82 Ϫ 53 ϭ ?.

Objective: Students develop the decomposition algorithm for • Present a second, similar problem. Team members work in
subtracting two-digit numbers. pairs to solve this problem. When pairs are finished, they
compare their work and clear any discrepancies.
Materials: Base-10 materials
• Present a third problem, which each member of a team
• With the team/pair/solo structure, four team members solves. Again, team members compare their answers.
solve the first problem, pairs solve the second problem,
and individuals solve the final problem. • Call on one team member to explain the solutions to each
of the problems. Different strategies may be presented
• Present a problem situation: “Ms. Hons packed 82 boxes for each solution. Probe for decomposition as an effective
of apples to sell at a fruit stand. At the end of the day, strategy.
53 boxes remained. How many boxes were sold during
the day?”

ence will be when both of them are 3 years older, In this case students look for a convenient number
5 years older, and 10 years older. Students will recog- to add or subtract from both numbers in the prob-
nize that the number of years between the two ages lem. The 5 in 2,305 appears to be an easy number to
remains constant. Ask them to suggest a rule for subtract, but adding 5 to each number also simpli-
what they see, for example, “If the same number is fies the computation.
added to both ages, the difference is still the same.”
Students who understand how this rule works are Think: Add 5 to both numbers
ready to use equal addition as a written algorithm 7,592 Add 5: 7,592 ϩ 5 ϭ 7,597
or mental computation strategy:
Ϫ2,305 Add 5: 2,305 ϩ 5 ϭ Ϫ2,310
• Hawthorn School had 678 students. When a new 5,287
school was built, 199 Hawthorn students were
transferred to it. How many students are enrolled Or subtract 5 from both numbers
at Hawthorn now? Subtract 5: 7,592 Ϫ 5 ϭ 7,587
Subtract 5: 2,305 Ϫ 5 ϭ Ϫ2,300
Subtracting 199 from 678 is seen as a hard problem by 5,287
many children, but subtracting 200 from 679 seems
simple. Increasing both 678 and 199 by 1 makes a Equal-addition reasoning also leads to a more
hard computation into an easier one. sophisticated type of equal-addition problem. If stu-
dents put 497 on the place-value mat and add 10 to
Traditional Expanded Equal Additions it, they could add 10 ones or 1 ten. Adding 100 to
any number could be done by adding 10 tens or by
5 16 18 500 ϩ 160 ϩ 18 Think: adding 1 hundred. In the Olympic torch problem
678 678 ϩ 1 ϭ 679 Add 1 both the total miles and the current mileage are ad-
Ϫ (199 ϩ 1) ϭ Ϫ200 Add 1 justed by 10. The total miles has 10 ones added to
Ϫ1 9 9 100 Ϫ 90 Ϫ 9 the ones place; the addend has 1 ten added to the
479 Subtract tens place:

4 7 9 400 ϩ 70 ϩ 9 ϭ 479 7592 Think: Th H T O
Ϫ2305 Add 10 ones 7 5 9 12
Students who understand the logic of equal ad- Add 1 ten
ditions find it a quick mental computation strategy Subtract Ϫ2 3 1 5
for some problems. 528 7

• The Olympic torch was carried across the United After the numbers are adjusted by adding 10 to each
States on a journey of 7,592 miles. At Columbus, of them, subtraction proceeds left to right or right to
Ohio, the runners had already covered 2,305 left, because the regrouping is already completed.
miles. How many miles were left?

238 Part 2 Mathematical Concepts, Skills, and Problem Solving

Students skilled with equal additions perform sub- pare students for front-end estimation and can be a
traction rapidly. Students who went to school in an- mental computation strategy.
other country might demonstrate the process for the
class. E XERCISE

See “Subtracting with Equal Additions,” one of Solve the following problems using the traditional,
many activities on the companion website, for an compensation, equal-addition, and low-stress algo-
example of how to introduce the equal-addition rithms. Write your thinking steps, and compare with
algorithm. a partner. Which strategy works best for you in each
problem?
Compensation with subtraction is similar to com-
pensation with addition. In the Hawthorn School 342 633 517
problem of losing 199 students to a new school, stu- ؊168 ؊287
dents might notice that 199 is close to 200, allowing ؊393 •••
them to subtract 200 and then add 1 back:

678 678 Mental Computation. Mental computation is
Ϫ199 Ϫ200 Think: Subtract 200 and add 1 used with estimation and rounding and instead of
algorithms to get exact answers. People look for
478 ϩ 1 ϭ 479 number combinations and multiples of 10, 100, or
1,000 that make calculations easy. Numbers that add
After inventing or learning about alternative al- to combinations and multiples of 10 are called com-
gorithms, students are able to consider whether de- patible numbers. For example:
composition, expanded notation, equal additions,
or compensation is the efficient way to approach • The votes for Geraldo for class president were 72,
the problem. Making decisions about the faster and 15, 36, 43, and 93.
easier computational approach is part of computa-
tional fluency. Rearranging numbers in the tens column gives 70 ϩ
30 and 10 ϩ 90 plus 40 ϭ 240. In the ones column,
The low-stress algorithm for subtraction is similar 2 ϩ 3 ϩ 5 ϭ 10 and 6 ϩ 3 ϭ 9, totaling 19. The mental
to the low-stress algorithm for addition with an ad- sum is 240 ϩ 19 ϭ 259.
ditional element of negative integers. After working
with positive and negative numbers on a number Mental calculations for subtraction use tech-
line or thermometer, intermediate-grade students niques from estimation and alternative algorithms.
should be comfortable with the idea of “below zero.” Several different approaches to mental calculations
Beginning at the left, subtraction is done for each are possible.
place value, including those resulting in a negative
value: • Yusef needed $75 to buy a DVD player. He had
saved $48. How much more did he need?
• The school carnival collected $5,162. They paid
$3,578 for supplies and concessions.

Round to nearest $5 and compensate:

5,162 $75 Ϫ $50 ϭ $25 $25 ϩ $2 ϭ $27
Ϫ3,578
Think: Add 2 to both numbers and subtract:
2,000 5,000 Ϫ 3,000 $77 Ϫ $50 ϭ $27
Ϫ400 100 Ϫ 500
60 Ϫ 70 Expand and subtract:
Ϫ10 2Ϫ8
Ϫ6 2,000 Ϫ 400 is 1,600; 1,600 Ϫ 10 is 1,590; $70 Ϫ $40 ϭ $30
1,584 1,590 Ϫ 6 is 1,584
$5 Ϫ $8 ϭ Ϫ$3 $30 Ϫ $3 ϭ $27

Many students find that a low-stress procedure is Think-aloud practice with mental computation
easier than decomposition because each place and estimation at the beginning of a class period
value is calculated by itself before combining them can be done by writing four addition or subtraction
for the final answer. Low-stress algorithms also pre- examples. Have students work in pairs and compare
their answers.

Chapter 12 Extending Computational Fluency with Larger Numbers 239

E XERCISE the traditional algorithm (Figure 12.8b). A lesson us-
ing books is described in Activity 12.3.
Work the following examples using compatible
numbers. Explain your thinking to another Traditional and alternative algorithms use the
student. distributive property; however, the steps can be re-
corded in different ways
16 ؉ 23 ؉ 7 ؉ 4 ‫____ ؍‬ •••
20 ؉ 392 ؉ 680 ‫____ ؍‬ • If Johnny bought 7 bags of oranges with 13
oranges in each bag, how many oranges would
246 ؉ 397 ؉ 3 ‫____ ؍‬ he have?
476 ؉ 385 ؉ 24 ‫____ ؍‬

Expanded Alternative Traditional

Multiplying and Dividing 7 ϫ 13 ϭ 7 ϫ (10 ϩ 3) 13 2 Think:
Larger Numbers ϭ 70 ϩ 21 ϫ7 Think: 13 7 ϫ 3 ϭ 21
ϭ 91 21 7 ϫ 3 ϫ7 Write 1 in ones place
Exploring Multiplication Algorithms. Multipli- 70 7 ϫ 10 91 Regroup 2 tens
cation and division of two- and three-digit num- 91 21 ϩ 90
bers are a logical extension of earlier work with the 7 ϫ 10 ϭ 70
operations. Following development of place value, 7 ϩ 20 ϭ 90
understanding of operations, and basic facts, stu- Write 9 in tens place
dents apply commutative, associative, and distribu-
tive properties with base-10 materials to explore tra- Think: 7 ϫ 10 ϭ 7 tens
ditional and alternative algorithms: Think: 70 ϩ 20 ϭ 9 tens

• Johnny bought 3 dozen bagels for the meeting. As multiplication numbers grow in size, the simplic-
Instead of getting 12 in a dozen, he got a baker’s ity of the alternative algorithm becomes even more
dozen of 13 bagels. How many bagels did he buy? obvious:

Modeling the story with materials, as in Figure 12.8a, • The scouts sold 15 dozen cookies in an hour. How
students see that 3 groups of 13 is the same as 3 many cookies were sold?
groups of 10 and 3 groups of 3:
Figure 12.9 shows each part of the multiplication
3 ϫ 13 ϭ 3 ϫ (10 ϩ 3) process with base-10 blocks:
(3 ϫ 10) ϩ (3 ϫ 3) ϭ 30 ϩ 9
15 ϫ 12 ϭ (10 ϩ 5) ϫ (10 ϩ 2)
Expanded notation using the distributive property ϭ (10 ϫ 10) ϩ (10 ϫ 2) ϩ (5 ϫ 10) ϩ (5 ϫ 2)
reflects the place-value model and sets the stage for
Students can model 15 ϫ 12 using blocks to show
the partial products from the alternative algorithm
(Figure 12.9).

Figure 12.8 Modeling (a) 3 baker’s dozens and (b) 7 baker’s dozens (b)

(a)

3 ϫ 13 ϭ 3 ϫ 10 ϩ 3 ϫ 3 7 ϫ 10 7ϫ3
ϭ 30 ϩ 9 70 ϩ 21
ϭ 39

240 Part 2 Mathematical Concepts, Skills, and Problem Solving

Alternative Transitional Traditional

12 12 10 Think:
ϫ15 Think: ϫ15 Think: 12 5 ϫ 2 ϭ 10
100 10 ϫ 10 ϫ15 Regroup 1 ten
10 5 ϫ 2 60 5 ϫ 10 ϭ 50 and add 10
20 10 ϫ 2 50 5 ϫ 10 ϩ120 Write 60
50 5 ϫ 10 20 10 ϫ 2 180 Think: 10 ϫ 12 ϭ 120
ϩ10 5 ϫ 2 100 10 ϫ 10 Write 120
180 180 Add 60 ϩ 120

10 5 materials, children should
10 develop confidence with
MISCONCEPTION mental calculations for
2 problems such as 20 ϫ
Notation for the tradi- 30 and 40 ϫ 800. These
100 ϩ 20 ϩ 50 ϩ 10 tional algorithm creates products are needed for
problems for many alternative and traditional
Figure 12.9 Array for 12 ϫ 15 showing distributive students. Teachers may algorithms. Because these
property suggest leaving spaces or problems involve multi-
putting X’s as placehold- ples of 10, they are called
The traditional algorithm requires short-term ers during multiplication, decade multiplication. De-
memory while performing other operations as stu- but these tend to weaken cade multiplication is crit-
dents switch back and forth between multiplica- mathematical understand- ical for success with multi-
tion and addition. The alternative algorithm records ing of multiplication by 10, plication, division, mental
each step with partial products. All the multiplica- 100, or larger multiples. computation, and estima-
tion operations are completed before addition is The partial sums found in tion of larger numbers, but
performed. Writing the entire number for each par- the alternative and transi- it is sometimes taught as a
tial product also keeps the place values lined up tional algorithms preserve rule without developing
without use of blank spaces or X’s. the value of numbers. If the reason behind it.
the traditional algorithm
Decade Multiplication. As children begin to work needs regrouping twice or Teachers should intro-
with larger numbers in multiplication, an important in both the ones and the duce multiplication with
step is building their knowledge and skill with mul- tens place, many children 10 and 100 in realistic situ-
tiples of 10 and 100. Because products greater than have difficulty keeping ations and as an extension
1,000 become cumbersome to draw or to model with numbers organized and of learning number facts
separated. Others cannot so that students see pat-
remember whether to terns of multiplication by
multiply and then add or 10 and 100:
to add and then multiply
the renamed tens and
hundreds. These proce-
dural problems relate
to poor understanding
of place value.

• A dime is 10 pennies; 2 dimes is 20 cents.
• Jaime had 7 dimes: 7 ϫ 10 ϭ 70 cents.
• Diego had 20 dimes: 20 ϫ 10 ϭ 200 cents.

Students can make a scale model of the school
grounds using an orange Cuisenaire rod ϭ 100 feet.
The front fence is 7 rods long: 7 ϫ 100 ϭ 700 feet.

Chapter 12 Extending Computational Fluency with Larger Numbers 241

ACTIVITY 12.3 Distribution (Representation and Reasoning)

Level: Grades 3–5 • Separate the books so that each set shows 10 books and
2 books. Hold up the 3 groups of 2 books. Ask: “How
Setting: Small groups or whole class many books are in these 3 sets of 2 books?” Show the
multiplication of 3 ϫ 2 in the algorithm.
Objective: Students multiply a two-digit number by a one-digit
number without regrouping. • Point to the 3 groups of 10 books. Ask: “How many
books are in 3 sets of 10 books?” Show the multiplica-
Materials: 36 books (preferably identical, such as a set of reading tion of 3 times 10 in the algorithm, and add the partial
texts), linking cubes for each child products:
12
• Arrange the 36 books into sets of 12. Display the three ϫ3
sets and tell the children, “I have 3 sets of 12 books. How 6
many books are there all together?” 30
36
• Have students consider the problem and discuss ways to
determine the answer. For example: Count the books; add • Children can also use Unifix cubes to illustrate combina-
12 ϩ 12 ϩ 12; skip count by 12’s; or multiply 3 ϫ 12. tions that demonstrate partial products and their sums:
6 ϫ 11, 3 ϫ 13, 4 ϫ 12.
• Write the multiplication sentence 3 ϫ 12 ϭ 36 and then
the vertical notation:

12
ϫ3
36

• Cloris read that a bridge is 3,000 feet long: 30 rods 50 ϫ 60 ϭ 3,000
ϫ 100 feet ϭ 3,000 feet. 50 ϫ 600 ϭ 30,000
500 ϫ 60 ϭ 30,000
Other illustrations use different multiples of 10 or 500 ϫ 600 ϭ 300,000
100:
From a study of situations like these, ask students
• Thirty classes at Upward Elementary each raised to propose a decade multiplication rule. After some
$50 for hurricane relief: 30 ϫ $50 ϭ $1,500. discussion they will state something like “Multiply
the numbers at the front and put zeros on the end.”
• The school sweatshirts cost $20, and 150 students This is a good start, but it needs more work. They
ordered one: 150 ϫ $20 ϭ $3,000. clearly see the pattern related to the number of ze-
ros. The more important idea is that the number of
• On the 100th day of school, students determined zeros is the number of tens being multiplied shown
how many minutes they had been in school: 300 in expanded notation using the associative and
minutes per day ϫ 100 days ϭ 30,000 minutes. distributive laws. Students also need to be aware
of situations when the multiplication of the leading
The calculator is an excellent learning tool for numbers results in a factor of 10. Students who have
students to explore a variety of decade products. an understanding of the number of tens being mul-
Have students work the following examples using tiplied are also setting a foundation for exponential
a calculator: notation:

2 ϫ 40 ϭ 80 30 ϫ 100 ϭ (3 ϫ 10) ϫ (1 ϫ 10 ϫ 10) ϭ (3 ϫ 1) ϫ
2 ϫ 400 ϭ 800 (10 ϫ 10 ϫ 10) ϭ 3 ϫ 1,000 ϭ 3,000
20 ϫ 40 ϭ 800
20 ϫ 400 ϭ 8,000 50 ϫ 400 ϭ (5 ϫ 10) ϫ (4 ϫ 10 ϫ 10) ϭ (5 ϫ 4) ϫ
200 ϫ 40 ϭ 8,000 (10 ϫ 10 ϫ 10) ϭ 20 ϫ 1,000 ϭ 20,000
200 ϫ 400 ϭ 80,000
Division problems with multiples of 10 and 100
5 ϫ 60 ϭ 300 also use expanded notation. What is sometimes re-
5 ϫ 600 ϭ 3,000

242 Part 2 Mathematical Concepts, Skills, and Problem Solving

ferred to as “canceling zeros” is actually dividing by Activity 12.4 illustrates multiplication with re-
multiples of 10 or 100: grouping in a cooperative group lesson. This format
encourages students to explore multiplication and
300 Ϭ 100 ϭ (3 ϫ 100) Ϭ (100) or to create alternative algorithms.
(3) ϫ (100 Ϭ 100) ϭ 3 ϫ 1 ϭ 3
Students should not devote extensive time to mul-
Students can make generalizations about multi- tiplying three-digit and larger numbers. All children
plying and dividing by 10’s and 100’s: should learn one or more algorithms that they can
use efficiently and accurately for these problems.
2 ϫ 10 ϭ 20 20 Ϭ 10 ϭ 2 Students who are efficient with alternative or tran-
2 ϫ 100 ϭ 200 200 Ϭ 100 ϭ 2 sitional algorithms may never need to learn a tradi-
tional algorithm. Other students prefer the compact
9 ϫ 30 ϭ 270 270 Ϭ 30 ϭ 9 and efficient nature of the traditional algorithm and
70 ϫ 10 ϭ 700 700 Ϭ 10 ϭ 70 seem to have no problems with the notation. Some
students switch from one algorithm to another de-
Extending Algorithms for Larger Numbers. By pending on the numbers. Learning different algo-
learning decade multiplication and division, chil- rithms gives all students power and flexibility with
dren develop mental computation strategies, es- multiplication of larger numbers, although in real
tablish skills for front-end estimation, and take an life they are more likely to use a calculator or to use
important step toward multiplication and division estimation. Several other multiplication algorithms
with larger numbers before they begin working with are interesting for students to explore. The historical
algorithms. Consider the following problem: algorithms in Activity 12.5 show that mathematics
has evolved over many centuries and that many cul-
• Each box contains 24 apples. Jamie has 47 boxes tures have contributed.
to sell. How many apples are being sold?
Introducing Division Algorithms. Division with
The distributive property of multiplication over larger numbers builds on ideas and skills developed
addition is the foundation for algorithms. A major for division with smaller numbers and place value.
difference between the traditional algorithm and To use division algorithms successfully, students
an alternative algorithm is notation. The alternative should understand both partitive and measurement
algorithm records a partial product for each multi- division situations, know the multiplication and divi-
plication, starting at the left with the largest place sion facts, including decade multiplication, and be
value. A transitional algorithm also records all the able to add and subtract accurately. When working
partial products but starts with multiplication in the with early division situations, students recognize
ones place. The traditional algorithm has all the that division is related to multiplication facts.
same steps but requires that students regroup and
remember whether they are multiplying or adding
during each step.

Alternative Transitional Traditional

47 47 Think: 12∕2
ϫ24 Think: ϫ24 4ϫ7 47
800 20 ϫ 40 4 ϫ 40 ϫ24
140 20 ϫ 7 28 20 ϫ 7 188
160 4 ϫ 40 160 20 ϫ 40 ϩ940
ϩ28 4 ϫ 7 140 1128
1128 ϩ800
1128

Chapter 12 Extending Computational Fluency with Larger Numbers 243

ACTIVITY 12.4 Multiplication with Regrouping (Representation)

Level: Grades 3–5 times 10 is 70, so 28 and 70 make 98.” Teams that used
algorithms might have representations such as these:
Setting: Cooperative groups
14
Objective: Students multiply two-digit numbers by one-digit num- ϫ7
bers with regrouping. 70
28
Materials: Place-value materials of students’ choice 98
7 ϫ 14 ϭ 7 ϫ (10 ϩ 4)
• Write a sentence and algorithm on the chalkboard: 7 ϫ 10 ϭ 70
7 ϫ 14 ϭ ?. 7 ϫ 4 ϭ 28
ϭ 98
14
ϫ7 • During discussion of models and algorithms, help students
to see that the number of ones is greater than 10. If the
• Allow time for each team to discuss how to determine standard algorithm is introduced, tens from the ones place
the product. When agreement is reached, each member are regrouped and added to the tens place in the product:
solves the problem using the procedures. Have team
members consult a second time to check their work. 2
14
• Have each team explain what it did. Teams should dem- ϫ7
onstrate how materials were used. For example, one team 98
might use squared paper. Call on a student to explain. For
example, a student may say, “First, we colored squares to • Repeat with examples such as 3 ϫ 26, 6 ϫ 12, and 4 ϫ 24.
show 7 times 14. Next, we cut the paper to show 7 times
4 and 7 times 10. We know that 7 times 4 is 28 and 7

• Twenty-eight marbles are shared equally by six Figure 12.10 Marbles used to model 28 ، 6
children. How many marbles will each child have?

When students divide 28 objects into 6 groups, they
place 4 objects in each group with 4 extra (Figure
12.10). They also record the word and number sen-
tences to represent this story.

• Twenty-eight marbles divided by 6 people ϭ 4
marbles each with a remainder of 4:

28 Ϭ 6 ϭ 4, remainder 4

Each child has four marbles, but there are not
enough for each child to have five. This situation is
partitive or sharing; however, the reasoning is simi-
lar in measurement, or repeated subtraction, divi-
sion situations.

• Twenty-eight marbles are put in gift bags with six
marbles in each bag. How many bags of marbles
can be made?

244 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 12.5 History and Multiplication Algorithms

Level: Grades 3–5 • Ask students to explain how the Egyptian method works.
What property is demonstrated in this method? (Answer:
Setting: Individual or group investigation Distributive property of multiplication over addition.)

Objective: Students explore historical multiplication algorithms to • Ask students to figure out how to multiply 13 ϫ 26 using
see how they work. the Egyptian method:

Materials: None *1 26

Lattice, or Gelosia, Multiplication 2 52

Multiplication using the lattice method is done in a lat- *4 104
tice, or gelosia. The lattice method is easy to use because
no addition is done until all multiplication is completed. *8 208
The method is demonstrated with the multiplication of
23 and 68: 13 338

• Make a lattice, and write one factor across the top and Russian Peasant Multiplication
the other down the right side.
The Russian peasant process, similar to the Egyptian pro-
• Ask students to multiply 3 ϫ 8, 3 ϫ 6, 2 ϫ 8, and 2 ϫ 6 cess, is based on doubling. However, a different method
and to write the product for each pair of numbers in the determines which partial products are kept.
appropriate cell of the lattice.
• Demonstrate the Russian peasant process for 48 ϫ 28.
• Add the numbers in each diagonal, beginning at the lower Write 48 in the first column and 28 in the second column.
right. The excess of tens in the diagonal that contains 6, In the next row the 48 is divided by 2 and 28 is doubled.
2, and 8 is regrouped and added in the next diagonal. The The division-by-2 and doubling process continues until 1
product is represented by the numbers outside the lattice, is at the bottom of the left column. Then add the doubled
beginning at the top left. numbers in the second column that appear next to odd
numbers in the first row. Doubled numbers adjacent to
• Ask students to compare the lattice to the alternative or even numbers are discarded.
traditional algorithms and to tell how place value is repre-
sented on the lattice. 48 ϫ 28

• Have students work individually or in groups to figure out 24 56
how this method works with larger numbers, such as 236
ϫ 498. 12 112

6 224

68 68 68 3 448

2 1 1 2 1 1 1 2 1 896
3 2 6 2 6
(a)
1 2 3 5 1 2 3 1,344
8 4 8 4
• If an odd number is divided by 2, the remainder of 1 is
(b) 64 dropped, as in the following example, 37 ϫ 57. Again,
(c) only the doubled numbers adjacent to odd numbers are
added to determine the product. Those next to even num-
498 bers are discarded.

1 8 1 1 2 37 ϫ 57
8 6

1 1 22 3 18 114
2 74

7 2 54 6 9 228
4 48
4 456
528
2 912
Egyptian Multiplication
1 1,824
The Egyptian process of multiplication is one of doubling
and adding partial products: 2,109

• Model the problem 7 ϫ 21 by writing two columns. In • The rationale for this process is more difficult to determine
the first column place 1, and in the second column 21. In than for Egyptian multiplication. Challenge students to see
the next row, double the 1 and the 21; in the third row, if they can figure out how it works. Can you?
double both numbers again. Stop doubling when any
combination of numbers in the left column adds to 7 or Extension
higher.
1 21 Napier’s rods, another multiplication process, was
invented by John Napier in 1617. Students should be
2 42 able to find several websites that describe Napier’s rods
and how to construct them. Challenge students to find
4 84 an explanation of Napier’s rods, or Napier’s bones, and
to make a set to demonstrate this early computational
7 147 device.

Chapter 12 Extending Computational Fluency with Larger Numbers 245

The divisor is subtracted from the dividend as many dents see meaning in the steps as the algorithm is
times as possible. completed.

• How many times did we distribute the six mar- Real-world situations help students understand
bles? (Answer: Four times.) division involving larger numbers just as they did
with basic facts. Children who use the traditional
• How many marbles were unshared? (Answer: algorithm of guessing, multiplying, subtracting,
Four marbles.) and bringing down often have difficulty predicting
reasonable numbers for starting the division. They
The repeated subtractions are shown in number also have difficulty lining up numbers and knowing
sentences and in the division bracket: how many numbers to bring down. The alternative
algorithm allows students to build the final answer
28 Ϫ 6 ϭ 22 28 Ϭ 6 ϭ 4, remainder 4 by multiplying any two numbers that the child finds
22 Ϫ 6 ϭ 16 easy instead of having to get the best or closest trial
16 Ϫ 6 ϭ 10 4, remainder 4 quotient. In either case children should have skill
10 Ϫ 6 ϭ 4 6ͤ28 and confidence in mental calculations, including
Ϫ24 decade multiplication. Without ready recall of ad-
dition, subtraction, and multiplication facts, they
4 cannot focus on the meaning of division with larger
numbers. For example:
Many division problems can be solved by inspec-

tion because they draw on knowledge of multiplica-

tion facts. Using multiplication facts to solve division

problems with dividends up to 100 is developed with

“think-back” flash cards. A child is shown the front • A farmer has 288 oranges to bag for market. If she
puts the oranges into 24 bags, how many oranges
of the card with the division algorithm and thinks will be in each bag?

of the closest fact associated with that division.

The back of the

8 79 9 card shows the How many times can I subtract 24 from 288?
8 72 nearest fact. If a

(a) (b) student is shown 24ͤ288 Think:
Ϫ240 10 ϫ 24 ϭ 240
Figure 12.11 Example of a think- the card in Fig- 48 10 288 Ϫ 240 ϭ 48
back flash card: (a) Front shows ure 12.11, the stu- Ϫ48 2 2 ϫ 24 ϭ 48
division algorithm; (b) back shows dent thinks of the 0 48 Ϫ 48 ϭ 0
the associated basic fact. related division

fact “72 divided 24 can be subtracted from 288 12 times.

by 8 equals 9.” Relate each numeral in the algorithm to the prob-
lem situation so that students understand each one’s
When a student cannot think back to the correct meaning. When talking through the problem, model
the actions by putting 10 oranges in each bag in the
basic fact, show the fact on the back of the card. Fre- algorithm, then place 2 more in each bag. The lad-
der algorithm uses the same thinking questions as
quent group and individual work with these cards the teacher and student record their work.

helps children develop skill in naming quotients. Students usually need several more examples
with guided practice or working with a partner be-
As the numbers increase in size, students have fore they gain confidence in the ladder method. The
alternative method gives students flexibility when
difficulty predicting appropriate trial quotients they cannot identify a trial quotient. They use any
reasonable quotient that is easy to multiply and sub-
through inspection. An alternative algorithm, called tract. Different students may find quotients by using
facts that they consider easier. The ladder algorithm
the ladder method, allows students to see each sub- can be simplified once the process is understood.

traction and gives them great flexibility in choosing

their trial quotients. Division is a more efficient way

to record the process than many subtractions.

Activity 12.6 shows how children use base-10

blocks to model division and relate the materials

to recording the algorithm. Because the parts of the

algorithm can be directly related to actions with

bean sticks or other manipulative materials, stu-

246 Part 2 Mathematical Concepts, Skills, and Problem Solving

Alternative Traditional

2 Think: 12 Think:
ϫ10 10 ϫ 24 ϭ 240 24ͤ288 1 ϫ 24 ϭ 24
24ͤ288 288 Ϫ 240 ϭ 48 28 Ϫ 24 ϭ 4
Ϫ240 2 ϫ 24 ϭ 48 Ϫ24 Bring down 8
48 Ϫ 48 ϭ 0 48 2 ϫ 24 ϭ 48
48 48 Ϫ 48 ϭ 0
Ϫ48 Ϫ48
0
0

Only the multiplier needs to be written down, either gallon). The fuel efficiency of the car was almost 26
on the side or above the bracket: miles per gallon.

• Sam drove 3,174 miles on 123 gallons of gas. What Enough work is needed so that students under-
was his mileage (miles per gallon)? stand the meaning and the thinking behind the al-
gorithm. However, extensive exercises with division
Despite using different number combinations, both of larger numbers are no longer seen as useful. In-
stead, more time should be spent on problem-solv-
students calculated the answer correctly. The re- ing situations with multiplication and division in

mainder 99 is interpreted as 99 (or about 0.8 of a
123

ACTIVITY 12.6 Division with Regrouping (Representation,
Problem Solving, and Reasoning)

Level: Grades 3–5
Setting: Pairs or small groups
Objective: Students use place-value materials to demonstrate divi-

sion with regrouping.
Materials: Place-value materials, such as bean sticks, Cuise-

naire rods, or base-10 materials, or bundled and loose tongue
depressors

• Present a situation: “Gloria put 45 apples into three
bags—the same number in each bag. How many apples
did she put in each bag?”

• Have students work with place-value materials to solve
the problem. Bean sticks illustrate the process here.

• Ask: “Could you separate the 4 tens into 3 equal-size
groups?” Allow time for students to determine that they
cannot do this.

• Ask: “What can you divide into 3 groups?” (Answer:
Divide 3 tens into 3 groups.)

• Ask: “What did you do next?” (Answer: Exchanged 1 ten
for 10 ones and divided 15 into 3 groups.)

• Ask: “How many apples are in each bag?” (Answer: 15.)

• Have pairs of students repeat with examples such as
56 Ϭ 4, 52 Ϭ 2, and 78 Ϭ 6. Observe and assist as
needed.

Chapter 12 Extending Computational Fluency with Larger Numbers 247

Student 1 Student 2 In some instances
exact calculations are
5 10 2 20 needed, but often estima-
10 10 3 3 tion, mental calculations,
10 5 20 2 and calculators can be
123ͤ3174 123ͤ3174 used instead of paper-
Ϫ1230 Ϫ2460 and-pencil procedures.
1944 714 Students who possess
Ϫ1230 Ϫ369 computational fluency
714 345 recognize which compu-
Ϫ615 Ϫ246 tational approach is best in different situations, de-
99 99 pending on the accuracy needed.
Problems with two- and three-digit numbers
which students use estimation, mental calculation, demonstrate the need for estimation to attain a fast
and/or calculators. Mental computation with multi- approximation. Estimation is most valuable for num-
plication and division is less frequent than with ad- bers in the thousands and larger or when several
dition or subtraction. Estimation and calculator use addends are considered. Procedures for front-end
with larger multiplication and division problems are estimation and rounding estimation are similar to
better strategies. alternative algorithms that start in the largest place
value for the first approximation. If more precision is
E XERCISE needed, students can adjust the answer to improve
the estimate by including the next place value. For
How many long-division problems do you think stu- example:
dents need to complete to show their understand-
ing of the algorithm? Look at a fifth- or sixth-grade • Students were tracking the energy usage of their
textbook to see how much practice is given. Is it school. Over four months the electrical usage was
2,345 kWh, 6,526 kWh, 3,445 kWh, and 3,152 kWh.
adequate? too little? too much? •••

Number Sense, Estimation, Intermediate-grade students make a front-end
and Reasonableness estimate by adding the thousands and adjust the es-
timate by adding the sum of the hundreds:
While students learn computational algorithms,
they are also developing number sense skills, such Front-End Adjusted Rounded
as estimation, rounding, and reasonableness. These
skills are particularly important with larger num- Add Add Round to
bers. For example: hundreds
thousands hundreds
• If one bank has deposits of $45,173, 893, and the 2,300
other bank holds $37,093,103, how much will be 2,345 2,000 300 6,500
deposited if they merge? 6,526 3,400
3,445 6,000 500 3,200
• The population of the largest city is 10,987,463, ϩ3,152 15,400
and the population of the second largest city is 3,000 400
6,423,932? How many more people live in the
larger city? ϩ3,000 ϩ100

• The hybrid car gets 37 miles per gallon. If the gas 14,000 ϩ1,300 ϭ 15,300
tank holds 16 gallons of gas, how many miles can
it go on a full tank? The front-end estimate is 14,000, and the adjusted
estimate is 15,300. Values in the tens and ones places
• The seven-day cruise costs $539. What is the cost are ignored because they add little to the sum. If stu-
per day for the vacation? dents round the addends to the hundreds place, the
estimate is 15,400. Any of these estimates serve as
reasonable comparisons for an answer using a cal-
culator. If the calculator answer is close to 19,000 or
1,500, the estimates would be clues that something
is wrong, When rounding, students determine how

248 Part 2 Mathematical Concepts, Skills, and Problem Solving

much accuracy is needed and round each number Flexible thinking is needed as students compare
to that place. Addends for the previous example and decide which result is best for different estima-
could be rounded to the nearest 1,000, 500, or 100. tion procedures—front-end, adjusted, and round-
ing. Continuing the pecan tree example, a student
When teaching estimation strategies, teachers might note that the front-end estimate of 8,000 is
model the process by thinking aloud. Students then low and that rounding both factors up gives 11,700,
think aloud in pairs or small groups to verbalize es- which will be high. The other three estimates are in
timation and mental computation processes. Com- between, so the student thinks that the best estimate
paring front-end estimation to adjusted and rounded is probably about 10,500.
estimation helps students determine whether one
method is better than the other for the purpose of A sample problem on estimation from the Texas
the problem. Estimation in subtraction also uses Assessment is shown in Figure 12.12. Children
front-end estimation and rounding techniques: choose the best estimate based on reasonable low
and high estimates.
• Lando wants to buy a car that costs $7,358. He
has a down payment of $2,679. How much money 26. Mr. Benjamin jogs for 33 minutes to 38 minutes every
will he borrow to complete the purchase? day. Which could be the total number of minutes that
Mr. Benjamin jogs in 4 days?
Front-End Adjusted Rounded
a. Less than 120 min
7,000 7,000 ϩ 300 Round to nearest
Ϫ2,000 Ϫ2,000 Ϫ 600 1,000 or 500 b. Between 120 min and 180 min

5,000 5,000 Ϫ 300 7,000 7,500 c. Between 180 min and 240 min
Ϫ3,000 Ϫ2,500
d. Between 240 min and 300 min
4,000 5,000
e. More than 300 min
The loan officer at the bank calculates the ex-
act amount being borrowed and adds taxes and Figure 12.12 Sample problem on estimation from the
registration fees using his computer. An estimate Texas Assessment for fifth grade
gives a sense of what is involved before starting
paperwork. Another multiplication concept, called factorials,
is shown in Activity 12.7, which is based on Anno’s
Both front-end estimation and rounding are ef- Magical Multiplication Jar. Children should see the
ficient ways to estimate products. Students can use magnitude of repeated multiplications.
either process, depending on the size of the number
and the precision. Front-end estimation might adjust E XERCISE
the answer by using the first two largest places. De-
cade multiplication is critical in making estimation Estimate each of the following products with front-
quick and easy. For example: end and rounding techniques. Which of these prod-
ucts can you calculate mentally?

• Mr. Johnson planted 125 pecan trees. After they 27 42 94 87 348
grow, he expects each tree to produce about 85
pounds of pecans. ؋8 ؋7 ؋16 ؋23 ؋75 •••

Front-End Adjusted Rounded Estimating quotients with larger numbers uses
the logic learned with the alternative division algo-
100 120 Up Down Up and rithm or by using multiplication. Decade multiplica-
ϫ80 ϫ80 130 120 down tion is an essential skill in estimation of division. For
8,000 8,000 ϫ90 ϫ90 130 example:
ϩ1,600 9,000 ϫ80
10,600 ϩ2,700 9,000 8,000 • The school PTA raised $7,348 at its carnival. They
11,700 ϩ1,800 decided to buy new printers for classrooms. If
10,800 ϩ2,400 each printer costs $324, how many printers can
10,400 they buy?

Chapter 12 Extending Computational Fluency with Larger Numbers 249

ACTIVITY 12.7 Factorials (Reasoning)

Level: Grades 3–5 • Give each student a large sheet of paper and crayons.
Reread the story, allowing time for sketches to be drawn
Setting: Small groups or whole class of the evolving scene.

Objective: Students model factorials, a sequence of multiplications • Have students study their sketches to see if they can de-
1ϫ2ϫ3ϫ4ϫ5.... termine the pattern that is developing. Have them discuss
their ideas about the pattern. Ask them if they can write
Materials: A copy of the book Anno’s Mysterious Multiplying Jar, a multiplication sentence to show the situations in the
by Masaichiro Anno and Mitsumasa Anno (New York: Philomel pictures.
Books, 1983).
• Tell students that the authors have represented the
Anno’s Mysterious Multiplying Jar “is about one jar and island’s growing number of objects. Turn to the page
what was inside it.” With this simple statement, Mit- where the display of dots begins. Ask: “Why didn’t the
sumasa Anno and son Masaichiro Anno, a writer-artist authors show dots for the boxes and for the jars in the
team, begin their tale about a fascinating jar and its boxes? Students may use calculators to determine the
contents, introducing the topic of factorials to intermedi- products of larger numbers.
ate- and middle-grade students. Inside the jar was a sea
of rippling water on which a ship appeared. The ship • Introduce the term and numerical representation of facto-
sailed to a single island. The island had two countries, rial. Children should come away from the activity with an
each of which had three mountains with four walled understanding that the factorial symbol—10!—is equal to
kingdoms on each mountain, and each kingdom had 10 ϫ 9 ϫ 8 ϫ 7 ϫ 6 ϫ 5 ϫ 4 ϫ 3 ϫ 2 ϫ 1, or 3,628,800,
five villages. Eventually students are asked to answer a very short way to write a very large number!
the question, “How many jars were in the boxes in the
houses in the villages in the kingdoms, on the mountains, • Students can create their own factorial stories and illus-
in the countries?” trate them. Their stories, pictures, and computations can
be displayed on a factorial bulletin board.
• Read the story, pausing to allow time for students to talk
about how many countries, mountains, and kingdoms
there are. Continue reading to the point where there are
10 jars in each box, and ask, “How many jars were in all
the boxes together?”

The teacher asks students whether the PTA could front-end estimation and rounding processes al-
buy 10, 20, 30, 50, or 100 printers. The students rea- low students to find a similar but simpler division
son that 10 printers would cost about $3,200; that 20 problem.
printers would cost twice as much, or about $6,400;
and that 30 printers would cost about $9,600. They E XERCISE
decide that the PTA could buy between 20 and 30
printers. Another estimation adjusts the numbers by What estimated quotients would be reasonable for
rounding the dividend and divisor to numbers that each of the following divisions? Tell a story that
are easy to divide. Different students find different might go with the numbers in the following prob-
number combinations for estimation: lems. Try both front-end estimation and rounding
methods or a combination to find numbers that
Ian: 7,500 divided by 500 is 15, and 7,500 divided by are easier to compute mentally. Compare the
250 is 30; the answer is halfway between 15 and 30 numbers you used with someone else’s estimation
or about 22. process.

Heather: 7,000 divided by 350 is 20, so we should 368 ، 442 is about? 1,268 ، 887 is about?
be able to buy a few more than 20 printers.
5,383 ، 677 is about? 15,383 ، 913 is about? •••
Sara: 7,200 divided by 600 is 12 and divided by 300
is 24. I think the answer is close to 25.

Students compare their answers and thinking to
see whether their estimates were reasonable. Both

250 Part 2 Mathematical Concepts, Skills, and Problem Solving

Take-Home Activities

Repeated Subtraction with a Calculator

Most calculators allow you to subtract repeatedly by setting up a problem such as this one:

597 Ϫ 19 ϭ ϭ ϭ ϭ

How many times do you think that you will have to touch the equal key to get the remainder to less than
19? How long do you think it will take you to reach that number?

Estimate the number of times you will subtract the divisor for the following problems:

238 divided by 41 238 Ϫ 41 ϭ ϭ Estimate Actual Number
571 divided by 16 571 Ϫ 16 ϭ ϭ _____ _____
1,200 divided by 63 1,200 Ϫ 63 ϭ ϭ _____ _____
4,594 divided by 425 4,594 Ϫ 425 ϭ ϭ _____ _____
9,007 divided by 113 9,007 Ϫ 113 ϭ ϭ _____ _____
15,073 divided by 743 15,073 Ϫ 743 ϭ ϭ _____ _____
_____ _____

Create problems for yourself and a friend to estimate and repeatedly subtract.

_____ divided by _____ _____ Ϫ _____ ϭ ϭ _____ _____
_____ divided by _____ _____ Ϫ _____ ϭ ϭ _____ _____
_____ divided by _____ _____ Ϫ _____ ϭ ϭ _____ _____

Tiling with Coins

Tiling your desk with quarters? You might use some of the existing artwork showing quarters covering a
surface.

25¢ How many quarters would it take to cover the
25¢ 25¢ top of a table or large book at your house? How
252¢255¢2¢522¢5252¢52¢5¢52¢22¢5525¢2¢52¢52¢52¢5¢5¢222¢55252¢¢52¢52¢52¢5¢5¢222¢55252¢¢52¢52¢52¢5¢5¢222¢55252¢¢52¢52¢52¢5¢52¢22¢5525¢2¢52¢52¢52¢5¢52¢22¢5525¢2¢52¢52¢52¢5¢252¢252¢55¢5¢222¢¢5255¢5¢¢22¢252525¢5¢252¢¢52¢5¢52¢22¢5525¢2¢52¢52¢52¢5¢5¢222¢55252¢¢52¢252¢552¢5¢¢5¢222¢55252¢¢522¢52¢552¢5¢¢5¢222¢55252¢2¢52¢552¢52¢¢5¢52¢22¢5525¢2¢52¢52¢52¢5¢52¢22¢5525¢2¢52¢52¢52¢5¢52¢2¢55¢22¢55¢2¢2525¢5¢¢ many quarters would you need to estimate the
number to cover the table? What could you use
Object being covered? _____ instead of real quarters?

What would be the value of the quarters
needed to cover the table?

How many dimes would it take to cover the
table? What would all the dimes be worth? How
many nickels would be needed to cover the
table? What would the nickels be worth?

Number of quarters? _____

Value of quarters? _____

Number of dimes? _____

Value of dimes? _____

Number of nickels? _____

Value of nickels? _____

If you could keep the coins needed to cover the floor of a room, would you rather cover it with quarters,
dimes, or nickels? Explain your reasons.

251

Summary to go with each of them. Work each example using

Traditional algorithms can be taught; however, many stu- both the traditional algorithm and one of the alter-
dents have difficulty with the traditional algorithms. The
algorithms are often confusing because of placement of native algorithms.
answers and because they involve switching from one
operation to another several times. The larger the num- a. 24 b. 64 c. 23 d. 536 Ϭ 24
bers become, the more complex the algorithm becomes
and the more it places a memory burden on many stu- ϩ 48 Ϫ26 ϫ18
dents who are trying to remember the steps.
5. Estimate the answers to the following examples.
Each of the operations has one or more alternative al-
gorithms that help students to develop their skill in com- What process did you use for estimation? Was your
puting with larger numbers. Many algorithms are based
on expanded notation, which emphasizes place value of strategy rounding or front-end estimation? Compare
numbers. Some algorithms allow students to adjust the
numbers in a given problem to make the computation your strategy with other students.
easier. Many alternative algorithms simplify computa-
tion by allowing students to write down partial answers a. 269 b. 3,879 c. 711 d. 1,826 Ϭ 35
as they work. In general, the alternative algorithms for
each of the operations build on understanding of the op- 924 Ϫ2,091 ϫ138
erations, preserve the meaning of numbers as students
work, are easy to model with base-10 materials, and re- 472
quire less memory of steps while students are working.
Many students are successful in computing with alterna- ϩ826
tive forms; others use the traditional algorithm readily.
6. Ask several fifth- or sixth-graders to estimate the an-
As students model problems and record answers, swers in Question 5 and to think aloud as they work.
they build skills with estimation, rounding, and men- Can you draw any conclusions about their skill with
tal computation. Several alternative algorithms parallel estimation strategies?
front-end estimation strategies because they are based Use a calculator or a computer spreadsheet to
on the numbers in the largest place value. Students can find the answers to the following problems.
compare the results of different estimation strategies to
decide what are reasonable answers to problems with 7. What is the total population of the 10 largest cities in
large numbers. This skill is also valuable when using the country (or your state, province)?
calculators and computers. Students should check cal-
culator answers against estimated answers to determine 8. Ms. Powell has donated a total of $348 to a library
whether their estimates were reasonable. Computational during the past 4 years. She has donated the same
fluency and flexibility means that teachers spend less amount of money each year. How much money has
time practicing paper-and-pencil algorithms and more Ms. Powell donated to the library in each of the past
time solving problems with larger numbers using a vari- 4 years? (Taken from the Fourth Grade Texas Assess-
ety of computational approaches. ment of Knowledge and Skills Release Test, 2006.)

Study Activities and Questions a. $82

1. Recall your own learning of algorithms for larger b. $87
numbers. Were they hard or easy for you to re- c. $352
member and perform? Did you learn or develop for d. $344
yourself any of the alternative algorithms? 9. Ted collected 22 pounds of aluminum cans. How
many ounces of aluminum cans did he collect?
2. Look at a current elementary textbook or teacher’s (Taken from the Sixth Grade Texas Assessment of
guide. Do they include alternative algorithms for Knowledge and Skills Release Test, 2006.)
students? If so, how do the materials present the a. 6 oz.
algorithms for children? b. 38 oz.
c. 352 oz.
3. Think of five or six ways that you computed answers d. 220 oz.
in the last week. Did you use the calculator? Did
you use estimation? Did you use a paper-and-pencil Praxis (http://www.ets.org/praxis/) The average num-
algorithm to get the answer? What led you to use ber of passengers who use a certain airport each
different approaches? year is 350,000. A newspaper reported the number
as 350 million. The number reported in the newspa-
4. Use place-value devices such as base-10 materials to per was how many times the actual number?
demonstrate the following examples. Write a story
a. 10

b. 100

c. 1,000

d. 10,000

NAEP (http://nces.ed.gov/nationsreportcard/) Amber
and Charlotte each ran a mile. It took Amber 11.79
minutes. It took Charlotte 9.08 minutes. Which num-
ber sentence can Charlotte use to best estimate the
difference in their times?

252

a. 11 Ϫ 9 ϭ c. 12 Ϫ 9 ϭ http://www.shodor.org/interactivate/activities/index.html
(see Clock Arithmetic)
b. 11 Ϫ 10 ϭ d. 12 Ϫ 10 ϭ
For websites to practice math facts, go to:
TIMSS (http://nces.ed.gov/timss/) A runner ran 3000
meters in exactly 8 minutes. What was his average Math Flash Cards: http://www.aplusmath.com/Flashcards
speed in meters per second?
Interactive factor trees: http://matti.usu.edu/nlvm/nav/
a. 3.75 d. 37.5 category_g_3_t_1.html

b. 6.25 e. 62.5 Interactive Flash Cards: http://home.indy.rr.com/lrobinson/
mathfacts/mathfacts.html
c. 16.0
Mathflyer (a space ship game that uses multiplication facts):
Technology Resources http://www.gdbdp.com/multiflyer/

There are many commercial software programs de- Math Facts Drill: http://www.honorpoint.com/
signed to help students with their number sense, recall
of number facts, and applications of number opera- Mathfact Cafe: http://www.mathfactcafe.com
tions. We list several of them here:
For Further Reading
How the West Was One ϩ Three ϫ Four (Sunburst)
Baek, J. (2006). Children’s mathematical understand-
Math Arena (Sunburst) ing and invented strategies for multidigit multiplication.
Teaching Children Mathematics 12(5), 242–247.
Math Munchers Deluxe (MEEC)
Classroom research shows teachers how children
Oregon Trail (Broderbund) think about multidigit multiplication, revealing miscon-
ceptions as well as understanding.
The Cruncher 2.0 (Knowledge Adventure)
Bass, H. (2003). Computational fluency, algorithms, and
Internet Game mathematical proficiency: One mathematician’s per-
spective. Teaching Children Mathematics 9(6), 322–327.
At http://www.fi.uu.nl/rekenweb/en, students may play a
variety of challenging mathematics games ranging from The purpose and value of alternative algorithms
number fact recall to spatial sense. In Broken Calculator is advocated for development of understanding how
students try to reach a given number by using the avail- numbers work.
able keys on a calculator. In this game not all the keys
are available. For example, in one game the ϩ and ϫ Computational Literary Theme Issue. (February 2003).
keys are missing, as are the 5, 7, and 9 keys. The task is Teaching Children Mathematics 9(6).
to reach 80 beginning with a value of 150.
This themed issue of Teaching Children Mathemat-
Find more games at http://www.bbc.co.uk/education/maths ics contains several articles that describe computa-
file/, http://www.bbc.co.uk/schools//numbertime/games// tional fluency and gives examples of strategies to build
index.shtml, and http://www.subtangent.com/index.php. operational fluency for students across the elementary
Internet Activity grades.

This activity is for students in grades 3–6. Students work Ebdon, S., Coakley, M., & Legrand, D. (2003). Mathemat-
in small groups to solve number puzzles on the Internet. ical mind journeys: Awakening minds to computational
The only material they need is a computer with Internet fluency. Teaching Children Mathematics 9(8), 486–493.
access. Have students go to http://nlvm.usu.edu/en/nav/
vlibrary.html and follow the links to the activity Circle 21. Teachers encourage flexible thinking about num-
This activity asks students to arrange numbers in each of bers and operations, and students consider alternative
the regions formed by overlapping circles so that each ways to represent solutions on mind journeys. Thinking
entire circle has a sum of 21. Have students solve three aloud is used in the classroom discussion.
of the puzzles and turn in the completed puzzles. Once
they have solved three puzzles, challenge each group to Fuson, K. (2003). Toward computational fluency in
create three original puzzles to use with the class. multidigit multiplication and division. Teaching Children
Internet Sites Mathematics 9 (6), 300–306.

For Internet sites that allow students to explore and Computational algorithms serve two purposes: com-
work with integers, go to the following websites: putation skill and understanding how operations work.

http://nlvm.usu.edu/en/nav/vlibrary.html (see Circle 21, Circle Whitenack, J., Knipping, N., Novinger, S., & Underwood,
3, Circle 99, Color Chips, and Rectangular Multiplication of G. (2001). Second graders circumvent addition and
Integers) subtraction difficulties. Teaching Children Mathematics
7(5), 228–233.
http://lluminations.nctm.org (see Voltage Meter)
Second-graders develop meaning for tens and ones
For explorations with modular systems go to the follow- in subtraction situations through stories, models, and
ing website: pictures.

CHAPTER 13

Developing
Understanding
of Common and
Decimal Fractions

hildren’s study of fractional numbers begins as early as
kindergarten and continues through middle school.
Fractional numbers and concepts related to them—in
the form of common fractions, decimal fractions, and
percentages—are encountered in everyday settings by

children and adults. Common fractions are used to express
parts of wholes and sets, to express ratios, and to indicate division. “One-
half of a pie” refers to part of a whole; “one-half of 12 apples” refers to part
of a set. If one apple is served for every two children, a ratio of 1:2 exists
between the number of apples and the number of children; this ratio is
also expressed as 12. Division, such as “2 divided by 4,” can be written as
the common fraction 42. Common fractions are an integral part of the English,
or common, system of measure, as indicated by quarter-inches, half-pounds,
and thirds of cups. Decimal fractions are used to express parts of wholes and
sets divided into tenths, hundredths, thousandths, and other fractional parts
of tenths. They are used to express money in our monetary system ($1.23
and $0.07) and represent various measurements in the metric system
(0.1 decimeter or 0.01 meter).

As with other numbers, children’s work with fractional numbers be-
gins with real-world examples, and representations of fractional numbers
are modeled with real materials and manipulatives. In all grades concrete
representations help children develop a clear understanding of these num-

253

254 Part 2 Mathematical Concepts, Skills, and Problem Solving

bers, their uses, and the mathematical operations associated with them. In
this chapter we focus on activities for developing foundational concepts of
fractional numbers expressed as common and decimal fractions and related
concepts. Activities featured in this chapter focus on concepts and skills
involving operations with fractional numbers. Although percent is commonly
linked with the study of fractions, we have chosen to consider percent in a
different chapter. Chapter 15 includes a full discussion of percent and ratio
and proportion.

The NCTM standards for number and operations suggest the following
standard for fractions at this level.

NCTMConnection

Understand numbers, ways of representing numbers, relationships among numbers,
and number systems
Pre-K–2 Expectations
In prekindergarten through grade 2 all students should
• understand and represent commonly used fractions, such as 14, 13, and 12.

In this chapter you will read about:
1 Uses of common and decimal fractions from everyday life commonly

studied in grades K–3
2 Real and classroom learning aids for representing parts of wholes

and sets to help children develop their understanding of common
and decimal fractions
3 Activities emphasizing equivalent common fractions and comparison
of unlike fractions
4 Materials and procedures for helping children learn how to round
decimal fractions
5 Ways to extend the concept of place value to include decimal
fractions
6 A take-home activity dealing with common fractions

Teaching children about common and decimal frac- children build knowledge of fractions also broad-
tions extends their understanding of number con- ens their awareness of the power of numbers and
cepts beyond knowledge about whole numbers. extends their knowledge of number systems. The
Knowledge of fractional numbers allows children concepts of common and decimal fractions devel-
to represent many aspects of their environment that oped in elementary school lay the foundations on
would be unexplainable with only whole numbers, which more advanced understandings and applica-
and it allows them to deal with problems involving tions are built in later grades.
measurement, probability, and statistics. Helping

Chapter 13 Developing Understanding of Common and Decimal Fractions 255

What Teachers Need to Know Table 13.1 suggests when various topics for com-
About Teaching Common
Fractions and Decimal Fractions mon and decimal fractions might be beneficially

The role of manipulatives in learning about both introduced to children. Notice that introductory
common fractions and decimal fractions is ex-
tremely important. At times, teachers are tempted to activities may stretch across several years in order
hurry students beyond manipulatives and concrete
models of fractional numbers into abstract compu- to build a foundational understanding. Introductory
tations. When students are moved along too quickly
from concrete models to abstract computations, activities are then followed by activities that main-
they never fully develop basic understandings of the
fractional concepts and relationships. Consequently, tain and/or extend understanding.
they advance their understanding of fractional num-
bers by rote memorization and not from any con- For example, the fact that common fractions
ceptual understanding. Knowledge about fractional
numbers gained in this way is fragile. Teachers need (21) and decimal fractions (0.5) represent the same
to allow children sufficient time to engage with number is not understood by many students. One
various models that represent fractions before they
move on to symbolic aspects of fractional numbers. reason the connection between the two types of
When students develop concepts about, and pro-
cesses with, fractional numbers slowly and carefully numerals may seem obscure is that instruction of
through activities with concrete materials and real-
istic settings, they avoid misconceptions that must common and decimal fractions is often completely
be corrected later. Children also construct mean-
ing of fractional numbers by interacting with peers separated. When students work with common frac-
and adults. During this process, their understanding
of fractions may not be identical to their teacher’s. tions at one time and decimal fractions at a different
Overemphasis of the teacher’s way of viewing these
new numbers may inhibit students’ progress in un- time, connections are often unclear or not made at
derstanding them. It is important, especially during
early work with fractional numbers, to allow stu- all. Another reason is that most people use the term
dents time to explore the meaning of these numbers
and to build their conceptual understanding. fraction when they refer to common fractions and

the term decimal when they refer to decimal frac-

tions. Using the terms common fraction and decimal

fraction helps students understand that both types

of numerals are used to represent fractional num-

bers. Both numerals refer to parts of units or sets;

the difference is that a common fraction represents

units or sets separated into any number of parts

(3 out of 5 or 3 ), whereas a decimal fraction repre-
5

sents units or sets separated into 10 parts or parts

that are powers of 10 (0.6 or 160; 0.03 or 1300). When
percent is used, the unit or set is separated into 100

parts. When the terms common fraction and decimal

The modern English term fraction was first used by Geoffrey
Chaucer (1300–1342), author of The Canterbury Tales. It has
the meaning “broken number” in Middle English.

TABLE 13.1 • Sequence for Fraction Topics in School

Topic K 12 3 4 56
M M MM
Basic understanding I I M I I MM
I I MM
Equivalent fractions I I I MM
I I IM
Improper fractions I

Mixed numbers

Ordering fractional numbers

I, topic introduced. M, topic maintained.

256 Part 2 Mathematical Concepts, Skills, and Problem Solving

fraction are used throughout the school years, chil- indicates the number of parts being considered at a

dren learn that both are representations of fractional particular time and is called the numerator (Figure
13.1a). For the common fraction 83, the whole has 8
numbers. equal-size parts, and 3 of the 8 parts are indicated

Mathematically, frac- (Figure 13.1b). Division of a unit into its parts is also

tional numbers are part of referred to as an area or geometric model.

the set of rational numbers

that can be expressed in

the form ab, where a is any 13
whole number and b is any 28

nonzero whole number.

Symbolically, fractional

numbers are expressed as (a) (b)
Figure 13.1 Fractions represent parts of a whole: (a) one-
common fractions ( 1 and half of a cake is represented by the common fraction 21;
2 (b) three-eighths of a pizza is represented by 83.
2
3 ), as decimal fractions

(0.5 and 0.6666. . .), and as

percents (50% and 66 2 %).
3

Five situations that give Set Partitioned into Equal-Size Groups

rise to common fractions When a collection of objects is partitioned into

are discussed in the fol- groups of equal size, the setting is clearly one that

lowing sections. involves division. When 12 objects are divided into

two equal-size groups, the mathematical sentence

Five Situations Represented 12 Ϭ 2 ϭ 6 describes the setting. The child thinks,
by Common Fractions
“How many cookies will each person get when a set

of 12 is divided equally between two people?” The

Unit Partitioned into Equal-Size Parts whole number 6 represents the amount of one of

Objects such as cakes, pies, and pizzas are frequently the two parts. A different interpretation of the same

cut into equal-size parts. When a cake is cut into four setting is to find 1 of a set of 12 objects, or 1 of 12 ϭ 6.
2 2
1
equal-size parts, each part is one-fourth of the entire The child thinks, “What is 2 of a set of 12?” Now the

cake; the common fraction 1 represents the size of whole number 6 refers to 1 of the set (Figure 13.2a).
4 2

each piece. Many measurements with the English

system of measurement require common fractions.

When you need a more precise measurement than

is possible with a basic unit of measure, such as an

inch, the object is subdivided into equal-size parts.

When an inch is subdivided into eight equal-size

parts, each part is one

of eight equal-size parts

made from the whole, or 1
8

of an inch.

The digits in a com-

mon fraction show this

part-whole relationship. In

the numeral 21, the 2 indi-
cates the number of equal-

size parts into which the (a) (b)

whole, or unit, has been

subdivided and is called Figure 13.2 Fractions represent parts of a set: (a) 6 is 1 of
2
the denominator. The 1 3
12; (b) 9 is 5 of 15.

Chapter 13 Developing Understanding of Common and Decimal Fractions 257

If 3 of 15 hamsters are brown, children must think of cepts. The following are examples of common situ-
5 ations that exhibit ratios.

first separating the 15 hamsters into five groups of

equal size. Each group of 3 hamsters relates to the • The relationship between things in two groups. In a

size of the original set of 15 hamsters so that each classroom in which each child has six textbooks,

group is 135, or 15, of the entire set. The denominator the ratio of each child to books is 1 to 6. This can
3
in 5 indicates the number of equal-size parts into be represented by the expression 1 to 6, 1:6, or by

which the set is subdivided (5), and the numerator the common fraction numeral 1 (Figure 13.4a).
6
indicates the number of groups being considered
• The relationship between a subset of things and
(3). If 3 of the 15 hamsters are brown, then there are
5 the set of which it is a part. When there are 3 blue-

9 brown hamsters (Figure 13.2b). covered books in a set of 10 books, the ratio of

Comparison Model blue-covered books to all books is 3 to 10, 3:10, or

Fractional relationships can also be represented as 3 (Figure 13.4b).
10

a comparison between two sets. Figure 13.3 shows

2 using the comparison method. The number of red
3

buttons compared to the number of green buttons is

2 (Figure 13.3a), as is the number of red cans com-
3

pared to the number of green cans (Figure 13.3b).

For 2 red buttons there are 3 green buttons, and for

2 red cans there are 3 green cans. In both cases the

numerator and the denominator are distinct.

In contrast to the part-whole model for fractions,

the fractional part is not embedded in the whole.

Counting out or removing the numerator (2 red

buttons) for examination will not affect the denomi-

nator (3 green buttons), because each part exists (a) (b)
10Ј
independently. This method of representing com-

mon fractions parallels the meaning of fraction as

a ratio.

Expressions of Ratios

The relationship or comparison between two num- 30Ј
bers is often expressed as a ratio. Although a full
discussion of ratio and proportion is presented in
Chapter 15, it is appropriate to briefly consider the
concept of ratio here, in contrast to fraction con-

(c) (e)

(a)

tabby tabby ¢ ¢ 25¢
treats treats ¢ 10¢
¢ 5¢
(b) cat ¢ ¢ ¢ ¢ 1¢
food

Figure 13.3 Fractions represent comparisons: (a) 2 as (d) (f)
3
2
many red buttons as green buttons; (b) 3 as many red

cans as green cans Figure 13.4 Fractions representing ratios: six examples

258 Part 2 Mathematical Concepts, Skills, and Problem Solving

• The relationship between the sizes of two things or 3 yd

two sets. When a 10-foot jump rope is compared

with a 30-foot jump rope, the ratio between the

two ropes is 10 to 30, 10:30, 1300, or 31. When a set
of 20 books is compared with a set of 30 books,

the expressions 20 to 30, 20:30, 3200, and 2 are used
3

(Figure 13.4c). 3
4
• The relationship between objects and their cost. If

the price of two cans of cat food is 69 cents, the Figure 13.5 Fractions representing division: 3 yards of

ratio between the cans of cat food and their cost cloth is cut into 4 equal-sized parts; each part is 3 yard
4
2
is 2 for 69, 2:69, or 69 (Figure 13.4d). long.

• The relationship between the chance of one event

occurring out of all possible events. When a regu-

lar die (die is singular for the plural dice) is rolled,

the chance of rolling a 4 can be expressed as 1 in

6, 1: 6, or 1 (Figure 13.4e).
6

• Ratio as an operator. In this case the ratio is a

number that acts on another number. When a toy
or model is built with a scale of 510, the ratio acts
as an operator between a measurement of the

model and the actual object (Figure 13.4f). If the Figure 13.6 Fractions representing division: 11 cookies
divided equally among 3 children; each child gets 131, or
actual object is 150 feet long, then the model is 323 cookies.

3 feet long (150 ϫ 1 ϭ 3). 43. A setting that illustrates the second sentence is
50 the equal sharing of 11 cookies by three children
(Figure 13.6). Division for the second sentence can
When children simply be expressed as 131. When the division is completed,
form a ratio between two the answer (3 with a remainder of 2) can be repre-
numbers, they will gener- sented as the mixed numeral 332, or each child’s fair
ally have little difficulty. It share of the 11 cookies.
is when ratios are used in
contexts that require pro- The term numerator is derived from the Latin term numeros,
portional reasoning that meaning “number,” and denominator is from the Latin term
difficulty can arise. In such denominaire, meaning “namer.” Thus the denominator names
settings, the tendency is the fraction (according to how many parts make up the
for children to use additive whole), and the numerator indicates the number of individual
reasoning and not multi- parts. (Bright & Hoffner, 1993)
plicative reasoning. See
Chapter 15 where ratio and
proportional reasoning is
discussed more fully.

Indicated Division

Sentences such as 3 Ϭ 4 The first European mathematician to use the familiar fraction
ϭ ? and 11 Ϭ 3 ϭ ? indi-
cate that division is to be bar was Leonardo of Pisa (c. 1175–1250), better known as
performed. Cutting a piece of cloth that is 3 yards
long into four equal-size pieces illustrates the first Fibonacci.
situation (Figure 13.5). Another way to indicate this
division is by using the common fraction numeral The horizontal fraction bar symbol ( 3 ) is called an obelus,
4

from the Greek word meaning “obelisk.” The diagonal frac-

tion bar symbol (3/4) is called a solidus. The term is derived

from the Latin term meaning “monetary unit.”

Chapter 13 Developing Understanding of Common and Decimal Fractions 259

Research for the Classroom •

An interesting research finding involves the difference or separating 1 of a licorice stick. Many children could not
3
between representing a common fraction with a set of dis-
represent any common fractions at all with continuous

crete objects and representing it with a continuous object. objects. Thus, although children may appear to have a

Hunting (1999) found that young children can represent good understanding of fraction representation when using

a common fraction of a set of marbles by setting aside discrete objects, they may require more experience with

some of the marbles, as for example, setting aside two common fractions before they can represent common frac-

out of a set of six marbles to represent 1 of the set. The tions with continuous objects such as a number line.
3

same children had great difficulty marking 1 of a rectangle
3

Introducing Common
Fractions to Children

Primary-grade children typically encounter com-

mon fractions through work with real objects and Image courtesy of ETA/Cuisenaire

models while learning about simple common frac-

tions such as halves, thirds, and fourths. Small sets

of objects can also be separated into equal-size

groups. Early on, common fraction numerals such

as 1 or 1 are introduced as names for common frac-
2 4

tions, but foundational understanding of fractions Fraction kit

continues throughout the primary grades. Children MULTICULTUR ALCONNECTION

label parts of wholes or sets as one-half and two- Antonio y Oliveres, a Spanish mathematician writing in
Mexico in the mid 1800s, first began the use of the solidus
thirds or refer to one part out of two parts or two parts (/) to represent fractions. The solidus is a popular alternative
to the common fraction bar because it allows printers to set
out of three parts. When students do begin to write type for fractions on a single line.

common fraction numerals, they should write their Partitioning Single Things

fractions with a horizontal bar, not a diagonal one. A Most children have experiences in which they share
parts of whole objects or collections of objects by
horizontal fraction bar will make future operations using fractional parts long before the concept of
common fractions is introduced in school. They
with fractions, especially multiplication and divi- help parents and others cut and share pizzas, cook-
ies, sandwiches, and myriad other items. These and
sion, much easier (see Chapter 14). As understand- similar common experiences can be illustrated on a
bulletin board to form a basis for discussing the pro-
ing of the concepts of common fractions for parts cess of cutting things into parts and sharing pieces
(Figure 13.7).
of units and parts of groups becomes established,
Squares, rectangles, circles, and other shapes
children will be able to work with other common cut from paper can extend real-life experiences dur-
ing introductory activities. Activity 13.1 shows one
fractions (fifths, sixths, eighths, and tenths) and will

learn to recognize and name the parts of numerals

such as 53, 26, and 38. When children use realistic set-
tings, stories, and models of common fractions, they

recognize that a given common fraction, such as

12, has many equivalent common fractions, such as 42,
63, and 48. The following photo shows typical commer-
cial models for elementary school children. Many

teachers have children use paper circles, squares,

rectangles, and triangles and drawings when com-

mercial materials are not available.

260 Part 2 Mathematical Concepts, Skills, and Problem Solving

We share many kinds of food. shape been cut?” (Answer: 4.) “How many parts

is my finger touching?” (Answer: 1.) Write 1 on the
4
chalkboard. Repeat a similar dialogue with 31, 21, and

other familiar fractions. Help children recognize that

the bottom numeral indicates the number of parts

PIZZA PIE CAKE into which the unit has been cut and that the top nu-

meral indicates one of the parts. Children typically

refer to the numerator as the “top number” and the

denominator as the “bottom number.”

MILK Accept Their Language in Early Work. Insisting
that primary school children use the terms numera-
ORANGE SANDWICH MILK tor and denominator may serve only to complicate
their understanding of fractions. As children mature
Figure 13.7 A bulletin board can be used to show how and meanings become established, introduce the
food is shared. terms denominator and numerator to identify the
two parts of a common fraction.
way to help young children use the share concept
to learn about one-half. Children can also fold pa- As work advances, children need activities that
per shapes to show fourths and eighths and to show extend beyond unit fractions. A unit fraction is one
thirds and sixths, as shown in Figures 13.8 and 13.9. in which the numerator is 1. Activities similar to Ac-
tivity 13.1 should be used to develop understanding
of fractions with numerators other than 1, such as 23,
24, and 43. (See Black-Line Masters 13.1, 13.2, and 13.3
for fraction circle and fraction strips templates.) See
“Quarters, Quarts, and More Quarters: A Fraction
Unit” on the companion website for an example of
a fraction unit that introduces simple common and
decimal fractions.

E XERCISE

Figure 13.8 Geometric regions to show fourths and Use fraction manipulatives to solve the following
eighths
problem: If 9 hamsters are 3 of the total number of
5

hamsters in a pet store, how many hamsters are

there? •••

Figure 13.9 Squares marked to show where to fold to Assessing Knowledge
make thirds and sixths of Common Fractions

Folding a shape to show three equivalent parts A quick way to assess children’s understanding that
is difficult, so fold lines should be marked to show common fractions represent fair-share, or equiva-
how to fold the shape during early experiences. lent, parts of a whole is by using shapes that show
When the idea of fair shares is well understood and both examples and nonexamples of the fractions.
the idea of cutting regions into equivalent pieces is Prepare some shapes that have shading showing
clear, numerals for common fractions can be intro- halves, thirds, or fourths and other shapes show-
duced. You might begin with a shape cut into four ing nonexamples of halves, thirds, or fourths (Fig-
equivalent pieces: “Into how many parts has this ure 13.10). You can gain a good idea of a child’s un-
derstanding by placing the shapes in an array and
directing the child, “Point to each shape that shows
41, 21, 13.” When a child correctly identifies all the halves,

Chapter 13 Developing Understanding of Common and Decimal Fractions 261

ACTIVITY 13.1 Introducing Halves (Representation)

Level: Kindergarten and Grade 1 children with unique folds have opportunities to show
and explain their work. It is easy to see that two parts are
Setting: Whole class the same size when a rectangle is folded along a center
line so the opposite edges come together. It is more dif-
Objective: Students develop the concept of one-half. ficult to see that pieces are the same size when a fold is
made along a diagonal or in some other way. It may be
Materials: Large circles, squares, and rectangles cut from news- necessary to cut along a fold line so that one piece can
print or colored construction paper be flipped or rotated to make it fit atop the second piece.
Help children see that even though a circle can be folded
• If space permits, have children sit on the floor in a semi- many times to show halves, the fold is always made in the
circle, with you at the opening (they can work at their same way.
tables or desks, if necessary).
• Develop understanding of the children’s knowledge by
• Nearly all young children have experiences with “fair saying, “Raise your left thumb if you can tell me what we
share” settings. Asking children to tell of their experiences call each part when we make two fair-share parts” (An-
will elicit comments that reveal the extent of their knowl- swer: one-half.) Discuss the meaning of one-half.
edge of the concept of sharing things.
• Reflect on their knowledge by asking children to name
• Give each child one of the shapes cut from newsprint or
colored construction paper. Ask them to name a type of times when they have used one-half. They might discuss
food each shape might represent (e.g., circle: pizza, torti-
lla, cake, pie, cookie; square: brownie, waffle; rectangle: such things as 1 of an hour, an apple, a candy bar, or a
cake, lasagna, candy bar). Tell them that they are to fold 2
each shape so that there are two fair-share parts. Chal-
lenge them to see if a shape can be folded in more than soft drink.
one way to form two fair shares.

• Use the folded shapes to develop new knowledge. Have
children discuss and show their fair shares. Be sure that

thirds, and fourths, verify the understanding by ask-
ing, “Are there any other fourths (halves, thirds)
shown on the shapes?” A child who is certain will
say no. Table 13.2 provides a scoring rubric for this
assessment. In Activity 13.2 children fold an equilat-
eral triangle into smaller shapes and compare the
area of each resulting part to the whole area of the
original triangle. In Activity 13.3 children use Cui-
senaire rods to explore part-whole relationships of
common fractions.

Figure 13.10 Shape cards for

testing understanding of

examples and nonexamples

of 1 , 31, and 1
2 4

262 Part 2 Mathematical Concepts, Skills, and Problem Solving

TABLE 13.2 • Scoring Rubric for Assessing Understanding of Unit Fractions

Inadequate Product or Solution Acceptable Product or Solution Superior Product or Solution

Doesn’t understand the concept Develops the concept Understands/applies the concept

Identifies few or no examples Identifies most but not all examples Identifies all shapes correctly
12, 31, 1 of 12, 31, or 14; is uncertain of some
or nonexamples of or 4 nonexamples

Unable to explain why a display Able to explain each shape as an Gives clear explanation of why
example of 21, 31, or 14; unsure about shapes are examples or nonexamples
is or is not a representation some nonexamples
21, 31, 1
of or 4

ACTIVITY 13.2 Fractions on a Triangle (Connection)

Level: Grades 2 and 3 6. Fold again, this time along
the new dotted line shown
Setting: Student pairs here.

Objective: Students identify various fractional parts of an equilat- 7 What shape is this new figure? If the original
eral triangle. triangle has an area of 1, what fraction area is this
new figure?
Materials: Two equilateral triangles (see Black-Line Master 13.4), 8. Take the other equilateral
scissors triangle and fold on the dot-
ted lines shown here so that
• Pair students by their order in your class roster. Pair the each vertex folds onto the
first student and the last student, the second student and center point.
the next-to-last student, and so forth.
9. What shape is this new fig-
• Pass out two equilateral triangle sheets, scissors, and ure? If the original triangle
a data table to each pair of students. Direct students has an area of 1, what frac-
to work through the folding steps given here. Be sure tion area is this new figure?
that students fill in the data table following each
question. Triangle Fractions Data Table

1. Cut out both equilateral triangles. Draw your Name the What fraction is
folded shape. shape. the shape compared
2. Fold on the dotted line
shown here so that the top to the original
angle of the triangle touches equilateral triangle?
the middle of the bottom
side. 1.

3. What shape is this new 2.
figure? If the original tri-
angle has an area of 1, what 3.
fraction area is this new
figure? 4.

4. Fold on the new dotted line
shown here to get another
shape.

5. What shape is this new
figure? If the original

triangle has an area of 1, what fraction area is new
figure?

Chapter 13 Developing Understanding of Common and Decimal Fractions 263

ACTIVITY 13.3 Cuisenaire Fractions

Level: Grades 1–3 • Suggest to students that there are rods that have equal-
length same-color combinations. Their task is to search for
Setting: Student pairs them and record their findings by making sketches of their
rods and writing out the fractions shown by the shorter
Objective: Students look to identify different Cuisenaire rods as rods.
fractional parts of longer rods.
• Post the results on the board. Once all students’ findings
Materials: Full set of Cuisenaire rods (10 white rods, 5 of each are posted, ask if students notice anything missing. The
other color) for each pair of students; overhead set of Cuisenaire red, black, green, and yellow rods have no same-color
rods combinations except white that match their lengths. As
it turns out, these are prime numbers. Although you may
• Display teal, green, red, and white rods on the overhead. not be ready to introduce such a concept or the vocabu-
lary, simply noting the fact that some lengths or numbers
• Ask students to speculate about which color rod is exactly cannot be divided up evenly into part-whole pieces (ex-
half the length of the teal rod. cept for unit pieces) will lay the foundation for later work
with prime and composite numbers.
• Once students have a chance to give their thoughts,
discuss how to be sure which rod is actually half. Probe for teal
using two same-color rods to line up with the teal rod for
an exact fit. Because two green rods are the same length green green
as a single teal rod, each green rod represents half the teal
rod (21t ϭ g, where t represents the length of the teal rod
and g represents the length of the green rod).

• Speculate aloud about whether there are other combina-
tions of same-color rods that are the same length as the
teal rod. Have students explore this possibility with their
partners, using their Cuisenaire rods. Children should find
that six white or three red rods are the same length as the
teal rod.

red red red

teal white white white white white white

green green (b) Green is—12 of teal
(a) Green is —12 of teal
Red is —13 of teal
White is —16 of teal

E XERCISE line to the unshaded portion, rather than to the en-
tire line segment. Thus they may see the part-part
Describe several common objects or settings to use relationship, or 12, instead of the part-whole relation-
during an introductory partitioning-a-whole activ- ship of 13. Although it can be beneficial to represent
ity and a partitioning-a-set activity, other than any mathematics concepts with different models,
the ones used in the text. Draw a simple picture it might be best to use the number line represen-
to illustrate a partitioning-a-whole setting and a tation after basic concepts are introduced and un-
derstood. One research study (Ball, 1993) indicates
partitioning-a-set setting. ••• that the only students who beneficially used a num-
ber line in their study of fractions were those who
Another example of a representation of a part-

whole relationship might be a number line. In the

number line shown in Figure 13.11, 1 is shaded. As
3

in the Unifix cubes example on page 265, some stu-

dents may compare the shaded portion of a number Figure 13.11 Number line representing 1
3

264 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 13.4 The Fraction Wheel

Level: Grades 1–3 If the answer is incorrect, quickly move to another student
for another answer.
Setting: Whole class
• Continue until a student gives the correct answer, and
Objective: Students develop their ability to identify common frac- then have another student explain why the correct answer
tions in an area model. matches the shaded part showing on the fraction wheel.

Materials: An angle wheel (see Black-Line Masters 13.5 and 13.6)

• Draw a circle on the board with 1 shaded. Ask a student • Show several unit fractions to the class (12, 31, 14, 51, 61, and 1 ),
4 8

volunteer to explain how much of the circle is shaded. and repeat the unit fractions as needed, so that each child

• Draw a second circle on the board, this time with 2 has an opportunity to give an answer and/or explain an
3
answer.
shaded. Again ask a student volunteer to explain how

much of the circle is shaded. • Once all the children have had an opportunity to give or

• Show the angle wheel with 1 shaded. Ask a volunteer to explain at least one answer, show a common fraction
4
between 1 and 1, such as 23. When you display 32, ask the
explain how much of the shaded part is showing. Be sure 2

that all the children understand that the shaded part of children how this new fraction is different from all the

the angle wheel shows part-whole fractions, then have a preceding fractions. Probe for the concept that this frac-

second student explain why the correct answer matches tion is greater than 12. That means that no unit fractions
1
the shaded part showing on the fraction wheel. Use only can be represented by shading that is greater than 2 the

unit fractions at first so that all the shaded portions are circle. Show several fractions greater than 1 (34, 23, 65, 78, . . .)
2

less than half of the circle. following the same procedure with the wheel as before.

• Quickly show a different angle, and call on a student to • As students explain answers, probe for the half-circle as
give the part-whole fraction that the shading represents. a benchmark to help determine the value of the common
fraction.

1 11 • Now display any of the preceding fractions on the angle
1 16 5 43
wheel, mixing all the common fractions used to this point.
8
The goal of the activity is to move quickly from one stu-

dent to another as they give estimates of the part-whole

0 fraction you display on the angle wheel. Although it is

not critical that students be able to discriminate 1 from 18,
6

the first few times you work with the angle wheel, all the

children should be able to use 1 (and possibly 1 and 43) as a
2 4

benchmark to help them make a reasonable estimate of

the part-whole fraction that the angle wheel displays.

Back Front

had already developed their foundational concep- Fraction stencils
tions of fractions and part-whole relationships. A
manipulative called the fraction wheel is the focus
in Activity 13.4. When students use two pieces to
model common fractions, they can manipulate and
even remove the part without affecting the whole.
The companion website activity “Mystery Fraction
Pieces” uses a circle to build children’s foundational
understanding of part-whole relationships. Fraction
stencils shown in the photo allow children to make
their own fraction representations. The companion
website activity “Tangram Fractions” explores frac-
tions in an area representation using tangram pieces.
“A Handful of Fractions” is another activity on the
companion website that stresses the set model of

Chapter 13 Developing Understanding of Common and Decimal Fractions 265

MISCONCEPTION

When children use manipulatives to model part-whole relation-

ship of fractions, they may use an area model (see Activity 13.4)

or a linear model to represent fractions. However, when children

are beginning to formulate fractions from either model, they may

form several different fractions from the same setting. Note the

Unifix cube train in Figure 13.12. A student trying to write what

fraction of the train is blue may write 2 and not 2 . Do you see why?
3 5

The child is comparing the part of the train that is blue (2) to

the remaining part of the train that is green (3) instead of to the Figure 13.12 Unifix cube train

entire length of the train (5). The child sees a part-part relation-

ship and not a part-whole relationship. One reason this happens

is that young children remove the blue cubes from the train and

then try to make sense of what remains. With the area and linear

models for part-whole relationships, once the numerator pieces

are removed, there is no longer any whole to use as a reference.

The whole no longer exists as a model.

Students can make a similar error with the array shown in

Figure 13.13. The fraction that represents the part-whole relation-

ship for green squares is 3 , but children who are just beginning to
8

express fractions might render it as a part-part relationship, or

3 . It is important to help children in these early stages to be sure
5

they do not develop such misunderstandings, which can become

difficult to break, and thus hold back their advancement in laying Figure 13.13 Unifix array

a foundation for fraction concepts. representing 3
8

fractions. A vignette on the website illustrates how such as 36, 23, and 49. Pictures cut from magazines can
California teacher Dee Uyeda had her third-graders be used to present real-world settings for introduc-
work in cooperative groups to further their under- ing these numerals. Later, paper shapes and num-
standing of fractions. ber lines can be used as representations of com-
mon fractions. The pizza problem discussed earlier
Partitioning Sets of Objects provides a real-world setting for looking at common
fractions with numerators greater than denomina-
The concept of a fractional part of a set should be tors. Children see that if the family of three gets one
introduced only after children demonstrate that pizza, the family of six must have two pizzas to have
they can conserve numbers, have a good grasp of the same amount of pizza per person. When one
whole numbers, and are skillful in counting objects pizza is cut into three equal-size parts, the common
in sets. Activity 13.5 provides a real-world setting for fraction is 33. When two pizzas are each cut into three
dealing with fractional parts of groups. Later, chil- equal-size pieces, the common fraction is 36. When
dren should be able to partition sets into equal-size two pies are each cut into six equal-size pieces,
groups without using fractional regions or markers the common fraction is 162. When two and one-half
as cues. As they work, children notice that not every cakes are cut into six equal-size parts, the common
set can be separated into equal parts with a whole- fraction is 165. Figure 13.14 illustrates each of these
number answer for each part. This realization sets common fractions modeled with pictures of popu-
the stage for understanding mixed numerals and lar food items. Common fractions with numerators
common fractions greater than 1. greater than the denominator have traditionally been
called improper fractions. But it is more meaningful
Fractional Numbers Greater than 1 to children to call them common fractions that are
greater, or larger, than 1. Fractional numbers greater
Many students seem to believe that all fractions are than 1 are sometimes converted to whole numbers
between 0 and 1. This is why students need opportu-
nities to deal with common fractions greater than 1,

266 Part 2 Mathematical Concepts, Skills, and Problem Solving

or to a combination of a whole number and a com-

mon fraction. The term mixed numeral refers to a

combination of a whole number and a fractional

number. When 13 cookies are divided fairly among

6 four people, the common fraction 13 can be used
3 4

12 to represent the result. This is an indicated division
6
interpretation of a common fraction. Twelve cookies
15
6 are divided into four groups of three cookies each,
Figure 13.14 Fractional numbers greater than 1 repre-
sented by common items of food and the remaining cookie is cut into four equal-size

parts. Each person gets three whole cookies and 1 of
4
another, or 314 cookies. The number 341 is read “three

and one-fourth.” This is because a mixed number is

composed of a whole number (3) and a common

fraction (41). When students read mixed numbers,
call their attention to the word and in the number

name. Have children explain why the word and is

critical in understanding the mixed number they are

reading. One last aspect of common fraction value

is worth mentioning here, in the form of a question:

ACTIVITY 13.5 The Fruit Dealer and His Apples (Communication)

Level: Grades 3–5 ner accepts or rejects the way the apples were grouped.
Setting: Cooperative learning
Objective: Students demonstrate strategies for finding nonunit A written record of the grouping is made. The students

parts of a collection of objects. alternate making groupings until they agree that all pos-

FRUIT sible groupings have been made and they have recorded
FOR
SALE all the groupings. Pair by pair, ask children to report on

one way they separated the apples. Some pairs may have

only two or three groupings; others may have all possible

groupings. As each pair reports, ask, “What part of 36

apples is each of your two groups? three groups? four

groups?” and so on. List groupings and fractional parts on

the chalkboard: 2 apples are 1 of 36, 3 apples are 1 of
18 12

36, and so on.

Materials: Green or yellow plastic beads or disks; small pieces of
paper to represent bags

• Organize children for a pairs/check cooperative-learning • Have children use the information on the chalkboard to
experience. deal with nonunit common fractions: “If you buy two
bags that each contain 6 apples, what part of the 36
• The activity develops as children use beads or disks to apples do you have?” Continue with other fractional parts
represent apples. Present this story: “A fruit dealer has 36 of 36, such as 43, 32, and 65. (Children’s level of understand-
Granny Smith apples. He wants to put his apples in bags ing of and ability to solve earlier problems will determine
with an equal number of apples in each bag. How many how many problems you present.)
different ways can he bag the apples so that each bag
contains the same number of apples?” • Discussion following each question enables children to
reflect on their learning by confirming the accuracy of
• One student in each pair groups the apples without con- their work.
sulting the partner. When the student is finished, the part-

Chapter 13 Developing Understanding of Common and Decimal Fractions 267

Is 1 Ͼ 31? Are you sure? Consider this conversation expressions out of context is not inappropriate and
2 is, in fact, useful for practicing many mathematical
operations, it is still important to stress that all com-
between two fourth-graders, who are comparing the mon fractions must be based on the same whole set
in order to be compared.
two fraction pieces shown in Figure 13.15.
Activity 13.6 is an Internet activity that uses vir-
Denyse: I still say one-half has to be bigger than tual manipulatives as area models of fractions to ex-
one-third. Remember that the more pieces you plore fractional numbers greater than 1.
need to make a whole, the smaller each piece is.
Alexa: I know, but look at this piece. It’s 31, but it’s a Introducing Decimal Fractions
lot bigger than this piece (21).
Denyse: You’re right. It is bigger. Hmm. Maybe it’s Decimal fractions are used to find parts of units and
if the bottom number is bigger, then the fraction is of collections of objects, just as common fractions
bigger. are. The difference between the two fractional num-
bers is that the denominator of a common fraction
Alexa: But that’s not what we did yesterday. can be any whole number except 0, whereas deci-
mal fractions are confined to tenths, hundredths,
Denyse: I know, but look at the two pieces. One- and other powers of 10.
third is bigger than one-half.
Teaching children about decimal fractions along
Alexa: I know. That’s what I said. with the study of common fractions is an integral
part of the mathematics curriculum in the primary
As you can see grades. The goal in the early grades is to lay a foun-
dation that enables older students to avoid miscon-
in Figure 13.15, ceptions and procedural difficulties. Children who
investigate the meaning of decimal fractions and
the 1 piece is cer- learn about them through activities with models will
3 have the understanding needed for more advanced
concepts and uses in later grades. An understand-
tainly larger than ing of decimal fractions and their relationship with
common fractions develops gradually, so work with
1 1 the 1 piece. What physical materials, diagrams, and real-world settings
3 2 2 is extended over a period of years.

is confusing these Children’s understanding of whole numbers and
common fractions forms the basis for their under-
students? They are standing of decimal fractions. Real-world examples
of things separated into tenths and hundredths are
comparing frac-

tional pieces from

two different-size

wholes. The 1 piece
3

Figure 13.15 Is 1 greater than 1 ? is from a larger
2 3

circle than is the 1
2
1
piece. The 3 piece is

larger than the 1 piece in the same sense that 1 of 300
2 3
1
(100) is larger than 2 of 100 (50). Many students have

this misconception because common fraction com-

parisons are done with abstract number representa-

tions, devoid of context. Although using numerical

ACTIVITY 13.6 Exploring Fractions (Internet Lesson)

Level: Grades 3–5 • Have students enter larger values into the numerator than
Setting: Pairs of students
Objective: Students use an Internet applet to explore improper the denominator and observe the resulting circle represen-
tation. Repeat for these four fractions: 181, 25, 47, 130.
fractions.
Materials: Internet access • Repeat with the rectangle representation.

• Go to http://illuminations.nctm.org/Activities.aspx? • Ask students to explain how to represent 7 using circles
gradeϭ2. Click on Fraction Models II. 3

• Have students use the circle representation of fractions. and using rectangles. Students can then use the applet to

check their answers.

268 Part 2 Mathematical Concepts, Skills, and Problem Solving

less common than are examples of common frac- A zero after the decimal point, as in 1.0, also has
tions. Metric units of measure, such as a meter stick, meaning. In 1.0 it indicates that a unit has been sepa-
can represent decimal fractions, as does our mon- rated into 10 parts and that all 10 parts are being
etary system. White and orange Cuisenaire rods considered; it is equivalent to the common fraction
also display a decimal relationship. Commercial or 1100, and the zero should not be omitted. The decimal
student-made materials are needed for individual point also indicates precision of measurement, in-
and group activities. Any commercial base-10 prod- dicating that 3.0 meters, for example, is accurate to
uct can serve as a model to represent decimal frac- the nearest tenth, in contrast to 3 meters, which may
tions. When base-10 materials are used, a large flat have been rounded to the nearest meter.
becomes a unit, a rod a one-tenth piece, and a small
cube a one-hundredth piece. MULTICULTUR ALCONNECTION

Any activities that use money can use currency and coins
from the native countries of minority students.

Activity 13.7 illustrates an introductory lesson us-
ing Cuisenaire rods. Construction paper can replace
the rods in this lesson. In Activity 13.8 a strip of pa-
per with units separated into 10 equal-size parts is
used to extend decimal fractions beyond 1. In Activ-
ity 13.9 a decimal fraction number line is used.

When commercial materials are unavailable, col- Flemish mathematician Simon Stevin (1542–1620) first used
ored construction paper with half-inch or centime- decimal fractions in his book La Thiende. When he wrote
ter squares can be laminated and cut to make an common decimals, Stevin used a small circle instead of a
activities kit. Each child might make a kit consisting decimal point. The word dime is derived from the title of the
of 10 square mats that are 10 units along each side, French translation of his book, La Disme.
20 strips that are 1 unit wide and 10 units long, and
150 one-unit squares. Introducing Hundredths

Introducing Tenths Children’s understanding of the decimal fraction
representation of hundredths is developed through
Activities involving different materials help children extension of activities with tenths. The hundredth
acquire a well-developed understanding of tenths pieces are included in kits for the new activities. An
and decimal notation for tenths. When children introductory lesson is shown in Activity 13.10. An ac-
learn to write whole numbers, a decimal point is tivity built around Cuisenaire rods and a meter stick
not part of the number. It is not needed because the is useful for helping students to understand tenths
whole number represents one or more whole units. and hundredths and to show how decimal fractions
Decimal numbers indicate that parts of units are are used to indicate parts of a meter. Let children
involved. The decimal point separates the whole- work in groups of three or four. Each group has a
number part of a numeral from the fractional part of meter stick, 10 orange rods, and 100 small cubes.
a numeral. When only a decimal part of a numeral First, the children align the 10 rods end to end along-
is written, it is common practice to write a zero in side the meter stick (Figure 13.16).
the ones place of the numeral, as in 0.3. The zero
helps make it clear that the numeral indicates a 10 20
decimal fraction. When no zero is written, it is pos-
sible to overlook the decimal point and misread the Figure 13.16 A meter stick and Cuisenaire rods used to
numeral. show tenths

Chapter 13 Developing Understanding of Common and Decimal Fractions 269

ACTIVITY 13.7 Introducing Tenths

Level: Grades 2–4 until 9 is reached and all common fractions have been
10
Setting: Whole class
written on the chalkboard.
Objective: Students are able to explain the meaning of decimal
tenths. • Introduce the decimal notation 0.1, and write it next to
the 110. Tell the children that both numerals are read as
Materials: Cuisenaire flats, orange rods (construction paper may “one-tenth.” Select students to write decimal fractions for
be used instead)
each of the other common fractions.

• When they are used with whole numbers, a large flat in a • Tell the students that 1.0 is the numeral to use when all
Cuisenaire set is considered to be 100, an orange rod 10, 10 parts are being considered. The decimal point and zero
and a white cube 1. Tell the children that for this lesson, indicate that the unit has been cut into 10 parts and that
each flat represents one unit, or 1. all 10 parts are being considered.

• Tell each child to cover a flat with orange rods. • Use money to help children understand how a dime

• Ask, “How many rods does it take to cover the unit shows one-tenth of a dollar. Display a dollar bill and ask,

piece?” Verify with the children that there are 10. Ask, “What coin is one-tenth of a dollar?” (Answer: dime.)

“What part of the unit piece is covered by one rod?” “What two ways can we write the value of a dime?”

Verify that it is 1 of 10, or 110. Write 1 on the chalkboard. (Answer: 10 cents or $0.10.) Ask, “Three dimes are what
10 part of a dollar?” (Answer: 130.) “What two ways can you
write 30 cents?” Repeat with other numbers of dimes.
• Ask volunteers to give names for two rods or strips (2 of
(Note: Do not use either a nickel or a quarter during these
10, or 120), three rods or strips (3 of 10, or 130), and so on,
early decimal fraction activities. Neither coin supports the

base-10 aspect of common decimals.)

• Summarize the lesson by pointing out that the common
fractions and the decimal fractions are both ways to des-
ignate the same quantity.

ACTIVITY 13.8 Fraction-Strip Tenths

Level: Grades 2–4 tell in which column to place their money. Write the dollar
Setting: Cooperative learning values beneath the words.
Objective: Students are introduced to mixed decimals and money
• Have students remove the dimes and mark an X in place
as an application of decimal fractions. of each one, then write the decimal numeral that indicates
Materials: One 3-unit-long fraction strip for each pair of children; a the number of tenths covered by X’s. Write the decimal
numerals alongside the corresponding dollar amounts,
die for each pair; plastic dimes or dime-stamped squares of paper and compare the two numerals. (The difference will be
the dollar sign and a zero in the hundredths place in each
• Organize the children as partner pairs. Give each pair a money numeral.)
fraction strip, a die, and replica dimes.
• This lesson presents a good opportunity to discuss ideas
1 related to the probability of events occurring (see Chapter
20). For example, you can discuss the smallest number of
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 squares (4) and the highest number of squares (24) that
could be covered. “What would have to occur if only
• Present these instructions: You will take turns to roll your four squares were covered?” (Answer: Four 1’s would
die four times. After each roll, the partner not rolling the be rolled.) “If 24 squares were covered?” (Answer: Four
die covers the strip, one square at a time, with enough 6’s would be rolled.) “Did this happen with any of you?”
dimes to equal the number showing on the die for that “Why are there more numbers in the middle column than
roll. When you have finished the four rolls, write numbers in the low or the high column?” (Answer: The likelihood
with a dollar sign to show the total value of the dimes on of getting four 1’s or four 6’s is much less than that of
your strip. getting a mixture of numbers. A mixture of numbers will
be closer to the middle.)
• When all have completed their rolls, call for attention,
then write the words “low,” “middle,” and “high” on the
chalkboard. In their pairs, students decide whether they
have a low, middle, or high amount of money and then

270 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 13.9 Number-Line Decimals

Level: Grades 2–4 do you stop when you start at 0.4 and move eight steps
to the right? Where do you stop if you start at 1.7 and
Setting: Cooperative learning move three steps to the left? Pairs write their questions on
one paper and their answers on the second paper.
Objective: Students are introduced to a decimal fraction number
line and lay a foundation for adding and subtracting decimal • Each pair exchanges papers with the other pair on its
fractions. team and answers the questions. When all questions are
completed, students meet in groups of four to check their
Materials: Duplicated copies of decimal numbers lines with tenths answers.
to 3.0, pencils, paper
• Students take turns reading the decimal numerals for their
• Organize children in pairs for a send-a-problem activity. answers. Each group is to resolve any situations in which
Each pair has a duplicated copy of the number line, two there are discrepant answers.
pieces of paper, and a pencil. Each pair is to write four
questions of the following type: Where do you stop when
you start at 0 and go seven steps along the line? Where

0 123

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

ACTIVITY 13.10 Introducing Hundredths

Level: Grades 3 and 4 • Ask, “What part of a flat is one white cube?” (Answer:
Setting: Whole class 1010.) Introduce the decimal notation 0.01. Ask, “How
Objective: Students demonstrate understanding of the concept of does this notation differ from the notation 0.1?” Help
children understand that the two numerals to the right of
decimal hundredths. the decimal point represent hundredths; in this case, it is
Materials: Cuisenaire flats, orange rods, and white cubes (con- one-hundredth.

struction paper can be used); cards containing numbers between • Present other decimal fractions for children to represent
0 and 1, such as 0.01, 0.23, 0.40, 0.57, 0.99 (a different number with the Cuisenaire materials: 0.15, 0.36, 0.86, 0.40.
on each card, one for each child)
• Review prior knowledge by having each student display a • Use money as a way to extend understanding of hun-
flat; then cover it with orange rods. Review the notation dredths. Ask, “What part of a dollar is one penny?” (An-
for decimal tenths. swer: 1010.) “What are two ways we can use money nota-
tion to show one cent?” (Answer: 1 cent and $0.01.) Have
• Instruct the children to remove one tenths piece from the the children write notations to show money amounts such
flat and cover it with white cubes. Ask, “How many white as 24 cents, 50 cents, 97 cents, 8 cents.
cubes cover one tenths piece?” (Answer: 10.) “What part
of a tenth is one white cube?” (Answer: 110.) Ask, “If it Variation
takes 10 white cubes to cover one orange rod, how many
will it take to cover all 10 rods?” (Answer: 100.) • Use a line-up cooperative-learning activity to extend
children’s thinking about decimal hundredths.
• Discuss the fact that 100 white cubes will cover the flat.
• Give each student a card containing a decimal number
with hundredths.

• At a signal, the children form a line that puts the numbers
in order from smallest to largest.

• When the order is correct, each student turns to the one
on either side and says, “My number is _____. It is larger/
smaller than your number.”

Chapter 13 Developing Understanding of Common and Decimal Fractions 271

Discuss that there are 10 rods and that their ends ten-thousandth has been cut into 10 parts to make
are at points along the meter stick that indicate hundred-thousandths.
decimeters. Next, the children align the 100 small
cubes side by side along the orange rods. Build on Some children may have initial difficulties read-
knowledge of the relationship of the cubes to rods ing decimal fractions because the name for the
to enable children to see that there are 100 small decimal fraction seems to be off by one. Consider
cubes and that each one represents one-hundredth 0.34. This decimal fraction is read as “thirty-four
(0.01) of the meter, or 1 centimeter. An orange rod hundredths,” although hundreds in whole numbers
is one-tenth (0.1) of the meter, or 1 decimeter. Ac- indicates three digits. Similarly, the decimal frac-
tivity 13.10 expands children’s knowledge of tenths tion 3.456 is read “three and four hundred fifty-six
and hundredths. Activity 13.11 uses a calculator to thousandths.” In this case, a three-digit decimal frac-
help children explore the base-10 aspects of deci- tion has the label thousandths, which, for children,
mal fractions.
MISCONCEPTION
Introducing Smaller Decimal Fractions
Some students think that the decimal point marks a symmetri-
When children learn about decimal fractions
smaller than hundredths, a large unit region marked cal location in a decimal fraction. Actually, the units position
into 1,000 parts can illustrate thousandths, but it is
impractical to make models to show 10,000 and in a decimal fraction is the point of symmetry. Can you see
100,000 parts. Older children who work with num-
bers smaller than thousandths can visualize that why?
each thousandth has been cut into 10 equal-size
parts to make ten-thousandths and then that each 432 1 .1 2 3 4

tens tenths

hundreds hundredths

ACTIVITY 13.11 Decimal Fractions on a Calculator (Reasoning and Proof)

Level: Grades 2 and 3 • Have students continue to press Image courtesy of Texas Instruments Inc.
the ϩ key until the display reads
Setting: Pairs of students 0.9, then ask student pairs to TI-10 Calculator
predict the display when they
Objective: Students use a calculator to develop an understanding press ϩ the next time. Discuss
of magnitude with decimal fractions. students’ conjectures as a class.

Materials: Student calculators • Allow students to press the ϩ
key to obtain 1.0. Ask student
The calculator provides the opportunity for students to volunteers to explain the result.
explore decimal fractions before they are able to use Have students continue to press
any common algorithms for operations using decimal the ϩ key until they reach 1.9.
fractions. Again ask students to predict
the display when they press ϩ
• Pass out a calculator to each student pair. again, and discuss as before.

• Ask students to enter 0.1 into the calculator. Allow time • Have students repeat using 0.2,
for students to locate the decimal point button. 0.3, and 0.4 in place of 0.1 in
their initial number sentence.
• Most calculators will display 0.1 even if students enter Ask them to predict how the dis-
“.1”. Explain that the 0 is used to emphasize that the play will read as they press the ϩ
decimal fraction is less than 1. Remind students to enter key, and then verify their predic-
all subsequent decimal fractions less than 1 with the lead- tion by using the calculator.
ing 0.

• Once all students have successfully entered 0.1 into their
calculators, have them clear the entry and this time enter
0.1 ϩ 0.1 ϭ. All should have 0.2 on their display.

• Now ask students to press the ϩ key again. This should
add 0.1 to the display of 0.2 for a new display of 0.3.

272 Part 2 Mathematical Concepts, Skills, and Problem Solving

evokes a four-digit whole number. In the case of Comparing Fractional Numbers
hundredths the number of hundredths will not ex-
ceed 99; we never reach 100. If there are more than Children compare whole numbers in many ways.
99 hundredths, then the result is a mixed number. It They match objects in one set with objects in a sec-
is good to acknowledge this apparent mismatch be- ond set and conclude that the one with excess ob-
tween the number of digits and the decimal fraction jects has a larger number than the other set. They
name as children are learning to read decimal frac- learn that larger numbers are to the right of smaller
tions and to help them understand why it is proper. ones on a number line, in numerical sequence, and
that the difference between any two consecutive
MULTICULTUR ALCONNECTION whole numbers is 1. They need to have similar expe-
riences to learn that fractional numbers can also be
Newspapers and magazines are good sources for dem- ordered by size. When children order and compare
onstrations of decimal fractions. Sports magazines, such whole numbers, they learn that there is a finite num-
as Sports Illustrated and Sports Illustrated for Kids, and ber of whole numbers between any pair of numbers.
newspaper sports sections contain many team and individual When they order and compare fractional numbers,
statistics from other countries and from the Olympics and they learn that there is an infinite number of frac-
World Cup soccer competitions. Monetary exchange rates tional numbers between any pair of numbers.
are also represented as decimal fractions. Students can find
examples and display them on a bulletin board or in a class Initial experiences comparing common and
book. Children can also record in their journals examples of decimal fractions come through investigations with
decimal fractions observed at home and other places outside models of various kinds. We discuss activities with
the classroom. three different models that are appropriate for sec-
ond-, third-, and fourth-graders, followed by more
Introducing Mixed Numerals abstract procedures suitable for older children.
with Decimal Fractions
Comparing Common Fractions
Whole numbers and decimal fractions form mixed
numerals in the same ways that whole numbers and Commercial kits and construction paper can be
common fractions do. Measurements made with
meter sticks often result in whole meters plus deci- used to model settings in which children compare
meters or centimeters. The measure of the length of
a room might be recorded as 4.3 meters. This means fractions whose numerators are 1. Models show that
that the room is 4 meters plus 3 decimeters long.
When children record measurements made with a 1 is more than 31, 41, or any other unit fraction for an
meter stick, explain that people read mixed numer- 2
als that contain decimal fractions in two ways. Al-
though a measurement of 4.3 meters is commonly object with a given size and shape. The patterns that
read as “four point three meters” rather than “four
and three-tenths meters,” the first reading hides the become apparent when models are arranged in se-
meaning of the number. As children begin to read
decimal fractions, avoid the common reading “four quence from smaller to larger or larger to smaller
point three.” When children read decimals by sim-
ply reading numerals and inserting “point” where help children order these common fractions.
the decimal appears, they mask the mathematical
value of the decimal fraction. The decimal fraction Fraction strips cut from colored construction pa-
2.4 is properly read as “two and four-tenths,” and
5.35 is read “five and thirty-five hundredths.” Read- per are used in Activity 13.12 to compare common
ing mixed numerals with a decimal fraction in this
manner helps children build an understanding of the fractions. The strips consist of a unit piece and half,
decimal fraction included with the whole number.
fourth, third, sixth, eighth, and twelfth pieces. Chil-

dren manipulate the pieces at their desks as they

complete the activity.

Comparing Common and Decimal
Fractions with Number Lines

Number lines marked with common fractions pro-
vide a more abstract way to compare fractional
numbers than do regions or strips. Children extend
their understanding by connecting their knowledge
of those models to the more abstract number lines.
Activity 13.13 provides a setting in which children
use communication and reasoning skills as they
compare common fractions on number lines.

Chapter 13 Developing Understanding of Common and Decimal Fractions 273

ACTIVITY 13.12 Fraction Strips (Representation)

Level: Grades 3–5 7. Name three common fractions that are equivalent to 12.
8. Name two common fractions that are equivalent to 142.
Setting: Cooperative learning
• Put a set of strips, four marking pens, and a question
Objective: Students demonstrate a strategy for comparing com- sheet together in a food storage bag for each team. Roll
mon fractions. a sheet of butcher paper for each team and secure with a
rubber band.
Materials: Multiple sets of fraction strips cut from colored con-
struction paper, four different colored marking pens, large sheets • Organize the children into team-project cooperative-
of butcher paper, masking tape, paper containing eight questions learning groups consisting of four children. Distribute one
similar to the following: bag of materials and a sheet of butcher paper to each
team.
1. How many 1 strips are as long as the 1 strip? How
2 • Team members rotate responsibilities as they answer the
1 eight questions. Each member is to use the strip material,
many 3 strips are as long as the 1 strip? Which is lon- if necessary, to complete two questions while the other
three serve as consultants. Answers are written on the
ger, a 1 strip or a 1 strip? butcher paper, with each student using a different colored
2 3 pen.

2. What is the shortest fraction strip in this set? Name • Tape the answer sheets side by side on the chalkboard or
a wall. Students check from their desks to see if there are
the strips that are longer than this strip. Use the frac- any discrepancies in answers. Discuss any discrepancies.

tion strips to put these common fractions in order,

beginning with the largest and ending with the small-

est: 81, 21, 31, 16, 41.

3. Which is longer, two 1 strips or two 1 strips?
2 3

4. Which is longer, two 1 strips or one 1 strip?
6 4

5. Which is shorter, two 1 strips or two 1 strips?
3 8

6. Use the strips to put these common fractions in order,

beginning with the smallest and ending with the larg-

est: 32, 36, 83, 34.

1 1 1
2 2
1
1 3 1
3 11 3
1 44
4 11 1
66 4
11 11
66 88 11
1 11 1 66
111 12 12 12 12
888 111
888
11 1 1
12 12 12 12 111 1
12 12 12 12

Children can also use a set of number lines to lar to those in Activity 13.14 to focus attention on
compare decimal fractions. Display three number comparisons:
lines on a large sheet of paper or overhead trans-
parency—one showing a unit, one showing tenths, • Which is more, 3 tenths or 27 hundredths?
and one showing hundredths—placed one above
the other so that the starting points are in a verti- • Name a number of tenths that is more than 80
cal line (Figure 13.17). You can provide leads simi- hundredths.

01 • What is the number of tenths that is equal to 70
hundredths?
0 1.0
• Which is more, 89 hundredths or 9 tenths?
0 1.00
An important concept about numbers—that there is
Figure 13.17 Number lines for comparing decimal no smallest fractional number—can be developed
fractions intuitively by using number lines like the ones in Ac-
tivity 13.16 and Figure 13.17. In the following vignette
a fourth-grade teacher uses both common fraction

274 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 13.13 Common Fractions on a Number Line (Communication)

Level: Grades 2–4 • Raise and discuss questions such as these:

Setting: Whole class 1. How many of the 1 segments match the length of
2
Objective: Students order and compare common fractions.
the unit segment?
Materials: Large sheet of paper showing number lines with only
whole-number locations marked, black marking pen, paper and 2. What is the shortest segment on the chart?
pencil for each student. The lines and marks must be visible to
all students; the set of lines can be displayed on an overhead 3. How many of the 1 segments are equivalent to a
projector. 16
1
8 segment?

4. Which is longer, a 1 segment or a 1 segment?
16 4

• Direct students’ attention to the top line, and point out 5. Which are shorter, two 1 segments or two 1
the unit segment. 8 4

segments?

• Go to the second line, and mark the point midway be- 6. What number of 1 segments are equivalent to a 1
8 2
1
tween 0 and 1. Darken and label the marks that show the segment? to a 4 segment? to a whole segment?

1 and 2 points on the line. 7. What is the order of these segments, from longest
2 2

• Say, “Raise your right hand if you can tell me the denomi- to shortest: 41, 83, 1392, 12, 34, 3320, 136?

nator for common fractions on the third line.” Darken and 8. Which fraction is nearer to 0: 1 or 116? 1 or 312?
8 4
label points for 14, 24, 34, 44, and 54.
9. Which fraction is nearer to 1: 5 or 1165? 7 or 156?
8 8

• Continue to the bottom line, where thirty-seconds will be 10. Which fraction is closer to 21: 1 or 1325? 3 or 87?
16 8
marked and labeled.
• Use questions like these to help children

develop generalizations about common

01 fractions: “What do you see about com-
0
0 mon fractions that are close to 1 on the
0
0 number line?” (Answer: Their top numbers
0
[numerators] are almost as large as their

bottom numbers [denominators].) “What

can you tell me about common fractions

that are close to 1 on the number line?”
2

(Answer: Their top numbers [numerators]

are about half as big as their bottom num-

bers [denominators].) “Common fractions

that are close to 0 can be recognized in

what way?” (Answer: They have a small top

number [numerator] and a large bottom

number [denominator].)

0

01
012

22
012345

44444
0 1 2 3 4 5 6 7 8 9 10 11

88888888888
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32
0

Chapter 13 Developing Understanding of Common and Decimal Fractions 275

ACTIVITY 13.14 Using Benchmarks to Order Fractions

Level: Grades 4–6 • Discuss where these fractions might be placed. Probe for
Setting: Small groups
Objective: Students develop their ability to order common using 1 and 1 (and possibly 1 and 43) as benchmarks.
2 4
fractions.
Materials: Ruler, paper, pencils • Write these common fractions on the board, and direct

• Draw a number line on the board like the one shown each group to place them on a number line like the one
here.
drawn on the board: 87, 12, 32, 32, 38, 14, 78, 21, 1290, 34, 13, 61.
• Ask for a few common fractions from a student
volunteer. • Ask for group volunteers to place one common fraction

01 on the number line you have posted on the board.

• As each group posts a common fraction, ask group mem-
bers to explain how they located it on the number line.
Again, probe for students’ use of benchmarks.

and decimal fraction number lines to conduct a dis- Lorrie: When I look at the points we’ve marked so
covery lesson designed to elicit children’s thoughts far, I see a pattern.
about numbers and infinity.
Teacher: Explain the pattern you see, Lorrie.
Teacher: I started at 21, which is midway between Lorrie: First, we marked 21, then 41, 18, 116, 312, 614.
0 and 1. Then I marked the midpoint between 0 and Roberto: I get it, the bottom number doubles each
12. What point is that? time.
Marcella: It’s 14. Liu: The next one will be 1218.
Teacher: What is the midpoint between 0 and 14? Teacher: What will the next one be?
Ben: It’s 18. Lorrie: It will be 2156.
Teacher: Yes, it is 18. Put your thumbs up if you Lin: Gee, the fractions are getting mighty small.
think you know the next point I’ll mark. [Teacher
Teacher: Let’s leave the common fraction lines
looks around to see who is predicting and calls on and look at the decimal fraction lines. The spaces
between hundredths are too small to separate into
a student.] 10 parts. Imagine that we can separate the space
between zero and one one-hundredth on the num-
Juanita: I think it will be 116. ber line into 10 equal-size parts. What would be the
size of each part?
Teacher: Why do you think it will be 116?
Carlos: One one-thousandth.
Juanita: Because 1 is a half of 18. If you had 2 you
16 16 Teacher: Good. So far, we see a pattern of tenths,
would have the same as 81. hundredths, thousandths. What is the next decimal
fraction for this pattern?
The teacher marks 1 and then 1 on the last num-
16 32 Steve: Ten-thousandths.

bered line and then asks, “What will be the name of Teacher: Does this pattern ever end?

the midpoint on the blank number line beneath the Al: No.

one that shows thirty-seconds?” Teacher: Can you explain why?

Alf: It will be 614. Al: We divided the second number line into 10
equal-size parts, and the third line into hundredths.
Teacher: I’ve run out of space for more points on If the line was bigger, one part of a hundredth line
could be cut into 10 parts to show thousandths.
these lines. Does that mean there are no fractional Even though the parts get too small to see and to

numbers between 1 and 0?
64

Salena: No.

Teacher: How do you know?

Morgan: There has to be a smaller one. It wouldn’t
make sense for them to just stop.

276 Part 2 Mathematical Concepts, Skills, and Problem Solving

show on the number lines, there is always a smaller Discussions of number concepts often lead chil-
decimal fraction than the last one we considered. dren to continue their investigations beyond 5112,
10124, and so on. They may also investigate patterns
Teacher: Good explanation. Now, who will summa- for 31, 15, or some other unit fraction. Some children
rize what we have discovered about common and may be interested in naming and writing the deci-
decimal fractions?
mal fractions for very small fractional numbers.
Lakeesha: There is no “smallest” common frac-
tion and no “smallest” decimal fraction. They go on Such children should be encouraged to write their
forever.
numerals and stories about them in their journals or
Teacher: Can anyone tell me what we call a se-
quence of numbers that never stops? learning logs.

Kareem: It’s called infinity. Equivalent Fractions

Teacher: That’s about right. A sequence of num- Materials used to help students understand common
bers that never stops is an infinite set. There is no
way to count the numbers. We say that the numbers fractions will also help them understand the mean-
can go on forever, or to infinity. Now, let’s look at
one other idea about common and decimal frac- ing of equivalent common fractions. Students can
tions. Do you believe we can count the common
fractions between 0 and 1? use identical-size shapes such as fraction circles to

Josh: I don’t think so, but I’m not sure. see that 1 is equivalent to 42, 36, and 48, as they stack
2
Teacher: Josh is right. Raise your hand if you can
explain to the class why we can’t count the com- pieces for fourths, sixths, and eighths on one-half
mon fractions between 0 and 1.
of the shape. Fraction strips (see Activity 13.12) and
Carlos: We saw that when you marked fractions
from 1 toward 0 they kept getting smaller but never number lines (see Activity 13.13) provide the means
stopped. So, I think that you can never stop stuffing
fractions between 0 and 1. If they don’t stop, there for additional study of equivalent common fractions.
will be no way to count them.
Students can work individually or in small groups
Teacher: That’s correct. We say that there is an
infinite number of common fractions between any to determine the equivalency of common fractions
pair of numbers. Do you think that is true of deci-
mal fractions? illustrated by each device. Encourage children to

Class: Yes! find the pattern that develops for an equivalent class

Teacher: You are thinking about some powerful of fractions. An equivalent class contains common
ideas here. You are learning something about what
infinity means. fractions that are names for a given part of a whole.

Different modes of representing numbers are not The equivalent class for 1 is 12, 42, 36, 84, 5 . . . . When
always evident to every child. One important goal 2 10
is to help children connect the different representa-
tions of numbers and make sense of them. A skill- children examine the common fractions in this set,
ful teacher helps children develop the higher-order
thinking skills needed to participate in discussions they see that the numerator of each successive nu-
that expand their thinking beyond the obvious. A
teacher’s skillful use of models, questions, responses, meral is one greater than the preceding numerator
and acceptance will encourage children to expand
their thinking, as did the children in the vignette. and that each denominator is two greater than the

denominator of the preceding numeral.

E XERCISE

Write a successive sequence of equivalent common
fractions for 13, 51, and 71. What pattern do you see for

each of your sets of equivalent fractions? •••

Ordering Fractions

When children have a good foundational under-
standing of the role of the numerator and denomina-
tor in a common fraction, they are able to order frac-
tions by magnitude without resorting to models or
a number line to compare them. One way they can
order fractions, as suggested in Activity 13.14, is to
use benchmarks. Children can easily tell if a fraction
is greater than 1, so they can use 1 as a benchmark

Chapter 13 Developing Understanding of Common and Decimal Fractions 277

to order the fractions 7 and 1115. Because the numera-
8
15 15
tor in 11 is larger than the denominator, 11 is larger

than 1 ( 1151 ϭ 11 ϩ 4 ). Common fractions can also
11 11
12. 2 95.
be ordered by comparison to Consider 5 and

Children can tell that 5 is greater than 1 by doubling
9 2

the numerator of each fraction and then comparing

the result to its respective denominator. In the case Figure 13.19 Comparing fractions when numerator and
denominator differ by the same amount
of 25, when the numerator is doubled (2 ϫ 2 ϭ 4)
2
the result is less than the denominator of 5, so 5 is

less than 12. In the case of 95, when the numerator is 3 3
11 14
doubled (2 ϫ 5 ϭ 10) the result is greater than the Notice that and have the same numerator.

denominator 9, so 5 is larger than 21. Using the method just discussed, we can conclude
9
3 3 8
When two common fractions have the same de- that 11 Ͼ 14 (Figure 13.19). That means that 11 needs a

nominator, they are easy to compare. For example, larger part to be equal to a whole, so 8 is the smaller
11

given the common fractions 3 and 75, it is easy for of the two fractions.
7
5
children to determine that 7 is the larger common

fraction. The reasoning is that 3 represents only 3
7
5 E XERCISE
parts out of 7, whereas 7 represents 5 parts out of 7
Put these fractions in descending order: 41, 53, 2117, 79,
(Figure 13.18a).
•••11000, 172.
Similarly, when the numerators are the same, chil-

dren can quickly determine the order of fractions.

To compare 3 and 83, children can reason that each
5

common fraction involves 3 parts out of the whole. Rounding Decimal Fractions

In the case of 35, there are 5 pieces to a whole, but A major goal of mathematics is for children to
3 reason as they work with numbers. One aspect of
for 8 there are 8 pieces to the whole. The size of the reasoning is the ability to judge whether answers
make sense. The ability to round decimal fractions
pieces is larger for 3 than for 38; therefore 3 Ͼ 3 (Fig- empowers children to make estimates to determine
5 5 8 whether their answers are reasonable.

ure 13.18b). This line of reasoning can also be used Number lines help children learn to round whole
numbers; they can also be used to learn how to
to compare two fractions, such as 8 and 1141. In this round decimal fractions. The number lines pictured
11 in Figure 13.17 can be extended to give children a
case both fractions are 3 parts short of a whole—131 model that shows a decimal number line that ex-
and 134, respectively. tends beyond the number 1. Paper adding machine
tape is a handy source of paper on which to make
(a) an extended number line. In Activity 13.15 chil-
dren use a cooperative-learning strategy to learn to
(b) round decimal tenths to whole numbers. A number
Figure 13.18 Comparing fractions with (a) the same line that is divided into hundredths can be used in
denominator and (b) the same numerator a similar way to show how to round hundredths to
tenths or to whole numbers.

To round a decimal hundredth to tenths, chil-
dren apply a rule similar to the one they used for
rounding tenths to whole numbers. For example, the
number 0.67 is rounded to 0.70 because it is closer
to 0.70 than 0.60 on the line. The number 0.23 is