GUIDING CHILDREN'S LEARNING OF MATHEMATICS

178 Part 2 Mathematical Concepts, Skills, and Problem Solving

Other Number Concepts Mr. Greene: What rule could we write about add-
ing even numbers?
Patterns Catasha: When you add two even numbers, the
answer is even?
Looking for patterns and relationships among num- Josh: What about adding three even numbers?
bers is fundamental to number sense and algebraic Dahntey: I tried three even numbers, and the
reasoning. Even before efficient algorithms, people answer is even.
studied number lore and the relationships among Mr. Greene: Can you find any examples when our
numbers. Many computational puzzles and games rule for adding even numbers does not work?
are based on relationships between numbers.
Other rules can be derived in a similar manner
Intermediate-grade students can explore a wider by giving examples.
variety of patterns, including increasing and de-
creasing patterns. They encounter these in many • The quotient of two even numbers is always even.
situations as they explore geometry and fractions.
The ancient Greeks thought of numbers as geomet- • The difference between an odd number and an
ric in nature. Square numbers are those numbers even number is an odd number.
that form squares, and cubic numbers form cubes.
For example, 4 tiles form a 2 ϫ 2 square and 9 tiles • The product of two odd numbers is an odd
make a 3 ϫ 3 square; 8 cubes form a 2 ϫ 2 ϫ 2 cube number.
and 27 cubes form a 3 ϫ 3 ϫ 3 cube.
• The sum of one odd number and one even num-
Square numbers: 1 4 9 16 25 36 49 ber is odd.
Cubic numbers: 1 8 27 64 125 216 343
E XERCISE
Another sequence, called the Fibonacci se-
quence, describes the growth of plants and other Give three examples that illustrate rules about
natural phenomena: 1, 1, 2, 3, 5, 8, 13, . . . subtraction of odd numbers from even numbers
and multiplication of two odd numbers. What rule
In the primary grades students recognize odd could you write about the multiplication of even
and even numbers. In the intermediate grades chil-
dren discover odd and even rules for number op- numbers? •••
erations. In the following vignette Mr. Greene pro-
vides examples and asks questions so students can Prime and Composite Numbers
find a rule that will help them think about number
combinations. Work with prime and composite numbers extends
understanding of factors, divisors, and multiples
Mr. Greene: I am going to put some addition encountered in the study of multiplication and di-
examples on the board. I want you to look at the vision. Some numbers have several factors and are
numbers and the answers to see if you can find a called composite numbers. Other numbers that
pattern or rule for each group. have only one set of factors—the number 1 and it-
self—are called prime numbers. Activity 10.12 al-
Group 1 lows students to investigate array patterns and fac-
tors with cubes and disks. A large classroom chart
42 8 12 26 shows numbers from 1 to 30 and identifies numbers
with only one set of factors and numbers with mul-
ϩ 4 ϩ 8 ϩ 14 ϩ 6 ϩ 18 tiple sets of factors.

8 10 22 18 44 A composite number is factored completely
when all the factors are prime numbers. When 18 is
Dahntey: I see a lot of 8’s and 4’s because 4 ϩ 4 is factored, it is expressed as 2 ϫ 3 ϫ 3; 36 is factored
8 and 8 ϩ 14 is 22. as 2 ϫ 2 ϫ 3 ϫ 3. Finding factors by examination is
easy when the product is one of the basic multiplica-
Josh: 4 and 8 are even numbers.

Catasha: I think all the numbers are even
numbers.

Emily: She is right. All the numbers are even, the
addends and the answers.

Chapter 10 Extending Number Concepts and Number Systems 179

ACTIVITY 10.12 Prime and Composite Numbers (Reasoning and Representation)

Level: Grades 3–6 (a)
Setting: Small groups
Objective: Students use arrays to find factors of numbers. 2x3
Materials: Tiles or other counting materials 3x2

• Have students put six tiles in all possible rectangular row 6x1
and column arrangements that they can find. Label the
arrays with the factors of 6. (b) 1x6
1x5
• When students understand the task, have them arrange
sets from 1 through 20 tiles in arrays. 5x1

• Put the results of the student exploration in a table listing
the factors for each number 1 through 20.

• Ask students to examine the table. “Some whole num-
bers—2, 3, 5, 7, 11, 13, 17—have only two arrays, such as
5: 1 ϫ 5, 5 ϫ 1.” “5 ϫ 1 and 1 ϫ 5 are really the same.”
“The arrays are just one line.” “Other numbers have
several arrays.” “They can be arranged in one or more
rectangular patterns as well as in straight lines.”

• Introduce the terms prime numbers for numbers that have
only one set of factors (1 ϫ 5 and 5 ϫ 1) and composite
numbers for numbers with more than one set of factors
(2 ϫ 3, 3 ϫ 2 and 1 ϫ 6 and 6 ϫ 1).

• Ask students what they notice about all the factors of
prime numbers.

Extension

Ask students whether the number 1 is prime or compos-
ite and why. Research the answer on the Internet.

tion facts and has only two factors, such as 4, 6, 15, can extend the sieve process for larger numbers.
and 63. However when factoring 12, children might Some students might search for twin primes, such as
name either 2 ϫ 6 or 3 ϫ 4, but they are not finished; 3 and 5, 5 and 7, 11 and 13, which have only one com-
because one of the factors is not prime, further fac- posite number between them. Challenge students to
toring is needed to find that the prime factorization find other twin primes between 100 and 300. Work
for 12 is 2 ϫ 2 ϫ 3. with the sieve is a good place to incorporate the cal-
culator to reduce the drudgery of calculation and
Factor trees are suitable for larger numbers. Fac- emphasize the mathematical pattern.
tor trees are created by expressing a composite
number in terms of successively smaller factors un- Integers
til all factors are prime numbers. The process is de-
scribed in Activity 10.13, where different factor trees Numbers used for counting discrete quantities
for 24 are developed. are called whole numbers. In other situations
numbers are needed that express values less than 0
Activity 10.14 shows how to find prime and com- as well. The temperature may be below 0 degrees
posite numbers not already known by students us- (Figure 10.7a); Death Valley is lower than sea level,
ing the sieve of Eratosthenes. Eratosthenes, a Greek which is considered 0 (Figure 10.7b); and a check
astronomer and geographer who lived in the third written on insufficient funds can put the checking
century B.C., devised a scheme for separating any account below 0, commonly called in the red (Fig-
set of consecutive whole numbers larger than 1 into ure 10.7c). A football team may lose yardage on a
prime and composite numbers. Interested students

180 Part 2 Mathematical Concepts, Skills, and Problem Solving

Figure 10.7 Negative integers (a) (b) Sea level 0 ft.

–100 ft.

100 –200 ft.
90 –280 ft.
80
70 (c) $50 Ϫ $60 ϭ – $10
60
50 0
40 (d)
30
20 Ϫ30 Ϫ20 Ϫ10
10
0

–10
– 20
– 30

play or penalty, and contestants go “in the hole” on move to the right and decrease to the left. When you
some television quiz programs. move past 0 to the left, the numbers become nega-
tive and their negative value increases. Negative
Positive and negative numbers are part of the temperatures get colder and colder as the tempera-
integer number system. Number lines can be ex- ture moves away from zero: Ϫ20 degrees is colder
tended to the left past 0. The same rules of number than Ϫ10 degrees. Similarly, negative bank balances
sequence, magnitude, and more than and less than get worse as the balance moves down from zero:
apply to negative integers. When comparing whole Ϫ$200 is worse than Ϫ$10.
numbers, children see that numbers increase as you

ACTIVITY 10.13 Factor Trees (Representation)

Level: Grades 3–5 • Ask why you did not use 24 ϫ 1 as a starting place.
Setting: Small groups
Objective: Students use a factor tree to find prime factors. • Ask students to make factor trees with other numbers
Materials: Chalkboard and chalk from 10 to 100, and post them on the board. Write prime
numbers that they find on one side of a poster or bulletin
• Write 24 on the chalkboard. Ask: “What are two numbers board and composite numbers on the other side.
that when multiplied have a product of 24?” Accept 1 ϫ
24, and write it to the side. Ask for another set of factors. 24 24 24
3ϫ8 2 ϫ 12 2 ϫ 12
• Write factors beneath their product. 3ϫ2ϫ4 2ϫ3ϫ4 2ϫ2ϫ6

• Ask if either of the factors can be factored again. 24 24 24
3ϫ8 2 ϫ 12 2 ϫ 12
• Complete one factor tree for 24. 3ϫ2ϫ4 2ϫ3ϫ4 2ϫ2ϫ6
3ϫ2ϫ2ϫ2 2ϫ3ϫ2ϫ2 2ϫ2ϫ2ϫ3
• Write 24 again, and ask if 24 has two other factors not
used in the first example. Complete the second factor
tree.

• Write 24 a third time, and ask for factors. Complete the
third factor tree.

• Ask what students notice about all the factor trees. (An-
swer: They all have the same factors even if they are writ-
ten in different orders. Every composite number has only
one set of prime factors, a rule called the fundamental
theorem of arithmetic.)

Chapter 10 Extending Number Concepts and Number Systems 181

ACTIVITY 10.14 Sieve of Eratosthenes (Reasoning)

Level: Grades 3–5 • Go to 2 on the chart, circle it, and cross out all the mul-
Setting: Small groups tiples of 2 up to 100.
Objective: Students identify composite and prime numbers.
Materials: Hundreds chart (see Black-Line Master 10.1) • Go to 3, circle it, and cross out multiples of 3 on the chart.

• Ask students to put a square around 1 on the chart • Go to 4, and ask why all the multiples of 4 are already
because 1 is a factor of all numbers including itself. It is crossed out. (Answer: They are all multiples of 2.)
neither prime nor composite.
• Go to 5. Ask students whether 5 is prime or composite,
1 2 3 4 5 6 7 8 9 10 and how they know. Circle 5, and cross out the multiples
11 12 13 14 15 16 17 18 19 20 of 5 that remain.
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 • Let students continue circling prime numbers and crossing
41 42 43 44 45 46 47 48 49 50 out multiples. Have them post their hundreds charts with
51 52 53 54 55 56 57 58 59 60 prime numbers circled and other numbers marked out.
61 62 63 64 65 66 67 68 69 70 Have them compare their charts to see if they agree or
71 72 73 74 75 76 77 78 79 80 disagree. After discussion, have students prepare a list of
81 82 83 84 85 86 87 88 89 90 all prime numbers under 100.
91 92 93 94 95 96 97 98 99 100
Variations

• Find prime numbers between 100 and 200, or larger.

• In 2003 the largest prime number was found to contain
2,090,960 digits. Search the Internet for the largest prime
number known.

• Look on the Internet for uses of prime numbers.

In elementary school teachers can introduce 3 red ؉ 3 blue ‫ ؍‬0
the idea of negative number situations with money, ؉3 ؉ ؊3 ‫ ؍‬0
football, temperature, and the number line. Activi-
ties provide background for understanding posi- ϩϪ
tive and negative integers and their symbols, such
as ϩ4 and Ϫ4. The use of these signs may lead to ϩϪ
confusion with addition and subtraction signs. For
this and other reasons, formal work with integers is ϩϪ
now recommended for middle school rather than
elementary school students. Some simple activities
based on pairing positive and negative numbers in-
troduce operations. Red chips are positive numbers
and blue chips are negative integers. The format for
positive and negative numbers should show super-
script plus and minus signs for positive and negative
to distinguish the signs from regular plus and minus
signs for addition and subtraction. Adding three red
and three blue chips creates a sum of zero because
each positive chip and negative chip pair has a sum
of zero. Students may call the chips matter and anti-
matter to show that they cancel each other out.

182 Part 2 Mathematical Concepts, Skills, and Problem Solving

Take-Home Activities

Take-home letters invite parents to extend and ics, 2000), students need to see how numbers are
support student learning. Both of these letters ask used in real life. The first letter suggests ways that
parents to engage children with numbers at home. parents can talk with students about numbers in
Because connections is one of the process stan- the grocery store, on television, and in games. The
dards in Principles and Standards for School Mathe- second letter asks parents to work with students to
matics (National Council of Teachers of Mathemat- complete an extension of a classroom project.

Numbers All Around

Dear Parents,

Your child is learning about counting and numbers at school. Numbers and
counting are also important in the child’s home life. Help your child see how
numbers are used every day by playing games, asking questions, and talking
about numbers. Here are some suggestions.

1. Numbers in the grocery store:
Ask which brand of bread is more expensive.
Find items that are on sale. Ask how much you can save.
Estimate how much you have bought, and compare it to the register total.

2. Numbers on television:
Ask which channels broadcast special shows.
Ask when certain programs begin and end.
Ask how much money people are winning on Wheel of Fortune or
Jeopardy.

3. Numbers in games:
Play card games, such as Battle, that compare number values.
Play card games, such as Gin, that match numbers or put numbers in
sequence.
Play dominoes. You can begin by matching dots and later introduce
scoring.
Play Yahtzee or Junior Yahtzee.

These simple activities can be fun for you and your child. Next month, we
will be having a math game night to introduce more games that you and your
child can play.

Chapter 10 Extending Number Concepts and Number Systems 183

Take-Home Activities

Big Number Hunt

Dear Parents,

For the past three weeks, the fourth- and fifth-graders have been studying
numbers larger than 10,000. Children have learned about millions, billions,
and trillions. Here are some examples we have found:

• Man wins $31,000,000 in lottery.
• The tanker spilled more than 4,000,000 gallons of crude oil; 16,000,000 gal-

lons are still on board.
• The airplane flies at 45,000 feet.

This week everyone is hunting for five large numbers in newspapers, on the
Internet, or from any other source to add to our big number collection. We
will use them for comparing and classifying, for making big number books,
and in rounding and estimating. You can work together with your child to find
big numbers. On Thursday we are planning a big number circus.

184

Summary Using the Technology Center

Number sense is an awareness of numbers in daily life The activities described here (courtesy of Texas Instru-
and an understanding of how they work. Children with ments, Dallas, Texas) are examples of how both pri-
many experiences develop an understanding of numbers mary- and intermediate-grade students can use calcula-
and of the Hindu-Arabic, or base-10, numeration system; tors to enhance their understanding of numbers.
they are able to think about numbers as well as compute
with them. The Hindu-Arabic numeration system is a Calculator Counting
place-value system based on multiples of 10 that can rep-
resent large numbers efficiently. The system, which was Students work with a calculator singly or in pairs. Tell
developed over centuries, also allows the use of efficient students that they are to figure out how to make the cal-
algorithms in computation. culator count by ones. When they have done this, have
them count by twos, fives, tens, or any other number.
After developing basic number concepts up to 20 with Give challenges: “Can you count by fives to 500 in a
objects, pictures, and numerals, children are ready to ex- minute?” Have one child time the other with the second
tend their knowledge to larger numbers in the hundreds or hand on a classroom clock or another timepiece. “Tell
thousands. Through modeling with Unifix cubes, bundled me how many numbers are in each of these sequences:
tongue depressors, base-10 and Cuisenaire materials, and 8, 15, 22, . . . , 113; 10, 19, 28, . . . , 118.” Young children
bean sticks, students can see the relative size of numbers can see numbers “grow” by entering a number—say,
and recognize the structure and rules of the Hindu-Arabic 453—then pressing ϩ, then 1, entering 100, pressing ϩ,
numeration system. The base-10 system accommodates then 1, and so on. Older children can do the same activ-
numbers of all sizes; millions and billions adhere to the ity with a number such as 43,482 and repeatedly adding
same rules as smaller numbers. They can represent large 1,000. Stop the activity periodically to discuss what hap-
numbers in both compact and expanded forms. pens. Ask: “How many times do you press the equals
key to go from 436 to 536 when you repeatedly add 10?
At the same time children learn the number system, How many hundreds are 10 tens?” “How much larger
they develop number awareness and number sense. is 23,481 after you add 100 ten times? Ten hundreds are
They become aware of the many uses of numbers in equal to how many thousands?”
their world and develop flexible thinking about using
numbers. Answers do not always need to be exact. Stu- Wipeout
dents learn that some answers can be “close enough” to
keep their meaning. Rounding and estimation are two This activity can be
skills that allow students to draw reasonable conclusions
about the precision needed for numbers. Other relation- used with place value
ships about numbers and the number system are also ex-
plored: number patterns, prime and composite numbers, in larger numbers.
and positive and negative integers.
Children work singly
Study Questions and Activities
or in pairs. The object
1. Number awareness and number sense are major
goals of instruction in numbers. How aware of num- of the activity is to
bers are you in everyday life? Do you notice how
much groceries cost? how much you tip a server? wipe out a number in
how long you wait on line? how much you save on a
sale? a particular place-

2. Use one of the base-10 materials to represent the fol- value position. For
lowing numbers:
example, when given
a. 128
a calculator showing
b. 478
the number 547, the
c. 397
student is to wipe out
d. 1,153
the 4 tens with one
If you do not have access to a set of commercial
materials, such as Cuisenaire rods or Unifix cubes, subtraction. To do
draw a diagram to show the numbers.
Courtesy of Texas Instruments this, the student must
3. List two examples of when you round numbers and
estimate. How comfortable and skilled are you with understand that the
rounding and estimating?
“4” represents 4 tens,

or 40: 40 is subtracted

from 547, leaving 507.

Numbers in any place-

TI-10 calculator value position other
than the extreme left

or right can be wiped

out. The size of numbers should increase as children

mature; older students can use numbers in the millions.

185

Teacher’s Resources Number sense. Reston, VA: National Council of Teachers
of Mathematics.
Burns, Marilyn. (1994). Math by all means: Place value,
grades 1–2. Sausalito, CA: Math Solutions (distributed by Soaring sequences: Thinking about large numbers. PBS
Cuisenaire Company of America). Video.

Fraser, Don. (1998). Taking the numb out of numbers. For Further Reading
Burlington, Canada: Brendan Kelly.
Fuson, K., Grandau, L., & Sugiyama, P. (2001). Achiev-
Verschaffel, L., Greer, B., & de Corte, E. (2000). Making able numerical understanding for all young children.
sense of word problems. Lisse, Netherlands: Swets and Teaching Children Mathematics 7(9), 522–526.
Zeitlinger.
Fuson and colleagues describe young children’s
Wheatley, G., & Reynolds, A. (1999). Coming to know developmental understanding of number and outline
number: A mathematics activity resource for elementary methods to enhance understanding.
school teachers. Tallahassee, FL: Mathematics Learning.
Lang, F. (2001). What is a “good guess” anyway? Teach-
Children’s Bookshelf ing Children Mathematics 7(8), 462–466.

Clements, Mike. (2006). A million dots. New York: Simon Lang presents procedures and activities that support
and Schuster Children’s Publishing. (Grades 1–4) estimation and reasonableness.

Cuyler, M. (2000). 100th day worries. Riverside, NJ: Ross, S. (2002). Place value: Problem solving and writ-
Simon & Schuster. (Grades 1–3) ten assessment. Teaching Children Mathematics 8(7),
419–423.
Schmandt-Besserat, D. (1999). The history of counting.
New York: William Morrow. (Grades 4–6) Ross describes strategy for assessing place value
through problem solving.
Schroeder, Peter, & Schroeder-Hildebrand, Dag-
mar. (2004). Six million paper clips: The making of a Sakshaug, Lynae. (1998). Counting squares. Teaching
children’s Holocaust memorial. Minneapolis: Kar-ben. Children Mathematics 4(9), 526–529.
(Grades 2–5)
The task of counting squares in pyramid shape leads
Schwartz, David. (1985). How much is a million? New to discoveries about number patterns.
York: Lothrop, Lee & Shepard. (Grades 2–6)
Taylor-Cox, J. (2001). How many marbles in the jar?
Schwartz, David. (2001). On beyond a million: An amaz- Teaching Children Mathematics 8(4), 208–214.
ing math journey. New York Dragonfly Books. (Grades
4–6) Estimation activities demonstrate different types of
estimation.
Technology Resources
Thomas, C. (2000). 100 activities for the 100th day.
Videotapes Teaching Children Mathematics 6(5), 276–280.
Both sides of zero: Playing with positive and negative
numbers. PBS Video. Thomas presents many activities to celebrate the
100th day of school.
Factor ’em in: Exploring factors and multiples. PBS
Video. Zaslavsky, C. (2001). Developing number sense: What
can other cultures tell us? Teaching Children Mathemat-
ics 7(6), 312–319.

Cultural differences in numerical representations
and language help students understand the numbers
we use.



CHAPTER 11

Developing
Number
Operations with
Whole Numbers

s elementary students work with objects, they encounter
problems that require combining and separating them.
They add the value of coins to pay for a snack; they re-
move animals from the farm diorama; they skip-count

the number of shoes in the class by 2’s; and they share
a bag of cookies equally among their friends. Realistic situations such as
these introduce number operations of addition, subtraction, multiplication,
and division. With a strong conceptual base through stories, models with
manipulatives, pictures, and symbolic representations, students build an un-
derstanding of how each operation works and learn the strategies that lead
to computational skill with basic facts. Building conceptual and strategic
understanding makes mastery of the facts a successful first step in computa-
tional fluency. Chapter 12 extends the number operations to larger numbers,
computational algorithms, and estimation strategies.

In this chapter you will read about:
1 Curriculum standards for number operations with whole numbers
2 The importance of problem solving in learning number operations
3 The situations and actions associated with addition and subtraction

and activities to model and develop the concepts
4 Properties of addition and subtraction and their application
5 Strategies and activities for learning addition and subtraction facts

187

188 Part 2 Mathematical Concepts, Skills, and Problem Solving

6 Situations and actions for multiplication and division and ways and
activities to model and develop the concepts

7 Properties of multiplication and division and their uses
8 Interpretation of remainders in division in different situations
9 Strategies for learning multiplication and division facts
10 Guidelines for developing accuracy and speed with number

combinations

Building Number Operations tions were often taught only as memorized facts for
each operation. Then a few dreaded story problems
Addition and subtraction situations are part of ev- were placed at the end of the chapter. Today, prob-
eryday life for children. Examples of the ways that lems are posed in realistic settings that require stu-
children use addition and subtraction every day dents to consider the relationships among numbers.
include: As they solve problems, children learn the four num-
ber operations and the basic facts that are critical
• Adding to determine the number of boys and girls for all computational procedures. When children
in the class. understand how numbers work together, they have
a foundation for large number operations using com-
• Determining the number of blue and red blocks putational strategies such as estimation and reason-
in a tower. ableness, whether they are working on paper or with
calculators.
• Paying for food at the market with bills and get-
ting change. What Students Need to Learn
About Number Operations
• Comparing one’s own allowance to a friend’s
allowance. The NCTM standards for number and operations
identify three broad expectations for students in
Multiplication and division events are also com- prekindergarten through grade 5 (National Council
mon in children’s lives. For instance, in kindergar- of Teachers of Mathematics, 2000):
ten each child gets a juice box and three crackers
during snack time and the total number of crack- 1. To understand numbers, ways of representing
ers must be computed; at the grocery store with a numbers, relationships among numbers, and
parent, children see that each orange costs 50 cents number systems.
and six oranges are purchased; or at home a pizza
is delivered and it is cut into eight pieces.

Experiences with combining, removing, and
sharing provide realistic contexts for number op-
erations. As children tell stories, draw pictures, and
write number sentences, they explore number pat-
terns and relationships leading to properties and
strategies for number combinations. Then students
extend their skill to computational procedures for
larger numbers and continue work on number sense
with rounding, estimation, and reasonableness. In
the elementary grades the goal for children is com-
putational fluency.

The approach to teaching number operations
and number sense has changed in contemporary
mathematics programs. In the past, number opera-

Chapter 11 Developing Number Operations with Whole Numbers 189

2. To understand meanings of operations and how 3. Developing accuracy and speed with basic
facts.
they relate to one another.
4. Extending concepts and skills for each opera-
3. To compute fluently and make reasonable tion with large numbers to gain computational
fluency.
estimates.
The first three steps to proficiency with num-
The complete standards are the following: ber operations occur chiefly in the primary grades
but continue throughout the elementary grades.
NCTM Standards for Number Table 11.1 illustrates how the concepts and skills
and Operations build over the elementary grades. Basic facts are
important for computational fluency, but knowing
Pre-K–2 Expectations when and where the operations are used to solve
In prekindergarten through grade 2 all students should: problems is equally important. In the upper grades
• develop a sense of whole numbers and represent and use of elementary school place-value concepts join
with understanding of addition and subtraction so
them in flexible ways, including relating, composing, and that children extend their understanding of whole-
decomposing numbers; number operations to larger numbers, including esti-
• understand various meanings of addition and subtraction mation, algorithms, calculators, and number sense.
of whole numbers and the relationship between the two Extending whole-number operations to larger num-
operations; bers is found in Chapter 12.
• understand the effects of adding and subtracting whole
numbers; When students understand operations, they are
• understand situations that entail multiplication and di- empowered to solve a wide variety of problems
vision, such as equal groupings of objects and sharing and gain confidence as they attempt more complex
equally; problems in later grades. Fluency and flexibility
• develop and use strategies for whole-number computa- with numbers extend students’ understanding of
tions, with a focus on addition and subtraction; numbers to algebraic situations.
• develop fluency with basic number combinations for ad-
dition and subtraction; Standards and grade-level expectations are guide-
• use a variety of methods and tools to compute, includ- lines for teachers. Individual children, however, learn
ing objects, mental computation, estimation, paper and number operations at different rates and in differ-
pencil, and calculators. ent manners. In a third-grade classroom one child
Grades 3–5 Expectations might count objects for addition, whereas another
In grades 3–5 all students should: might estimate sums in the millions. Teachers face
• understand various meanings of multiplication and the constant issue of balancing the expectations
division; of the curriculum and the needs of individual
• understand the effects of multiplying and dividing whole children.
numbers;
• identify and use relationships between operations,
such as division as the inverse of multiplication, to solve
problems;
• understand and use properties of operations, such as the
distributivity of multiplication over addition;
• develop fluency with basic number combinations for multi-
plication and division and use these combinations to men-
tally compute related problems, such as 30 ϫ 50;
• develop fluency in adding, subtracting, multiplying, and
dividing whole numbers.

Developing proficiency with number operations E XERCISE
proceeds through four interrelated phases:
Compare the elementary mathematics standards
1. Exploring concepts and number combinations in your state curriculum with another state’s cur-
through realistic stories, with materials, and riculum standards. Describe how your state stan-
through representations of situations using pic- dards are similar to or different from the NCTM
tures and number sentences. standards. Do you find a sequence of skill devel-
opment similar to the phases described in this
2. Learning strategies and properties of each op-
eration for number combinations. chapter? •••

190 Part 2 Mathematical Concepts, Skills, and Problem Solving

TABLE 11.1 • Sequence of Concepts and Skills for Addition and Subtraction

Concepts Skills Connections

Number concepts 1–100 Rote counting Addition and subtraction
Rational counting with objects in sets
Representing numbers with pictures and numerals

Numbers 1–1,000 Representing numbers with base-10 materials Algorithms
Exchange rules and games Estimation
Regrouping and renaming

Numbers larger than 1,000 Representing numbers with numerals Algorithms
in the base-10 system

Learning names of large numbers and realistic
situations for their use

Visualizing larger numbers

Operation of addition Stories and actions for joining sets Problem solving,
Representing addition with materials, pictures, multiplication

and number sentences

Operation of subtraction Stories and actions for separating sets Problem solving, division
Stories and actions for whole-part situations Problem solving, fractions
Stories and actions for comparison Problem solving,

situations measurement
Stories and actions for completion Problem solving,

situations open sentences
Representing subtraction with materials,

pictures, and number sentences

Basic facts for addition Finding and using strategies Estimation and
and subtraction for basic facts reasonableness

Recognizing arithmetic rules and laws Algebraic rules and relations
Achieving accuracy and speed with basic facts Mental computation

Addition and subtraction Story situations and actions with larger numbers Problem solving
operations with larger Developing algorithms with and without regrouping Computational fluency,
numbers
with materials (to 1,000) and symbols mental computation
Estimation Reasonableness
Using technology: calculators and computer Problem solving,

reasonableness

What Teachers Need to Know • Heather has 37 books in her library. She receives
About Addition and Subtraction five books as presents for her birthday.

As teachers begin their work with addition and sub- • The school collection for earthquake victims was
traction, they introduce students to stories that illus- $149 on Friday and $126 on Saturday.
trate the situations, meanings, and actions associ-
ated with the operations. The natural question is how many or how much is
in the joined set. In each situation the total, or sum,
Addition is the action of joining two or more sets. is found by combining the number of pencils, the
number of books, or the amount of money collected
• Juan has four red pencils and three blue pencils. on Friday and Saturday. The numbers related to each
set being joined are addends.

In contrast to addition with one action, subtrac-
tion is used to solve four problem situations.

1. Takeaway: Removing part of a set.

2. Whole-part-part: Separating a set into subsets.

Chapter 11 Developing Number Operations with Whole Numbers 191

3. Comparison: Showing the difference between • Deidre has 17 stuffed animals. Fourteen are bears.
two sets. • Sally has 48 snapdragons. Thirty of them are

4. Completion: Finding the missing part needed to white. The rest are yellow.
finish a set. • Darius counted 114 people at the family reunion.
Takeaway subtraction is used when part of an
Forty-nine had his same last name.
original set is moved, lost, eaten, or spent. Whole-part-part subtraction identifies membership
• Jeff collects wheelie cars. He had 16, but traded 3 in two subgroups that are included in the whole
group. The number in one part of the whole is
of them for a new track. known, and the question posed is how many belong
in the other part. In whole-part-part stories no items
• Janyce had $94 saved from birthday money. She are removed or lost, as in takeaway situations.
spent $33 on new shoes, $14 for a new top, and
$27 for a new skirt. Comparison subtraction, not surprisingly,
compares the size of two sets or the measure of two
• Jamal had 36 cookies. He gave two cookies each objects. The quantity of both sets or measurement
to 12 friends. of both objects is known.
In takeaway situations the question or problem • In the NBA game the shortest player is 5 feet,

asks how much is left or how many are left after 3 inches, and the tallest player is 7 feet, 6 inches.
part of the set is removed. The answer is called the
remainder. 7’6’’

Whole-part-part subtraction identifies the 23 5’3’’
size of a subgroup within a larger group. The whole
group has a common characteristic, but parts or 18
subgroups have distinct characteristics.
• The class has 25 children. Fourteen are girls. • On Friday, $149 was collected for flood victims;
on Saturday, $126 was collected.

• The circus put on two performances. The matinee
was attended by 8,958 people, and the evening
performance attracted 9,348 people.

The comparison question in subtraction asks how
much larger or how much smaller one set is than the
other. “What is the difference?” is another way of
expressing comparison between the two sets.

Completion is similar to comparison in that two
sets are being compared. However, in completion
situations the comparison is between an existing set
and a desired set or between an incomplete set and
a completed set.

192 Part 2 Mathematical Concepts, Skills, and Problem Solving

• Saundra has saved $9. The CD she $ • Throw six beanbags in the basket. Now throw
wants costs $16. $ two more. Count how many beanbags are in the
$ basket now.
• Shaeffer is collecting state quarters. He $
already has 23. $ • You have nine leaves in your box. When you take
$ three out, how many are left in the box?
• Mr. Lopez is making lemonade for the $
party. The recipe calls for 3 cups of • One Unifix tower has nine red blocks, and an-
water for each can of concentrate. He other tower has six green blocks. Which one is
used 6 cans of concentrate and has put taller? How much taller is it?
in 10 cups of water so far.
• We have 12 napkins and 15 children. How many
The question for completion stories is how $ more napkins do we need?
Many children’s books engage students with
many more are needed or how much more $
is needed. Completion is subtraction be- numbers in active ways as they hear the stories,
count pictures, and search for pictures that comple-
cause the total $ ment the text. A walk-on number line models addi-
tion and subtraction kinesthetically. Addition is a
MISCONCEPTION and one addend $ forward movement on the number line, and subtrac-
are known, and $ tion reverses the action.
Many students have dif- the other ad- $
ficulty with the missing- dend repre- $ Introducing Addition
addend form. First, the sents the part $
problem is subtraction of the set that is Addition is a process of combining sets of objects
but is written as addition. needed or miss- and is introduced through story situations that pose
Second, many children a problem to be solved. After hearing the number
do not understand the problem, children act out the story with physical ob-
jects and find the results.
meaning of the equal sign ing. Completion $ • I had two apples in a bowl. Then I put in three
as balancing the two sides problems can
of the number sentences. oranges.
Students who have had be written in After actions with physical models are developed,
experience with balanc- the same stories can be represented with simple
subtraction form, for ex- pictures or diagrams. Then the situation is recorded
symbolically with numerals and mathematical
ing scales will be more ample, 9 Ϫ 5 ϭ ?, but are signs.
• Two apples and three oranges are five pieces of
comfortable with adding sometimes written as ad-
numbers to either side of dition sentences, 5 ϩ ? ϭ fruit: 2 ϩ 3 ϭ 5.
the equation to achieve 9, called the missing-
balance. addend form. This process is repeated with different stories. Stu-
dents gain understanding and confidence with ad-
Developing Concepts
of Addition and Subtraction

Understanding of addition and subtraction concepts
and procedures develops through children’s infor-
mal experiences with numbers. During play, chil-
dren share cookies, count blocks, compare heights
and distances, complete sets, and classify objects by
attributes. Without rich mathematical experiences
children have a weak foundation for mathemati-
cal concepts and skills. Early childhood teachers
provide many informal experiences that help build
children’s experiential background for numbers and
operations. Children’s intuitive understanding of ad-
dition and subtraction builds on counting skills to
10 and beyond.

Chapter 11 Developing Number Operations with Whole Numbers 193

ACTIVITY 11.1 Solving Problems with Addition (Representation)

Level: Grades K–2 • Repeat with other familiar objects, such as books and
pencils.
Setting: Small group or whole class
• Have children work in pairs with math box materials to
Objective: Students demonstrate addition by joining objects con- share “joining” stories. Take turns making up stories.
tained in two or more groups.
• Introduce the combining board for addition. Students put
Materials: Stuffed animals (or other suitable objects), books, math the objects in the two rectangles and pull them down into
box materials the larger rectangle.

• Begin with a story about some stuffed toys: “I like to col-
lect stuffed animals. I recently went looking for them at
garage sales. I found these two at one house (show two
animals) and these three at another house (show three
more animals). How many stuffed animals did I find?”

• Ask students to tell how they determined the answer.
They might say, “I counted,” “I counted two more, begin-
ning at 3,” “I just looked and knew.”

ACTIVITY 11.2 More Cats (Representation and Communication)

Level: Grades K–2 • As students are ready, write addition sentences as each
new cat or cats arrive (e.g., 6 ϩ 2 ϭ 8).
Setting: Whole group
• Have children tell a progressive story, real or imaginary,
Objective: Students model addition. about pets (e.g., “I had one frog, then two alligators
followed me home.”). Each child can add new animals to
Materials: So Many Cats! by Beatrice Schenk de Regniers, illus- the story. Children may model stories using pictures or
trated by Ellen Weiss (New York: Clarion Books 1985); pictures of counters and write addition sentences.
cats; magnetic numbers and symbols
Extension
So Many Cats! describes how a family collected cats.
They started with one, but more cats arrived, and none • Invent new stories about cats that come or leave. Ask the
could be resisted. After each new arrival, the cats are children to model how many cats there are as you tell the
counted again. The book ends with yet another cat at the story. Begin with one or two cats arriving or leaving at a
window. time, and progress to larger numbers. Discuss how many
cat eyes or cat paws are in the house with the addition or
• Read through the book once. On the second reading have subtraction of cats as a foundation for multiplication.
children count each time new cats appear. Pictures of cats
can be placed on the board rail to show the number of
cats.

• Talk about the progressive accumulation of cats. “They
had three, and three more came.” “They had six, and two
more came.”

dition represented in physical, pictorial, oral, and Schenk de Regniers, illustrates addition as more and
written forms that will lead to images of addition more cats arrive at a house.
and mental operations.
Concrete materials and realistic situations for ad-
Activity 11.1 presents addition situations with fa- dition and subtraction allow the teacher to draw on
miliar objects. Story problems introduce concepts the environment and experience of students from
by having students act out the situation described. varying backgrounds. Stories can be personalized
Activity 11.2, based on So Many Cats! by Beatrice with familiar names and situations. Some students

194 Part 2 Mathematical Concepts, Skills, and Problem Solving

advance quickly to more symbolic representations; Many other activities demonstrate addition.
others remain at the concrete level longer. Children Beans painted different colors are good for explor-
need not be hurried to represent addition and sub- ing number combinations (Figure 11.2a). One child
traction with symbols because they can use num- puts seven beans painted red and blue in a cup and
ber combinations cards, such as 2 ϩ 3, to label the dumps them out on a paper saucer. The second
joined sets. child finds and reads the number sentence card for
the combination shown by the red and blue beans:
After modeling addition stories and actions with 4 ϩ 3 ϭ 7. Dominoes also represent addition com-
concrete materials, introduce the addition sentence binations up to 6 ϩ 6 (Figure 11.2b). Double domi-
2 ϩ 3 ϭ 5, read “two plus three equals five.” When noes represent addition facts up to 12 ϩ 12. With
written below the word sentence, the plus sign takes Cuisenaire rods a child can select one length rod
the place of “and” and the equal sign substitutes and build all the two-rod combinations of the same
for “are” or “equals.” Working with objects, pictures, length. Combinations for the dark green rod are
and number cards prepares students to write addi- shown in Figure 11.2c. Sentences based on color
tion sentences alone or with a partner. One child (yellow ϩ white ϭ dark green) or number sentences
might separate beads strung on a card or wire into (5 ϩ 1 ϭ 6) describe the combinations. As children
two sets, and the other would respond “Three plus are exploring the concept of addition and recording
five equals eight” or write “3 ϩ 5 ϭ 8” (Figure 11.1). their findings, the teacher can introduce the terms
addend and sum.

(a)

3+5=8

Bead cards for representing addition
combinations

Magnetic objects and a pan balance are sug- (b)
gested in Activity 11.3 for modeling addition and in-
troducing the addition sentence with the equal sign. (c)
This activity also develops the algebraic concept of Figure 11.2 Addition demonstrations
equality. Although adults understand that the equal
sign means balanced or equal, many children do
not understand that the equal sign means that the
number combinations on each side of the equal
sign are the same. Their understanding is more “do
it now” rather than “is the same as.” In Activity 11.4
the teacher demonstrates writing addition sentences
with the equal sign.

Chapter 11 Developing Number Operations with Whole Numbers 195

ACTIVITY 11.3 Introducing the Equal Sign (Problem Solving and Representation)

Level: Grades K–2 number sentence cards or write number sentences for
their discoveries.
Setting: Small groups or whole class
• Place the pan balance in a learning center for students to
Objective: Students represent joining objects in two or more sets experiment with different combinations.
with the addition sentence.
• An inexpensive number scale can also be purchased that
Materials: Counting objects such as bears; pan balance; magnetic has weighted numerals.
board objects, such as animal outlines or flowers; large magne-
tized numerals and equal sign for the magnetic board Children can explore balance by hanging 3 and 4 on one
side of the scale and balancing it with 6 and 1. Make
• Place several uniformed-weighted objects, such as small number sentence cards or a whiteboard available for
counting bears, on one side of a pan balance. Ask children recording.
what is needed to make the two sides level. They will
answer, “Put some bears on the other side.” Extension

• Put bears on the other side of the pan balance one at a • Add the medium-size and large bears from sets of count-
time so that gradually the balance is level. ing bears to encourage exploration of equivalence by
weight. This introduces the element of variability into the
• Count the bears on each side to establish equality of process because the bears have different weights. One
number. large bear is equal to or weighs the same as three small
bears.
• Introduce the equal sign to children, and explain that the
equal sign indicates that the same amount is on both
sides of the equal sign in the equation.

• Ask children to make up and share other addition stories
with objects or with the pan balance. They can select

ACTIVITY 11.4 Addition Sentence (Representation)

Level: Grades K–2 • Place a plus sign between the 3 and the 4, and read
Setting: Whole class and pairs “Three cars and four trucks.” Ask what the plus sign
Objective: Students represent joining objects in two or more sets means.

with the addition sentence using an equal sign. • Ask how many vehicles were on the highway. Place an
Materials: Various counters, such as interlocking cubes or two- equal sign and a 7 to complete the addition sentence,
and read “Three cars and four trucks are equal to seven
color counters; combining board from Activity 11.1; number vehicles.” Ask students what the equal sign means. (An-
sentence cards or whiteboards with markers. swer: 3 plus 4 is the same as 7.) Reply, “Seven is another
name for three plus four.”
• Place three cars and four trucks on the magnetic board
and tell a story about them. “Three red cars and four • Reduce the number sentence to 3 ϩ 4 ϭ 7, and read
trucks were on the highway.” Tell the students that a “Three plus four equals seven.”
number sentence represents the same story. Have one
student put a 3 beneath the cars and a 4 beneath the • Ask students to work together to create more number
trucks. stories and sentences. They may use the combining board
from Activity 11.1. Ask them to find cards that match the
addition stories they are telling or to write number sen-
tences on their whiteboards.

196 Part 2 Mathematical Concepts, Skills, and Problem Solving

Place-value material activities (discussed in Introducing Subtraction
Chapter 10) can continue when operations are in-
troduced. Number combinations with sums greater Subtraction, like addition, is introduced with real-
than 10 should not present any great difficulty as istic stories and is modeled with real objects and
children count to 20, 30, and more. Children see materials such as chips, blocks, and Unifix cubes.
that the numbers 11–19 are combinations of ten and Takeaway stories are often the first of four subtrac-
ones represented with Cuisenaire rods, bundles of tion actions presented.
stir sticks, and bean materials. With Cuisenaire rods
a dark green (dg) rod and a black (bk) rod make a • I have six books. If I give two to Nia, how many
train equal in length to a train made with an orange books will I keep?
(o) rod and three white (w) rods (Figure 11.3).
As children act out this story, ask how many books
Figure 11.3 Cuisenaire train for 13 each has. Use of varied stories, objects, and num-
bers allows students to describe their actions and
dark green ϩ black ϭ orange ϩ white ϩ white ϩ white the results in many contexts.
dg ϩ bk ϭ o ϩ 3w
6 ϩ 7 ϭ 10 ϩ 3 • Eight elephants were on the shore; three waded
into the lake for a drink. How many were left on
When six loose beans and eight loose beans are the shore?
joined, ten beans are exchanged for one bean stick
and four loose beans remain. Children can also illustrate stories with simple pic-
tures and record the results with a number sentence
Teachers assess whether students understand (8 Ϫ 3 ϭ 5). As children make up new stories for
the concept of addition by observing and question- subtraction, they see the relation between action
ing children as they work with materials, draw pic- and notation.
tures, tell stories, and write number sentences. A
class checklist or rating scale in Figure 11.4 keeps a Subtraction is the inverse operation for addi-
weekly record of student progress over time. tion; takeaway subtraction “undoes” addition. Other
inverse operations can be modeled by opening and
E XERCISE closing a door or by turning the lights on and off. Ac-
tivities for subtraction typically follow introduction
Based on Figure 11.4, what conclusions can you of addition, but the connection between operations
draw about each child’s conceptual understanding is easily modeled as sets are joined and separated.
of addition? What does this information suggest for The inverse relationship between addition and sub-
traction can be modeled with stories, objects, and
instruction? ••• pictures.

• Six adult elephants and two baby elephants were
at the watering hole. The baby elephants got full
and left the watering hole. How many were still
drinking?

1—not yet Tell Addition Story 2—developing Draw Pictures 3—proficient
Student Name 1222 Model Addition 1223 Write Number Sentence
Lilith 1222 1122
Anthony 1223 2222 1223 1112
Veronica 1223 1112 1233 1111
Josephina 1233 1223 1333 1223
Sandy 1233 1233
1333 1233

Figure 11.4 Class assessment checklist on addition and number sentences

Chapter 11 Developing Number Operations with Whole Numbers 197

A large domino drawn on a file folder demon- Whole-part-part subtraction involves a whole set
strates the inverse relationship and resulting num- that is divided into subsets by some attribute.
ber sentences by folding back one side of the folder
at a time. • Bill saw eight airplanes fly overhead. Five of the
planes were painted red. How many of the planes
were not red?

2ϩ5ϭ7 7Ϫ5ϭ2 7Ϫ2ϭ5

Children become familiar with mathematics Plastic airplanes or counters or triangles can be
symbols and develop an understanding of inverse used to represent the eight airplanes with five of
operations as they connect the actions and their them red. Children talk through the problem by de-
meanings. The plus sign indicates joining and is scribing what they see. “I see eight airplanes and
read “plus”; the minus sign indicates separation in five are red; three are not red.” Sorting activities
the takeaway story and is read as “minus.” are also a good time to highlight whole-part-part
subtraction.
Takeaway is one of four subtraction situations
and should not be used as the name for the mi- • You have 15 buttons and 8 are gold. How many
nus sign. The other subtraction situations are are not gold?
whole-part-part, comparison, and completion. Just
as with addition and takeaway, children’s expe- • There are 22 boys and girls here today. Let’s count
riences provide context for stories that are told, the boys. How many girls are here?
acted out, modeled, and recorded. When all types
of subtraction stories are developed, students gain After several examples, children determine that
a broader understanding of subtraction. Students they can write a subtraction number sentence for
who have only been introduced to takeaway stories the stories. The sum is the whole and the addends
are frustrated when confronted by other subtraction are the two parts. Activity 11.5 describes how to
types in a textbook or on tests.

ACTIVITY 11.5 How Many? (Representation)

Level: Grades K–2 • Following the initial presentation, read the book again
and ask an additional question: “How many clouds are
Setting: Small groups or whole group not big and fluffy?”

Objective: Students model whole-part-part subtraction. • Ask students to tell the total number of clouds (8), the
number of big, fluffy clouds (3), and the number of clouds
Materials: How Many Snails? by Paul Giganti, illustrated by Donald that are not big and fluffy (5).
Crew (New York: Greenwillow, 1988)
• Write the number sentence 8 Ϫ 3 ϭ 5. Ask what each
How Many Snails?—a picture counting book with pat- number represents.
terned text—has two-page pictures of dogs, or snails, or
clouds with different characteristics. Children are asked • After reading several more pages, ask students to make
to count the number of dogs, the number of spotted their own picture for a page for a classroom book.
dogs, and the number of spotted dogs with their tongues Depending on their level, children may draw a picture
out. The idea of whole-part-part is introduced. or include the repetitive text on their own page. Share
picture pages with the group, post them on the bulletin
• The first encounter with the text will probably be as a board, and make a classroom book.
counting book. The picture and text invite student partici-
pation; read one display at a time with students count-
ing the clouds (the whole group) and big, fluffy clouds
(subgroup).

198 Part 2 Mathematical Concepts, Skills, and Problem Solving

introduce whole-part-part subtraction using the six play dollar bills or counters on a magnetic board.
book How Many Snails? by Paul Giganti. Ask, “How many dollars does Miguel need for the
CD?” (Answer: 6.) Ask, “How many dollars does
Children have daily experiences comparing ages, Miguel have?” (Answer: 9.) Let students count-on,
amounts, and lengths. Their language already in- “1, 2, 3” or “7, 8, 9,” as you place the three more dol-
cludes comparison words such as older, younger, lars or counters on the board. Finally, ask how many
bigger, smaller, taller, shorter, more, and less. Pre- were added. (Answer: 3.) Activity 11.6 describes the
school children are often asked whether one set is counting-on strategy to solve completion situations.
bigger or smaller. Such comparison experiences and
vocabulary provide the background for comparison Balancing is another strategy for the comple-
subtraction. tion type of subtraction. Put nine disks on one side
of a pan balance and six on the other side (Fig-
• Antoinette has 12 baseball cards; her brother has ure 11.5). Ask student to estimate how many more
9. How many more cards does Antoinette have disks balance the scale, and count as you add disks.
than her brother? The number sentence can be written either as sub-
traction, 9 Ϫ 6 ϭ 3, or as addition, 6 ϩ 3 ϭ 9. The
23 A 45 10 43 12 W 11 26 2 5 19 missing addend is an algebraic equation form that
shows that something is missing and needs to be
44 20 21 16 22 S 33 24 D balanced: 6 ϩ ? ϭ 9.

Show two sets of cards, and ask which set has more Completion subtraction with a balance scale
cards in it (the set with 12 cards) and which has
fewer (the set with 9 cards). Matching the cards one As important as modeling addition and sub-
to one shows three unmatched cards in the larger traction with manipulatives is, activities alone do
set. Count each set and ask, “How many more cards not guarantee meaningful learning. Like any tool,
does Antoinette have? How can we write a subtrac- manipulatives are the means rather the ends of an
tion sentence to show how many more cards Antoi- activity. Discourse between students, and with a
nette has.” Write two statements: teacher, is essential for scaffolding mathematical
ideas. Mathematical conversations encourage stu-
Antoinette’s cards Ϫ brother’s cards ϭ difference dents to transfer learning from short-term to long-
12 Ϫ 9 ϭ 3 term memory. Drawing the subtraction stories is
an important problem-solving strategy that helps
The fourth subtraction situation, completion, students understand how subtraction works. Each
is similar to comparison, but children often find it story situation demands a slightly different picture.
more difficult to understand. In comparison children Students may need help to draw the story rather than
see two existing sets, but completion compares an the number sentence. In takeaway or whole-part-
existing set with a desired set. The existing set is in- part subtraction the student draws the total number
complete and the problem or aim for the student is and identifies a part that is removed or a part that
to complete the set. belongs to a subgroup of the total. For comparison
• Miguel has saved $6 for a CD. The CD costs $9. and completion problems children model or draw
• Ari can get a free kite by saving nine lids from her two sets and show how they compare or measure
with each other. Students may want to draw both
favorite yogurt. She already has six lids. the sum and the addend. Help students depict the
story first; then write the number sentence as a sum-
Counting-on is a successful approach for comple- mary of the story situation.
tion problems. After telling the story of Miguel, put

Chapter 11 Developing Number Operations with Whole Numbers 199

ACTIVITY 11.6 Counting On (Problem Solving and Reasoning)

Level: Grades 1–3 blank. Ask what the five stands for, and the three. (An-
swer: She has five; she needs three more.) Use the card to
Setting: Small groups illustrate how to count on from 5: “6, 7, 8.”

Objective: Students use counting-on strategies to learn subtraction • Have students model the situation with counters or beans.
facts. Each child counts five showing one color and three of
another color.
Materials: Counting-on cards, counters, or painted lima beans
• Follow the action by having children say the appropriate
• The counting-on strategy is one way to have children sentence for the combination: “She wants eight, she
conceptualize how many more of something are needed. has five, she needs three more; eight minus five equals
It is best used when the difference between two numbers three.”
is relatively small, such as in 9 Ϫ 2 ϭ 7.
• Repeat with similar stories, counting-on cards, and
• Tell a story: “Janell has $5, but the CD she wants costs manipulatives.
$8. How much more money does she need?” Show a
counting-on card with eight circles, five filled in and three

Source: Adapted from Fennell (1992, p. 25).

“5, . . . , 6, 7, 8” Extension

Part I know • With beans (or other objects) and small cups, label each
cup with numerals 1 through 18. Put the corresponding
number of beans in each cup. Remove a few beans from
the cup, and put them on the table. Ask how many more
counters are in the cup to complete the number in the set.

• Six children were at the party. Two went home. Vertical Notation
If the children start by writing 6 Ϫ 2, the picture they
draw may not accurately show the story. Pictures are Early in learning number operations, mathematical
a problem-solving strategy that students use to un- sentences are recorded horizontally, like a word sen-
derstand the problem and develop a plan. Starting tence. The horizontal form has two benefits:
with addition and subtraction situations, children
learn that accurate representation of the story is im- 1. The order of numerals in a sentence is the same
portant (Figure 11.6). as the verbal description and reinforces the
meaning of a problem story. Joining three
Incorrect representation of 6 Ϫ 2 apples and four apples is described as “I had
as a takeaway problem. three apples and got four more. I now have
seven apples.” The mathematical sentence is
3 ϩ 4 ϭ 7.

2. Horizontal notation is later used for formulas
and algebraic expressions.

Making the transition is not difficult when attention
is paid to the meaning and order of the stories being
recorded. Compare two addition situations.

Correct representation of 6 Ϫ 2 • Four parents were watching soccer practice. Then
as a takeaway problem. six more joined them. 4 ϩ 6 ϭ ?

Figure 11.6 Children’s drawings of subtraction • Six parents were sitting in the stands watching
soccer. Four more joined them. 6 ϩ 4 ϭ ?

200 Part 2 Mathematical Concepts, Skills, and Problem Solving

When the horizontal number sentence and its mean- What Teachers Need to Know
ing are established, students are introduced to the About Properties of Addition
vertical, or stacked, notation form. The number sen- and Subtraction
tences 4 ϩ 6 ϭ 10 and 6 ϩ 4 ϭ 10, respectively, are
written in stacked form: As students work with addition and subtraction,
the properties of the operations should be empha-
46 sized because they are so important in learning the
ϩ6 ϩ4 basic facts. The commutative property of addi-
10 10 tion states that the order of the addends does not
affect the sum.
Preserving the order of the numbers from hori-
zontal sentences to stacked forms reinforces the Commutative law: 2 ϩ 3 ϭ 3 ϩ 2
logic of the story. A magnetic board with moveable n1 ϩ n2 ϭ n2 ϩ n1
numbers allows for easy translation of the numbers
from horizontal to vertical notation, as seen in Activ- Subtraction is not commutative: 7 Ϫ 6 is not equal
ity 11.7. to 6 Ϫ 7.

Order in subtraction sentences is more critical The associative property of addition applies
because subtraction is not commutative. to three or more addends. Addition is a binary op-
eration involving two addends. When working with
• Four parents were watching soccer practice. Two three or more addends, two are added, then an-
had to leave early. 4 Ϫ 2 ϭ 2. other addend is added, and another, until the sum
is determined. If the problem shows the associative
4 property, the order in which pairs of addends are
Ϫ2 added does not change the sum. Subtraction is not
associative.
2
Associative law: 8 ϩ (2 ϩ 3) ϭ (8 ϩ 2) ϩ 3
Although 2 Ϫ 4 is a valid sentence with negative in- a ϩ (b ϩ c) ϭ (a ϩ b) ϩ c
tegers, it does not describe this story.

Simple addition and subtraction problems can
be presented in either form; however, as students
work with larger numbers, alignment of place values
shows the advantage of vertical notation.

ACTIVITY 11.7 Vertical Form (Representation)

Level: Grades 1 and 2 (Answer: No.) Ask what has changed. (Answer: The
arrangement.)
Setting: Small groups or whole class
• Transform the number sentence from horizontal to vertical
Objective: Students use horizontal and vertical notation for also. Put the numerals alongside the shapes with the plus
addition. sign to the left of the bottom numeral. Read the sentence:
“Two plus four equals six.”
Materials: Magnetic board shapes; numerals; ϩ, Ϫ, and ϭ signs
• Tell other stories and have students represent them with
• Tell the following story: “Two children were playing kick magnetic shapes and numbers in both horizontal to verti-
ball and four more joined the game.” Arrange two sets cal form.
of shapes in a horizontal row on a magnetic board. Ask
children to use magnetic numerals and symbols to write Assessment
the number sentence for the objects. Read the sentence:
“Two plus four equals six.” • Ask children to draw pictures for an addition and a sub-
traction story, and record the numbers for the problem in
• Rearrange the shapes to form a vertical column. Put horizontal and vertical form.
two shapes at the top and the other four beneath.
Ask whether the number of shapes has changed.

Chapter 11 Developing Number Operations with Whole Numbers 201

of the strategies students are learning. Some facts

are learned with several strategies so that students

choose the best strategy for their own learning.

The commutative law for addition means that

the order of the addends does not change the sum:

When adding five addends, such as 4 ϩ 5 ϩ 3 ϩ 6 3 ϩ 6 and 6 ϩ 3 have the same sum. Because each
ϩ 7, the commutative and associative laws work to-
gether to make addition easier by finding combina- fact is paired with its reverse, students who under-
tions of 10: 4 ϩ 6 is 10; 7 ϩ 3 is 10; 10 ϩ 10 ϩ 5 is 25.
Looking for easy combinations of 10, 100, or 1,000 is stand the commutative law can use knowledge of
a mental computation strategy called compatible
numbers. one fact to learn its partner. The strategy is illustrated

Zero is the identity element for both addition by rotating a Unifix cube train or a domino. “Turn-
and subtraction because adding or subtracting zero
does not change the sum. When working with con- around facts” or “flip-
crete materials, students model stories with zero and
find that zero does not change the sum regardless of flop facts” are other
the size of the number.
ϩ 0 12 3 4 5 67 8 9

0 names for the commu-

1 tative facts in addition.

2 Figure 11.8 shows the

3

symmetrical relation

4

5 of commutative facts.

6 Students should also

Identity element: 5ϩ0ϭ5 72 Ϫ 0 ϭ 72 7 understand that the
nϩ0ϭn nϪ0ϭn
8

9 commutative law does

not apply to subtraction

Learning Strategies for Addition Figure 11.7 Blank addition because 9 Ϫ 6 does not
and Subtraction Facts table equal 6 Ϫ 9.

As children gain understanding and confidence The identity element
with addition and subtraction, the emphasis of in-
struction moves toward learning number combina- ϩ 0 1 2 3 4 5 6 7 8 9 of zero for addition is
tions. Children who learn addition and subtraction
conceptually with realistic stories and materials 0 demonstrated by put-
have already encountered most or all combinations. 1 ting three cubes in one
Learning strategically is more efficient than memo-
rization alone and has longer term benefits for num- 2
ber sense. Instead of learning 100 addition and 100
subtraction combinations as isolated facts, children 3 hand and leaving the
learn a strategy, generalization, or rule that yields
many facts. Learning properties and rules empha- 4 other hand empty. Put-
sizes how numbers work and fosters number sense
and mental computation. 5 ting the hands together

The 100 basic addition facts are all the combina- 6
tions of single-digit addends from 0 ϩ 0 ϭ 0 to 9 ϩ 9
ϭ 18, although some teachers or state standards ex- shows that adding
tend number combinations up to 12 ϩ 12 ϭ 24. The
subtraction facts are the inverse of the 100 addition 7
facts. The basic facts are shown on the addition-sub-
traction table in Figure 11.8. As students learn a new 8 “three plus no more” is
rule or generalization, they can fill in the 10 ϫ 10 ad-
dition and subtraction fact table (Figure 11.7, Black- 9 three. After several ex-
Line Master 11.1). The fact table becomes a record
Figure 11.8 Addition table amples, ask students to
with commutative facts develop a rule for add-
ing zero. They will say

“adding zero doesn’t

ϩ 0 12 3 4 5 67 8 9 change the answer” or
0 similar phrasing. The
1 “plus 0” rule accounts
2

3 for 19 facts on the addi-
4 tion table (Figure 11.9).

5

6 Counting-on is a

7 particularly effective

8 strategy for adding

9

one or two, and some

Figure 11.9 Strategy of plus students may even use

0, plus 1, and plus 2 counting-on for adding

202 Part 2 Mathematical Concepts, Skills, and Problem Solving

three. Model the “plus 1” rule with four cubes in one Near doubles or neighbor facts include com-
hand and one cube the other. After seeing further
examples, such as 5 ϩ 1, 8 ϩ 1, and 2 ϩ 1, students binations such as 8 ϩ 9 and 7 ϩ 6 that may be trouble-
will generalize that “adding 1 is just counting to the
next number.” The “plus 2” and “plus 3” rules are ex- some for some children. The neighbor strategy builds
tensions. The “plus 1” strategy yields 17 basic facts,
and the “plus 2” rule gives 15 more facts (Figure on the doubles strategy. Display six red counters and
11.9). With only three strategies of “plus 0,” “plus 1,”
and “plus 2,” students have strategies for almost half six blue counters, and ask for the total. Place one
the addition facts.
more red counter, and ask how many there are now.
The associative law is an important strategy be-
cause it encourages students to think flexibly about After similar examples using a double fact plus one,
combinations and to use combinations that they
already know as building blocks. If students know many children will notice, “It’s just one more than
4 ϩ 2 but have problems with 4 ϩ 3, they split the fact
and recombine it as 4 ϩ 2 ϩ 1. Activity 11.8 is a les- the double fact.” Ask children to show how neighbor
son on the commutative and associative properties.
facts line up beside the
Most children learn the double facts, such as 1 ϩ
1 ϭ 2 and 4 ϩ 4 ϭ 8, with ease. Perhaps repetition ϩ 0 1 2 3 4 5 6 7 8 9 doubles on the addition
is a musical cue because many children almost sing
the double combinations. The 10 double facts oc- 0 chart (Figure 11.10).
cupy a diagonal in the addition table (Figure 11.10).
1

2 Some children do not

3 make the connection

4 immediately and need
5 additional time and ex-

6

7 perience with the near

8 doubles. Activity 11.9 in-

9 troduces near doubles.

Figure 11.10 Doubles, near “Make 10” empha-
doubles, and combinations sizes combinations with
of 10 a sum of 10 that fill the

ACTIVITY 11.8 Commutative and Associative Properties (Representation)

Level: Grades 1–3 • Have students repeat with other numbers of beans.

Setting: Small groups or whole class • Ask students to generalize what they found out. (Answer:
The sum is the same, no matter which number is first.)
Objective: Students demonstrate the commutative and associative You may decide to tell them they have discovered the
properties of addition. commutative property of addition.

Materials: Beans and paper plates, Cuisenaire rods, a sheet of • Ask how they could show the commutative property with
number lines for each child Cuisenaire-rod trains and with the number line.

• Give each student a paper plate with a line dividing it in
half. Also give each student 20 beans. Ask students to put
five beans in one half of the plate, and three in the other
half. Then have them combine the beans on one side.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

• Ask for the total number of beans. (Answer: 8.) Ask which Extension
of the students put three beans with five and which put
five beans with three. Ask if the answer was the same. • The same materials and procedures can be changed
slightly to demonstrate and develop the associative prop-
• Say: “Tell me about the two number sentences.” erty. Children place three sets of beans on a plate show-
ing thirds. Move the plates around in various sequences,
and write the number sentences such as 3 ϩ 4 ϩ 2 and
2 ϩ 4 ϩ 3. Students can combine any two sets first and
then add the third addend. Ask which combinations are
easiest for them to recall.

Chapter 11 Developing Number Operations with Whole Numbers 203

ACTIVITY 11.9 Near Doubles (Reasoning)

Level: Grades 1–3 • Have each child make a set of double facts cards on 4 ϫ 6
index cards.
Setting: Small groups
• Hold up a near-double card, such as 5 ϩ 6. Say: “Look at
Objective: Students use double number combinations to find a your double cards and find one that is almost the same.
near-double strategy. Show me that card.” Some children will hold up 5 ϩ 5;
others might hold up 6 ϩ 6.
Materials: Double cards for each child, set of near-double cards
(double dominoes can be used), worksheet for each child • Ask students how their card is similar to 5 ϩ 6. (Answer:
5 ϩ 5 is 10, so 5 ϩ 6 will be one more, or 11; or 6 ϩ 6 is
56 5 12, so 5 ϩ 6 is one less.)
_ϩ_6_ _ϩ_6_ _ϩ_5_
• Ask what strategies were used for changing the double
• Ask students which addition facts are doubles. When they fact to a new fact. (Answer: The first suggests a “double
answer 1 ϩ 1, 2 ϩ 2, 3 ϩ 3, and so forth, ask why they plus 1” combination. The second card suggests a “double
are called doubles. (Answer: Because both addends are minus 1” combination.) Either strategy is useful, and stu-
the same number.) dents should be encouraged to consider them both.

• Ask students to find another double-fact card and near-
double facts that are related to it.

Assessment

• On a fact worksheet, ask students to circle double facts in
blue and near doubles in orange.

opposite diagonal in Figure 11.10. Students have had When students begin working with more than two
many experiences with tens in place-value activities. addends, the make-ten and ten-plus strategies are
The Exchange game from Chapter 10 requires stu- particularly useful because they look for compatible
dents to count 10 as they trade up and down. Work numbers totaling 10.
with pennies and dimes provides many experiences
for sums of ten. The tens frame in Figure 11.11 is also 7 ϩ 6 ϩ 3 ϭ 7 ϩ 3 ϩ 6 ϭ 10 ϩ 6 ϭ 16
good for developing the make 10 strategy.
Rearranging the numbers to make a more difficult
Figure 11.11 Tens frames problem into an easier one continues the theme of
thinking about how numbers relate.
“Ten plus” is based on the associative property
and combines “make 10” with counting-on. For 8 ϩ Activity 11.10 uses the tens frame (Black-Line
5 the child thinks how to make 10 starting with 8 and Master 11.2) to demonstrate a visual strategy for ten-
renames 5 as 2 ϩ 3. The problem becomes 8 ϩ 2 ϩ plus combinations.
3, or 10 with 3 extra.
“Ten minus 1” is used for fact combinations with
8 ϩ 5 ϭ ___ 8 ϩ (2 ϩ 3) ϭ 10 ϩ 3 ϭ 13. 9 as an addend. With the tens frames shown in Fig-
ure 11.11, ask, “What is the sum of 10 plus 4?” When
students respond 14, remove one counter from the
10: “What is the sum of 9 plus 4?” After several exam-
ples, give students examples starting with 9 ϩ 7 and
ask them to explain their thinking. The tens frame is
also useful to show the 10 minus 1 strategy.

Understanding addition and subtraction begins
conceptually through activities with concrete mate-
rials and examples and continues with the develop-
ment of properties, strategies, and rules for the num-
ber combinations. Knowing the properties and rules

204 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 11.10 Making Ten with the Tens Frame

Level: Grades 1–3
Setting: Small groups or student pairs
Objective: Students use the add-to-ten strategy for finding sums

greater than 10.
Materials: Tens frames on overhead or board, magnetic shapes,

math boxes

• Tell a story that has a sum greater than 10, such as “Juan
had seven dimes and then got six more dimes. How many
dimes did he have?” Model the problem 7ϩ 6 ϭ ? on the
tens frame. Use one color of counter on one frame and a
different color on the other.

• Ask: “Does anyone see a way to rearrange the coun-
ters to make the answer easy to see?” Ask a student to
explain while moving three counters from one tens frame
to another tens frame. Ask if they could have moved the
counters the opposite way.

• Ask: “How much is 10 and 3 more?” Ask whether this is
an easier way to think about adding the two numbers.

• Tell several other stories for pairs of students to model
with the tens frame.

• Ask students to generalize a rule for the sums they have
been working with. (Answer: First, make ten. Then see
how many are still in the other tens frame and add that to
the ten.)

• Have children work in pairs with counters and tens frames
to model several other problems.

supports students’ thinking while they are learning These stories make good “think-aloud” examples for
the facts. If they cannot remember a fact, they can students.
reconstruct the fact. Strategies develop mental com-
putation and number sense and are a critical step in When students understand that subtraction is
learning the facts. the inverse operation for addition through many
activities, they find that each pair of addition facts
Strategies for Subtraction also yields a pair of subtraction facts. A “fact family”
Many subtraction strategies are counterparts of ad- involves the commutative law of addition and the
dition strategies. Counting-down strategies for sub- inverse relationship between addition and subtrac-
traction are shown by walking backward on the tion to produce four related facts. A triangle flash-
number line or removing items from a set. Students card (Figure 11.12) shows how three numbers form
will say, “It is just the number before.” This leads to four number sentences.
strategies of “minus 1” and “minus 2.” Students can
also recognize situations in which the difference 946
between the sum and the known addend is one
or two. 45 3 1 4 2

• Juan’s dog had 10 puppies. He found new homes 4ϩ5ϭ9
for nine of the puppies: 10 Ϫ 9 ϭ 1. 5ϩ4ϭ9
9Ϫ5ϭ4
• We want to buy a book that costs $7.00, but we 9Ϫ4ϭ5
have only $5.00: 7 Ϫ 5 ϭ 2.
Figure 11.12 Triangle flash cards

Chapter 11 Developing Number Operations with Whole Numbers 205

A child covers one number on the flash card and Developing Accuracy and Speed
asks the partner for the number sentence that com- with Basic Facts
pletes the relationship of addition or subtraction.
If students do not develop useful fact strategies, they
Students describe the identity property of the revert to less efficient methods, such as finger count-
“minus 0” strategy as not taking anything away, so ing, marking a number line, knocking on the desk,
that they still have what they had at the beginning. or drawing pictures. Although these behaviors are
useful in developing the concept, continued reliance
• I have six oranges and gave none of them away: on them inhibits quick, confident responses with
6 Ϫ 0 ϭ 6. number facts. Many children move from concrete to
abstract symbolic representations quickly and give
Children sometimes generalize incorrectly. For ex- up concrete materials; others need them for longer
ample, when the problem involves subtracting a periods of time. However, teachers should avoid let-
number from itself, the zero rule sometimes causes ting students become dependent on physical strate-
difficulties. gies. Thinking strategies for number operations are
an important link between understanding concrete
• I have six oranges and gave six away: 6 Ϫ 6 ϭ 0. meaning and achieving accuracy and ready recall
of number facts.
If a child confuses subtraction facts involving
zero, act out several stories to show different situa- Activities that focus on accuracy and speed with
tions. Ask children to generate a rule for a number basic facts occur after children understand con-
subtracted from itself as “you don’t have any left.” cepts and symbols for operations and have devel-
This conclusion has an algebraic generalization of oped strategies for many facts. The goal of practice
n Ϫ n ϭ 0. Activity 11.11 describes an activity in activities is ready recall—knowing the sum or differ-
which students imagine the missing number in a ence with accuracy and appropriate speed. Without
subtraction situation. ready recall of basic facts, various algorithms, es-
timations, and mental computation with numbers
larger than 100 become laborious and frustrating.

ACTIVITY 11.11 Subtracting with Hide-and-Seek Cards (Reasoning)

Level: Grades 1–3 • Cover both portions by folding flaps over them. Have stu-
dents identify the total number (12). Uncover one part of
Setting: Student pairs or small groups the picture. Have students identify the amount seen (e.g.,
6) and the number still “hiding” (6). Ask students to say
Objective: Students use the hide-and-seek strategy for learning and write the number sentence.
subtraction facts.
• Have children work with several different cards.
Materials: Teacher-made hide-and-seek cards for several sub-
traction combinations (picture different objects on cards), math Source: Adapted from Thompson (1991, pp. 10–13).
boxes

• Have students work in pairs with math box materials. Ask
students to place a set of objects on a plate (e.g., 11 ob-
jects). Have one student cover part of the set with paper
or a paper plate cut in half.

• Ask: “How many do you see?” and “How many are hid-
ing?” The students identify both numbers and say, for
example, “Eleven minus five equals six.”

• Students repeat with other combinations of objects, tak-
ing turns covering objects on the plate.

• Show students a hide-and-seek card with numerals on
flaps folded back. Have students identify the “whole”
(e.g., 12).

206 Part 2 Mathematical Concepts, Skills, and Problem Solving

How much emphasis to place on basic facts is with a kitchen timer and keep their own record
a continuing issue in mathematics education. Many of facts learned rather than facts missed. When
adults remember speed demands and “timed tests” children set their own reasonable time limits, they
as a negative emotional experience that tainted are more motivated to see learning the facts as a
their attitude toward mathematics for the rest of personal accomplishment.
their lives. Some children enjoy the challenge of tak-
ing timed tests and seeing their progress on public The goal is for all children to have ready recall
display, whereas others shut down emotionally and of facts so that they can move forward in their com-
cognitively. Competition between students and pub- putational fluency. Many computer programs offer
lic displays that show which students have learned individual practice routines set for individual speed
their facts may seem benign, but many adults re- and different challenges as well as report scores and
call them as a discouraging advertisement of their progress over sessions for the student and teacher.
failures. Card games such as bingo, memory, and matching
provide variety in practice.
Knowing the negative impact of timed tests and
other drill situations, effective teachers should care- 0 6 18 9 15
fully consider methods for encouraging accuracy
and speed with basic facts. The following guidelines 4 3 11 8 7
suggest how practice might be handled in a more
positive manner. 5 7 Free 17 16

• Develop accuracy with facts before speed. Speed 6 15 13 2 4
is the secondary goal.
10 1 12 14 6
• Expectations for speed will be different for each
child. The time required to master the facts When students make flash cards for the facts as
differs from student to student. Students should they learn them, they can practice alone or with a
not be compared to each other, but they can be partner.
encouraged to monitor their own progress and
improvement. • If you can say the answer before you flip the flash
card over, put the card in your fast stack. If there
• Help children develop strategies that make sense is a delay, put that card in a second stack to work
to them. Begin with the easy strategies and facts on later.
such as the “plus 1” and “minus 1” rules and then
move to more complex strategies and facts, such This activity is also used as an assessment tech-
as fact families or combinations for 10. nique in Activity 11.12. Children identify the facts
they know and the ones they are working on. The
• Review already learned facts, and gradually add personal set of flash cards can be taken home to
new ones. Have students record their mastery practice with siblings or adults.
of the strategies and facts in a personal log or
table. Self-assessment puts students in charge of their
learning. They may keep personal improvement
• Keep practice sessions short, perhaps five charts that show how many facts they answered cor-
minutes. rectly in 5 minutes, then 4 minutes, and 3 minutes.
They can mark the combinations they know on an
• Use a variety of practice materials, including addition table or write entries in their journals: “I am
games, flash cards, and computer software. Flash very fast with the 9s. I know 9 ϩ 6 is 15 because it is
cards encourage mental computation, whereas 1 less than 10 ϩ 6.”
written practice pages may slow down thinking
due to focus on writing. Effective teachers also watch for signs of frustra-
tion or confusion. If students know only a few facts,
• Avoid the pressure of group timed tests. Instead they are frustrated by practice and avoid it. If stu-
allow children to test themselves individually dents revert to finger counting, tally marks, or con-

Chapter 11 Developing Number Operations with Whole Numbers 207

ACTIVITY 11.12 Assessing with Flash Cards

Level: Grades 2–5 • Note on a checklist or addition table which addition facts
are in each stack.
Setting: Individuals
• Add new cards as new strategies are learned, and check
Objective: Student accuracy and speed with subtraction facts is for progress.
assessed, and self-assessment is encouraged.
397
Materials: Commercial flash cards or child-made flash cards made _ϩ_3_ _ϩ_7_ _ϩ_5_
with index cards. Put the number combination on the front with
the answer on the back so that when the flash card is flipped, the This assessment encourages students to take responsibil-
answer is right side up. ity for learning the facts and also demonstrates progress
as the green stack gets bigger. The assessment takes only
• As students learn a strategy, have them make their own set a couple of minutes and clearly identifies problems and
of flash cards or choose the flash cards for that strategy. difficulties.

• Let students work through the flash cards with partners.

• After practice, do a quick assessment with the flash cards
that the students have. Have students place the cards in
three stacks: I know the answer fast; I know the answer
but have to think about it; I don’t know the answer yet.

• Have the students mark cards green for the first stack,
yellow for the second, and red for the third.

ACTIVITY 11.13 Practice Addition and Subtraction Number Facts with Calculators

Level: Grades 1–4 • Start each pair of students simultaneously. An easy way
to time the activity is to record elapsed time in seconds on
Setting: Pairs the chalkboard or overhead projector: 8, 10, 12, 14, . . . .
Each student notes the time it took her or him to finish
Objective: Students build accuracy with number facts, and they the task. Discuss the results to help students see how
compare the speed of a calculator with thinking. knowing the facts speeds their work.

Materials: Set of student- or teacher-made fact cards, calculator Variations

Use this activity with children who are reasonably profi- • Use subtraction, multiplication, and division facts.
cient with basic facts for addition and subtraction.
• Use three or four single-digit addends.
• Students work in pairs with two identical sets of 15 fact
cards. One student is the Brain, and the other is the But-
ton. The Brain recalls the answers for the fact combina-
tions and writes them on a piece of paper.

• The Button must key each combination into a calculator,
then write the answers. As the activity begins, ask: “Who
will complete the work first, the Brain or the Button?”

crete strategies, they may be indicating weakness in might ask students to assess their understanding of
understanding or strategies. In both cases students strategies and recall of facts. Activities for review and
need work with concepts and strategies before maintenance should be based on student strengths
working on accuracy and speed. Even when stu- and needs. Activity 11.13 uses a calculator to provide
dents have become proficient with facts, they need practice with basic facts. A benefit of this activity is
to refresh their knowledge of strategies and recall of that it points out the advantage of knowing the facts:
facts. At the beginning of the school year a teacher the brain usually wins over the button.

208 Part 2 Mathematical Concepts, Skills, and Problem Solving

Research for the Classroom •
mon instructional techniques emphasize counting strate-
Arthur Baroody and his colleagues Meng-Lung Lai and gies that may inhibit learning of facts beyond sums of 10.
Kelly Mix have compiled a review of the literature on the Overemphasis on counting may limit investigation of other
development of young children’s number and opera- more useful and generative patterns. Students—in particu-
tions sense (Baroody et al., 2005). Based on this research, lar, students with learning disabilities—are trapped by inef-
Baroody (2006) advocates a number sense approach for ficient strategies that limit their understanding of relation-
students who are learning number combinations. He ships between numbers and their combinations. Students
proposes helping students to find patterns that connect should build a variety of strategies that create groups of
number combinations by starting with the idea that the related facts such as fact families. Although practice has a
same number can be represented in many forms; 8 can be role in learning number combinations, it should be based
renamed 1 ϩ 7, 2 ϩ 6, 5 ϩ 3, and 4 ϩ 4. Flexible thinking on reasoning strategies that become automatic rather than
about numbers allows students to compose and decom- on drill of isolated facts.
pose them in ways that make learning facts easier: 8 ϩ 5
ϭ 8 ϩ 2 ϩ 3 ϭ 10 ϩ 3. Baroody also suggests that com-

What Students Need to Learn E XERCISE
About Multiplication and Division
Compare your state mathematics curriculum with
As with addition and subtraction, the goal of in- the NCTM standards for learning multiplication and
struction in multiplication and division of whole division. Does the state curriculum emphasize learn-
numbers is computational fluency—knowing ing the operations by using realistic situations and
when and how to compute accurately when solv- materials and by solving problems? Does your state
ing problems. Experiences that children have with curriculum address how technology is used in learn-
modeling, drawing, and representing addition and
subtraction problems in everyday situations provide ing multiplication and division? •••
a model for work with multiplication and division.
Understanding multiplication and division begins in What Teachers Need to Know
kindergarten and the primary grades when children About Multiplication
skip-count, share cookies at snack times, or make and Division
patterns with linking cubes. Such activities set the
stage in second and third grade for directed teach- Adults who use multiplication and division in a
ing/thinking lessons about multiplication and divi- variety of settings in their everyday life may not
sion situations, actions, and strategies for learning realize the varied situations and meanings for the
number combinations. operations.

The NCTM curriculum standards referring to Multiplication Situations,
whole-number operations emphasize that they Meanings, and Actions
should be taught in meaningful ways. Table 11.2 out-
lines an instructional sequence for multiplication Multiplication has three distinct meanings in real-
and division that addresses number concepts, con- world situations:
nections between operations, and computational
fluency with larger numbers. Children who under- 1. Repeated addition: The total in any number of
stand how numbers and operations work make con- equal-size sets.
nections between whole numbers and fractions, see
relationships among the four basic operations, and 2. Geometric interpretation: The number repre-
grasp topics in number theory such as divisibility. sented by a rectangular array or area.

3. Cartesian product: The number of one-to-one
combinations of objects in two or more sets.

Chapter 11 Developing Number Operations with Whole Numbers 209

TABLE 11.2 • Development of Concepts and Skills for Multiplication and Division

Concepts Skills Connections

Number concepts Skip counting Multiplication and division
1–100 Recognizing groups of objects
Thinking in multiples

Numbers 1–1,000 Representing numbers with Algorithms
base-10 materials
Estimation
Exchange rules and games Mental computation
Regrouping and renaming

Numbers larger Expanded notation Alternative algorithms
than 1,000 Learning names for larger

numbers and realistic situations
for their use
Visualizing larger numbers

Operation of Stories and actions for joining Multiples, factors
multiplication equal-sized sets: repeated addition
Area measurement
Stories and actions for arrays and
area: geometric interpretation Probability and
combinatorics
Stories and actions for Cartesian
combinations Problem solving

Representing multiplication with
materials, pictures, and number
sentences

Operation of Stories and actions for repeated Fractions
division subtraction division
Divisibility rules
Stories and actions for partitioning Problem solving
Representing division with materials,

pictures, and number sentences

Basic facts for Developing strategies for basic Estimation and
multiplication facts reasonableness
and division
Recognizing arithmetic properties Algebraic patterns and
Achieving accuracy and speed with relations

basic facts Mental computation
Achieving accuracy and speed with Estimation, mental

multiples of 10 and 100 computation

Multiplication and Story situations and actions with Problem solving
division with larger larger numbers
numbers Computational fluency,
Developing algorithms with mental computation
and without regrouping using
materials (to 1,000) and symbols Reasonableness
Problem solving,
Estimation
Using technology: calculators reasonableness

and computer

Repeated addition, the most common multi- 4 packages of 3 juice boxes
plication situation, involves finding a total number
belonging to multiple groups of the same number. • Diego had five quarters in his pocket.

• Johnny had four packages of juice boxes. Each 25 cents ϩ 25 cents ϩ 25 cents ϩ 25 cents
package had three juice boxes. ϩ 25 cents ϭ 125 cents

3 ϩ 3 ϩ 3 ϩ 3 ϭ 12 juice boxes 5 ϫ 25 cents ϭ 125 cents
4 ϫ 3 ϭ 12 juice boxes

210 Part 2 Mathematical Concepts, Skills, and Problem Solving

5 quarters Combinations of blouses and pants

The geometric interpretation is represented as • Kara had two kinds of cake (yellow and choco-
a row-and-column arrangement and is also called a late) and three flavors of ice cream (vanilla,
rectangular array. Area is one example of the geo- strawberry, and chocolate mint). Two kinds of
metric meaning of multiplication. cake paired with three flavors of ice cream gives
six possible dessert choices: 2 ϫ 3 ϭ 6.
• Paolo arranged the chairs in four rows with five
chairs in each row: 4 ϫ 5 ϭ 20 chairs.

4 rows of 5 chairs Combinations of cake and ice cream

• Gina measured the room. It was 9 feet by 10 feet. If Kara offered four types of toppings (chocolate, hot
fudge, butterscotch, berry), the number of possible
dessert combinations would be 2 ϫ 3 ϫ 4 ϭ 24. The
important limitation on this is that only one choice
can be made in each category.

Numbers multiplied together are called factors.
The result of multiplication is the product.

9 feet x 10 feet Division Situations, Meanings,
and Actions
The third type of multiplication, Cartesian
cross-product or combinations, shows the total Division is the inverse operation of multiplication.
number of possibilities made by choosing one op- In multiplication two factors are known and the
tion from each group of choices. product is unknown. In division the total or prod-
uct is known, but only one factor is known. The
• Emma hung up five blouses and three pants to total or product is called the dividend, and the
see all the outfits she could make. Five blouses known factor is the divisor. The unknown factor is
paired with three pants makes 15 possible outfits: called the quotient. There are two types of division
5 ϫ 3 ϭ 15. situations:

1. Repeated subtraction, or repeated measurement:
How many groups of the same size can be sub-
tracted from a total?

Chapter 11 Developing Number Operations with Whole Numbers 211

2. Sharing, or partitioning: The total is equally dis- 24 cookies, serving size 5
tributed among a known number of recipients. 24
_Ϫ_5_
Because division is the inverse operation of mul- 19
tiplication, a multiplication situation can illustrate _Ϫ_5_
the difference between the two types of division. 14
_Ϫ_5_
Multiplication 9
• Three boys had four pencils each. How many _Ϫ_5_
4
pencils did they have?
24 Ϭ 5 ϭ 4 remainder 4
3 groups of 4 ϭ 12
Figure 11.13 Repeated subtraction, or measurement
When the known factor is the number of pencils division
for each boy, the question is, How many boys can
get pencils? The pencils are being measured out in Figure 11.13 shows how Marisa measured the cook-
groups of 4. ies into groups of five. Marisa can serve four people
and has four extra cookies.
Repeated Subtraction
• John had 12 pencils. He wants to give four pen- 24 cookies divided by 5 cookies
ϭ 4 people at the party plus 4 extra cookies
cils to each of his friends. How many friends can
John give pencils to? 24 Ϭ 5 ϭ 4, remainder 4

12 Ϫ 4 Ϫ 4Ϫ 4 ϭ 0 Subtract 4 three times. In an earlier example Paolo arranged 20 chairs
12 Ϭ 4 ϭ 3 with 5 chairs in each row. How many rows?

When the known factor is the number of boys, the 20 chairs in rows of 5 chairs ϭ 4 rows
question is how many pencils each will get.
When the number of groups is known but the
Sharing size of each group is not known, the division situa-
• John has 12 pencils and 3 friends. How many pen- tion is called partitive, or sharing. The action in-
volves distributing, or sharing, the number as evenly
cils can John give to each of his friends? as possible into the given number of groups. Parti-
tive division asks how many are in each group.
The pencils are shared one at a time among the
friends. John gives each friend one pencil, a second, • Kathleen dipped 19 strawberries in chocolate. If
a third, and then a fourth pencil before exhausting she is filling three gift boxes, how many strawber-
the pencils. ries go in each box?

12 Ϭ 3 ϭ 4 Kathleen distributes the 19 strawberries one at a
time into three boxes.
When the known factor is the number belonging
to each group, the unknown factor is the number 19 strawberries separated into 3 boxes ϭ
of groups. This division situation is called repeated 6 strawberries in each box with 1 extra strawberry.
subtraction, or repeated measurement. Measure-
ment division asks how many groups of a known 19 Ϭ 3 ϭ 6, remainder 1
size can be made. Measurement division is called
repeated subtraction because the number in each If division is done with numbers that can be
group is repeatedly subtracted from the total. evenly divided, the result is two whole-number fac-
tors. But division does not always work out evenly
• Marisa baked 24 cookies and wants to serve 5
cookies to each person at her party. How many
people can attend, including herself?

212 Part 2 Mathematical Concepts, Skills, and Problem Solving

in real life. The groups or objects that are not evenly size groups. Addends for addition can be any value,
divided are called remainders. Depending on the but in multiplication all the addends must have the
situation, remainders can change the interpretation same value. Teachers help students find and chart
of the answer by rounding up or down, making frac- common objects found in equal-size groups.
tions, or ignoring the remainder altogether.
• Groups of 2: Eyes, ears, hands, legs, bicycle
• If 4 children can go in each car, how many cars wheels, headlights.
are needed for 21 children?
• Groups of 3: Triplets, tripod legs, tricycle wheels,
• If you have 26 cookies for seven children, how juice boxes.
many cookies does each child receive?
• Groups of 4: Chair legs, car tires, quadruplets.
• A farmer has 33 tomatoes and is packing them 4
to a box. How many packages are needed? • Groups of 5: Fingers, nickels, pentagon sides.

In each of these cases, the remainder has a specific • Groups of 6: Soft drinks, raisin boxes, hexagon
meaning or multiple possibilities. Helping students sides, insect legs,
consider the meaning of remainders is an important
aspect of teaching division. • Groups of 7: Days of the weeks, septagon sides,
spokes on wheels.
E XERCISE
• Groups of 8: Octopus legs, octagon sides, spider
What is the calculated answer for each of the re- legs.
mainder stories? What is a realistic way to interpret
the remainder in each case? Is there more than one • Groups of 9: Baseball teams, cats’ lives.

way to consider the remainder? ••• • Groups of 10: Fingers, toes, dimes.

Developing Multiplication • Groups of 12: Eggs, cookies, soft drinks, dodeca-
and Division Concepts gon faces.

Many experiences with counting, joining, and sepa- • Groups of 100: Dollars, metersticks, centuries,
rating objects and groups of objects in preschool football-field lengths.
and first grade should prepare students for multi-
plication and division. During the second and third Children represent the equal groups with objects,
grade children’s earlier informal experiences enable pictures (Figure 11.14), and numbers.
them to invent and understand multiplication and
division from realistic problems. Just as for addition Patterns are another way that students represent
and subtraction, multiplication and division follow multiplication. Linking-cube trains are arranged
an instructional sequence that moves from concrete in patterns of two, three, four, or other numbers of
to picture to symbol to mental representations and cubes. Children counting a three-cube pattern and
from simple to complex problems that stimulate emphasizing the last object in each sequence learn
children’s thinking. As children learn how multipli- to skip-count.
cation and division work, they uncover basic num-
ber combinations, properties, and strategies. One, two, three, four, five, six, seven, eight, nine, . . .

Introducing Multiplication Three, six, nine, . . .

Repeated-addition multiplication extends children’s Children often skip-count by 2’s, 5’s, and 10’s,
experiences with counting and addition. Through and the multiplication facts for these numbers are
skip-counting by 2’s, 3’s, 5’s, and 10’s and working among the easiest for children to learn. Skip count-
with groups of buttons, small plastic objects, or ing by 3, 4, 6, 7, and other numbers helps students
similar materials, children have established a fun- learn those facts as well. Skip counting with the
damental idea for multiplication—adding equal- calculator is easy. Many calculators allow students
to key in 6, press the ϩ key, and press the ϭ key
repeatedly. The display will show 6, 12, 18, 24, and
so forth. The MathMaster calculator keeps track of
the number of groups of 6’s on the left side of the
display and the total number of cubes on the right
side (Figure 11.15).

Chapter 11 Developing Number Operations with Whole Numbers 213

(a) 4 sets of 2 eyes (b) 3 chairs have 12 legs (c) 6 ladybugs have 36 legs
Figure 11.14 Multiples

Focusing on groups instead of individual items Students write stories, draw pictures, and write
helps children move to multiplicative thinking. In- number sentences to record multiplication situa-
stead of counting each insect leg, students can think tions. For example, in Figure 11.16 children write
of six legs for each insect, and four insects have “6, the addition sentence 7 ϩ 7 ϩ 7 ϭ 21 to represent
12, 18, 24 legs.” Thinking in multiples is quicker the story. Students notice that both the picture and
for determining the total in equal-size groups than
counting each item. Activity 11.14 illustrates one way
to encourage thinking in multiples.

16 7 ϩ7 ϩ 7
2 12 3 ϫ 7 ϭ 21
3 18
4 24 Figure 11.16 Markers used to illustrate the
5 30 multiplication sentence 3 ϫ 7 ϭ 21

Figure 11.15 Numbers and group
display on MathMaster calculator

ACTIVITY 11.14 Repeated Addition (Representation)

Level: Grades 2–4 • Jaime has eight boxes of motor oil. Each box contains
four cans of oil. How many cans of oil does Jaime have?
Setting: Groups of four
• Each team member models one story with objects and
Objective: Students describe the relationship between multiplica- shares the results with team members.
tion and repeated addition.
• Ask children to explain their solution using different tech-
Materials: Counting cubes niques: repeated addition, skip counting, and multiplica-
tive thinking. Write addition sentences and multiplication
• Each team of four students is given four different multipli- sentences for the problems on the board.
cation word problems.
• Sadie buys gum in packages that have seven sticks each. • Have teams make up problems to send to other teams to
When she buys four packages, how many sticks does solve.
she get?
• Conrad has five packages of trading cards. How many
cards does he have if each package holds five cards?
• Jawan has three packages of frozen waffles. How many
waffles does he have if each package has eight waffles
in it?

214 Part 2 Mathematical Concepts, Skills, and Problem Solving

the addition sentence show three 7’s. The 3 is the Students also show understanding by arranging
number of sets, and 7 is the number in each set. disks or tiles into arrays. When exploring rectangular
The answer, 21, tells the total number of objects in arrays, students discover that some numbers can be
3 sets of 7. arranged in only one array; these numbers are prime
numbers. Other numbers (composite numbers) can
7 ϩ 7 ϩ 7 ϭ 21 be arranged in several ways to show whole-number
3 groups of 7 ϭ 21 factors, such as 16 tiles: 1 row of 16, 2 rows of 8, or 4
rows of 4 (Figure 11.17).
3 ϫ 7 ϭ 21
Figure 11.17 Arrays of 16 tiles
This situation can be used to introduce the sym-
bols for multiplication. The multiplication sentence Arrays also illustrate the commutative property
for “three groups of seven” is written 3 ϫ 7 ϭ 21 be- of multiplication, as in Figure 11.18. Library book
neath the addition sentence. Reading the number pockets and index cards in Figure 11.19 show arrays
sentence as “three groups of seven equals twenty-
one” emphasizes the multiplier meaning of the 3. Figure 11.18 Commutative property of multiplication:
The first factor is called the multiplication operator 2 ϫ 4 array, 4 ϫ 2 array
or multiplier because it acts on the second factor,
the multiplicand, which names the number of ob-
jects in each set. Activity 11.14 shows how coopera-
tive groups explore and share story problems that
model repeated addition and multiplication.

Because children are already familiar with
stacked notation for addition, both the multiplica-
tion sentence in horizontal and stacked notation can
be introduced, with one difference. In the horizontal
notation the first number is the multiplier; in the ver-
tical notation the multiplier is the bottom number.
Both forms are read “three groups of seven.” The
order of factors has meaning when related directly
to a multiplication story and when algorithms are
introduced.

The geometric interpretation of multiplication
shows the graphic arrangement of objects in rows
and columns called rectangular arrays. Arrays
are seen in the desk arrangement in some class-
rooms, ceiling and floor tiles, cans at the grocery
store, window panes, shoes boxes on shelves, and
rows of a marching band. As students identify arrays
in their world, they can draw pictures for the bul-
letin board and write journal entries to share with
classmates.

(a) (b) (c)

Figure 11.19 Library book pocket and index cards can be
used to show arrays.

Chapter 11 Developing Number Operations with Whole Numbers 215

of bugs with 5 as a factor: One row of 5 bugs (a); 2 (a) Cutouts
rows of 5 bugs is 10 (b); 3 rows of 5 bugs is 15 (c). (b) Cubes
As each row is revealed, students skip-count. Stu-
dents can make index cards for other multiplication
combinations.

Area measurement of rectangles is related to
multiplication arrays. In an informal measurement
activity, children cover their desktop or book with
equal-size squares of paper, sticky notes, or tiles.
They soon discover that they only need to cover
each edge of the rectangle to compute the total
number needed to cover the top by multiplying. Col-
oring rectangles on a piece of centimeter grid pa-
per also shows arrays. Two number cubes generate
the length of the sides. The length of each side and
the total number of squares create a multiplication
sentence (Figure 11.20). Area measurement is dis-
cussed further in Chapter 18.

2 by 3 3 by 2 3 by 5 5 by 3

2ϫ3ϭ6 3 ϫ 5 ϭ 15 (c) Letter code
3ϫ 2ϭ6
B–B B–T B–G B–R B–Y
5 ϫ 3 ϭ 15 R–B R–T R–G R–R R–Y
G–B G–T G–G G–R G–Y
4 by 6 6 by 4
(d) Lattice

4 ϫ 6 ϭ 24

6 ϫ 4 ϭ 24

Figure 11.20 Geometric interpretation of multiplication:
area measurement

The third multiplication interpretation is combi- Figure 11.21 Four representations of Cartesian
nations, or the Cartesian cross-product, which gives combination
the number of combinations possible when one
option in a group is matched with one option from sentations, while the same combinations are shown
other groups. by color cubes. More symbolic representations of
combinations of shirts and pants are letter-coded
• Tracy has three shirts (red, blue, white) and five combinations and a lattice diagram showing 15
pairs of pants (blue, black, green, khaki, brown). points of intersection.
How many outfits are possible?
Additional stories provide exploration of other
Combinations of three shirts and five pairs of pants combinations with objects, drawings, diagrams, or
are represented in four different ways in Figure 11.21. symbols.
Cutouts of shirts and pants are the most literal repre-

216 Part 2 Mathematical Concepts, Skills, and Problem Solving

• In an ice cream shop, Ari chooses from vanilla, E XERCISE
chocolate, or strawberry ice cream and either
cone or cup for container. Create stories for the multiplication sentence 3 ؋ 5
‫ ؍‬15 that illustrate repeated addition, geometric
Six choices are possible so long as only one choice arrays or area, and Cartesian products. Draw a pic-
is possible from each category. ture for each story. Can you extend your stories for
3 ؋ 4 ؋ 5? How would you show three factors with
Vanilla Cup, Vanilla Cone, Chocolate Cup, Choco-
late Cone, Strawberry Cup, Strawberry Cone an array or combinations? •••
3 kinds of ice creams ϫ 2 types of containers ϭ
6 options Introducing Division
3ϫ2ϭ6
Division is the inverse operation for multiplica-
• With four flavors of ice cream and three choices tion. Multiplication is used for situations when the
of container (cup, waffle cone, or sugar cone), factors are known to calculate a product. For divi-
how many different treats could Tonya buy? sion the product, or total, is known but only one
factor is known. Children need experience with
Children can make up their own menus for choices two division situations—measurement and parti-
in the ice cream shop (1 flavor from 31, 1 cone from tive—illustrated by realistic stories and modeled
4, 1 topping from 6) or other situations that involve with manipulatives.
making choices from a set of options such as pizza
(crust, toppings) or automobiles (color, engine, inte- • Multiplication: I am making breakfast for four
rior). A cooperative learning activity in Activity 11.15 people. If I cook each of them two eggs, how
involves creating color combinations for custom- many eggs do I need?
ordered bicycles.
• Measurement, or repeated subtraction, division: I
have eight eggs. How many people can be served
if I cook two eggs for each person?

ACTIVITY 11.15 Color Combinations for Bicycles (Representation)

Level: Grades 2–4 trim worker distributes the trim colors (yarn). The fourth
member of each team is the trim painter; the trim painter
Setting: Groups of four applies the trim (tapes the yarn to the paper).

Objective: Students model concepts of multiplication. • Each team has 15 minutes to complete the samples.
Before teams begin work, team members plan how to
Materials: Pieces of red, silver, gold, and black paper; six different organize different combinations of colors.
colors of yarn; clear tape
• After 15 minutes, ask the teams to report and show their
• Present this situation: Each team of four is a work crew in color samples and the number of combinations. Ask stu-
a bicycle manufacturing plant. The company’s designer dents whether all the combinations are likely to be equally
has decided that the plant will produce bicycle frames popular.
in four colors: red, silver, gold, and black. Each bicycle
will have one color of trim painted on it, chosen from six • Ask the teams to determine the number of combinations
possibilities. The task will be to create color samples for for five frame colors and three trim paints and for four
the bicycles before making them. How many different frame colors and five trim paints. During discussion of the
bicycles can be made from four frame colors and six trim new situations, ask students if there is a way to know
paints? the number of different combinations without making
the samples. What is the advantage of making all the
• Assign a role to each team member. One student is the samples?
manager; the manager reads the directions and keeps
the team on track. The second student is the layout
worker; the layout worker organizes the bicycles (colored
papers) by color. The third student is the trim worker; the

Chapter 11 Developing Number Operations with Whole Numbers 217

In measurement division the total (8 eggs) and the for children. As soon as students see how division
size of each group (2 eggs) are known. The number is working, teachers can introduce situations with
of groups (people) is unknown. Children act out the remainders.
problem by placing two plastic eggs each on one
plate, two plates, three plates, and four plates until • Natasha bought a package of 14 pencils. If she
all the eggs are gone. Repeated subtraction, another gave four pencils to each friend, how many
name for measurement, is modeled by students re- friends would get four pencils? Four pencils are
moving two eggs, then two more, then another two, put into three cups with two extra pencils.
and finally two more.
NIA NILES NATASHA
• Partitive, or sharing, division: I have eight eggs to
divide equally among four people. How many 14 Ϫ 4 ϭ 10 10 Ϫ 4 ϭ 6 6Ϫ4ϭ2
eggs can I cook for each person?
14 Ϭ 4 ϭ 3 remainder 2
In a partitive division story the total number of eggs
(8) and the number of people (4) are known, but the From their actions students see that 14 is the orig-
number of eggs for each person is unknown. Chil- inal group of pencils, 4 indicates the size of each
dren share eggs by putting one egg on each person’s group of pencils being subtracted, and 3 represents
plate, then another for each person, until all the eggs the number of groups. Two pencils are not in a cup
are gone (Figure 11.22). Sharing is another name for because they are not a complete group of 4. The
partitive division. teacher asks students what they might do with the
two extra pencils.
Measurement division: 8 eggs are distributed 2 each
to 4 plates. Division stories of both types are acted out with
Partitive division: 8 eggs are distributed one at a materials and represented with pictures; the teacher
time to 4 plates. introduces the division sentence and symbols. As
children see repeated subtraction, they write three
Measurement stories are preferred by many subtraction sentences. After 4 is subtracted three
teachers because subtraction is a well-known model times, 2 pencils are left undivided. The horizontal
division sentence is introduced to show the same
8Ϭ2ϭ4 information. The division sentence is read “Four-
teen divided by three equals four with a remainder
(a) of two.”

12 Ϭ 4 ϭ 3 Students act out familiar situations of sharing
items such as cookies to illustrate partitive division.
13 13 13 13 If they begin with 17 cookies, they can share the
2 2 cookies with three people, four people, five people,
22 or six people. After students have acted out several
sharing situations and represented the actions in
pictures or diagrams, record their work with a divi-
sion sentence:

17 Ϭ 5 ϭ 3, remainder 2.

(b) The Doorbell Rang, by Pat Hutchins (New York:
Harper Collins, 1986), is a picture book that models
Figure 11.22 (a) Measurement division; (b) partitive sharing (Activity 11.16). Each time the doorbell rings,
division

218 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 11.16 Sharing Cookies (Problem Solving, Connection, and Representation)

Level: Grades 2–4 • Have students discuss what happens as the number of
people increases.
Setting: Groups of four
• Tell each team to create a story about sharing things and
Objective: Students model and describe the meaning of partitive prepare to act it out (allow approximately 5 minutes).
division situations.
• Each team acts out its story while other teams observe.
Materials: “Cookies” cut from construction paper, other manipula- After each story, discuss what was demonstrated by the
tive materials selected by teams actors.

• Read The Doorbell Rang, by Pat Hutchins (New York: • End the lesson by giving each group a small bag of cook-
Harper Collins, 1986) through once. ies or other treats, which they must share equally among
themselves.
• Tell the teams of four that they are going to act out the
story. Give the students the construction paper cookies.
Read the book again, and have each team model the situ-
ations described as the story progresses.

more guests come to the party, and the cookies are Working with Remainders
shared to accommodate all the guests.
Introducing remainders early through division sto-
When division is introduced with materials, most ries allows students to see remainders as a natural
children recognize that division and multiplication event. They find that some situations involve num-
are inverse operations. Multiplication joins sets, just bers that divide without a remainder and that some
as addition does; division separates, just as subtrac- do not. When remainders are included in examples,
tion does. Children also learn mathematical vocab- children discuss the meaning of remainders in dif-
ulary associated with division. The number being ferent situations and develop options for working
divided is the dividend, the number by which it is with them.
divided is the known factor or divisor, and the an-
swer is the unknown factor or quotient. • The basket had 25 apples in it. If 25 apples are
shared equally by three children, how many
The meaning of the dividend never changes; it apples will each have?
always tells the size of the original group. But the
roles of divisor (the known factor) and quotient (the 25 Ϭ 3 ϭ 8 remainder 1
unknown factor) are interchanged, depending on
the situation. In measurement the divisor tells the Children may suggest, “Eat 1 apple yourself and don’t
size of each group, and the quotient tells the number tell,” “Cut it up so that everybody gets some,” “Put it
of groups. In a partitive situation the divisor tells the back in the box,” “Give it to grandmother,” and so
number of groups, and the quotient tells the size of forth. Sometimes the remainder can be divided into
each group. Children also recognize that a dividend equal parts; other times the remainder is ignored or
in division is related to a product in multiplication. might call for an adjustment of the final answer up
As children develop number sense about division, to the next whole number. Talking about remainders
they think about the meaning of the remainder in helps students understand that the remainder needs
context rather than by rule. to be considered in the calculated answer and its
meaning.
E XERCISE
Various examples illustrate the different mean-
Write a measurement story and draw a picture ings for the remainder.
for the sentence 34 ، 5 ‫ ؍‬6, remainder 4. Write
a partitive story and draw a picture for the same

sentence. •••

Chapter 11 Developing Number Operations with Whole Numbers 219

• We have 32 children in our class. If we play a The computed answer for 52 Ϭ 6 is 8, remainder
game that requires three equal-size teams, how 4, or 8.6667. Acting out or modeling this situation
many players will be on each team? shows that four children are without transportation
after eight vans have been filled. One more van will
The computed answer is 10 children on each team be needed for the four remaining children. A total
with 2 children left over. A calculator answer is 10.67 of nine vans is needed, so the number is raised to
or 1023. However, cutting children into parts is not the next whole number. More discussion about the
reasonable. Students may decide that the practical fractional treatment of remainders is included in
solution requires two teams of 11 and one team of Chapter 14.
10. They might decide that three equal teams of 10 is
better and assign two students tasks such as keeping What Teachers Need to
score, managing equipment, or refereeing. Students Know About Properties of
can recognize the difference between a computed Multiplication and Division
answer and a practical answer.
An identity element is a number that does not change
• Sixty-eight apples are to be put into boxes. If each the value of another number during an arithmetic
box holds eight apples, how many boxes are operation. Multiplying by 1 does not change the
needed? value of a number, so 1 is the identity element for
multiplication. For example, consider 12 ϫ 1 ϭ 12.
MISCONCEPTION The paper-and-pencil an- The identity element for division is also 1. Consider
swer is 8, remainder 4, 12 Ϭ 1 ϭ 12.
Many tests, such as the and a calculator answer
National Assessment of will be 8.5. The physical The commutative property of multiplication,
Educational Progress answer to the question is similar to the commutative property of addition, is
(NAEP), report that eight full boxes and an- illustrated when the order of factors is switched but
proper interpretation of other half-box. The practi- the product is the same.
remainders in division cal answer could be eight • A fruit stand has gift boxes of four apples. How
problems causes errors if the remainder is disre-
by many students even garded or nine if all the many apples are in six boxes?
through middle and high apples need to be placed
school. Even if students in boxes even if they are (a) 6 boxes of 4 apples
can compute an answer not full. The remainder
correctly, they do not con- might be ignored if only • A fruit stand has gift boxes of six apples. How
sider how the remainder the number of full boxes many apples are in four boxes?
is used in a problem con- is considered, or students
text. Introducing remain- may suggest that they (b) 4 boxes of 6 apples
ders in stories and asking need nine boxes, even
students to consider though one box is not Division is not commutative, as students should rec-
how remainders should completely full. ognize when they consider 35 Ϭ 5 and 5 Ϭ 35.
be treated establishes a
foundation for reasoning The associative property for multiplication
about their meanings. is similar to the associative property for addition.
When three or more factors are multiplied, the order
• Three cans of cat food cost $0.98. How much will
one can cost?

When $0.98 is divided by 3, the answer is $0.3266667.
However, students should understand that the price
will be rounded up to $0.33 when only one can is
bought.

• Our grade has 52 children who will ride to camp
in minivans. If each van can carry six children
and their gear, how many vans will be needed?

220 Part 2 Mathematical Concepts, Skills, and Problem Solving

in which they are paired for computation does not derstanding the inverse relationship between mul-
affect the product. tiplication and division connects the multiplication
facts to division facts. In fact, when students are act-
• Robin was taking inventory at the grocery store. ing out division problems, they often discover this
The cans of green beans were stacked three cans connection.
across, two cans high, and six cans back.
Learning number facts through strategies means
The associative property allows factors 3 ϫ 2 ϫ 6 to that students are going beyond memorization; they
be grouped for multiplication in several ways: understand the fundamental rules and properties
for multiplication and division that support men-
(3 ϫ 2) ϫ 6 ϭ 6 ϫ 6 tal calculations and number sense. Some multipli-
2 ϫ (3 ϫ 6) ϭ 2 ϫ 18 cation and division strategies are similar to strate-
3 ϫ (2 ϫ 6) ϭ 3 ϫ 12 gies for addition and subtraction facts, so children
(6 ϫ 2) ϫ 3 ϭ 12 ϫ 3 should be familiar with them. Because many facts
(n1 ϫ n2) ϫ n3 ϭ n1 ϫ (n2 ϫ n3) can be learned with a variety of strategies, children
can adopt the strategy that works best for them. The
The associative property, like the commutative prop- multiplication chart is a good way to organize facts
erty, applies to multiplication but not to division. as students learn strategies.

The distributive property of multiplication Skip counting is the first valuable strategy be-
and division over addition allows a number to be cause the “verbal chain” of multiples (2, 4, 6, 8, . . .
separated into addends and multiplication or divi- and 5, 10, 15, 20, . . .) is a familiar sequence. Most chil-
sion to be applied or distributed to each addend. dren learn the facts for 2, 5, and 10 easily. Learning
skip counting for other facts by counting multiples
• The stools cost $27 each. Ms. Turner wanted to on the hundreds chart supports the meaning of mul-
buy four of them. tiplication. On the multiplication chart (Black-Line
Master 11.3) students can fill in the multiplication
In the case of 4 ϫ $27, the multiplier 4 can be ap- facts for 2’s and 5’s.
plied to 20 ϩ 7, to 25 ϩ 2, or to 15 ϩ 12:
The commutative law for multiplication states that
4 ϫ (20 ϩ 7) ϭ (4 ϫ 20) ϩ (4 ϫ 7) ϭ 80 ϩ 28 ϭ 108 the order of the factors does not change the prod-
4 ϫ (25 ϩ 2) ϭ (4 ϫ 25) ϩ (4 ϫ 2) ϭ 100 ϩ 8 ϭ 108 ucts so that each multiplication fact has a mirror
4 ϫ (15 ϩ 12) ϭ (4 ϫ 15) ϩ (4 ϫ 12) ϭ 60 ϩ 48 ϭ 108 fact: 6 ϫ 4 ϭ 4 ϫ 6. Activities with arrays illustrate
the commutative law graphically. When children
The distributive property of division over addition recognize that the commutative law gives them
also means that the dividend can be broken into pairs of equivalent facts, they can use this strategy
parts that are easier to calculate. For example, when for learning facts.
39 is divided by 3, 39 can be thought of as 30 ϩ 9 or
as 24 ϩ 15 and division can be performed on each
addend:

39 Ϭ 3 ϭ (30 ϩ 9) Ϭ 3, or (30 Ϭ 3) ϩ (9 Ϭ 3) ϭ 10 ϩ 3 ϭ 13
39 Ϭ 3 ϭ (24 ϩ 15) Ϭ 3, or (24 Ϭ 3) ϩ (15 Ϭ 3) ϭ 8 ϩ 5 ϭ 13

Learning Multiplication and 4 rows of 6 6 rows of 4
Division Facts with Strategies
Multiplying with 0 is modeled with no cubes in
As students work with equal-size groups, arrays, one cup, no cubes in two cups, up to no cubes in
area models, and Cartesian product situations, they nine cups. “How many are in one group of zero?
become familiar with the multiplication of number two groups of zero? nine groups of zero? a million
combinations. Learning 2 ϫ 5 ϭ 10 is not difficult groups of zero?” The question can also be asked,
for children who understand two groups of five and “How many are in zero groups of five? no groups of
have experience skip counting 5, 10, 15, . . . . Un-

Chapter 11 Developing Number Operations with Whole Numbers 221

17? zero groups of a thousand?” Students develop 7ϫ6
the rule that “multiplying with zero gives you zero” ϭ7ϫ3ϩ7ϫ3
or similar phrasing. ϭ 21 ϩ 21

Multiplication with 1 introduces the identity ele- Figure 11.23 Using the distributive property to learn
ment for multiplication. It is also easy to model with multiplication facts
language similar to that used for multiplication by
zero. Put one cube in several cups. “How many is Division Facts Strategies
two groups of one? six groups of one? ninety-nine
groups of one?” The commutative situation is also When students understand that division and mul-
presented. “How many are in one group of six? one tiplication are inverse operations, students use the
group of 50? one group of a million?” Reasoning from multiplication facts to learn division facts through
the model leads most children to the realization that activities that focus on a product and its factors. A
“multiplying by 1 doesn’t change the number.” factor tree introduces students to a missing factor.
By drawing a factor tree, they can see that the fac-
Multiplication by 2 connects skip counting 2’s and tors for 24 could be 6 ϫ 4, 8 ϫ 3, 12 ϫ 2, or 24 ϫ 1.
relates to double facts in addition. Because 3 ϩ 3 is While looking at the various trees, the teacher can
6, two groups of three are six. This pattern is easy to ask, “If 24 is a product and 4 is a factor, what is the
illustrate with linking cubes, and children recognize missing factor?”
that “a number multiplied by 2 is the same as adding
a number twice.” 24 24 24

The squared facts are those facts found when 6 3 12
a number is multiplied by itself, such as 4 ϫ 4 or 7 24 24 24
ϫ 7. The geometric interpretation makes a strong vi-
sual impression that the square facts are also square 14 8
arrays. The near squares, or square neighbors,
occupy the spaces on either side of the square facts Other properties and rules are also useful in
and can be thought of as one multiple more or one learning division facts.
multiple less than the square fact. If 8 ϫ 8 is 64, then
7 ϫ 8 is 64 Ϫ 8 ϭ 56 and 9 ϫ 8 is 64 ϩ 8 ϭ 72. Skip Zero in division can be used in two ways. When
counting by 8’s provides good background for this the total being divided is zero, the quotient is also
mental computation. zero. Children enjoy the absurd notion of dividing
zero objects into groups.
Multiplying with 9 as a factor can be developed
with several strategies. One strategy involves multi- • I had no oranges in my basket. I divided them
plying by 10’s—which students have already learned among seven friends. How many oranges did
using skip counting. If 10 ϫ 9 is 90, then 9 ϫ 9 is 90 each get? Each friend gets 0 oranges.
minus 9. If 10 ϫ 5 is known to be 50, then 9 ϫ 5 is 50
minus 5. Other interesting patterns are explored in
Activity 11.17.

The distributive law of multiplication provides
another strategy for learning multiplication facts
by breaking an unknown product into two known
products. If the answer to 7 ϫ 6 is unknown, the
problem can be distributed as (7 ϫ 3) ϩ (7 ϫ 3)
or 21 ϩ 21 (Figure 11.23). This strategy will become
important as children work with algorithms in inter-
mediate grades.

However, zero is never used as a divisor. Stories il-
lustrate the logical absurdity of dividing by zero.

222 Part 2 Mathematical Concepts, Skills, and Problem Solving

ACTIVITY 11.17 Putting on the Nines (Reasoning)

Level: Grades 2–5 For the math fact 4 ϫ 9, ask students to bend down the
fourth finger. Tell them the three fingers to the left of the
Setting: Small groups or whole class bent finger are worth ten each and the six fingers to the
right of the bent finger are worth one each. Ask students
Objective: Students explore the multiplication facts with 9 to find the math fact for 36. When you bend the fourth finger
patterns in the products. down say, “4 ϫ 9 is 36.” Ask students to see if the finger
calculator works for all the 9 facts starting with the first
Materials: Linking cubes finger. After students are able to multiply with their fin-
ger calculator, ask them why the finger calculator works
• Ask students to use linking cubes to make rods of 10 and for 9’s but not for other facts.
write the number facts for 10 in a chart.
4ϫ9
• Ask students to make rods of 9 with the linking cubes Bend down 4th finger from the left (represents the “4” in
and write the number facts for 9 next to the 10’s facts in the equation)
a chart. Three fingers before bent finger represent 3 tens
Six fingers after the bent finger represent 6 ones
Rods 10’s 9’s 4 ϫ 9 ϭ 36
1 10 9
2 20 8ϫ9
3 30 18 Bend down 8th finger (represents the “8” in the equation)
4 40 27 Seven fingers before bent finger represent 7 tens
5 50 36 Two fingers after the bent finger represent 2 ones
6 60 45 8 ϫ 9 ϭ 72
7 70 54
8 80 63 Although the basic multiplication facts go through 9 ϫ 9,
9 90 72 including the facts through 10 ϫ 10 is common because
100 81 the 10’s are easy and important for later computa-
10 90 tion. Many teachers also encourage students to learn
number combinations for multiples of 11 and 12. The
• Ask students to look for patterns and relationships combinations for 11 are easy using the 11’s pattern and
between the two lists of facts. They may suggest several skip counting: 11, 22, 33, 44, 55, 66, and so on. When
different patterns in the facts: students learn the multiplication facts for 12, they should
• The digits in each of the 9 facts all add up to 9. notice the relationship between the 12 facts and facts
• A product can be matched with another product that already learned for 6.
has the same numbers but reversed, such as 81 and 18.
• You can count to get the facts for 9. Write 9, 8, 7, and
so forth in the ones place; then start at the bottom and
write 9, 8, 7, and so forth in the tens place.
• The 9 facts are less than the 10 facts.
• The difference between the 10 facts and the 9 facts is
the number of rods. Some students may express the dif-
ference with a number sentence: n ϫ 9 ϭ (n ϫ 10) Ϫ n.

• Ask students which patterns help them remember the
facts for 9. Their answers will vary depending on the pat-
tern that they notice and use.

Extension

The facts for 9 can also be shown with ten fingers. Tell
students that you will show them how to multiply by 9
using their finger calculator. Ask them to hold both hands
up so that they can see their palms. Model this action.

Chapter 11 Developing Number Operations with Whole Numbers 223

• I had seven oranges and divided them between In Activity 11.18 children fill in the multiplication
two friends. How many oranges did each friend table to show 19 facts with 0 as a factor, 17 identity
get? facts (1 as a factor), and 15 double facts (2 as a fac-
tor). These three strategies supply 51 quick and easy
• If I divided seven oranges between one friend, multiplication facts. When the facts for 5 are also
how many would the friend get? placed on the table, only 36 facts remain. Various
strategies might be used to simplify these 36 facts,
• If I divided seven oranges between no friends, such as pairing facts using the commutative law
how many would the friends get? (e.g., 7 ϫ 9 and 9 ϫ 7). The associative law is also
useful for facts involving 6, 7, and 8 when students
Dividing any amount into zero groups is implausible know easier facts. The squared facts, such as 3 ϫ 3
and impossible; in mathematics, division by zero is
undefined.

ACTIVITY 11.18 Find the Facts (Reasoning)

Level: Grades 3–5 found with the identity element of 1. (Answer: 17 more
Setting: Small groups facts—those that have 1 as a factor, excluding those that
Objective: Students identify the easy facts and the difficult multi- have 0.) Add 19 and 17 to show that by knowing the role
of 0 and 1 in multiplication, students know more than
plication facts. one-third of the basic facts.
Materials: Multiplication table through 9 ϫ 9
• The third row and third column of cells have 2 as a factor.
• Put a blank multiplication table on an overhead transpar- Ask students about skip counting by 2’s, or doubles. Write
ency (see Black-Line Master 11.3). the 15 products for these combinations in the table. Point
out that the easy combinations with 0, 1, and 2 contain
ϫ0 1 2 3 4 5 6 7 8 9 more than half the basic facts.

0 • Combinations with 5 as one factor offer little difficulty to
most children because of skip counting by 5’s. Add these
1 to the table. They contribute another 13 facts.

2 • Most students probably already know the remaining com-
binations for cells in the upper left-hand part of the table:
3 3 ϫ 3, 3 ϫ 4, 4 ϫ 3, and 4 ϫ 4.

4 • Ask how many facts are not yet written in. Ask which of
those facts the students already know. Different children
5 may already know some of these facts. This process high-
lights the 20 or so facts that usually take extra work, and
6 students can concentrate on them in their practice.

7 ϫ0 1 2 3 4 5 6 7 8 9

8 000 0 00 000 00

9 101 2 34 567 89

• Point out the factors across the top and down the left 2 0 2 4 6 8 10 12 14 16 18
side; products will be at the intersection of two factors.
3 0 3 6 9 12 15
• Begin with 0 on the left side as a factor: “If we multiply
each number in the top row by 0, what are the prod- 4 0 4 8 12 16 20
ucts?” (Answer: They are all 0.) Fill in the cells after each
question: “Where else will there be products that are 0?” 5 0 5 10 15 20 25 30 35 40 45
(Answer: Down the first column of cells.) “How many
multiplication facts have 0 as a factor?” (Answer: 19.) 6 0 6 12 30

• Use 1 as a factor and ask: “What is the product when 7 0 7 14 35
we multiply each number in the top row by 1?” (Answer:
Each product is the same as the number in the top row.) 8 0 8 16 40
“Where else do we find factors that work this way?”
(Answer: When numbers in the left-hand column are 9 0 9 18 45
multiplied by 1 in the top row.) Ask how many facts are

224 Part 2 Mathematical Concepts, Skills, and Problem Solving

and 7 ϫ 7, are on a diagonal on the multiplication 46 58
chart. Ten square facts and 18 neighbor facts mean 1 3
that almost a quarter of the facts can be learned
with this strategy. 9 ؋8 7

A table also demonstrates the relationship be- 02
tween multiplication and division facts. Factors are
on the left side and top row, with products in the Front
center of the table; for every product the two factors
can be identified by tracing up and to the left. For 48 63 24 64 40 48 32 8
Ϭ 8, find 48 in the table and trace left to the 8 and up
to the 6 or left to 6 and up to 8. 79 56 ، 8 72
(a) 16 0
Building Accuracy and Speed with
Multiplication and Division Facts Back
(b)
Knowledge of multiplication and division facts is
expected of elementary school students. When stu- Figure 11.24 Two types of flash cards for practic-
dents understand the operations and have strategies ing multiplication and division facts
for learning and remembering the number combina-
tions, they are ready to work on ready recall. 9.” The partner card in Figure 11.24b shows all multi-
plication facts with 8 as a factor on the front and all
Accuracy is the first priority in developing ready related division facts with 8 as a divisor on the back.
recall and speed, which varies among children. Students take turns putting a pencil or fingertip in a
Practice activities for multiplication and division notch next to a number, such as 9. One partner says
follow the same guidelines presented for addition “9 ϫ 8 ϭ 72” or “8 ϫ 9 ϭ 72” and is checked by the
and subtraction facts; practice should be short, fre- partner on the opposite side of the card. For division
quent, and varied in method, with an emphasis on the partner on the division side might say “72 Ϭ 8 ϭ
personal improvement. Classroom procedures used 9” and be checked.
for practicing number combinations can support
confidence and motivation, or undermine them. Many drill games and activities for improving ac-
curacy and speed are available on the Internet and
Many practice activities can be used, including in commercial packages. Student improvement has
computer software, flash cards, and partner games. been reported as a result of using these materials.
Consider triangle flash cards (Figure 11.24a). When
children pair up to play, one child covers the product E XERCISE
63, the other child multiplies the factors to find the
product: 7 ϫ 9 ϭ 63. For division one child covers What role, if any, do you think timed tests should
one factor such as 9, and the partner says, “63 Ϭ 7 ϭ play in developing accuracy and speed with the

facts? •••

Chapter 11 Developing Number Operations with Whole Numbers 225

Take-Home Activities find a pattern that will help you make arrange-
ments that work? Write your solutions in your jour-
Take-home activities show parents that the cur- nal as you work so that you can share them with
riculum includes more than practicing computa- classmates and your teacher tomorrow.
tion with algorithms. The two accompanying
activities require reasoning to determine cor- Number Puzzles
rect solutions. The challenge of each activity will
intrigue many parents as well as students. They are Below are some algorithms in which letters have
good cooperative-learning activities for adults and been substituted for numbers:
students at home.

Addition with Number Squares ABA XYZ BOX YOU
؊ACD ؊MNO ؊OUT ؊FOR
Work this puzzle with other people in your home.
EAB OZR ORB USA
• Cut out nine 1-inch squares of paper.
• Put the numeral 1 on one square of paper, the A letter stands for the same number each time it
appears in the algorithm.
numeral 2 on the next, and so forth until each
square is numbered (1 to 9). See how many of these puzzles you can solve. It is
• Arrange the squares so that you have a large possible to have more than one solution to a given
square with 3 three-place numbers and so puzzle.
that the sum of the numbers made by the
squares in the top and middle rows equals the Make one addition and one subtraction puzzle
number made by the squares in the bottom and bring them to school tomorrow for a friend to
row. solve.

Can you arrange the squares in more ways than
one and get a correct answer each time? Can you

226 3. Write a subtraction question and draw a simple pic-
ture or diagram for each of the following situations.
Summary Label each situation with the type of subtraction it
illustrates.
Curriculum standards emphasize student understand-
ing of four arithmetic operations, when and how to use a. Mr. Ramirez had 16 boys in a class of 25 children.
the operations to solve problems, and accurate recall
of basic number facts. Development of number opera- b. Joan weighed 34 kilograms, and Yusef weighed
tions begins while children are counting, classifying, and 41 kilograms.
comparing sets of concrete objects. During the concep-
tual development phase of number operations, children c. Mrs. Bennett bought two dozen oranges and
tell stories, model problems, draw pictures, and write served 14 of them to girls after soccer practice.
number sentences as they construct meanings and ac-
tions for addition, four subtraction situations (takeaway, d. Mr. Hoang had 18 mathematics books for 27
whole-part-part, comparison, completion), three multi- students.
plication interpretations (repeated addition, geometric
interpretation, Cartesian product), and two division e. Diego had 34 cents. The whistle costs 93 cents.
types (measurement, or repeated subtraction, division;
partitive, or sharing, division). During this time, students 4. Write stories to illustrate the three multiplication in-
explore all the number combinations and properties of terpretations. Draw pictures and diagrams for each.
the operations. Are there any similarities in your diagrams for the
three interpretations?
As students understand each operation, they find
strategies and properties to organize the number com- 5. Identify each of the following situations as measure-
binations. Strategies based on arithmetic properties ment or partitive. Write the question that goes with
and rules about how numbers work make learning facts the situation, and draw a picture for one problem of
more meaningful and easier. Instead of memorizing, each type of division. Then write a number sentence
students learn how numbers work as a foundation for for the picture. What would you do with the remain-
number sense and estimation. Finally, students need der, if any, in each of these cases?
time to develop ready recall of number combinations.
Practice with number combinations might include flash a. Geraldo shared 38 stickers with six friends so
cards, partner games, computer games, and worksheets. that each got the same number of stickers.
Because individual children differ in their progress with
number facts, personal improvement and success are b. Juan had 25 chairs. He put six of them in each
crucial elements to a successful practice program. Over- row.
emphasis on speed, such as timed tests, is cited as a ma-
jor cause of mathematics anxiety. c. Mr. Hui imported 453 ornaments. He packaged
24 ornaments in each box to ship to retailers.
When students understand each of the operations,
have learned meaningful strategies for learning number d. Ms. Krohn divided $900 evenly among her four
combinations, and have developed ready recall of facts, children.
they have foundational skills for computational fluency.
They will extend these skills as they add, subtract, mul- 6. Do you remember learning about remainders? Do
tiply, and divide larger numbers with a variety of algo- you agree or disagree with the idea to introduce
rithms, through technology, and with number sense remainders early in learning about division? Why?
such as estimation and checking for reasonableness.
7. Look at an elementary school textbook, and see
Study Questions and Activities how addition and subtraction are introduced. Do
the teaching/learning activities in the teacher’s
1. Tell stories that illustrate addition and each of the manual use stories, pictures, and manipulatives to
four types of subtraction. Model each story with model the operations? Does the textbook emphasize
counters, draw a picture of the story, and write a the development of strategies for learning number
number sentence for it. Which of the stories were facts?
easiest for you? Were any more difficult?
8. Obtain a practice page of 100 addition or subtrac-
2. Using counters or interlocking cubes, model the tion facts, and take a 3-minute timed test. How many
commutative property, the associative property, and did you complete? How many of the completed an-
the identity element for addition. Can you think of swers were correct? How did you feel at the end of
real-life situations that illustrate ideas similar to com- the test? Discuss with fellow students their memories
mutativity, associativity, or identity? of timed tests. What are the possible harmful as-
pects of this practice? What ways can you think of to
overcome the pressure and anxiety many students
feel from such tests?

227

Teacher’s Resources Also, programs published by Tabletop Press and the
Nectar Foundation are helpful.
Baroody, A., and Coslick, R. (1998). Fostering children’s Internet Game
mathematical power: An investigative approach to At http://www.fi.uu.nl/rekenweb/en there are a number
K–8 mathematics instruction. Mahwah, NJ: Lawrence of interesting games. Make Five allows students to play
Erlbaum Associates. alone or with a partner. The object of the game is to
capture five squares in row on a 10 ϫ 10 grid. Children
Fosnot, C., & Dolk, M. (2001). Young mathematicians at capture a square by solving the problem in the square.
work: Constructing number sense, addition, and subtrac- Students can select addition facts, subtraction facts, or
tion. Westport, CT: Heinemann. multiplication facts for their grid problems.

Kamii, C., & Housman, L. (2000). Young children rein- Find more games at
vent arithmetic: Implications of Piaget’s theory (2nd ed.).
Williamston, VT: Teachers College Press. http://www.bbc.co.uk/education/mathsfile/

Kirkpatrick, J., Swafford, J., and Findell, B. (Eds.). http://www.bbc.co.uk/schools//numbertime/games//index.shtml
(2001). Adding it up: Helping children learn mathematics.
Washington, DC: National Academy Press. http://www.subtangent.com/index.php

Children’s Bookshelf Internet Activity
This activity is for children in grades K–1. Students play
Butler, M. Christina. (1988). Too many eggs. Boston: a game of electronic concentration where the cards
David R. Godine. (Grades PS–2) contain a digit, the numeral word, or a number of dots.
Students can turn over any two cards to try to find a
Calmenson, Stephanie. (1984). Ten furry monsters. New matching pair. The object here is to find all the match-
York: Parents Magazine Press. (Grades PS–3) ing pairs in the fewest number of turns.

Chorao, Kay. (1995). Number one number fun. New York: Students will find Concentration at http://illuminations
Philomel Books. (Grades K–2) .nctm.org/Activities.aspx?grade‫؍‬1&grade‫؍‬4. Demonstrate
how to play the game and then allow pairs of students
Chwast, Seymour. (1993). The twelve circus rings. San the opportunity to play a game. They should keep track
Diego: Gulliver Books, Harcourt Brace Jovanovich. of how many turns they used, and keep a list of the
(Grades K–4) matching pairs of cards.

de Regniers, Beatrice Schenk. (1985). So many cats! Internet Sites
New York: Clarion Books. (Grades PS–3) For virtual manipulatives to practice number facts, go to

Edens, Cooper. (1994). How many bears? New York: http://nlvm.usu.edu/en/nav/vlibrary.html (see Base Blocks,
Atheneum. (Grades K–3) Base Blocks—Addition, Base Blocks—Subtraction,
Number Line Arithmetic, Number Line Bounce, Number
Matthews, Louise. (1980). The great take-away. New Line Bars, Abacus, and Chip Abacus)
York: Dodd, Mead. (Grades K–2)
http://Illuminations.nctm.org (see Five Frame, Ten Frame,
McMillan, Bruce. (1986). Counting wildflowers. New Electronic Abacus, and Concentration)
York: Lothrop, Lee & Shephard. (Grades PS–2)
http://www.arcytech.org/java/ (see BaseTen Blocks and
Moerbeck, Kees, & Dijs, Carla. (1988). Six brave explor- Integer Bars)
ers. Los Angeles: Price/Stern/Sloan. (Grades PS–3)
For sites to practice math facts, go to:
Viorst, Judith. (1978). Alexander who used to be rich last
Sunday. New York: Aladdin/Macmillan. (Grades 1–3) Math Flash Cards: http://www.aplusmath.com/Flashcards

Technology Resources Interactive factor trees: http://matti.usu.edu/nlvm/nav/
category_g_3_t_1.html
Commercial Software
There are many commercial software programs de- Interactive Flash Cards: http://home.indy.rr.com/lrobinson/
signed to help students with their number sense, recall mathfacts/mathfacts.html
of number facts, and applications of number opera-
tions. We list several of them here. Mathflyer (a space ship game that employs multiplica-
tion facts): http://www.gdbdp.com/multiflyer/
How the West Was One ϩ Three ϫ Four (Sunburst)
Math Arena (Sunburst) Math Facts Drill: http://www.honorpoint.com/
Math Munchers Deluxe (M.E.E.C.)
Oregon Trail (Broderbund) Mathfact Cafe: http://www.mathfactcafe.com
The Cruncher 2.0 (Knowledge Adventure)