528 Part 2 Mathematical Concepts, Skills, and Problem Solving
MULTICULTUR ALCONNECTION likely to show after it is rolled. Therefore the chances
of rolling any given number are one chance in six.
The contexts for probability situations should reflect multi- When the probability of an event is quantified, it can
cultural diversity. In Chapter 2 a classroom exploration of be expressed in three different ways: as a common
the lottery lasted several days as students simulated lottery fraction, as an equivalent decimal fraction, and as an
processes with increasing complexity. But a study of the lot- equivalent percent. When probability is expressed
tery might also consider lotteries in other countries. Different as a common fraction, it takes the general form:
games of chance from other cultures provide a vehicle for
probability settings that can engage students with diverse Number of times the specific event occurs
backgrounds. (See The Multicultural Game Book, by Louise Pϭ
Orlando, 2006.)
Number of times all events occur
Thus the chances of getting a 5 on a die is ex-
Extending Probability pressed as 16, because there is one chance of rolling
Understanding a 5 out of a total of six possible outcomes. In a few
rolls of a die one number may occur more frequently
than the other five, but after many rolls the probabil-
Children in grade 3 and ity is that each number will occur equally, and each
higher are able to reason
more thoughtfully about number will be rolled 1 of the number of total rolls.
probability situations. 6
They are able to move
MISCONCEPTION beyond simply assigning Probabilities from both historical and mathemat-
events as likely or unlikely
Children without prob- to using principles of ical situations are expressed in the same ways. The
ability experiences will probability to compute or
often reason that every predict the probability of probability of getting heads when a coin is tossed is
possibility, no matter how an event. The NCTM stan-
unlikely, has an equal dards for older children 1 out of 2 and is expressed as 21, 0.50, or 50%. If rain
chance of happening. are as follows: has occurred 980 times during the 31 days of July
They develop understand-
ing through recording the for the past 40 years, the historical probability is 980
results of many probability 1,240
activities.
(1,240 ϭ 31 ϫ 40), or 0.79, or 79%, that it will rain in
July this year. Because probability is expressed in
common fraction, decimal fraction, or percent form,
it is critical that students have a working knowledge
of these different representations before they begin
Develop and evaluate inferences and predictions that are to compute theoretical probabilities. Students with-
based on data
Grades 3–5 Expectations out knowledge of fractional representations may
In grades 3–5 all students should:
• propose and justify conclusions and predictions that are perform the proper calculations to compute a prob-
based on data and design studies to further investigate ability, but they will not be flexible in their interpre-
the conclusions or predictions.
Grades 6–8 Expectations tation of any fractions.
In grades 6–8 all students should:
• use observations about differences between two or more Students discover the probability of getting a cer-
samples to make conjectures about the populations from
which the samples were taken; tain number by rolling a die and keeping a tally of the
• make conjectures about possible relationships between
two characteristics of a sample on the basis of scatter- results. Because a small number of rolls—say, 36—
plots of the data and approximate lines of fit;
• use conjectures to formulate new questions and plan new with one die may not produce experimental results
studies to answer them.
that match the mathematical, or theoretical, prob-
Number Times ability of getting each number,
students need to make a large
1 number of rolls. If each child in
2 a class rolls a die 36 times, com-
The standard suggests that computing numerical 3 bining all the tallies shows that
probability is appropriate for elementary school chil-
dren in grades 3 and above, using various manipula- 4 the experimental, or empirical,
tives and activities to generate data. Experimenting
with dice is a common classroom probability activ- 5 probability of the die comes
ity. With one die, six outcomes are possible. If a die
is fair, or unbiased, any of the six sides is equally 6 close to, or approximates, the
Results theoretical probability. Figure
of rolling a die 36 20.1 shows the results of rolling
times one die 36 times. The table in
Chapter 20 Investigating Probability 529
Figure 20.2 shows the results of using a computer Probability investigations and computations can
program to roll one die 500 times. Notice that the re-
sults for 500 rolls more closely match the theoretical be adapted to the interests and mathematical ma-
probabilities than do the results for only 36 rolls.
turity of upper elementary
Figure 20.2 Results of rolling one die using the program
at http://www2.whidbey.net/ohmsmith/webwork/ MISCONCEPTION grade students. In simple
javascript/diceroll.htm investigations with spin-
Children tend to think that ners, dice, and coin, stu-
The results of the computer program in Figure the results of a sampling,
20.2 illustrate the Law of Large Numbers. The Law
of Large Numbers stipulates that the greater the no matter how small, dents keep records and
number of events or trials that occur for a particular will accurately reflect an discuss the results of their
situation, the closer to theoretical probability the re- entire group. experiments. When chil-
sults will be. When students collect experimental or
empirical data from dice rolls, coin flips, and other dren use a five-colored
events, unless they record a large number of events,
it is unlikely that the empirical results will match spinner (see photo), they
theoretical results.
find that the likelihood of
As the data of the computer program show, for
many trials the results closely mirror theoretical re- getting a certain color is
sults. Students need a large number of events before
any conclusions can be drawn about probabilities. 1 in 5, 51. With the spinner
Students who are just beginning to gather data for numbered 1–6 shown in
probabilities tend to draw conclusions from the data
of few events. For example, a child who picks three the photo, an event such
blue marbles in a row from a bag of marbles may de-
clare that all the marbles in the bag are blue, when as having the spinner stop
in fact six are blue and four are green.
on 4 is 1 in 6, 16. Students
will know that chances
of winning many popular
board games are equal for
each player when they understand the mathemati-
cal probabilities associated with the dice or spin-
ners that control moves on the board.
Probability Investigations
Coins provide many opportunities for probability in-
vestigations. Students can put a coin in a soft plastic
cup and shake it before dumping it onto a mat. For
each coin event, or trial, the result is either a head or
a tail. Children keep a tally of the number of heads
and tails for 40 trials, as shown in Figure 20.3. For
each flip the chances of a head or a tail are equal.
Over the long run the number of heads and tails will
be about equal, or close to 50-50 (50%-50%). In the
unlikely event that they are not, children should con-
Courtesy of NASCO
Probability devices
530 Part 2 Mathematical Concepts, Skills, and Problem Solving
sider the reasons. They should consider the ques- dice (see the six dice at the top center of the photo
tion of whether the coin is a fair one. A coin, or any on p. 529). Children should record data in tables as
other device, is fair when the observed chances ap- they work. As a further extension, students could
proximate the mathematical chances. If, however, a compute the theoretical probability of rolling each
head on a coin occurs many more times than tails, sum and compare that number with the classroom
the coin might be bent or nicked or made unfair in results (empirical probability). In Activity 20.5 stu-
some other fashion. If that is not the case, does the dents mark a blank number cube for a board game
way the coin is tossed contribute to the disparity? All they design.
possible reasons should be considered. In the com-
panion website activity “Dice Data” students predict Probability experiments can be varied by using
the results of rolling a single die 60 times and then different objects and by varying the number of tri-
do the rolling. als. A coin-tossing experiment using one, two, three,
four, and five coins leads to an interesting pattern as
Figure 20.3 Tally of coin flips the number of outcomes increases. When one coin
is tossed, only two possible outcomes can happen:
Older students work with more complex prob- heads or tails. When two coins are tossed, the out-
ability experiments. The work with a pair of dice comes can be two heads (HH), two tails (TT), or
in Activity 20.4 introduces them to the idea that one head and one tail. However, because each coin
different probabilities exist for events occurring in can land independently, there are two ways to get
the same activity. The table here shows the possible one head and one tail, so there are four possible
sums for rolling a pair of dice. outcomes: TT, TH, HT, and HH. Thus the chance
of tossing two coins and getting two tails (or two
123456 heads) is 41, or 25%. With three coins, eight outcomes
1 234567 are possible: HHH, HHT, HTH, HTT, THH, THT, TTH,
2 345678 and TTT. The combinations for one, two, and three
3 456789 coins are shown in a triangular arrangement in Fig-
4 5 6 7 8 9 10 ure 20.4. After children have worked out combina-
5 6 7 8 9 10 11 tions for one, two, and three coins, ask them to look
6 7 8 9 10 11 12 for a pattern to see if they can predict the combina-
tions for four coins. If they cannot, list combinations
After building such a table, the teacher asks why 6, for four coins, enter their numbers into the triangle
7, and 8 have more occurrences than 2 and 3, or 11 in Figure 20.4, and look for a pattern again. Soon the
and 12. Many children will claim that there are only students will see that each number in a lower line is
three ways to roll a 7 because, for example, they per- the sum of adjacent numbers above it in the previ-
ceive 6 ϩ 1 and 1 ϩ 6 as a single possibility. One ous row. For example, the 3 in row 3 is the sum of
reason they do so is that the commutative law for the two numbers above it in row 2 (1 and 2). Extend
addition postulates that 6 ϩ 1 ϭ 1 ϩ 6. Using two the pattern to 10 coins and see what happens to the
different colored dice can help children understand number of possibilities.
why 6 (red die) ϩ 1 (green die) is different from 1
(red die) ϩ 6 (green die). The arrangement of numbers in Figure 20.4
is called Pascal’s triangle, named after the 17th-
An extension of this activity can be made by
using three or four regular dice or by investigating 1 coin 1 H, T
results with pairs of tetrahedral and icosahedral 2 coins 11 HH, HT, TH, TT
3 coins 12 1 HHH, HHT, HTH, THH,
13 3 1 HTT, THT, TTH, TTT
4 coins
5 coins
Pascal’s triangle
Chapter 20 Investigating Probability 531
ACTIVITY 20.4 Tossing Dice
Level: Grades 4–6 • As a follow-up to the activity, have two children calcu-
late the totals for each column for all the tally sheets and
Setting: Whole class prepare a bar graph showing the class totals. Compare
this graph with the students’ graphs. Does the classroom
Objective: Students determine the probability of an event. graph align with the mathematical probabilities more
closely than the students’ graphs?
Materials: Two dice for each pair of children, tally sheet for each
pair of children, pencils, centimeter or half-inch squared paper, • Some computer programs can be used for a similar
marking pens, calculator, large sheet of 1-inch squared paper activity.
• Students work in pairs. Give each team of children a • The following tables and graphs show results of 100 and
pair of dice. The first task is to determine what sums are 500 “rolls” of dice with the program found at http://nces
possible when two dice are rolled and the numbers are .ed.gov/nceskids/probability/.
totaled. Have the children guess which of the sums will
be more frequent or less frequent when rolling two dice.
(Students with experience rolling one die often guess that
all the numbers will have the same frequency.)
• The children roll the dice 50 times and tally each sum as
it occurs. One child rolls the dice 25 times while the other
records; then they exchange roles.
• When all pairs are finished and have counted tallies for
each sum, discuss their results. Ask why some numbers
came up more frequently than others. Consider the ways
the dice can make 2 and the ways they can make 7. Ask if
any pairs had one number come up more often than any
middle-range numbers, say, 3 or 10. This provides an op-
portunity to discuss how small samples, such as 50 rolls of
the dice, do not always produce the expected outcome.
• Each pair of children uses an Internet program, if available,
to make a bar graph showing the results of their tosses.
Have students make their graph on squared paper if an
Internet program is not available.
century mathematician Blaise Pascal, who studied • When six coins are tossed, what is the probabil-
and wrote about it. You can create problem cards ity of getting three heads and three tails? What
to guide students’ investigation of many interesting percent of the possible outcomes is this?
questions based on Pascal’s triangle. Here are some
examples: • What is the probability of getting four heads and
four tails when eight coins are tossed? What per-
• What is the sum of each row of the triangle? cent of the possible outcomes is this?
Can you describe the sequence of numbers that
includes each sum? • Will the percent of possible outcomes for five
heads and five tails when 10 coins are tossed be
532 Part 2 Mathematical Concepts, Skills, and Problem Solving
greater or less than the percent for four heads and Combinations
four tails when 8 coins are tossed? Why do you
think this is so? Pascal’s triangle is useful for determining all the pos-
• When you toss seven coins, what is the probabil- sible combinations of feasible events in a probability
ity of getting three heads and four tails? What is
the probability of getting four heads and three situation. It is impossible to determine the probabil-
tails? Is each of these probabilities the same per-
cent of the total outcomes? How do you explain ity of any event without determining all the conceiv-
this?
able combinations of possible events. Older students
E XERCISE
can find all possible combinations for an event by
Complete Pascal’s triangle to include tosses for 10
using a probability tree. The probability tree shown
coins. Now answer the preceding questions. •••
in Figure 20.5 was drawn by Sal, a fifth-grade stu-
French mathematician and philosopher Blaise Pascal (1623–
1662) is considered one of the cofounders of probability dent. The tree shows all the possible combinations
theory. Pascal began his explorations into probability as the
result of the request of a friend, Antoine Gombard. Gombard of outfits that he can wear using the following ar-
had a question about how to divide up the stakes of a dice
game when the game was cut short. Pascal’s exploration of ticles of clothing: three pairs of pants (tan, gray, and
his friend’s query led him to write to another mathematician,
Pierre de Fermat, and their correspondence established the black), three shirts (gray, tan, and white), and two
principles of probability.
pairs of shoes (black and tan). This probability tree
Pierre de Fermat (1601–1665), cofounder of probability
theory with Pascal, was a solicitor for the French government shows 18 possible outfit combinations with these
by day and an amateur mathematician by night. But what a
mathematician! He has been called Prince of the Amateurs three articles of clothing. In his probability tree Sal
because of his refusal to publish any of his mathematics dis-
coveries; yet he had a hand in the development of probability, has highlighted the single combination of clothing
calculus, number theory, coordinate geometry, and plane
geometry. that will result in an outfit that is all tan. If Sal’s lit-
tle sister were to randomly select his clothing for a
school day, there is a 1 probability that he would be
18
dressed in all tan. Students will also discover that
the number of combinations can be expressed as a
multiplicative relationship:
(number of pants choices) ϫ (number of shoe
choices) ϫ (number of shirt choices) ϭ number of
different outfits
Students who are beginning their study of combina-
tions in probability require much experience listing
ACTIVITY 20.5 Designer Number Cubes
Level: Grades 4–6 • The game will use a single designer number cube for mov-
ing the game piece. Each group will design a new number
Setting: Small groups cube. Their designer cube will not show each number
from 1 to 6 but will show some numbers more than once,
Objective: Students design their own number cubes for a new and some numbers not at all. Each number cube must
board game. contain at least three numbers on the six faces, and no
more than two numbers can have the same chance of
Materials: Blank number cubes, colored markers, poster board coming up. Students might use different colors for each
number.
• Ask students what games they have played that use dice
or number cubes. Discuss the chances of rolling a 3 with • Students display their finished game boards to the class.
a single number cube. How does the chance of rolling a 3 They describe their number cube and show the chances of
compare to the chance of rolling a 6? a 1? rolling each number represented on their number cube.
• Describe to students a board game that involves moving a • Have groups exchange game boards and number cubes
game piece along a 25-square-long path. A few squares and play each other’s games.
can contain commands such as “Move ahead,” “Lose a
turn,” or “Move back.” Each group will design a game
board with a 25-square path from start to finish.
Chapter 20 Investigating Probability 533
Eventually, middle T
A
school students learn how
C
to compute combinations
T
in various situations. They A
T
also study permutations,
C
which are a type of combi-
A
nation in which the order
T
of the individual items is C
A
important, such as the let-
C
ters in a word. For example,
T
consider the three letters C,
A
A, and T. There is only a sin- C
gle combination with these Figure 20.6 Probability
tree: Permutations of
three letters: C, A, and T (in CAT
any order). Six possible per-
mutations can be formed
with these same three let-
ters. The probability tree in
Figure 20.6 shows the six
possibilities: CAT, CTA, ACT,
ATC, TCA, and TAC.
Sampling
Figure 20.5 Student-drawn probability tree: Outfit Sampling, another probability technique, involves
combinations choosing one or a few items from a larger group and
drawing conclusions about the larger group based
combinations using a probability tree. Activity 20.6 on the sample. Public opinion surveys, for example,
asks students to use a probability tree to determine are based on the opinions of sometimes fewer than
the total number of different breakfast meals that 1,000 people. Because the survey is taken from a
are possible from a fast-food restaurant. sample of people who are carefully selected to be
representative of the entire country, such surveys
MULTICULTUR ALCONNECTION are reasonably accurate representations of the
opinions of the entire population. Children can use
Although Pascal’s triangle bears the name of Blaise Pascal, colored candies, marbles, beads, or blocks for sam-
he was not the first to explore its properties. Chinese math- pling experiments.
ematician Chu Shih-Chieh wrote about the same triangle in
1303, more than 300 years before Pascal made his discovery Activity 20.7 describes a simple sampling process
of it. The triangle bears Pascal’s name because he investi- that can be used to estimate the colors of marbles
gated its properties more deeply. in a bag or box. Activity 20.8 is a similar activity, in
which students use a cumulative sampling strategy.
They find that by combining the results of several
independent samples, they can make a better esti-
mate about the entire group. As with experimental
probability results, the larger the sample, the better
the resulting data are for making predictions.
Geometric Probability
In addition to dice, coins, and shoes, students can
gather data from area relationships between geomet-
ric figures. Children have an intuitive understanding
534 Part 2 Mathematical Concepts, Skills, and Problem Solving
ACTIVITY 20.6 Probability Tree (Representation)
Level: Grades 3–5 1. You go to Henrietta’s Hamburger Heaven every
morning and eat a different breakfast combination.
Setting: Groups of three students How many mornings would it be before you had the
exact same breakfast again?
Objective: Students build a probability tree to solve a real-life
problem. 2. What is the chance that someone across the room
eating breakfast has ordered the same combination
Materials: Pencil and paper, markers, poster board that you ordered for breakfast?
• Relate to students that Henrietta’s Hamburger Heaven has 3. Henrietta is thinking of adding tea as a beverage. If
decided to offer breakfast meals. The plan is to have the she does add tea to the breakfast menu, how many
following foods: different breakfast combinations would there be?
• Three kinds of breakfast sandwiches: egg and cheese;
ham, egg, and cheese; and bacon, egg, and cheese 4. The tea idea didn’t work out. Instead of adding tea
• Four types of breads: English muffin, bagel, croissant, as a drink selection, Henrietta drops the tea and
and biscuit adds another bread: toast. Now how many break-
• Two drinks: coffee and orange juice fast combinations are there?
• Have groups select a reporter, a recorder, and a resident 5. The toast idea didn’t work either. Now Henrietta is
artist for the activity. going to add another sandwich: sausage, egg, and
cheese. How many different breakfast combinations
• Each group is to build a probability tree to display all these are there now?
data and to answer the questions that follow.
6. Using the answers to the last four questions, explain
• Once groups have designed a probability tree that they how many different ice cream cones you could
all agree is correct, have each group copy the tree onto a order from Iggy’s Ice Cream Igloo. Iggy offers 12
poster board. flavors, plain or sugar cones, and nuts, sprinkles, or
chocolate bits as a topping. You may make a new
• Groups are to use their probability tree to answer each probability tree, or just part of one, to help you get
of the questions and to explain how they used the tree the total number of different sundaes.
to get their answers. Once all groups have completed the
questions, have groups take turns with the questions and
use their probability tree to explain their answers to the
rest of the class.
ACTIVITY 20.7 Sampling (Communication)
Level: Grades 4–6 • After 50 draws, have the students study the total for each
color and guess how many of each color are in the bag.
Setting: Groups of four Remember, there are 100 objects in the bag!
Objective: Students conduct and explain a simple sampling • Use the totals to create percentages: 14 blue beads in
investigation.
50 draws ϭ 14 ϭ 28%, or 28 blue beads of 100; 12 red
Materials: Bags of 100 plastic beads or chips of three or four 50
different colors, one bag for each group of four students; record 12
sheet for each group; pencils. Prepare two or three bags with the beads in 50 draws ϭ 50 ϭ 24%, or 24 red beads in 100;
same number of beads or chips of each color and the rest with
different numbers of beads or chips of each color. 24 yellow beads in 50 draws ϭ 24 ϭ 48%, or 48 yellow
50
• Use the roundtable cooperative plan. The children number
off: 1, 2, 3, 4. Child 1 has the bag, and child 2 has the beads in 100.
tally sheet and pencil. Child 1 draws an object from the
bag, notes its color, and replaces it while child 2 makes a • Share results of the draws. Do any of the groups believe
tally mark for its color on the record sheet. that they may have the same number of beads or chips of
each color in the bag? Do they believe that each bag has a
• Drawing and tallying, the children rotate through the different number of beads or chips of each color?
group after each draw until 50 draws have been made.
• Have children remove and count each color of bead or
Color Tally sheet chip. How accurate were the results of the samplings? Did
of marbles Count any bags contain the same number beads or chips of each
color? If so, were they the same ones predicted during the
Blue earlier discussion? Are the children satisfied that drawing
50 objects enabled them to make accurate predictions?
Red
Total To extend the activity, tell children to take a bag home to
Yellow 14 repeat the process by drawing more than 50 or fewer than
12 50 objects and tallying the results. Discuss whether the
24 accuracy decreases when fewer objects are drawn. Does it
increase when more are drawn? Does there appear to be a
number beyond which accuracy does not increase?
Chapter 20 Investigating Probability 535
ACTIVITY 20.8 Multiple Drawing Sampling
Level: Grades 5 and 6 work in a roundtable format. Child 1 has a bag, child 2
Setting: Groups of four has a record sheet, child 3 has a calculator, and child 4
Objective: Students use multiple drawings to estimate the contents has a record sheet. Child 1 draws five objects, and child 2
writes the number of each color on a record sheet. Child
of bags. 3 uses the calculator to determine the percent that each
Materials: Bag with 100 plastic beads of four colors, two record color is of the total, and child 4 writes the percent on a
record sheet.
sheets for each team, calculators
• Functions rotate until each child has performed each role
Mixed beads twice. After each draw, students calculate a running total
for each color and use it to estimate the fraction for each
color of bead.
Red Green Black Yellow
TRIAL 1
TRIAL 1 %
TRIAL 2
CUMULATIVE %
TRIAL 3
CUMULATIVE %
TRIAL 4
• Tell the children that each bag contains 100 beads of four • After five beads have been drawn four times, each team
different colors. Their task is to estimate the number of estimates the number of beads and the percent of each
each color. They will draw five beads at a time without color.
looking and return the beads after each draw. Children
• Repeat the process to make a total of eight drawings, and
again estimate the number and percent of each color.
• Count the actual objects to check the estimates. Follow
up with a discussion similar to the one described in Activ-
ity 20.7.
of targets and bull’s-eyes. The bull’s-eye is the small- of the template. In both cases the area marked 1 is
twice as large as the area marked 2. With a sufficient
est part of the target and carries the highest number number of spins, this 2:1 ratio would be approached
with experimental data. Activity 20.9 extends the
of points. The largest parts of a target (the easiest to concept of geometric probability on a spinner by
having students design their own spinner templates.
hit) carry the fewest points. In Activity 20.10 students use an interactive activity
on the Internet to explore probability on a spinner.
Spinners that children use to play board games
are an example of geometric probability. In the spin-
ner pictured in Figure 20.7a, all the numbers have an
equal chance of coming up because each number
has a region that is equal in size to all the other in-
dividual regions. The spinners in Figures 20.7b and
20.7c show that 1 will come up twice as often as 2, a
relationship that could be predicted by visual inspec- 12 1 1
3 432 3
tion or by gathering data from a number of spins. In 3
(a) 1 52
Figure 20.7b the area of region 1 is 1 of the area of 3 2
4 (b)
21
1
the spinner template, and the area of region 2 is 8 of (c) (d)
the template. In Figure 20.7d the area marked 1 is 1
4
the area of the template, and the area marked 2 is 1 Figure 20.7 Probability spinners
8
536 Part 2 Mathematical Concepts, Skills, and Problem Solving
ACTIVITY 20.9 Spinner Probability
Level: Grades 4–6 equal chance of coming up. They should use three to six
numbers or colors for the spinner template.
Setting: Small groups
• Once a group draws a new spinner template, have them
Objective: Students design their own templates for spinners, pre- predict how often the spinner will land on each number
dict probabilities, and test their predictions. or color in 100 spins. Each group should build identical
templates of their spinner for each person in the group.
Materials: Spinners with blank templates, paper and colored Then the group should make 100 spins and collect data to
pencils verify their predictions for the template.
• Show students a spinner from a board game (or from • Have groups take turns displaying their templates to the
a game you have designed) with all numbers having an class. Each group can challenge classmates to predict
equal chance of coming up (see Figure 20.7a). about how often each number or color will come up in
the 100 spins, based on visual inspection of the spinner.
• Have students discuss why this is a fair spinner. Probe for
students to recognize that equal areas on the spinner have
an equal chance of coming up.
• Challenge students to design two different spinner tem-
plates on which the numbers (or colors) do not have an
Students can make their own spinner with a pa- E XERCISE
per clip. Students can use a pencil to secure one end
of the paper clip spinner and spin the paper clip to Describe the probabilities shown in Fig-
replicate a spinner. In Chapter 14 the black-line mas-
ters that show fraction circles can also be used as ure 20.7. •••
templates for spinners.
Expected Value
Expected value is a topic that can have much ap-
peal to older children. They may be familiar with the
television show Wheel of Fortune or carnival games
of chance that use a spinning wheel. Once students
have ample experiences designing spinners for prob-
ability explorations, they are able to use the princi-
ples of probability to explore the expected value of
specific spinners. Consider the spinner shown here.
Chapter 20 Investigating Probability 537
ACTIVITY 20.10 Design Your Own Spinner (Internet Activity)
Level: Grades 5 and 6 • Ask students to design a spinner with three colors so that
Setting: Groups of three students the spinner has an equal probability of landing on each
Objective: Students design a spinner according to predetermined color.
probabilities. • Check on each group’s design, and then have students
Materials: Internet access check their spinner design by recording the data for
30 spins.
• Go to http://nlvm.usu.edu/en/nav/vlibrary.html and follow
the links to Spinners in 3–5 Data and Probability. • Discuss the role of the Law of Large Numbers in the data
that result. (These data will not always show a 10-10-10
• Demonstrate how to change the spinner by increasing or result.) Combine the data from all groups to obtain an
decreasing the share of each color and by increasing and experimental result that is closer to the theoretical predic-
decreasing the number of colors in the spinner. tion of 1᎐3 for each color.
• Show how to “spin” the spinner and record the results.
This spinner will pay out to the player the amounts $6.00 to play, they will likely raise some funds using
this spinner.
shown in each sector of the spinner. Expected value
Students can explore expected value using spin-
concepts enable students to determine the expected ners that they have produced from earlier activities.
They can also design original spinners with different
payout for the spinner. Naturally any prediction such payouts for each sector and determine the break-
even price to use the spinner in a game of chance.
as this assumes a large number of spins, so the Law Students might then test their findings by recording
the results of spinning their spinner. Again, the Law
of Large Numbers is a factor in making any predic- of Large Numbers requires that students collect data
from a large number of spins. In this case perhaps
tion. It is impossible to predict the result of a single 50 spins will produce enough data to support the
students’ computations.
spin. Rather, the probabilities shown here specify
Both Activities 20.9 and 20.10 used spinners to
the tendency of the spinner for many spins. develop probability concepts. In Activity 20.9 stu-
dents built their own spinners, and in Activity 20.10
For this spinner the chance of landing on each students used an Internet program to simulate us-
ing a spinner. Activity 20.11 is an assessment activity
sector is that probes students’ understanding of probability
in a geometric setting. Note that the dimensions of
$4 ϭ 1 $6 ϭ 1 $8 ϭ 1 the figures in the game board are not given. We sug-
2 4 8 gest making a game board that resembles the game
board that is shown but does not duplicate it. One
The probability of landing in each sector can be of the tasks of the activity is to assign an appropri-
ate number of points according to the probability of
combined with the posted winnings for each sec- landing a penny in each figure. With no specific di-
mensions given for any of the figures, students must
tor to determine the expected value or expected use empirical probability to assign points to each
figure. (In this case the chances of a penny landing
winnings for playing a game of chance with this anywhere on the target is totally random, in contrast
to a game of darts or pitching pennies, in which
spinner: players aim for one particular region of the target.)
As in all the other probability activities, the chance
(1 ϫ $4) ϩ (1 ϫ $6) ϩ (1 ϫ $8) ϩ (1 ϫ $8)
2488
ϭ $2 ϩ $1.50 ϩ $1.00 ϩ $1.00
ϭ $5.50
In the long run, with many spins, the expected
value or winnings per spin is $5.50. Thus, if a chari-
table organization were to hold a fundraiser and in-
clude a game using this spinner to raise funds for
their charity, the price of each spin would have to
exceed $5.50. Any lower price to play would result
in the charity paying out more money than the game
earns. A charge of $5.50 to play would mean that the
game would break even in the long run and not earn
any funds. So, if the organization charges at least
538 Part 2 Mathematical Concepts, Skills, and Problem Solving
Research for the Classroom •
ple, when asked whether there was a greater number of
Research has confirmed the persistence of characteristics two-person groups or eight-person groups possible out of
similar to the gambler’s fallacy in children who are learn- a total of ten people, students responded that more two-
ing probability principles (e.g., Watson & Moritz, 1999; person groups were possible, although this is untrue by a
Shaughnessy, 1992, pp. 470–472). Individuals who believe large factor. The reason is that these students could easily
the gambler’s fallacy predict that a probability event will be compose many different two-person groups in a short
influenced by past events—that the coin or slot machine time compared to the time needed to construct eight-
has a memory. For example, if a coin has come up heads person groups. Thus they persisted in their conclusion that
10 times in a row, a believer in the gambler’s fallacy will more two-person groups would be possible. Children will
reason that the next flip of the coin has to come up tails predict that they will select their favorite color marble out
because “tails is due.” This reasoning is similar to that of of a bag, even if there are few of those marbles compared
individuals who play a single slot machine for a large num- to the other colors.
ber of tries, believing that eventually the machine has to
pay out. Research with students drawing colored balls out These research results and many others underscore the
of a bag revealed that they expected a 50-50 distribution effects of personal experiences and beliefs that result in
of the two colors. They persisted in their belief through misconceptions. Students will bring to school experiences
many trials of drawing the balls out of the bag (and then with specific events, ranging from board games with dice
replacing them), anticipating that the next draw and sub- and spinners to personal preferences regarding colors.
sequent draws would equalize the colors they had already These outside influences will prejudice their thinking about
drawn, as if the marbles or the bag had a memory of the probability. Some students may even believe that God or
preceding events. their mental powers will influence a coin flip or a roll of
the dice. The classroom teacher must be aware of these
Related research has found that students will make factors and work to help students leave their misconcep-
predictions of the likelihood of an event based on how tions behind.
common the event is to them or how easily the event can
be duplicated or simulated (Shaughnessy, 2003). For exam-
ACTIVITY 20.11 Geometric Probability (Assessment Activity)
Level: Grades 3–5 Just one thing: A player has to pick up the penny with his
or her eyes closed and then drop it on the game board.
The game board you see here is designed for tossing Thus a player cannot really aim for any part of the board.
pennies. Here are the rules for playing the game. Use what you know about areas and probability to give
the chance of landing on each part of the game board.
• If the penny lands off the game board, it doesn’t count,
and the player gets another turn. 1. What are the chances that the penny will land in the
rectangle? in the square? in the triangle? on the game
• If the penny lands on the board but misses the figures, the board but not in any figure?
player still scores 1 point.
2. Are the points on the game board OK? Do you think
1 4 that the points should be different to make the game
2 board fairer? How would you change the game board,
and why?
3. Suppose that you designed a new game board with a
circle, square, and triangle. What would be true about
the size of these three figures if the chances of drop-
ping a penny in each of them were equal?
3
Chapter 20 Investigating Probability 539
of an individual event occurring is totally random. carefully so that the data will properly represent
If events were not totally random, they could not be the actual event. Activity 20.12 describes a simula-
predicted over the long run by theoretical probabil- tion activity for students. In this simulation rolling a
ity and the results would not be effective for predict- single number cube represents finding toy prizes in
ing future results unless the same nonrandom fac- random boxes of cereal.
tors, such as aiming at a particular part of the target,
were reproduced exactly. Probability simulations with dice or similar manipulatives
that are also used for games of chance are called the Monte
Simulations Carlo method, after the famous Casino de Monte Carlo in
Monaco. Simulations were first used by physicists working on
Simulations are used to determine the probability the Manhattan Project during World War II. The simulations
of an event when there are no data from previous were used to develop shields and dampers for nuclear bombs
events, it is not possible to gather experimental data and reactors, something that could not be done by direct
of the event, or the theoretical probability is impos- experimentation.
sible to compute. Simulations must be constructed
ACTIVITY 20.12 A Probability Simulation (Connections)
Level: Grades 3–5 cereal to buy in order to collect all six action figures. (An-
swer: There are six action figures and six numbers on the
Setting: Small groups faces of the number cube.)
Objective: Students conduct a simulation to explore the probability • As a class, assign one number from the number cube to
of a real-life situation. each action figure. Be sure that each group’s recorder
writes out which figure each number represents for
Materials: One number cube per student the group to use as a reference. Because each number
represents an action figure, one roll of the number cube
• Describe to students how and why simulations are used can represent opening a box of cereal. Whatever number
to determine the chance of an event happening. Stress comes up will represent the specific action figure in that
that simulations are sometimes necessary because it is box.
impossible to gather experimental data from the event to
determine the probabilities involved. • Have each student roll a number cube until each num-
ber on the cube has come up. Be sure that each student
• Ask students what prizes or toys they have collected from records the results of each roll, using tally marks in a table
cereal boxes or fast food meal packs. Probe for students like the one shown here. When all the numbers have
to tell how many different individual figures or toys there come up, students should count the number of times they
were in a particular collection they mention. rolled the number cube.
• Have students select a reporter and a recorder for their 123456
group.
• Have the group reporter tell the group’s results, and write
• Describe to students a promotion for a new cereal, Break- the results on the board. Ask students to think of the
fast Bash. The Breakfast Bash manufacturers have put one number of rolls that seems to best represent the overall
of six different action figures in each cereal box: Handy results for the class. Is it the mean, median, or mode of
José, Sharp Simone, Mighty Miguel, Lucky Luisa, Super the data? Ask students how to evaluate outliers such as
Sam, and Crafty Crissy. Ask students to estimate in their 7 rolls or 32 rolls.
groups about how many boxes of cereal they would have
to buy to be likely to collect all six action figures. • Ask students about the role of the Law of Large Num-
bers in this simulation. One classroom set of data is not
• Relate that one way to find out would be if they all really sufficient to make a prediction about the number
bought many boxes of cereal and recorded the data. Have of boxes. More data are needed.
students discuss why that strategy is impractical.
• Because it is impractical to collect the experimental data,
explain to students that using a number cube can replace
actually buying boxes of cereal.
• Ask students what it is about a single number cube that
relates to the problem of deciding how many boxes of
540 Part 2 Mathematical Concepts, Skills, and Problem Solving
Take-Home Activities students in the class do not have access to a VCR or
digital video recorder at home, they might work
This take-home project involves sampling data to with a student who has a VCR or complete the
predict how many minutes of commercials run in project in school using a tape you have made.
an hour of television time. Students will need a
television and VCR to complete this project. If some
Dear Parents,
Your daughter/son will be using a television and VCR to conduct a probability
experiment. They will record one hour of evening television and then analyze
the taped material. Their data findings will be combined with data from other
students in the class to predict the amount of commercial content in a typical
hour of network television. This will help them to understand how sampling is
used in mathematics to make predictions.
Analyze the recorded material with your child. Have them explain what they
are doing and how the data they record will be used.
If I can answer any questions, please contact me at school.
1. Have your child record one hour of any network television station. Be
sure your hour runs during the evening, between 5 and 11.
2. Have your child start the program from the beginning, and then press
fast forward. After a short time, your child should stop the tape and note
whether there is a commercial or a program playing.
3. Have your child press fast forward again and again stop the tape. What is
playing now, a commercial or a show?
4. Your child should repeat this process 30 times. Help your child keep a
careful record of commercials and programs that are playing at each
stop. If the program runs out, your child can move back to the beginning
and continue the process.
5. With your child, use the data results to predict how many minutes out of
the 60 minutes are commercials.
6. Time how long the commercials actually last in the hour you taped.
7. Compare your prediction to the actual length of commercial time.
8. Have your child bring the data to class on Wednesday. We will combine
data to determine what is a reasonable estimate of commercial time in an
hour of typical network evening television.
9. Be sure your child reports back to you the results of the class data and
analysis.
541
Summary separate times, what is the probability that tails will
result on exactly two of the tosses?
Younger children first learn about probability when they a. Ϫ29
describe real-life events as impossible or certain, likely b. Ϫ38
or unlikely. Older children conduct probability experi- c. Ϫ49
ments to develop important problem-solving skills and d. Ϫ23
an understanding of chance. They perform experiments,
record and organize data, examine data for patterns, es- NAEP (http://nces.ed.gov/nationsreportcard/) There
tablish numerical odds, and make predictions from their are 15 girls and 11 boys in a mathematics class. If a
observations. student is selected at random to run an errand, what
is the probability that a boy will be selected?
Probabilities are derived from analysis of previous a. Ϫ246
events and from mathematical sources. Historical prob- b. Ϫ1216
ability is based on accumulated data that pertain to par- c. Ϫ1256
ticular events, such as rainfall or appliance reliability. d. Ϫ1115
Mathematical probability deals with the mathematics of e. Ϫ1151
chance, such as flipping coins and rolling dice. Children’s
work with probability is largely exploratory; formal in- TIMSS (http://nces.ed.gov/timss/) In a school there
struction may be limited to mature students in the upper were 1,200 students (boys and girls). A sample of 100
elementary grades. However, a teacher models correct students was selected at random, and 45 boys were
terminology so that children can develop a vocabulary found in the sample. Which of these is most likely to
about probability. be the number of boys in the school?
Study Questions and Activities a. 430
1. Have a roommate or family member make and give b. 500
you a collection of objects in a bag or box similar
to one of the collections in Activity 14.7. Use the c. 540
method explained in Activity 14.7 to determine the
characteristics of the set of objects. Compare your d. 600
predictions with the actual objects in the bag or
box. Using Children’s Literature
2. Examine the local or state lottery. Find out the Holtzman, Caren. (1997). No fair! New York: Scholastic.
chances of winning the grand prize. How many (Grades 2–4)
tickets would you have to purchase to be assured of Probability as it relates to fairness is the theme of this
having a 50% chance of winning? book. Two central characters, David and Kristy, play
games against each other. As they play, several issues
3. Explain why one mathematician has called lot- arise. Who gets to choose the game? Do they each have
teries “a tax on people who do not understand a fair chance of winning? The story examines probabil-
mathematics.” ity in the context of fair games.
4. Monitor the time it takes you to brush your teeth for Cushman, Jean. (1991). Do you wanna bet? Your chance
the next few days. Use the data to predict how long to find out about probability. New York: Clarion.
you will need to brush your teeth the next time you (Grades 4–6)
brush. This book is a collection of problem situations that in-
volve probability. Probability with coins, playing cards,
5. Some reports suggest that the chance of dying from the weather, and sampling are some of the ideas in the
a bee sting in the United States is greater than the problem stories. A changing cast of characters is con-
chance of dying from a snakebite. Does this seem fronted with a problem to solve, the answer to which is
reasonable to you? How are such probabilities found by the principles of probability (e.g., Given a 60%
determined? chance of rain tomorrow, should the friends still plan
a hike for the next day?). The solution for each story
6. There is a 60% chance of rain tonight. Discuss with a problem immediately follows the story presentation in
classmate whether you would still plan on going to the text and is explained at the same level as the story
an outdoor concert. itself. This book could almost serve as a self-tutorial in
basic probability principles.
Praxis (http://www.ets.org/praxis/) A two-sided coin is
unfairly weighted so that when it is tossed, the prob-
ability that heads will result is twice the probability
that tails will result. If the coin is to be tossed three
542
Teacher’s Resources There are other Internet sites for probability activities
listed in this section. Some websites provide data tables
Bright, G., Frierson, D., Tarr, J., & Thomas, C. (2003). along with the experiments, and some also build graphs
Navigating through probability in grades 6–8. Reston, to display the resulting data.
VA: National Council of Teachers of Mathematics. Internet Sites
Burrill, G., & Elliot, P. (Eds.). (2006). Thinking and rea- For coin flipping simulations see:
soning with data and chance: Sixty-eighth yearbook. Res-
ton, VA: National Council of Teachers of Mathematics. http://www.shodor.org/interactivate/activities/chances/index.html
Carter, Rick. (1996). Chance, statistics, and graphs: http://www2.whidbey.net/ohmsmath/webwork/javascript/
Handling data. Reston, VA: National Council of Teachers cointoss.htm
of Mathematics.
For adjustable spinners and dice simulations see:
Chapin, Suzanne, Koizol, Alice, MacPherson, Jennifer,
& Rezba, Carol. (2003). Navigating through data analy- http://www.shodor.org/interactivate/activities/spinner/index.html
sis and probability in grades 3–5. Reston, VA: National
Council of Teachers of Mathematics. http://www2.whidbey.net/ohmsmath/webwork/javascript/
spinner.htm
Cuomo, Celia. (1993). In all probability: Investigations in
probability and statistics—Teachers guide, grades 3–6. http://nlvm.usu.edu/en/nav/vlibrary.html
Berkeley: University of California Press.
A virtual representation of the cereal prize simulation
Peterson, Ivars. (1998). The jungles of randomness: A (Activity 20.12) can be found at
mathematical safari. New York: Wiley. http://www.mste.uiuc.edu/users/reese/cereal/default.html.
Sheffield, Linda, Cavanaugh, Mary, Dacey, Linda, Fin- Three Internet sites that enable students to use geom-
dell, Carol, Greenes, Carole, & Small, Marian. (2002). etry contexts to solve probability problems are:
Navigating through data analysis and probability in http://www.explorelearning.com/index.cfm?methodϭcResource
prekindergarten–grade 2. Reston, VA: National Council .dspResourcesForCourse&CourseIDϭ233
of Teachers of Mathematics.
http://www.mste.uiuc.edu/activity/estpi/
Weaver, Jefferson Hale. (2001). What are the odds? The
chances of extraordinary events in everyday life. Am- http://www.mste.uiuc.edu/activity/rocket/
herst, NY: Prometheus Books.
Internet Game
Children’s Bookshelf The game Fish Tank, available at http://www.bbc.co.uk/
education/mathsfile/, challenges students to move red
Barrett, Judi. (1978). Cloudy with a chance of meatballs. and yellow fish in and out of an aquarium to match the
New York: Atheneum. (Grades 4–7) given probabilities of randomly capturing a red fish.
The game has three levels of difficulty and sound effects
Burns, Marilyn. (1975). The I hate mathematics book. to heighten children’s interest. The sound can be turned
Boston: Little, Brown. (Grades 5–8) off for children who become distracted.
Cushman, Jean. (1991). Do you wanna bet? Your chance For Further Reading
to find out about probability. New York: Clarion. (Grades
4–7) Abaugh, F., Scholten, C., & Essex, K. (2001). Data in the
middle grades: A probability web quest. Mathematics
Franco, Betsy, & Salermo, Steven. (2004). Counting Teaching in the Middle School 7(2), 90–95.
our way to the 100h day. New York: Simon & Schuster.
(Grades K–3) In this article Abaugh and colleagues describe how
to use Internet-based materials and applets to introduce
Holtzman, Caren. (1997). No fair! New York: Scholastic. principles of probability.
(Grades 2–4)
Aspinwall, L., & Shaw, K. (2000). Enriching students’
Morgan, Rowland. (1997). In the next three seconds. New mathematical intuitions with probability games and tree
York: Lodestar. (Grades 3–8) diagrams. Mathematics Teaching in the Middle School
6(4), 214–220.
Technology Resources
Aspinwall and Shaw present several games for
Computer Software students to explore using tree diagrams in order to build
We have used two computer software programs from students’ intuitions in probability.
the Internet in this chapter:
Coffey, David, & Richardson, Mary. (2005). Rethinking
http://www2.whidbey.net/ohmsmith/webwork/javascript/diceroll fair games. Mathematics Teaching in the Middle School
.htm and http://nces.ed.gov/nceskids/probability/. 10(6), 298–303.
Coffey and Richardson examine the fairness of a
common matching game and use a probability tree to
determine the game’s fairness.
543
Ewbank, W., & Ginter, J. (2002). Probability on a bud- Penner and Lehrer describe a project in which first-
get. Mathematics Teaching in the Middle School 7(7), and second-graders design a model for a fair game of
280–283. chance using experimentation and revisions to achieve
a fair model.
Ewbank and Ginter present several exploration les-
son plans designed to help students understand prob- Phillips-Bey, Carol. (2004). TI-73 calculator activi-
ability. The experiments involve cheaply made manipu- ties. Mathematics Teaching in the Middle School 9(9),
latives and are easily incorporated into a middle school 500–508.
classroom.
Phillips-Bey describes how to use the technology of
Frykholm, Jeffrey. (2001). Eenie, meenie, minie, moe the TI-73 calculator to simulate rolling dice. The results
. . . Building on intuitive notions of chance. Teaching of the simulation are then used to determine the fair-
Children Mathematics 8(2), 112–115. ness of various dice games.
Frykholm summarizes research on probability in the Tarr, James. (2002). Providing opportunity to learn
early grades, offers several activities in probability for probability concepts. Teaching Children Mathematics
younger children, and suggests some potential topics in 8(8), 482–487.
probability for action research.
This article provides an overview of the NCTM stan-
Nicholson, Cynthia. (2005). Is chance fair? One student’s dards for probability for grades K–2 and 3–5, including
thoughts on probability. Teaching Children Mathematics connections to other mathematics topics.
11, 83–89.
Weist, L., & Quinn, R. (1999). Exploring probability
Nicholson describes students’ typical misconcep- through an even-odds dice game. Mathematics Teaching
tions about probability as revealed in the thinking of in the Middle School 4(6), 358–362.
one fifth-grade student.
In this article Weist and Quinn describe probabil-
Penner, Elizabeth, & Lehrer, Richard. (2000). The ity explorations using several types of dice. Students
shape of fairness. Teaching Children Mathematics 7(4), develop game strategies based on probability principles
210–214. and the data they generate.
Appendix A
NCTM Table of Standards and Expectations
545
Number and Operations 546 Appendix A
Standard Pre-K–2 Grades 3–5
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In prekindergarten through grade 2 all students should—
all students to— In grades 3–5 all students should—
Understand numbers, ways of repre- ■ count with understanding and recognize “how many” in sets of
senting numbers, relationships among ■ understand the place-value structure of the base-ten number system
numbers, and number systems objects; and be able to represent and compare whole numbers and decimals;
Understand meanings of operations and ■ use multiple models to develop initial understandings of place value and ■ recognize equivalent representations for the same number and gener-
how they relate to one another ate them by decomposing and composing numbers;
the base-ten number system;
Compute fluently and make reasonable ■ develop understanding of fractions as parts of unit wholes, as parts
estimates ■ develop understanding of the relative position and magnitude of whole of a collection, as locations on number lines, and as divisions of whole
numbers;
numbers and of ordinal and cardinal numbers and their connections;
■ use models, benchmarks, and equivalent forms to judge the size of
■ develop a sense of whole numbers and represent and use them fractions;
in flexible ways, including relating, composing, and decomposing ■ recognize and generate equivalent forms of commonly used fractions,
decimals, and percents;
numbers;
■ explore numbers less than 0 by extending the number line and through
■ connect number words and numerals to the quantities they represent, familiar applications;
using various physical models and representations; 14᎐, 1᎐3 , ■ describe classes of numbers according to characteristics such as the
nature of their factors.
■ understand and represent commonly used fractions, such as
and 1᎐2. ■ understand various meanings of multiplication and division;
■ understand the effects of multiplying and dividing whole
■ understand various meanings of addition and subtraction of whole
numbers and the relationship between the two operations; numbers;
■ identify and use relationships between operations, such as division as
■ understand the effects of adding and subtracting whole numbers;
■ understand situations that entail multiplication and division, such as the inverse of multiplication, to solve problems;
■ understand and use properties of operations, such as the distributivity
equal groupings of objects and sharing equally.
of multiplication over addition.
■ develop and use strategies for whole-number computations, with a
focus on addition and subtraction; ■ develop fluency with basic number combinations for multiplication
and division and use these combinations to mentally compute related
■ develop fluency with basic number combinations for addition and problems, such as 30 ϫ 50;
subtraction;
■ develop fluency in adding, subtracting, multiplying, and dividing whole
■ use a variety of methods and tools to compute, including numbers;
objects, mental computation, estimation, paper and pencil, and
calculators. ■ develop and use strategies to estimate the results of whole-number
computations and to judge the reasonableness of such results;
■ develop and use strategies to estimate computations involving fractions
and decimals in situations relevant to students’ experience;
■ use visual models, benchmarks, and equivalent forms to add and sub-
tract commonly used fractions and decimals;
■ select appropriate methods and tools for computing with whole
numbers from among mental computation, estimation, calculators, and
paper and pencil according to the context and nature of the computa-
tion and use the selected method or tools.
Number and Operations
Standard Grades 6–8 Grades 9–12
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In grades 6–8 all students should— In grades 9–12 all students should—
all students to—
Understand numbers, ways of repre- ■ work flexibly with fractions, decimals, and percents to solve problems; ■ develop a deeper understanding of very large and very small numbers
senting numbers, relationships among ■ compare and order fractions, decimals, and percents efficiently and find and of various representations of them;
numbers, and number systems
their approximate locations on a number line; ■ compare and contrast the properties of numbers and number systems,
■ develop meaning for percents greater than 100 and less than 1; including the rational and real numbers, and understand complex num-
■ understand and use ratios and proportions to represent quantitative bers as solutions to quadratic equations that do not have real solutions;
relationships; ■ understand vectors and matrices as systems that have some of the
■ develop an understanding of large numbers and recognize and appro- properties of the real-number system;
priately use exponential, scientific, and calculator notation; ■ use number-theory arguments to justify relationships involving whole
■ use factors, multiples, prime factorization, and relatively prime numbers numbers.
to solve problems;
■ develop meaning for integers and represent and compare quantities
with them.
Understand meanings of operations and ■ understand the meaning and effects of arithmetic operations with frac- ■ judge the effects of such operations as multiplication, division, and
how they relate to one another tions, decimals, and integers; computing powers and roots on the magnitudes of quantities;
■ use the associative and commutative properties of addition and multi- ■ develop an understanding of properties of, and representations for, the
plication and the distributive property of multiplication over addition to addition and multiplication of vectors and matrices;
simplify computations with integers, fractions, and decimals;
■ develop an understanding of permutations and combinations as count-
■ understand and use the inverse relationships of addition and subtrac- ing techniques.
tion, multiplication and division, and squaring and finding square roots
to simplify computations and solve problems.
Compute fluently and make ■ select appropriate methods and tools for computing with fractions and ■ develop fluency in operations with real numbers, vectors, and matrices,
reasonable estimates decimals from among mental computation, estimation, calculators or using mental computation or paper-and-pencil calculations for simple
computers, and paper and pencil, depending on the situation, and ap- cases and technology for more-complicated cases;
ply the selected methods;
■ judge the reasonableness of numerical computations and their
■ develop and analyze algorithms for computing with fractions, decimals, result.
and integers and develop fluency in their use;
Appendix A
■ develop and use strategies to estimate the results of rational-number
computations and judge the reasonableness of the results;
■ develop, analyze, and explain methods for solving problems involving
proportions, such as scaling and finding equivalent ratios.
547
Algebra 548 Appendix A
Standard Pre-K–2 Grades 3–5
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In prekindergarten through grade 2 all students should— In grades 3–5 all students should—
all students to—
■ sort, classify, and order objects by size, number, and other ■ describe, extend, and make generalizations about geometric and nu-
Understand patterns, relations, properties; meric patterns;
and functions
■ recognize, describe, and extend patterns such as sequences of sounds ■ represent and analyze patterns and functions, using words, tables, and
Represent and analyze mathematical and shapes or simple numeric patterns and translate from one represen- graphs.
situations and structures using algebraic tation to another;
symbols ■ identify such properties as commutativity, associativity, and distributiv-
■ analyze how both repeating and growing patterns are generated. ity and use them to compute with whole numbers;
Use mathematical models to
represent and understand ■ illustrate general principles and properties of operations, such as com- ■ represent the idea of a variable as an unknown quantity using a letter
quantitative relationships mutativity, using specific numbers; or a symbol;
Analyze change in various contexts
■ use concrete, pictorial, and verbal representations to develop an under- ■ express mathematical relationships using equations.
standing of invented and conventional symbolic notations.
■ model problem situations with objects and use representations such as
■ model situations that involve the addition and subtraction of whole graphs, tables, and equations to draw conclusions.
numbers, using objects, pictures, and symbols.
■ investigate how a change in one variable relates to a change in a sec-
■ describe qualitative change, such as a student’s growing taller; ond variable;
■ describe quantitative change, such as a student’s growing two inches in
■ identify and describe situations with constant or varying rates of change
one year. and compare them.
Algebra
Standard Grades 6–8 Grades 9–12 Appendix A 549
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In grades 6–8 all students should—
all students to— In grades 9–12 all students should—
Understand patterns, relations, ■ represent, analyze, and generalize a variety of patterns with tables,
and functions graphs, words, and, when possible, symbolic rules; ■ generalize patterns using explicitly defined and recursively defined
functions;
Represent and analyze mathematical ■ relate and compare different forms of representation for a relationship;
situations and structures using algebraic ■ identify functions as linear or nonlinear and contrast their properties ■ understand relations and functions and select, convert flexibly among,
symbols and use various representations for them;
from tables, graphs, or equations.
Use mathematical models to ■ analyze functions of one variable by investigating rates of change, inter-
represent and understand ■ develop an initial conceptual understanding of different uses of cepts, zeros, asymptotes, and local and global behavior;
quantitative relationships variables;
■ understand and perform transformations such as arithmetically
Analyze change in various contexts ■ explore relationships between symbolic expressions and graphs of lines, combining, composing, and inverting commonly used functions, using
paying particular attention to the meaning of intercept and slope; technology to perform such operations on more-complicated symbolic
expressions;
■ use symbolic algebra to represent situations and to solve problems,
especially those that involve linear relationships; ■ understand and compare the properties of classes of functions,
including exponential, polynomial, rational, logarithmic, and periodic
■ recognize and generate equivalent forms for simple algebraic expres- functions;
sions and solve linear equations.
■ interpret representations of functions of two variables.
■ model and solve contextualized problems using various representations,
such as graphs, tables, and equations. ■ understand the meaning of equivalent forms of expressions, equations,
inequalities, and relations;
■ use graphs to analyze the nature of changes in quantities in linear
relationships. ■ write equivalent forms of equations, inequalities, and systems of equa-
tions and solve them with fluency—mentally or with paper and pencil
in simple cases and using technology in all cases;
■ use symbolic algebra to represent and explain mathematical
relationships;
■ use a variety of symbolic representations, including recursive and para-
metric equations, for functions and relations;
■ judge the meaning, utility, and reasonableness of the results of symbol
manipulations, including those carried out by technology.
■ identify essential quantitative relationships in a situation and determine
the class or classes of functions that might model the relationships;
■ use symbolic expressions, including iterative and recursive forms, to
represent relationships arising from various contexts;
■ draw reasonable conclusions about a situation being modeled.
■ approximate and interpret rates of change from graphical and numeri-
cal data.
Geometry 550 Appendix A
Standard Pre-K–2 Grades 3–5
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In prekindergarten through grade 2 all students should— In grades 3–5 all students should—
all students to—
Analyze characteristics and properties of ■ recognize, name, build, draw, compare, and sort two- and three- ■ identify, compare, and analyze attributes of two- and three-
two- and three-dimensional geometric dimensional shapes; dimensional shapes and develop vocabulary to describe the
shapes and develop mathematical argu- attributes;
ments about geometric relationships ■ describe attributes and parts of two- and three-dimensional shapes;
■ investigate and predict the results of putting together and taking apart ■ classify two- and three-dimensional shapes according to their proper-
Specify locations and describe spatial ties and develop definitions of classes of shapes such as triangles and
relationships using coordinate geometry two- and three-dimensional shapes. pyramids;
and other representational systems
■ describe, name, and interpret relative positions in space and apply ideas ■ investigate, describe, and reason about the results of subdividing, com-
Apply transformations and use symme- about relative position; bining, and transforming shapes;
try to analyze mathematical situations
■ describe, name, and interpret direction and distance in navigating space ■ explore congruence and similarity;
Use visualization, spatial reasoning, and and apply ideas about direction and distance; ■ make and test conjectures about geometric properties and relationships
geometric modeling to solve problems
■ find and name locations with simple relationships such as “near to” and and develop logical arguments to justify conclusions.
in coordinate systems such as maps.
■ describe location and movement using common language and geomet-
■ recognize and apply slides, flips, and turns; ric vocabulary;
■ recognize and create shapes that have symmetry.
■ make and use coordinate systems to specify locations and to describe
■ create mental images of geometric shapes using spatial memory and paths;
spatial visualization;
■ find the distance between points along horizontal and vertical lines of a
■ recognize and represent shapes from different perspectives; coordinate system.
■ relate ideas in geometry to ideas in number and measurement;
■ recognize geometric shapes and structures in the environment and ■ predict and describe the results of sliding, flipping, and turning two-
dimensional shapes;
specify their location.
■ describe a motion or a series of motions that will show that two shapes
are congruent;
■ identify and describe line and rotational symmetry in two- and three-
dimensional shapes and designs.
■ build and draw geometric objects;
■ create and describe mental images of objects, patterns, and paths;
■ identify and build a three-dimensional object from two-
dimensional representations of that object;
■ identify and draw a two-dimensional representation of a three-
dimensional object;
■ use geometric models to solve problems in other areas of mathematics,
such as number and measurement;
■ recognize geometric ideas and relationships and apply them to other
disciplines and to problems that arise in the classroom or in everyday
life.
Geometry
Standard Grades 6–8 Grades 9–12
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable
all students to— In grades 6–8 all students should— In grades 9–12 all students should—
Analyze characteristics and properties of ■ precisely describe, classify, and understand relationships among types ■ analyze properties and determine attributes of two- and three-
two- and three-dimensional geometric of two- and three-dimensional objects using their defining properties; dimensional objects;
shapes and develop mathematical argu-
ments about geometric relationships ■ understand relationships among the angles, side lengths, perimeters, ■ explore relationships (including congruence and similarity) among
areas, and volumes of similar objects; classes of two- and three-dimensional geometric objects, make and test
conjectures about them, and solve problems involving them;
■ create and critique inductive and deductive arguments concerning
geometric ideas and relationships, such as congruence, similarity, and ■ establish the validity of geometric conjectures using deduction, prove
the Pythagorean relationship. theorems, and critique arguments made by others; use trigonometric
relationships to determine lengths and angle measures.
Specify locations and describe spatial ■ use coordinate geometry to represent and examine the properties of
relationships using coordinate geometry geometric shapes; ■ use Cartesian coordinates and other coordinate systems, such as navi-
and other representational systems gational, polar, or spherical systems, to analyze geometric situations;
■ use coordinate geometry to examine special geometric shapes, such as
regular polygons or those with pairs of parallel or perpendicular sides. ■ investigate conjectures and solve problems involving two- and three-
dimensional objects represented with Cartesian coordinates.
Apply transformations and use symme- ■ describe sizes, positions, and orientations of shapes under informal
try to analyze mathematical situations transformations such as flips, turns, slides, and scaling; ■ understand and represent translations, reflections, rotations, and dila-
tions of objects in the plane by using sketches, coordinates, vectors,
■ examine the congruence, similarity, and line or rotational symmetry of function notation, and matrices;
objects using transformations.
■ use various representations to help understand the effects of simple
Use visualization, spatial reasoning, and ■ draw geometric objects with specified properties, such as side lengths transformations and their compositions.
geometric modeling to solve problems or angle measures;
■ draw and construct representations of two- and three-dimensional
■ use two-dimensional representations of three-dimensional objects to geometric objects using a variety of tools;
visualize and solve problems such as those involving surface area and
volume; ■ visualize three-dimensional objects and spaces from different perspec-
tives and analyze their cross sections;
■ use visual tools such as networks to represent and solve problems;
■ use geometric models to represent and explain numerical and algebraic ■ use vertex-edge graphs to model and solve problems;
■ use geometric models to gain insights into, and answer questions in,
relationships;
■ recognize and apply geometric ideas and relationships in areas outside other areas of mathematics;
■ use geometric ideas to solve problems in, and gain insights into, other
the mathematics classroom, such as art, science, and everyday life.
disciplines and other areas of interest such as art and architecture.
Appendix A
551
Measurement 552 Appendix A
Standard Pre-K–2 Grades 3–5
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In prekindergarten through grade 2 all students should—
all students to— In grades 3–5 all students should—
■ recognize the attributes of length, volume, weight, area, and time;
Understand measurable attributes of ■ compare and order objects according to these attributes; ■ understand such attributes as length, area, weight, volume, and size
objects and the units, systems, and ■ understand how to measure using nonstandard and standard units; of angle and select the appropriate type of unit for measuring each
processes of measurement ■ select an appropriate unit and tool for the attribute being measured. attribute;
Apply appropriate techniques, ■ measure with multiple copies of units of the same size, such as paper ■ understand the need for measuring with standard units and become
tools, and formulas to determine clips laid end to end; familiar with standard units in the customary and metric systems;
measurements
■ use repetition of a single unit to measure something larger than ■ carry out simple unit conversions, such as from centimeters to meters,
the unit, for instance, measuring the length of a room with a single within a system of measurement;
meterstick;
■ understand that measurements are approximations and how differ-
■ use tools to measure; ences in units affect precision;
■ develop common referents for measures to make comparisons and
■ explore what happens to measurements of a two-dimensional shape
estimates. such as its perimeter and area when the shape is changed in some way.
■ develop strategies for estimating the perimeters, areas, and volumes of
irregular shapes;
■ select and apply appropriate standard units and tools to measure
length, area, volume, weight, time, temperature, and the size of angles;
■ select and use benchmarks to estimate measurements;
■ develop, understand, and use formulas to find the area of rectangles
and related triangles and parallelograms;
■ develop strategies to determine the surface areas and volumes of rect-
angular solids.
Measurement
Standard Grades 6–8 Grades 9–12
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In grades 6–8 all students should— In grades 9–12 all students should—
all students to—
■ understand both metric and customary systems of measurement; ■ make decisions about units and scales that are appropriate for problem
Understand measurable attributes of ■ understand relationships among units and convert from one unit to situations involving measurement.
objects and the units, systems, and
processes of measurement another within the same system;
■ understand, select, and use units of appropriate size and type to mea-
Apply appropriate techniques,
tools, and formulas to determine sure angles, perimeter, area, surface area, and volume.
measurements
■ use common benchmarks to select appropriate methods for estimating ■ analyze precision, accuracy, and approximate error in measurement
measurements; situations;
■ select and apply techniques and tools to accurately find length, area, ■ understand and use formulas for the area, surface area, and volume of
volume, and angle measures to appropriate levels of precision; geometric figures, including cones, spheres, and cylinders;
■ develop and use formulas to determine the circumference of circles ■ apply informal concepts of successive approximation, upper and lower
and the area of triangles, parallelograms, trapezoids, and circles and bounds, and limit in measurement situations;
develop strategies to find the area of more-complex shapes;
■ use unit analysis to check measurement computations.
■ develop strategies to determine the surface area and volume of selected
prisms, pyramids, and cylinders;
■ solve problems involving scale factors, using ratio and proportion;
■ solve simple problems involving rates and derived measurements for
such attributes as velocity and density.
Appendix A
553
Data Analysis and Probability 554 Appendix A
Standard Pre-K–2 Grades 3–5
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In prekindergarten through grade 2 all students should—
all students to— In grades 3–5 all students should—
Formulate questions that can be ad- ■ pose questions and gather data about themselves and their
dressed with data and collect, organize, surroundings; ■ design investigations to address a question and consider how data-
and display relevant data to answer collection methods affect the nature of the data set;
them ■ sort and classify objects according to their attributes and organize data
about the objects; ■ collect data using observations, surveys, and experiments;
Select and use appropriate statistical ■ represent data using tables and graphs such as line plots, bar graphs,
methods to analyze data ■ represent data using concrete objects, pictures, and graphs.
and line graphs;
Develop and evaluate inferences and ■ describe parts of the data and the set of data as a whole to determine ■ recognize the differences in representing categorical and numerical
predictions that are based on data what the data show.
Understand and apply basic concepts of data.
probability ■ discuss events related to students’ experiences as likely or unlikely.
■ describe the shape and important features of a set of data and compare
related data sets, with an emphasis on how the data are distributed;
■ use measures of center, focusing on the median, and understand what
each does and does not indicate about the data set;
■ compare different representations of the same data and evaluate how
well each representation shows important aspects of the data.
■ propose and justify conclusions and predictions that are based on data
and design studies to further investigate the conclusions or predictions.
■ describe events as likely or unlikely and discuss the degree of likelihood
using such words as certain, equally likely, and impossible;
■ predict the probability of outcomes of simple experiments and test the
predictions;
■ understand that the measure of the likelihood of an event can be repre-
sented by a number from 0 to 1.
Data Analysis and Probability
Standard Grades 6–8 Grades 9–12
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable In grades 6–8 all students should—
all students to— In grades 9–12 all students should—
Formulate questions that can be ad- ■ formulate questions, design studies, and collect data about a character-
dressed with data and collect, organize, istic shared by two populations or different characteristics within one ■ understand the differences among various kinds of studies and which
and display relevant data to answer population; types of inferences can legitimately be drawn from each;
them
■ select, create, and use appropriate graphical representations of data, ■ know the characteristics of well-designed studies, including the role of
Select and use appropriate statistical including histograms, box plots, and scatterplots. randomization in surveys and experiments;
methods to analyze data
■ find, use, and interpret measures of center and spread, including mean ■ understand the meaning of measurement data and categorical data, of
Develop and evaluate inferences and and interquartile range; univariate and bivariate data, and of the term variable;
predictions that are based on data
■ discuss and understand the correspondence between data sets and ■ understand histograms, parallel box plots, and scatterplots and use
their graphical representations, especially histograms, stem-and-leaf them to display data;
plots, box plots, and scatterplots.
■ compute basic statistics and understand the distinction between a
■ use observations about differences between two or more samples to statistic and a parameter.
make conjectures about the populations from which the samples were
taken; ■ for univariate measurement data, be able to display the distribution,
describe its shape, and select and calculate summary statistics;
■ make conjectures about possible relationships between two character-
istics of a sample on the basis of scatterplots of the data and approxi- ■ for bivariate measurement data, be able to display a scatterplot,
mate lines of fit; describe its shape, and determine regression coefficients, regression
equations, and correlation coefficients using technological tools;
■ use conjectures to formulate new questions and plan new studies to
answer them. ■ display and discuss bivariate data where at least one variable is
categorical;
Appendix A
■ recognize how linear transformations of univariate data affect shape,
center, and spread;
■ identify trends in bivariate data and find functions that model the data
or transform the data so that they can be modeled.
■ use simulations to explore the variability of sample statistics from a
known population and to construct sampling distributions;
■ understand how sample statistics reflect the values of population
parameters and use sampling distributions as the basis for informal
inference;
■ evaluate published reports that are based on data by examining the
design of the study, the appropriateness of the data analysis, and the
validity of conclusions;
■ understand how basic statistical techniques are used to monitor process
characteristics in the workplace.
(Standard continues on page 554) 555
Data Analysis and Probability 556 Appendix A
Standard Grades 6–8 Grades 9–12
Instructional programs from prekinder- Expectations: Expectations:
garten through grade 12 should enable
all students to— In grades 6–8 all students should— In grades 9–12 all students should—
Understand and apply basic concepts of ■ understand and use appropriate terminology to describe complemen- ■ understand the concepts of sample space and probability distribution
probability tary and mutually exclusive events; and construct sample spaces and distributions in simple cases;
■ use proportionality and a basic understanding of probability to make ■ use simulations to construct empirical probability distributions;
and test conjectures about the results of experiments and simulations; ■ compute and interpret the expected value of random variables in simple
■ compute probabilities for simple compound events, using such methods cases;
as organized lists, tree diagrams, and area models. ■ understand the concepts of conditional probability and independent
events;
■ understand how to compute the probability of a compound event.
Used with permission from Principles and Standards for School Mathematics, copyright © 2000 by the National Council of Teachers of Mathematics, Inc. All rights reserved.
Appendix B
Black-Line Masters
Full-sized versions of the black-line masters shown
in this appendix are available on the text’s compan-
ion website—www.thomsonedu.com/education/kennedy.
1 2 3 4 5 6 7 8 9 10 BLM 11.2 Tens frame
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
BLM 10.1 Hundreds chart
؉012 34 56 78 9 ؋012 34 56 78 9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
BLM 11.1 Blank addition chart BLM 11.3 Blank multiplication chart
557
558 Appendix B 4 111
BLM 13.1 Fraction circle 1 55 2215
BLM 13.2 Fraction circle 2 1 11
6 3 33 6
7 1111 1
4444
81 11 1 1 18
6 66 6 6 6
3 11111111 1
4 88888888 4
111111111111
2 12 12 12 12 12 12 12 12 12 12 12 121
3 3
BLM 13.3 Fraction strips
3 2 3
BLM 13.4 Equilateral triangles5558
1082
BLM 13.5 Fraction wheel 11
Appendix B 559
BLM 13.6 Fraction wheel 2
Set 1 BC BLM 17.1 Isometric dot paper
A
1
2
Set 2 BC 3 5
A 6 4
Set 3 BC 78
A
BLM 13.7 Decimal circles BLM 17.2 Sorting shapes 1
560 Appendix B 10
9
25 26
11 28 27
13 15 29
12 30 31
BLM 17.3 Sorting shapes 2 14 32
BLM 17.5 Sorting shapes 4 33
16 17 35
18
19 20
23 24 34
22 21
BLM 17.4 Sorting shapes 3 BLM 17.6 Sorting shapes 5
Appendix B 561
BLM 17.7 Geoboard pattern
BLM 17.9 Nets for platonic solids: tetrahedron, cube, and
octahedron. Fold along dotted lines and tape together
to form each solid.
BLM 17.8 Geoboard record keeping BLM 17.10 Nets for platonic solids: regular dodecahe-
dron. Cut out both shapes and fold along dotted lines.
The two shapes fit together to form a single regular
dodecahedron.
562 Appendix B
ba
c
BLM 17.13 Pythagorean puzzle A
BLM 17.11 Nets for platonic solids: regular icosahedron.
Fold on dotted lines and glue, using the tabs.
AB
ab
c
C
BLM 17.12 Tangram template. Copy onto card stock and BLM 17.14 Pythagorean puzzle B
laminate before cutting out.
Appendix B 563
70 80 9 0 100 110
120
60 130
50
140
40
30 150
20 160
10 170
0 180
cut here
cut here
BLM 18.1 Cylinder and cone template for volume rela- BLM 18.3 Angle wheel. Cut to the center of both circles
tionship. Print on card stock before cutting out. along the indicated radius. Face the blank sides of both
circles against each other. Fit the centers of the two
circles together along the two cut radii.
BLM 18.2 Cube and pyramid template for volume rela-
tionship. Print on card stock before cutting out.
564 Appendix B
BLM 19.1 Centimeter grid BLM 19.3 Bar graph template
BLM 19.2 Inch grid BLM 19.4 Circle graph
Glossary
A Cartesian cross-product A multiplication strategy that in-
volves the number of one-to-one combinations of objects in
acute triangle A triangle whose angles are all less than 90 two or more sets. It shows the total number of possibilities
degrees. Compare with obtuse triangle and right triangle. made by choosing one option from each group of choices.
Also called combinations. Compare with geometric interpreta-
addends In addition, the numbers being added to arrive at tion and repeated addition.
a sum.
Cartesian geometry See coordinate geometry.
algorithm A step-by-step procedure for calculation.
classification Matching groups of objects sharing common
alternative algorithm An algorithm that is different from the characteristics, or attributes. Classification is an important
traditional or conventional algorithm used most frequently in skill in all subject areas. Also called sorting or grouping or
a culture. categorizing.
analytic scoring rubric A detailed assessment of overall classroom action research The process in which teachers
quality of an assignment or task. The analytical scoring rubric conduct research in their classrooms by asking and answer-
lists several performance objectives and gives a rubric for each ing questions about student learning and effective teaching.
performance indicator. Compare with holistic scoring rubric. Teachers describe, explain, and try methods to improve their
students’ learning. They assess student learning by collecting
anecdotal record A written note about what a student did data, drawing conclusions, and changing their techniques to
and said. increase learning.
array In the geometric interpretation of multiplication, the closed questions Questions that call for short, specific
graphic arrangement of objects in rows and columns. answers.
associative property of addition The order in which ad- cognitive-developmental theories Theories such as those
dends are added does not change the sum. Applies to three or of Jean Piaget, Richard Skemp, and Jerome Bruner that pro-
more addends. See also commutative property of addition. pose levels of successively more complex intellectual under-
standing or conceptualization.
associative property of multiplication When three or more
factors are multiplied, the order in which they are paired for combinations See Cartesian cross-product.
computation does not affect the product. See also commutative
property of multiplication. commutative The property in which the order of the numbers
in a mathematical function does not matter. For example, addi-
B tion is commutative; subtraction is not.
bar graph A graphical representation of numerical data that commutative law of addition See commutative property of
uses vertical or horizontal bars. Bar graphs are used to display addition.
categorical data.
commutative law of multiplication See commutative prop-
base The number to which a percent is applied. Example: 50% erty of multiplication.
of 60 (the base) is 30. See also percent and percentage.
commutative property of addition The order of addends
behaviorism A theory of learning that focuses on observable does not affect the sum. Also called commutative law of addi-
behaviors and on ways to increase behaviors deemed positive tion. See also associative property of addition.
and decrease behaviors deemed undesirable.
commutative property of multiplication When the order
bell curve See normal distribution. of factors is switched, the product is the same. Also called com-
mutative law of multiplication. See also associative property of
benchmark (1) Specific grade level learning target provided multiplication.
by state and local schools. (2) In estimation, a referent that
gives students a comparison unit or amount to use for an esti- comparison subtraction Compares the size of two sets or
mate. Without a benchmark, students have a hard time making the measure of two objects. The quantity of both sets or mea-
a reasonable estimate and refining it. surement of both objects is known. The comparison question
in subtraction asks “how much larger” or “how much smaller”
bimodal When there are two modes for a set of data. one set is than the other. “What is the difference” is another way
of expressing comparison between the two sets. Compare with
box-and-whisker plot A graphical representation of the me- completion subtraction, takeaway subtraction, and whole-part-
dian, the first and third quartiles, and the extreme values of a part subtraction.
data set. The data for a box-and-whisker plot are fairly easy to
obtain, consisting of the high and low data points and three compatible numbers A mental computation strategy that
medians. looks for easy combinations of 10, 100, or 1,000.
C compensation An alternative approach to calculation often
used in addition and subtraction in which the original num-
cardinal number A number that tells how many objects are
in a set. Compare with nominal number and ordinal number.
565
566 Glossary
bers are adjusted to make an easier calculation and then cor- or ones place. From any place in the system, the next position
rected, or compensated, in the final answer. to the left is 10 times greater, and the position to the right is
one-tenth as large. This characteristic makes it possible to rep-
completion subtraction Comparison between an existing resent whole numbers of any size as well as decimal fractions
set and a desired set, or between an incomplete set and a com- with the system. The metric system for measurement and the
pleted set. The question for completion is “how many more United States monetary system are decimal systems.
are needed?” or “how much more is needed?” Compare with
comparison subtraction, takeaway subtraction, and whole-part- decomposition The traditional subtraction algorithm, com-
part subtraction. monly called borrowing, which involves regrouping or trading
down from a larger place value to a smaller place value before
composite number A number that has several factors. Com- subtraction.
pare with prime number.
denominator In a common fraction, the number of equal-
computational fluency The ability to compute with confi-
dence and ease in a variety of situations using the appropri- sized parts into which the whole, or unit, has been subdivided.
ate operation and computational strategy and determining
whether the result is reasonable and useful. For example, the “2” in 1 is the denominator. Compare with
2
concrete operational stage Third stage of Piaget’s devel-
opmental stages. Occurs around ages 7 to 12 when children numerator.
master the underlying structure of number, geometry, and
measurement. Work with concrete objects is the foundation for density A measure of the mass to volume ratio of a substance
developing mathematical concepts represented by pictures, or object. The higher the mass to volume ratio, the greater the
symbols, and mental images. Compare with formal operational density will be.
stage, preoperational stage, and sensorimotor stage.
density property of numbers There is always another num-
congruent The property in which one object fits exactly on ber between any two numbers.
the other. If all points on one line segment match exactly with
all points on another segment, they are congruent. Any two difference The degree or amount by which one set differs
plane or space figures are congruent when all their corre- from another set.
sponding points match exactly. Compare with similar.
directed teaching/thinking lesson Structured, but interac-
constructivism A learning theory that holds that learning is a tive, method of teaching mathematics in which manipulatives,
result of building or constructing meaning from many experi- discussion, and performance of tasks encourage student think-
ences; individuals construct their understanding from physi- ing, for small groups or the whole class.
cal and social interactions through the formation of schema—
complex cognitive relationships. discrete Quantitative in nature; countable. Compare with
continuous.
continuous Qualitative in nature. Continuous properties can
be subdivided into progressively smaller and smaller units that distributive property of multiplication and division over
can take on an infinite number of values; to make quantitative addition The operations of multiplication and division may
comparisons about them, they must be measured. Compare be carried on a sum or on addends of that sum.
with discrete.
dividend In division, the number being divided.
cooperative group learning A grouping strategy that is de-
signed to increase student participation by capitalizing on the divisor In division, the number by which the dividend is di-
social aspects of learning. vided; the known factor. Compare with quotient.
coordinate geometry Using a coordinate grid formed by E
two number lines—one extending horizontally and the other
vertically—to locate objects on the grid. Also called Cartesian equal-addition algorithm An alternative algorithm for sub-
geometry. traction that is based on the fact that adding or subtracting the
same amount to the sum and one addend will not change the
counting A number concept in which a sequence of words is difference between the two numbers.
related to increasing number: 1, 2, 3, 4, 5, . . . .
equal participation One of the basic principles for imple-
counting-on A particularly effective learning strategy for ad- menting cooperative learning successfully, in which team
dition. Following examples of the “plus 1” rule such as 5 ϩ 1, members have equal opportunity to participate; an activity is
8 ϩ 1, and 2 ϩ 1, students will generalize “adding 1 is just count- not dominated by one member, nor can members choose not
ing to the next number.” “Plus 2” and “plus 3” are extensions. to participate. In traditional instruction, interchanges are often
between the teacher and one student at a time. Compare with
criterion-referenced test A standardized test that provides individual accountability and positive interdependence.
diagnostic information for teachers because it shows which
concepts and skills each student has mastered instead of com- equilateral triangle A triangle with three equal, or con-
paring students to each other as in a norm-referenced test. Mas- gruent, sides. Compare with isosceles triangle and scalene
tery is determined by the number of items answered correctly triangle.
for each performance objective and the number of objectives
mastered. Compare with norm-referenced test. estimation A reasonable guess, hypothesis, or conjecture
based on numerical information. Compare with rounding.
D
even numbers The set of numbers divisible by 2 with no re-
decimal system A number system in which place value is mainder. Compare with odd numbers.
based on powers of 10. The starting position is called the units
F
factors Numbers multiplied together.
flexible grouping The teacher uses different instructional
groups appropriate to the task or situation.
Glossary 567
formal operational stage Fourth stage of Piaget’s develop- of the team effort. Every student on the team is responsible for
mental stages. Occurs around ages 11 to 13 when more sophis- learning the content and skill, and the team is accountable for
ticated ways of thinking about mathematics, including propor- all the team members’ being successful. Compare with equal
tional reasoning, propositional reasoning, and correlational participation and positive interdependence.
reasoning, begin and continue to develop during the teenage
years and into adulthood. Enables children and adults to form information processing A cognitive theory that uses a
hypotheses, analyze situations to consider all factors that bear computer as a metaphor for learning. Instinctual behavior is
upon them, draw conclusions, and test them against reality. similar to read only memory (ROM) in a computer; it is prepro-
Compare with concrete operational stage, preoperational stage, grammed. Random access memory (RAM) is short-term mem-
and sensorimotor stage. ory the computer receives and stores temporarily, but does not
retain when turned off. Learning becomes permanent when
fundamental theorem of arithmetic Every composite num- new ideas and experiences are transferred from short-term
ber has only one set of prime factors. memory and stored in long-term memory for later retrieval and
use. Information is stored in the computer’s permanent mem-
G ory based on the needs and choices of the computer user.
geometric interpretation A multiplication strategy that integer A positive or negative whole number, including 0.
shows the graphic arrangement of objects in rows and col-
umns, called rectangular arrays. Compare with Cartesian cross- inverse operation For addition; take-away subtraction
product and repeated addition. “undoes” addition. Division is the inverse operation of
multiplication.
greatest common factor The common factor with the largest
value. isosceles triangle A triangle with two congruent sides. Com-
pare with equilateral triangle and scalene triangle.
H
iteration In direct measurement, applying a unit of measure
heuristic A general strategy or procedure that applies to all one or more times to an object being measured.
problem solving in mathematics and parallels the scientific
method. L
Hindu-Arabic numeration system A base-10 system that de- learning center An area in the classroom set up with materi-
veloped in Asia and the Middle East. It is the number system als that students may use in their learning.
that we use today.
learning modalities Learning styles. Do learners prefer to
histogram A type of bar graph that represents the frequency process information visually, auditorily, kinesthetically, or in
of data that have been grouped into several intervals. Histo- some combination? Do learners prefer to learn individually
grams are used to display numerical data, rather than categori- or in groups, with quiet or background noise, in structured or
cal data, as with bar graphs. open-ended assignments?
historical probability One of two types of probability, histor- least common multiple The common multiple with the
ical probability is based on information accumulated over time smallest value.
about a wide range of topics. These empirical data can then be
used to predict the likelihood of certain events. Compare with Likert-type scale A rating scale that indicates the levels of
mathematical probability. performance for a specific concept or skill on a number line.
holistic scoring rubric A judgment of overall quality of an line A set of points that extends infinitely. A line can be curved
assignment or task. Compare with analytical scoring rubric. or straight.
I line graph A graphical representation that shows patterns,
trends, and relationships.
identity element A number that does not change the value of
another number during an arithmetic operation. The number line plot A graphical representation of each item of data as
0 is an identity element for both addition and subtraction be- an individual entry.
cause adding or subtracting 0 does not change the value of a
number. The number 1 is an identity element for both multipli- line symmetry Symmetry that produces mirror images that
cation and division because multiplying or dividing by 1 does can be “folded” along the symmetry line onto one another.
not change the value of a number. Each side of the symmetry is a mirror image of the other. Com-
pare with point symmetry.
idiocentric Self-centered. Refers to children’s perspective on
the world, including their understanding of space. lowest common denominator The common denominator
with the smallest value.
inclusion Rather than being placed in isolated special edu-
cation settings, more and more students with disabilities are M
being included in the regular classroom, sometimes with in-
structional support from aides or special education teachers. “make 10” A strategy used in addition based on finding num-
Teachers work with students with a variety of cognitive abili- ber combinations for 10 such as 8 ϩ 5 ϭ 10 ϩ 3. See also com-
ties, learning styles, social problems, and physical challenges. patible numbers.
individual accountability One of the basic principles for im- manipulatives Sometimes called “objects to think with,” in-
plementing cooperative learning successfully, in which team clude a variety of objects—for example, blocks, scales, coins,
members are held accountable for contributions and results rulers, puzzles, and containers.
mastery assessment See summary assessment.
mathematical probability One of two types of probabil-
ity, mathematical probability is based on the mathematics of
568 Glossary
chance, or the arithmetic probability of an event’s occurring. scores from the lowest score of 1 percentile to the highest of 99
“What is the chance a six will come up when this die stops roll- percentile. Compare with criterion-referenced test.
ing?” “What is the chance of drawing a yellow marble from the
bag of colored marbles?” Compare with historical probability. numeral A written symbol that represents a number, for ex-
ample, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the Hindu-Arabic numera-
mean A measure of central tendency used to describe the tion system.
middle value of a set of data. The mean is computed by sum-
ming all the data in a set and then dividing that sum by the numerator In a common fraction, the number of parts being
number of data items. Also called mean average. Compare
with median and mode. considered at a particular time. For example, the “1” in 1 is the
2
mean average See mean.
numerator. Compare with denominator.
meaning theory See constructivism.
O
measurement division When the known factor is the num-
ber belonging to each group and the unknown factor is the object graph A graphical representation in which actual ob-
number of groups. Measurement division asks “How many jects are part of the graph. Compare with picture graph.
groups of a known size can be made?” Also called repeated
subtraction because the number in each group is repeatedly object permanence The ability of children to learn to recog-
subtracted from the total. Compare with partitive division. nize persons and things and to hold mental images when the
people or things can no longer be seen. Essential for recalling
median A measure of central tendency used to describe the past experiences to connect with new experiences.
middle value of a set of data. The median is the middle number
in a data set. Compare with mean and mode. obtuse triangle A triangle having one angle larger than 90
degrees. Compare with acute triangle and right triangle.
mode A measure of central tendency used to describe the
middle value of a set of data. The mode is the number or re- odd numbers The set of numbers not divisible by 2 evenly,
sponse that appears most frequently. Compare with mean and and which cannot be organized into pairs. Compare with even
median. numbers.
multiple intelligences A theory, first proposed by Howard open-ended questions Questions that ask students to ex-
Gardner in 1983, that asserts that people are capable, or even plain their reasoning.
gifted, in different ways and that no single description of intel-
ligence exists. Gardner initially identified seven intelligences: order A basic thinking/learning skill that involves the ar-
linguistic, logical/mathematical, spatial, body/kinesthetic, mu- rangement of objects, events, and ideas. Order has a begin-
sical, interpersonal, and intrapersonal. An eighth intelligence ning, middle, and end, but placement within the order may be
has since been added: naturalist. arbitrary. Compare with sequence and seriation.
multiplicand The second factor in a multiplication sentence ordinal number A number that is used to designate location
such as 3 ϫ 7 ϭ 21 that names the number of objects in a set. in a sequence. Compare with cardinal number and nominal
number.
multiplier The first factor in a multiplication sentence such as
3 ϫ 7 ϭ 21. It is called the multiplication operator or multiplier outlier A piece of data that is widely distanced from other
because it acts on the second factor. data on either the high or the low end of a distribution.
N overlearning Learning, practicing, and drilling the same
math fact, procedure, or algorithm many times to achieve the
near doubles An addition strategy that relates an unknown ability to use it without prompting.
sum such as 7 ϩ 8 to a known sum that is a double such as
7 ϩ 7 or 8 ϩ 8. P
near squares A multiplication strategy that relates an un- partitive division When the number of groups is known but
known product such as 4 ϫ 5 to a known product that is a the size of each group is not known. The action involves dis-
square such as 4 ϫ 4 and 5 ϫ 5. tributing, or sharing, the number as evenly as possible into the
given number of groups. Partitive division asks “How many are
net A two-dimensional shape that can be cut, folded, and in each group?” Also called sharing division. Compare with
taped to form a geometric solid. measurement division.
nominal number A number that is used to identify or name, percent The rate applied to the base. Example: 50% (the per-
although it may code other information. Compare with cardi- cent) of 60 is 30. See also base and percentage.
nal number and ordinal number.
percentage The amount or quantity derived when a percent
nonproportional materials Place-value materials that do is applied to a base. Example: 50% of 60 is 30 (the percentage).
not show value with their size. Money is the most familiar non- See also base and percent.
proportional material. For example, dimes are not 10 times as
large as pennies, and dollar coins or bills are not 10 times as performance indicator Specifies how the student demon-
large as dimes. Compare with proportional materials. strates the performance objective. The objective in instruction
becomes the performance indicator in assessment. The per-
normal distribution A graphic display of the number of stu- formance indicator describes what the teacher expects to see
dents at each percentile rank. Also called a bell curve. the student do, say, write, or demonstrate the objective. It may
include situation and the criterion for success.
norm-referenced test A standardized test that compares the
scores of students who have taken the test to each other. The performance objective Describes the expected student
norm-referenced standardized test is reported in percentile learning so that teachers and students understand what they
are working toward. Performance objectives guide what the
teacher teaches, what students are to learn, and what is going
to be assessed during and after instruction.
Glossary 569
picture graph A graphical representation in which drawings or the product, is the focus. Such problems enable students
or pictures are used to represent the actual objects in the data to apply their algorithms in real-life situations, and appreciate
set. Compare with object graph. how empowering such algorithms are, in contrast to sterile
rote computations performed from a set of skill problems on a
place value In the Hindu-Arabic number system, a pattern of worksheet. Compare with process-oriented problem.
increasing/decreasing powers of 10 on which whole numbers
are based. proportional materials Place-value materials that show
value with their size. For example, when the manipulative
plane figure A shape that has two dimensions, such as a piece representing one unit is determined, the tens piece is 10
square, triangle, rectangle, circle, and ellipse. times larger, and the hundreds piece is 10 times the tens unit.
Compare with nonproportional materials.
Platonic solid See regular solid.
Q
point A location in space that has no dimension.
quotient In division, the result; the unknown factor. Compare
point symmetry Symmetry found as an object is rotated with divisor.
about its center, resulting in the same image being formed sev-
eral times. Also called rotational symmetry. Compare with line R
symmetry.
range A statistical measure obtained by subtracting the low
polygon A closed figure with no curved lines. Polygons are from the high.
named for the number of sides they have: triangle (3), quadri-
lateral (4), pentagon (5), hexagon (6), heptagon (7), octagon rational counting Meaningful counting that begins when
(8), enneagon (9), decagon (10), . . . . children connect number words to objects such as apples,
blocks, toy cars, or fingers. Compare with rote counting.
polyhedron A space figure made up of plane faces. Poly and
hedron are Greek terms meaning “many” and “seat” (formed by reflection A mirror image created when an object is reversed
two flat surfaces), so polyhedron means “many flat surfaces.” across a reference line; commonly called flip. Compare with
rotation and translation.
positive interdependence One of the basic principles for im-
plementing cooperative learning successfully, in which team regrouping and renaming (1) Exchange between place val-
success is achieved through the successes and contributions ues. (2) The traditional, or conventional, algorithm for addi-
of each member. Compare with equal participation and indi- tion, commonly referred to as “carrying.”
vidual accountability.
regular polygon A polygon with congruent sides and con-
practice In a constructivist sense, providing many rich and gruent angles, such as a square, equilateral triangle, or regular
stimulating experiences with a concept or skill rather than us- hexagon.
ing repetition of the same experience.
regular solid A polyhedron with congruent faces, that is, with
preoperational stage Second stage of Piaget’s developmen- the same number of faces meeting at each vertex. There are
tal stages. Occurs around ages 2 or 3 to 6 or 7 when children five regular solids, also called Platonic solids: (1) hexahedron,
gradually change from being egocentric and dominated by or cube; (2) tetrahedron; (3) octahedron; (4) dodecahedron;
their idiosyncratic perceptions of the world. They begin to be- and (5) icosahedron.
come aware of feelings and points of view of others in their
world. Children develop symbol systems, including objects, remainder (1) In take-away situations, the answer to the ques-
pictures, actions, and language, to represent their experience. tion or problem that asks “how much is left” or “how many are
Compare with concrete operational stage, formal operational left” after part of a set is removed. (2) In division, the groups or
stage, and sensorimotor stage. objects that are not evenly divided.
prime number A number that has only one set of factors— repeated addition A multiplication strategy that involves the
the number 1 and itself. Compare with composite number. total number belonging to multiple groups having the same
number; related to skip counting. Compare with Cartesian
principle of compensation The smaller a measurement cross-product and geometric interpretation.
unit is, the more of these units will be required to measure an
attribute. repeated subtraction division See measurement division.
problem-based learning Classroom investigations. Thorp right triangle A triangle having a 90-degree angle. Compare
and Sage describe problem-based learning as “focused, expe- with acute triangle and obtuse triangle.
riential learning (minds-on, hands-on) organized around the
investigation and resolution of messy, real-world problems.” rotation A transformation resulting from an object being
turned about a fixed point; commonly called turn. Compare
problem-based projects or investigations Students inves- with reflection and translation.
tigate a problem (initiated by teacher or student) by exploring,
applying concepts and skills, expanding ideas, and drawing rotational symmetry See point symmetry.
conclusions.
rote counting A memorized list of number words, such as 1,
process-oriented problem Focuses on the process students 2, 3, 4, . . . , that provides prior knowledge for number concepts.
employ to solve a problem. For example, a calculator frees stu- Compare with rational counting.
dents from computational boredom and allows them to con-
centrate on the process they use to find the answer. They can rounding A way to express vital information about a num-
think about the problem itself rather than the computations the ber without being unnecessarily detailed. Compare with
problem requires. Compare with product-oriented problem. estimation.
product The result of multiplication. running record A handy way to keep track of student prog-
ress. A running record is made by taping index cards on a clip-
product-oriented problem Concentrates students’ efforts on board and jotting down anecdotal notes or individual check-
getting the answer by using familiar algorithms. The answer, lists about each student.
570 Glossary
S symmetry The property of a shape in which there is an ex-
act reflection of form on opposite sides of a dividing line or
scaffold To frame student instruction that supports new skills plane.
and concepts based on student experiences. Teachers pre-
assess or learn about the background knowledge of their stu- T
dents and plan their instruction accordingly.
take-away subtraction Used when part of an original set is
scalene triangle A triangle with no congruent sides. Com- moved, lost, eaten, or spent. In take-away situations, the ques-
pare with equilateral triangle and isosceles triangle. tion or problem asks “how much is left” or “how many are left”
after part of the set is removed. The answer is called the remain-
scope The range of topics, concepts, and skills to be taught. der. Compare with comparison subtraction, completion subtrac-
tion, and whole-part-part subtraction.
scoring rubric Description added to each number in a
rating scale to create a uniform understanding for levels of terminating decimal A fraction whose quotient eventually
performance. terminates when converted to a decimal with the division
algorithm.
sensorimotor stage First stage of Piaget’s developmental
stages. Occurs between birth and age 2 or 3. Foundations for tessellation Tiling pattern made when a surface is completely
later mental growth and mathematical understanding are de- covered with one geometric figure or a combination of figures
veloped at this stage. Compare with concrete operational stage, in a repeating pattern.
formal operational stage, and preoperational stage.
test bias When items on a test provide an advantage or a dis-
sequence (1) The organization of knowledge and skills across advantage due to content or wording that is more familiar to
grade levels. (2) A basic thinking/learning skill that involves one group than another. Minority students and those learning
the arrangement of objects, events, and ideas. In sequence, English may experience test bias, although students who speak
order has meaning. Compare with order and seriation. English also interpret questions differently depending on geo-
graphic, cultural, and linguistic backgrounds.
seriation A basic thinking/learning skill that involves the ar-
rangement of objects, events, and ideas. Arrangement is based tracking Grouping students by ability and keeping students
on gradual changes of an attribute and is often used in mea- together for long periods. Tracking frequently works to the det-
surement. Compare with order and sequence. riment of lower-ability students with instruction that deempha-
sizes concepts and overemphasizes isolated skills. Tracking
sharing division See partitive division. also isolates the lower-ability students from the modeling and
support of more proficient students. The lower-track students
similar The property in which two or more objects have the often have the least qualified and least experienced teachers
same shape, but differ in size; the corresponding sides are pro- and the least opportunity for a full and balanced curriculum.
portional. Compare with congruent.
traditional algorithm The algorithm that is most commonly
simultaneous interaction A feature of cooperative learning used and taught in a culture.
that involves several students at the same time.
transitivity Transferring the measurement characteristic of
skip counting Counting in groups of 2, 3, 4, 5, . . . . Skip count- an object to another object in order to make a comparison.
ing encourages fast and flexible counting and is connected to
multiplication and division. translation Transformation in which an object is relocated in
a plane; commonly called slide. Compare with reflection and
social constructivism Cognitive learning theory that em- rotation.
phasizes the importance of social interactions in developing
concepts. U
space figure A shape that has three dimensions, such as unit fraction A fraction in which the numerator is 1.
boxes and blocks, balls and world globes, cones, pyramids,
and eggs. W
spatial perception See spatial sense. whole number A number used for counting discrete
quantities.
spatial sense The ability to perceive objects in relation to one
another and to oneself; the ability to mentally change the ori- whole-part-part subtraction Identifies the size of a sub-
entation of an object in relation to other objects or to oneself. group within a larger group. The whole group has a common
Also called spatial perception. characteristic, but parts or subgroups have distinct characteris-
tics. Whole-part-part subtraction identifies membership in two
squared facts When a number is multiplied by itself, such as subgroups that are included in the whole group. The number in
4 ϫ 4 or 7 ϫ 7. one part of the whole is known, and the question posed is “how
many belong” in the other part. Compare with comparison sub-
stem-and-leaf plot A graphical representation that displays traction, completion subtraction, and takeaway subtraction.
numerical data by using the leftmost digit of the data as a stem,
and the remaining digits as leaves on the graph. Stem-and-leaf Z
plots are especially useful in comparing raw data.
zone of proximal development The gap between current
stimulus-response (S-R) theory Drill and practice of facts knowledge or skill and the desired knowledge or skill. Students
and mathematical procedures is based on a belief that repeti- are able to learn within their zone of proximal development. If
tion establishes strong bonds. students do not have the requisite background for learning, the
teacher provides experiences that develop the foundation for
sum In addition, the total found when adding numbers. successful learning.
summative assessment Assessment following instruction to
determine if students have mastered the concepts and skills
and can apply them in problem-solving situations. Summative
assessment provides accountability for students and for teach-
ers. Also called mastery assessment.
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Index
Abacus, 79 National Council of Teachers of Mathemat- Area
A-B-C pattern representations, 18 ics (NCTM) standards, 138, 358, 373 of circles, 467–468
Abstraction principle, 147 English system, 442
Accountability patterns, 117–118 estimation of, 465–466
and thinking, 358–359 extending concepts about, 465–470
assessment and, 93 formulas, inventing, 466–470
individual accountability, 70 See also Equations; Functions; Patterns; manipulatives for, 76
Accuracy Variables metric system, 443
in addition, 205–207 multiplication arrays and, 215
in division, 224 Algebra tiles, 379 of parallelograms, 466
in measurement, 441 Algorithms perimeter compared, 468
in multiplication, 224 teaching measurement of, 451–452
in subtraction, 205–207 for addition, 231–235 volume, conversion to, 472
Acres, 466 al-Khowarazmi and, 144
Action research, 56 compensation algorithm, 234–235 Area models for common fractions, 265
Act-it-out strategy, 118–120 defined, 145 Ares, 466
Acute angles, 404–405, 453 division algorithms, 242–247 Arrays in multiplication, 214–215
protractors for measuring, 475, 477 fractional numbers, computing, 294–297 Art
Acute triangles, 419 learning disabled students and, 38
Addends, 190–191 limited English proficiency (LEP) students mathematics and, 23
associative property of, 200–201 multicultural activities, 33
introduction of term, 194 and, 36–37 Asian Americans. See Ethnicity
missing-addend form, 192 multicultural activities, 33 Assessment
for multiplication, 212 multiplication algorithms, 239–247 accountability and, 93
Addition, 13–14, 188, 190–192 notation for, 240 addition, checklist for, 196
accuracy in, 205–207 for subtraction, 235–240 of angle measurement, 478
alternative algorithms, 233 for visually impaired students, 39–40 base-10 numeration system understand-
calculators, practicing with, 207 See also Alternative algorithms
compensation algorithm, 234–235 Alhambra Palace, symmetry in, 422 ing, 165–167, 169
developing concepts of, 192 Aljandre, S., 88 central tendency measures, 516
double facts, 202 al-Khowarazmi, 144, 362 collecting information, 96–99
estimation and, 176 All possibilities, accounting for, 127–129 common fractions, knowledge of,
flash cards for, 206, 207 Almanacs for large number searches,
on hundreds chart, 232–233 260–265, 280
identity element for, 201 171–172 creating tasks, 96
introducing, 192–196 Alternative algorithms, 233 defined, 94
large numbers, strategies for, 231–239 diagnostic assessment, 99
learning strategies, 201–204 ladder algorithm, 245–246 during instruction, 102–104
materials for modeling, 193–194 mental computation and, 238 at end of instruction, 105–107
near doubles, 202–203 for multiplication, 240 formative assessment, 102–104
problem-solving with, 193 for subtraction, 236–237 for fractions, 280
properties of, 200–201 Ambrose, R., 30, 56 geometric probability, 536
repeated addition, 208, 209–210 Analog clocks. See Clocks of limited English proficiency (LEP)
sentence, 194 Analysis stage of geometric understanding,
sequence of skills for, 190 students, 36
speed in, 205–207 396–397 multiple assessments, checklist for, 103
subtraction, inverse relationship with, Analytic rubrics, 99 multiples table with proportions, 341
Anasi the Spider: A Tale from the Ashanti performance indicators, 95–96
196–197 performance objectives, 94–95
See also Common fractions; Commuta- (McDermott), 32 planning for, 94–101
Anderson, G., 83 polygon understanding, 404, 420
tive property of addition; Decimal And Meatballs for All (Burns), 363 portfolios for, 106
fractions Anecdotal records, 97 preassessment, 101–102
Addition sentence, 194 Angles, 403–405 questions, 95
African Americans. See Ethnicity understanding of, 60
Ainsworth, S., 177 degrees, 476 writing as, 104–105
Airline schedules, 479 expanding concepts, 474–478 See also Grading; Quizzes; Recording in-
Aksu, M., 294 manipulatives for, 76
Algebra, 353–354 measurement of, 453, 474–478 formation; Self-assessment; Standard-
and al-Khowarazmi, 362 misconceptions about, 474 ized tests; Tests
content standards, 12, 17–19 See also Triangles Assessment principle, 6, 8
extending thinking, 373–378 Annenberg Media, 90 Associative law, 200–201, 202
Anno, Masichiro, 249 for multiplication, 219–220
Anno, Mitsumasa, 249 Attribute blocks, 141
Anno’s Counting Book, 146–147 Auditory patterns, 360
Anno’s Magical Multiplication Jar, 248–249
Applets, 87–88
for fractions, 283–284
Approximation, 440–441
577
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GUIDING CHILDREN'S LEARNING OF MATHEMATICS
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GUIDING CHILDREN'S LEARNING OF MATHEMATICS
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