Using tape diagrams to solve problems · 6. John bought a bag of marbles. ¼ of the marbles were...

© 2015 National Council of Teachers of Mathematics

www.nctm.org/profdev

Using tape diagrams to solve problems

1. Jay and Nick have 2000 stickers altogether. If Jay has 600 more stickers than Nick, how many

stickers does Jay have?

2. Gus’s and Ike’s combined running distance this week was 48 miles. If Gus ran three times as far

as Ike, how many miles did Ike run?

3. Gus and Ike are playing with toy cars. The ratio of Gus’s cars to Ike’s cars is 7 to 3. Gus gives Ike

14 cars, so now they each have the same number of cars. How many cars do they each have

now?

4. Eileen paid $8.25 for a book and a comic. The book cost twice as much as the comic. Find the

cost of the book.

5. Mr. Jones gave ¼ of a sum of money to his wife. Then he divided the remainder equally among

his 4 children. If each child received $600, find the sum of money.

6. John bought a bag of marbles. ¼ of the marbles were blue, 1/8 were green and 1/5 of the

remainder were yellow. If there were 24 yellow marbles, how many marbles did he buy?

7. Dunkin Donuts sold 2/3 of their donuts in the morning and 1/6 in the afternoon. They sold 200

donuts altogether. How many donuts did they have left?

8. If 2/3 of a number is 12, what is the value of ½ of the number?

9. At a sale, Mrs. Brown bought a fan for $140. This was 70% of its usual price. What was the

usual price of the fan?

10. Ten glasses of water can fill 5/8 of a bottle. How many more glasses of water are needed to fill

up the bottle?

278 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 20, No. 5, December 2014/January 2015

wWhat better way to interest students in mathematics than using a Super Bowl® commercial? A Prudential® insurance commercial aired during the Super Bowl in 2013 was the impetus for our lesson (see it here on You-Tube™: http://www.youtube.com/watch?v=IsNiKGMSHUQ). In the commercial, four hundred people were polled on “How Old Is the Oldest Person You’ve Known?” and each was given a sticker to place on a larger-than-life dot plot marking the age of the oldest person they knew. We re-created this activity for sixth-grade and seventh-grade students to engage them in collecting meaningful data, creating organized data displays, and

analyzing and interpreting the data to draw generalizations. We also posed a series of questions about measures of central tendency, the shape of data, the interquartile range, absolute mean de-viation, representativeness, predictions, and more. Our activity was tested in the sixth-grade and seventh-grade classes taught by one of the authors.

LESSON DESCRIPTIONOverviewOur activity incorporates a model adapted from the GAISE report (Franklin et al. 2005), which rec-ommends that students engage in statistical problem-solving tasks in which they—

Sarah B. Bush, Karen S. Karp,

Judy Albanese, and Fred Dillon

A Super Bowl commercial became the impetus for engaging students in a meaningful data collection project.

OLDESTYou’ve Known

The

PERSON

Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 20, No. 5, December 2014/January 2015 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 279

1. formulate questions; 2. collect data; 3. analyze data; and 4. interpret results (p. 11).

We emphasize the latter three because the Prudential commercial provided the initial question, which launched the activity: How old is the oldest person you’ve known?

Collecting the Data We wanted students to work with one data set consisting of subsets of data, so that we could ask students questions that required them to make comparative inferences among three sample groups. Therefore, students

needed to collect three pieces of data. First, they needed the age and name of the oldest person they knew. When students asked their parent or guardian and then their grandparent or other older relative the same question, three data points resulted. Our only stipula-tion was that the “oldest person you’ve known” must be someone who was still living and who was known person-ally. Students or family members were not allowed to choose celebrities or historical fi gures. Students used a data collection sheet (see fi g. 1) and had fi ve days to submit their three data points. Then each student made a prediction of the mean for the oldest person from each group (see fi g. 2).

We anticipated that the student data set would have a spread and measure of center that was smaller than the other two sets of data (parent and grandparent). We also predicted that the grandparent data would have a small spread and a much higher measure of center than either of the other data sets.

Creating a Dot PlotAfter students fi nalized predictions about the average ages, they created a dot plot similar to the one that appeared on the television commer-cial. Students wrote the ages from their own, their parents’, and their grandparents’ oldest person choices FUN

STO

CK

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INK

STO

CK


on purple, pink, and blue sticky notes, respectively, and placed them on the wall-size graph. Because the graph was so large, it was prepared before-hand with ages 54–106 years along the x-axis and with frequencies on the y-axis. We felt comfortable prepar-ing the graph ahead of time because students had already shown profi-

ciency in creating graphs with appro-priately labeled axes, scales, and titles. We carefully spaced the intervals far enough apart so that sticky notes would not overlap.

Analyzing and Interpreting the Data The next day, students watched the commercial. We explained that the commercial was the catalyst for this activity and although the subject mat-ter involved saving for retirement, our focus was on using the dot plot with their data to analyze and interpret the results of their data. We hoped that students would generate links to real-world situations that were similar to the retirement scenario.

Students were eager to examine their data and discover if their predic-tions (see fig. 2) were accurate. They used an activity sheet (see fig. 3) containing questions that centered on grade-level statistics concepts and worked in mixed-ability groups of three or four.

Question 1, complete the table, asked students to calculate measures of central tendency for each of the three groups and for the combined data set. Some groups were assigned the mean and range, and other groups, the median. Students walked up to the dot plot to observe the data and calculate these measures. We no-ticed that some students were initially unsure about counting the same age more than once when there was more than one data point for the same age. Student groups then shared their de-scriptive statistics with the class. After every student had completed the table, we were ready for data analysis.

Using responses from question 1 and their prior knowledge, students worked on questions 2–10. Students also strategically selected tools and resources to use, such as calculators, the dot plot, the vocabulary word wall, and the teachers.

Immediately, students noticed that the range of ages for the parent group was narrower than for the other two groups (as prompted in question 2). One student wrote, “The parents know more people that are in the same age group while everyone else is spread apart, that says they [students and grandparents] know a wider vari-ety of peoples’ ages.” Another student wrote, “The ranges are differing from grandparents at 40, parents at 21, and students at 43. The parents have more clustered data, where the students are spread out.” Some students used terms such as “dense” and “scattered” to de-scribe the data. Hearing these student answers gave us an opportunity to re-view mathematical vocabulary of devi-ation including “spread” and “range.” We required that students use precise terminology in their explanations with their groups. It was surprising that the range was so large for the grandpar-ent group. This situation provoked a rich discussion, particularly about one value being lower than the other blue

Fig. 1 Students kept track of information in a data collection sheet.

How Old Is the Oldest Person You’ve Known? Data Collection Sheet

Age Name How Do You Know This Person?

Student

Parent/guardian

Someone else two generations older (grandparent, neighbor, etc.)

Fig. 2 Students predicted the average of three data sets.

How old do you think the average age will be for the three groups? Write your predictions here:

Student average: ________________

Parent/guardian average: ___________

Grandparent/other older relative average: ____________

Students counted to the median.

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sticky notes. We discussed the concept of an outlier and how it affects the mean and the range.

In examining responses for ques-tion 3, students determined that our dot plot looked like a combination of “skewed left” or “mound shaped.” We heard a variety of thoughtful conver-sations in groups, such as this one:

Student 1: I think it is skewed left because there are more [points] to the right.

Student 2: I think mound shaped because the end [student points to the data on the right of the graph] is making it a mound.

Student 3: I think it is skewed right or mound.

Student 1: Why do you think it is skewed right?

Teacher: Another word we use when describing mound-shaped graphs is symmetry. Does that change your opinion about the shape of this graph? Why?

To answer this question, the students discussed how the language of “skewed left” and “skewed right” could be confusing because it is the opposite of the direction that one may expect. Although student groups were not satisfied describing the data using just one shape, they realized that this graph did not meet the requirement of line symmetry for a mound shape. Therefore, they decided that the best descriptor was “skewed left.” These conversations engaged students in one of the Common Core’s Standards for Mathematical Practice (SMP) as they explained their thinking and justified their reasoning to others (SMP 3).

For question 4, student groups used the dot plot to help them find the interquartile range for the combined data set. They first counted in from the ends to find the median of all data points. Next, they identified the upper and lower quartiles by finding the

median of the lower half of data (from the least value to the median) and the median of the upper half of data (from the median to the greatest value) us-ing this counting method. Students explained that the difference of the first and third quartile represented the interquartile range.

The responses for questions 7 and 8 showed not only that students under-

stood the algorithm for calculating the mean absolute deviation (MAD) but also that they understood the concept of the MAD.

Question 7 asked students to find the MAD (average distance of the data points from the mean) for each group. Previously, students had discussed the definition of mean absolute deviation and had practiced finding the MAD

How Old Is the Oldest Person You’ve Known?

1. Complete the table.

Sample Mean Median Range

Students

Parents/guardian

Grandparents/other older relative

Combined

2. Are there differences in the ranges of the three groups? If so, describe the differences.

3. Examine the dot plot for the entire data set. Describe the shape of the data using the words uniform, mound-shaped, skewed left, or skewed right.

4. What is the interquartile range for the entire data set?

5. Are there any values for the entire data set that can be considered outliers? If so, what are they? How did you determine whether they were outliers or not?

6. Which measure of central tendency describes the data most accurately? Why?

7. Find the mean absolute deviation for each group. What information do you learn by comparing the three statistics?

8. Do you think the results of your data collection are representative of students at other schools? Why or why not? Would it be representative of students in other countries? Why or why not?

9. What overlap do you notice on the dot plot among the three colors (groups)? What does this overlap represent?

10. What was the difference between your predicted average (before plotting the data) and the actual average for each group? Are you surprised? Why or why not?

Fig. 3 Students recorded their data analysis on an activity sheet.


for small data sets. Students were ini-tially challenged by this larger data set (31 data points for each group; 93 combined). Students knew they must start by recalling the mean for that data set (referring back to question 1).

Next, students decided to work in groups of three at the dot plot, with one student calling out a data point, another student quickly computing the difference between each individual data point and mean, and a third student recording that difference to keep a running log. Students knew to take the absolute value of the differences (or subtract the smaller number from the larger number) because the distance could not be negative. After students recorded all the differences between the data points and the mean, they went back to their desks and found

the mean of those differences, thus calculating the MAD for their data sat. Another option to confi rm this value was to ask students to make a list of data-point values below and above the mean, then sum each set to see if they were equal. Because the means above and below summed to zero, we used the absolute value. When asked what the MAD told them about their data, one student said, “It’s the average dis-tance each one is away from the mean.” Students realized that MAD repre-sented the average number of years that the reported age for each data point was from the mean. After deep pondering, another student explained that he just realized the MAD was probably lowest for the parent group because its narrower range might be a predictor of how large or how small

the MAD would be. Students were not only calculating the MAD but also made conjectures about what could cause the MAD to be a smaller or a larger number, such as occurred in its connection to the range of the data set or how an outlier can affect the MAD.

We wrote question 8 to build cultural connections and encourage students to think about the gener-alizability of their data. Students considered many factors when think-ing about their responses and asked questions about the ages of students at other schools, whether they were students in our city or elsewhere, and whether they should think about other countries. We received the fol-lowing responses to question 8:

For other schools yes because if they do the same grade they are most likely the same age as us. For other countries no because their life expectancy might not be as long.

In other schools kids would be our age so yes. But in other countries no because the death rate could be different.

We felt that this question required students to consider the problem’s context, thus aligning well with SMP 4: Model with mathematics. As students considered the context and the idea of asking the same question of students in other countries, they explained that life expectancies might be different internationally. Addition-ally, students discussed the idea that a sample at an individual school was relatively small and not random and that this could cause the means to vary.

The fi nal question asked students to think about their original predic-tions of the mean for student, parent, and grandparent groups and compare that prediction with the actual results. Students provided thoughtful respons-es, such as those in fi gure 4. Most

(a)

(b)

Fig. 4 These thoughtful student responses were in answer to question 10, “What was the difference between your predicted average (before plotting the data) and the actual average for each group? Are you surprised? Why or why not?”


students were close on at least one of their predictions. Some students pre-dicted the average for the grandparent group would be much higher than it actually was, because they assumed that their grandparents were likely to know the oldest people. This was in line with our original thoughts, so we were surprised about the results for the three groups, as well.

AN EXTENSION: EXPLORING BOX PLOTS The final extension task asked students to create three box plots representing the student, parent, and grandparent data. Instead of making the plots with paper and pencil or technology, stu-dents stood in for the data (see fig. 5).

We taped a number line to the floor that used a limited domain from 50 to 105 years to align with our data, using 5-year intervals as the scale, and asked students to look at their original data collection sheets (see fig. 1) to recall their data points. First, students lined up silently on the number line ac-cording to the data point of the oldest person they knew (student data set). Students used gestures to communicate with one another as they identified others’ numbers and figured out their position on the number line. Students who had the same data point were confused about where to stand. They acknowledged that for their specific data point to be accurately represented on the number line, students with the same values should line up behind, rather than beside, one another. We have also seen this activity completed in which students stood beside one another, rather than behind, but we respected the students’ reasoning.

We asked students how they could find the median, and they said they needed to count toward the middle. We gave small flags to the two individu-als who represented the smallest and greatest data points, and students called “pass” as they systematically moved

(a)

(b)

(c)

Fig. 5 Students held a rope to show the box plots for student (a), parent (b), and grandparent (c) data.

The finished dot plot provided an instant data-collection visual.

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B. B

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them toward the middle until the fl ags met at the median. When there were two middle numbers at the completion of the passing, we asked students what to do. After discussion, they suggested averaging those two numbers.

Next, we asked students to follow a similar fl ag-passing process to deter-mine the upper quartile and lower quartile. This time, the fl ags moved from the middle back to the ends, with students on each end passing additional fl ags inward toward the middle. The two quartiles were de-termined when each set of fl ags met.

Box Plot Questions1. Examine the length of the whiskers of the three data sets. What does each of their lengths tell you about the data?

2. Which box is the shortest? What does that tell you about the data points inside the box?

3. What is the most surprising thing you notice when you compare the three box plots? What does that tell you about the data?

(a)

(b)

Fig. 6 Question 3 of the box plot activity sheet was the impetus for students’ responses in (a) and (b).

Students noticed that the numbers from the lower quartile to the upper quartile were the same as the “box” part of a box plot. Students were then given a rope. The students at the 1st and 3rd quartile changes (and every-one in between) were asked to lift the rope. The remaining students held the rope at their waist, thus representing the “whiskers” of the plot. The biggest challenge that students had with mak-ing the box was when there were two middle numbers at the upper or lower quartile, causing both students to want to raise the rope. However, the

actual upper and lower quartile fell between those two students, which confused some students. After the hu-man graph was complete for the stu-dent data, we took a photograph (see fi gs. 5a–c). We repeated this process for the parent and grandparent data.

We displayed all three photographs on the overhead projector. Students were given an activity sheet, which asked them to make comparative inferences about the three differ-ent groups, using the three box plots displayed (see fi g. 6).

For the fi rst question, most students agreed that the data from the student and grandparent box plots were more spread out, with some students attributing this to their larger ranges (from their previous work). Other students mentioned that outliers caused the length of the whiskers. On question 2, students explained that the parent data made the shortest box. Student responses were similar to this student comment, “The parents’ box is the smallest, which tells us that the range is small, the data is clustered, and that the ages of the people that the parents

When determining measures of central tendency, some students were unsure about counting the same age more than once.

SAR

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B.

BU

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Sarah B. Bush, [email protected], is an assistant professor of mathematics education at Bellarmine University in Louisville, Kentucky. She is a former middle-grades mathematics teacher who is interested in interdisci-plinary and relevant and engaging math activities. Karen S. Karp, [email protected], is a profes-sor of mathematics edu-cation at the University of Louisville in Kentucky. She is a former member of the NCTM Board of Directors and a former

president of the Association of Mathemat-ics Teacher Educators (AMTE). She con-tinues to work in classrooms to support teachers of students with disabilities in their mathematics instruction. Judy Albanese, jalbanese@stleonard louisville.org, is a middle-grades math-ematics teacher at St. Leonard School in Louisville, Kentucky. She seeks to develop her students’ conceptual understanding of mathematics by implementing instruc-tion and activities that are engaging and relevant to her students. Fred Dillon, [email protected], is a mathemat-ics teacher from Strongsville, Ohio, and a former member of the NCTM Board of Directors and MTMS Editorial Panel. He is interested in helping teachers use engaging tasks and student involvement in their classrooms.

know is similar.” When asked what surprised them, students had a variety of interesting responses (see fig. 6). Most students originally thought that the grand-parent box plot would have been more clustered (and include older ages) because they should have friends their age and older. However, some students noted that perhaps some of their grandparents’ friends may have passed away, thus meaning that they might have known fewer older people. Students also thought that the parent data set would have been more spread out and found it strange that it was so tightly clus-tered. However, they were unable to infer possible causes.

STUDENTS: WE “WERE THE DATA”This activity provided an engaging and relevant context that allowed students to use their knowledge of statistics to collect, display, analyze, and interpret data in meaningful ways. Students who took part in this activity were excited to “be the data,” survey two different generations about the oldest person they knew, and compare the three sets of data by answering a variety of questions.

This lesson provided a relevant ave-nue to involve sixth-grade and seventh-grade students in multiple CCSSM content standards. What was most important was this activity’s student-centered focus because students were doing the work of thinking, model-ing, justifying, and reasoning and were authentically engaged in several mathematical practices. Specifically, students modeled with mathematics as they worked within this real-life context and were personally connected to the data and represented the data in multiple ways.

We placed a strong focus on asking students to explain their reasoning; to justify their answers, considering the data and the context

of the problem; and to discuss their thoughts in their groups, thus helping them construct arguments and cri-tique the reasoning of others. Finally, we continually reinforced the idea of being precise in all of their spoken, written, and calculated work, which emphasized the importance of giving attention to precision in all math-ematical endeavors. Students found ways to connect to their families and the real world as they gathered information for an engaging learning experience.

CCSSM Practices in ActionSMP 3: Construct viable arguments

and critique the reasoning of others. SMP 4: Model with mathematics. SMP 6: Attend to precision.

Statistics and Probability, both sixth-grade and seventh-grade domains: 6.SP.1, 6.SP.2, 6.SP.3, 6.SP.4, and 6.SP.5; 7.SP.3 and 7.SP.4

REFERENCESCommon Core State Standards Initiative

(CCSSI). 2010. Common Core State Standards for Mathematics. Wash-ington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards .org/wp-content/uploads/Math_ Standards.pdf

Franklin, Christine, Gary Kader, Denise Mewborn, Jerry Moreno,

Roxy Peck, Mike Perry, and Richard Scheaffer. 2005. Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report. Alexandria, VA: American Statistical Association.

YouTube. 2013. Prudential—Stickers. http://www.youtube.com /watch?v=IsNiKGMSHUQ

Any thoughts on this article? Send an email to [email protected].—Ed.

To referee manuscripts or review books and products for this journal, visit

www.nctm.org/mtms or email [email protected] for information.

MTMS NEEDS YOU!

Algebra Magic

Instructions Pictures Alice Ben Carla

Think of a number.

8 3

Add 6.

14 9 21

Multiply by 2.

42

Subtract 2.

26

Divide by 2.

13 20

Subtract your original number.

5

If = 33, what is ?

If = 30, what is ?

If = 15, what is ?

If = 34, what is ?

© 2015 National Council of Teachers of Mathematics www.nctm.org/profdev

Algebra Magic

Instructions Pictures Dej Eli Fran

Think of a number.

7

Multiply by 6.

18

Add 8.

26 50

25

9 30

Divide by 3.

10

Match each picture of a trick on the left with a set of instructions on the right.

1. a. Think of a number. Multiply by 3. Add 4.

2. b. Think of a number. Multiply by 4. Add 3.

3. c. Think of a number. Add 3. Multiply by 4.

4. d. Think of a number. Add 4. Multiply by 3.


Algebra Magic Make a separate algebra magic trick with at least five steps that will meet one of the bullets listed below:

• Final result is one more than the original number. • Final result is 0. • Uses all four operations. • Result is same, whether steps are done backwards or forward.

Instructions Pictures Giam Hiroshi Ian

Think…


Algebra Magic

Instructions Pictures Description of Pictures

Abbreviation

Think of a number.

a bucket

Add 6.

a bucket and 6 b + 6

Multiply by 2.

2b + 12

Subtract 2.

Divide by 2.

Subtract your original number.

Match each abbreviation on the left with a set of instructions on the right. 1. 2b + 10 a. Think of a number. Multiply by 2. Add 5. 2. 5b + 2 b. Think of a number. Multiply by 5. Add 2. 3. 5b + 10 c. Think of a number. Add 2. Multiply by 5. 4. 2b + 5 d. Think of a number. Add 5. Multiply by 2.


Algebra Magic

Instructions Pictures Description of Pictures

Abbreviation

Think of a number.

a bucket

b + 2

Multiply by 3.

2b + 6

Divide by 2.

3

If + + 7 = + 10, then = __________. If 2b + 7 = b + 10, then b = __________.

If + + 1 = 11, then = __________.

If 3 + 1 = + 11, then = __________. If 4b + 5 = 3b + 7, then b = ________.


Solving Equations Using the Cover up Method

1. 5(𝑥 − 10) = 15

2. 3(𝑥 + 10) = 15

3. 3 + 𝑥10

= 15

4. 18𝑥

+ 12 = 15

5. 34 − 2𝑥+62

= 4

6. 34 − 2𝑥+62

= −4

7. 21 = 12 + 3𝑥8

8. 12 = 21 + 3𝑥8

Solving Equations Using the Cover up Method


9. 5 + 𝑥6

= 17

10. 5 + 6𝑥

= 17

11. 5 − 𝑥6

= 17

12. 5 − 6𝑥

= 17

13. 3 = 12𝑥+1

14. 3 = 𝑥+112

15. 3 = 12𝑥+7

16. 3 = 𝑥+712


Solving Equations Using Strip Diagrams

Solve the following problems using the Strip Diagram Method. Consider a variety of ways to set-up the algebraic equation to satisfy the solving of the problem.

1. There are 50 children in a dance group. If there are 10 more boys than girls, how many girls are there?

2. Ann has three times as much money as Brenda. Brenda has $200 less than Carol. Carol has $50 more than Ann. Find the total amount of money that Ann, Brenda, and Carol have.


Solving Equations Using Strip Diagrams

3. The ratio of Sam’s money to Mel’s money was 4:1. After Sam spent $26, Sam had $2 less than Mel. How much money did Sam have at first?

4. John is four times as old as his son. If their total age 10 years ago was 60, find their present ages.

5. A box contains a total of 200 blue, yellow, and orange beads. Twice the number of blue beads is 10 more than the number of yellow beads. There are 50 more yellow beads than orange beads. How many blue beads are there in the box?




Integers

Represent the following story problems on a number line. Then write an equation that aligns with your number line.

1. John walks 2/3 of a mile on Wednesday. He walks ¾ of a mile on Thursday. How far did he walk on both days?

2. Over night it gets to 23 degrees below zero in Antarctica. During the day it

warms up 17 degrees. What is the temperature at the end of the day?

Represent the following story problems on a number line. Then write two equations – one that involves addition and one that involves subtraction.

3. Joyce has $43. She spends $17 on the new Justin Bieber CD. How much does he have now?

4. At 6 PM, it’s 15 degrees above zero. Overnight it cools down 19 degrees.

What is the temperature after it cools down?

5. At 6 PM, it’s 3 degrees below zero. Over night it cools down 9 degrees. What is the temperature after it cools down?



Represent the following story problems on a number line. Then write two equations – one that uses the missing addend interpretation and one that uses the difference

interpretation.

6. At 8 AM, the temperature in Wisconsin is 61 degrees. At 3 PM, the temperature is 80 degrees. What is the difference in temperatures?

7. At noon a marine biologist measures the temperature of a lake to be 46

degrees. At midnight he measures the temperature of the lake to be 38 degrees. What is the difference in temperatures?

8. A turtle is hunting for fish 4 feet beneath the surface of the lake. After his

hunt, he decides to sun on a rock 3 feet above the surface of a lake. What is

the difference between the turtle’s starting position and ending position?

9. A turtle is swimming 4 feet beneath the surface of the lake. He spies a fish to eat and dives to 17 feet beneath the surface of the lake. What is the

difference between the turtle’s starting position and ending position?

Create story problems that would align with the following equations:

10. (-3) – (-2) = (-1)

11. (-3) – (-4) = (+1)

84 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 20, No. 2, September 2014

cConsider the following story problem about money, which we posed to 40 seventh graders during an inter-view that involved a variety of integer-related tasks: Yesterday, you borrowed $8 from a friend to buy a school T-shirt. Today, you borrowed another $5 from the same friend to buy lunch. What’s the situation now?

Ian Whitacre, Jessica Pierson Bishop, Randolph A. Philipp, Lisa L. Lamb, and Bonnie P. Schappelle

CR

AZY

DIV

A/T

HIN

KST

OC

K

We asked the students who in the story would owe money to whom and how much. We also asked each stu-dent to write an equation to describe the situation. How would you expect students to respond?

Many of the seventh graders wrote 8 + 5 = 13 to describe the story. Evelyn explained that the numbers in her equation represented amounts of money borrowed from her friend. Her

DollarsSense:&&Dollars&Dollars&&Sense:&Sense:Sense:&Sense:&

Students’ Integer Perspectives

Copyright © 2014 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Vol. 20, No. 2, September 2014 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 85

A story problem about borrowing money may be represented with positive or negative numbers and thought about in different ways.

Learn to identify and value these different perspectives.

equation showed that she now owed her friend $13. A few students wrote equations involving negative numbers, such as –8 + –5 = –13. Most students who did not initially write equations involving negatives nonetheless were able to interpret these equations.

We found that students had differ-ent ways of relating these equations to the story. For example, Chrissy said that –8 + –5 = –13 described the story

because negative numbers represented money that you owe. Anh said that this same equation, –8 + –5 = –13, also described the scenario because negative numbers represented money that you took from your friend.

When we began talking with students, we viewed borrowing money as a context for the topic of integers. We expected that students who had studied integers would use

negative numbers to represent money owed. We were surprised to fi nd that students often did not use negative numbers to describe this situation and that they had different ways of relating equations to the story. At the same time, students’ responses made sense. In this article, we describe the sense that students made of the money problem and how they related it to equations.

Sense:


SIGNED NUMBERSWe describe ideas related to integers (both positive and negative) and how students used them in relation to a story problem. Before instruction on integers, which typically occurs in middle school (CCSSI 2010), the no-tion of sign is rarely if ever discussed. Therefore, for much of their child-hood, many students live in a world in which numbers have no signs. We count 1, 2, 3—not positive 1, positive 2, positive 3. Then students are told that some numbers are negative and that familiar numbers like 3 are actually positive. Further-more, these signed numbers behave in strange ways that contradict previous generalizations that students have made. We confound students further with the information that addition can make smaller and that subtraction

can make larger. Students’ mathemati-cal worlds have therefore been turned upside down. This explains why students often learn to operate with integers without having a conceptual understanding.

INVESTIGATING STUDENTS’ UNDERSTANDINGThe Common Core’s Standards for Mathematical Practice emphasize modeling real-world situations math-ematically, making sense of problems, and relating symbols to contexts (CCSSI 2010). Overall, the empha-sis is on students doing mathematics with understanding. Contexts such as money, elevation, and temperature

are commonly found in the integer sections of middle school textbooks. In fact, in a review of 18 fifth-grade and sixth-grade textbooks, Whitacre et al. (2011) found that the context of money was associated with integers in all but one textbook. Given the preva-lence of this context, we were inter-ested in how students thought about relationships between story problems that involved money and arithmetic equations.

To investigate students’ thinking, we interviewed 40 seventh graders. These interviews were conducted in spring 2011 after the students had studied integers in grades five, six, and seven. The interviews consisted of a variety of tasks, many of which explic-itly involved negative integers. One task asked students to relate equations to the money problem. We found that

students who had studied integers had different ways of relating equations to this story problem.

STUDENTS’ RESPONSESGiven the story problem about money, all 40 students answered correctly that the borrower would owe the friend $13. The majority (80 percent) wrote the equation 8 + 5 = 13, which seemed to reflect their thinking about the problem. One such student, Evelyn, explained her thinking this way:

Okay. So yesterday I borrowed $8 from my friend to buy a school T-shirt. So I have $8 from my friend [points to the 8 that she wrote].

And then today I borrowed $5 from the same friend, and I bought a lunch. And then plus 5 that I borrowed from her [circles the 5 with her pen] equals 13 dollars that I borrowed from her [circles the 13 with her pen]. And what’s the situation? I owe her 13 dollars [points to the 13].

Evelyn’s explanation demonstrates that she knew very well who owed money to whom. To represent the situation, adding whole numbers suf-ficed for her. She did not need nega-tive numbers to solve this problem or to represent it with an equation. Stu-dents who reasoned like Evelyn were perfectly capable of making sense of the story and representing it with an equation without invoking negatives.

We, along with many teachers, have often used such contexts as borrowing money in integer instruction. Our in-struction and the majority of textbook examples focused on how students could use negative numbers to repre-sent the story problem. Therefore, dur-ing an interview that focused on these numbers, we were surprised that only 20 percent of students, after substantial instruction, wrote an equation involv-ing negative numbers. The students who did use negative numbers wrote –8 + –5 = –13 or –8 – 5 = –13. (One student wrote L8 + L5 = L13, in which L stood for his friend Liam. We group this response with those that involve negative numbers because the student used a symbol to indicate directional information.) Students explained equa-tions involving negatives in one of two ways:

1. Some thought that negative numbers represented money borrowed from and now owed to the friend.

2. Others thought that negative numbers represented money taken away from the friend.

We count 1, 2, 3—not positive 1, positive 2, positive 3. Then students are told that some numbers are negative and that familiar numbers like 3 are actually positive.


These interpretations were echoed in the responses of students who wrote 8 + 5 = 13 initially and then were shown equations involving negative numbers. What follows is elaboration on these interpretations.

STUDENTS’ INTERPRETATIONS OF EQUATIONSAfter students wrote their own equations to describe the story, regardless of what they had written, interviewers showed them the equa-tions 8 + 5 = 13, –8 + –5 = –13, and –8 – 5 = –13, explaining that these equations had been written by other students. Interviewers asked whether the student thought that these equa-tions described the story and why or why not. In most cases, students had written 8 + 5 = 13 and were then asked to reason about equations involving negative numbers.

Two of the 40 students thought that negative numbers had nothing to do with the story and that these equations did not make sense. The other 38 students were able to inter-pret equations involving negatives in relation to the story. (Eight of the stu-dents had written equations like these, whereas the other 30 students had written 8 + 5 = 13.) Students thought about the relationship between nega-tive numbers and the story in two distinct ways. We refer to these as the Negatives as Debt perspective and the Negatives as Loss perspective.

Negatives as DebtChrissy expressed this perspective, writing 8 + 5 = 13 to describe the story. When shown –8 + –5 = –13, she offered her interpretation:

Let’s see: –8 + –5 = –13. Well, since you owe the person, the negative rep-resents that. So, you have –8. Then you add another –5 for the next day, which equals –13, which is what you owe the friend.

Chrissy had not used negative numbers in her own equation, but she was able to interpret –8 + –5 = –13 in relation to the story. She said that the negative numbers represented money owed to the friend. Therefore, for Chrissy, the given equation described the story. Chrissy interpreted the equation in a way that is common in textbook presentations (Whitacre et al. 2011). She viewed negative numbers as appropriate for representing money owed. In this interpretation, nega-tive numbers make sense to describe the situation from the borrower’s side because the borrower owes money. One-half of the 38 students who were able to relate equations involving nega-tives concurred with Chrissy, using the Negatives as Debt perspective. The other half of the students analyzed it differently.

Negatives as LossAnh expressed this loss perspective. Like Chrissy, she wrote 8 + 5 = 13 to describe the story. When Anh was shown –8 + –5 = –13, she read the equation out loud and gave this interpretation:

–8 + –5 = –13. I would think that’s sort of right because you subtracted $8 from your friend, and then you subtracted 5 again. So, you took—so, negative’s like you took—so, you took $13 from your friend.

Anh’s sensible interpretation of this equation in terms of the story prob-lem was distinct from Chrissy’s. For

Anh, negative numbers indicated that “you took” money from the friend. “You took” $8 and $5, so a total of $13 was taken from the friend. From this Negatives as Loss perspective, nega-tives make sense from the lender’s, rather than the borrower’s, side be-cause the lender lost money. Although the Negatives as Loss perspective differs from the more conventional Negatives as Debt perspective, it is important to be aware of and recog-nize this perspective because one-half of the 38 students used the Negatives as Loss perspective.

SUMMARY OF STUDENTS’ RESPONSESAll the seventh graders in our study were able to solve the story problem about money. However, only 20 percent wrote an equation involv-

ing negative numbers, whereas 80 percent wrote the equation 8 + 5 = 13. When asked to interpret equations involving negative numbers in relation to the story, 95 percent of the 40 students were able to do so. However, they did so in two dis-tinct, but equally common, ways: by thinking of Negatives as Debt and by thinking of Negatives as Loss.

Many middle school mathemat-ics textbooks expect students to use negative numbers to represent stories involving money and other contexts, and textbooks may privilege either one or the other reasoning path (Whitacre et al. 2011). We believe that the students we interviewed had

In a review of 18 fifth-grade and sixth-grade textbooks, Whitacre et al. (2011) found

that the context of money was associated with integers in all but one.


reasonable and mathematically valid ways of representing the situation of borrowing money from a friend. Each described a correspondence between their equation and the story; they just did this in different ways. Tables 1 and 2 summarize students’ responses.

ACCOUNTING FOR DIRECTIONAL INFORMATIONWhen students used or interpreted negative numbers, they described the magnitudes and signs of the numbers as having distinct meanings in the story: The magnitudes conveyed how much money was involved, whereas the signs conveyed directional infor-mation, such as who had borrowed money from whom. When students wrote and explained 8 + 5 = 13, they seemed to be thinking of the num-bers as representing the amount of money involved but not the direction of borrowing or owing. That direc-tional information was something that students, like Evelyn, knew from the story and made a point to explain. However, the students did not seem to think of the equation as

doing the job of conveying direc-tional information; they seemed to see the 8, 5, and 13 in the equation as having magnitude but not sign. (Note that Evelyn and all but two of the students were capable of inter-preting negative numbers in relation to the story. However, they did not use negatives or show evidence of reasoning about numbers as signed until they were asked to interpret equations like –8 + –5 = –13.)

NOTATIONAL CONVENTIONS Looking beyond mathematics class, we find that different notational conventions are used to describe situa-tions involving money in our everyday lives. If you make a $100 purchase us-ing a debit card, your bank statement may show the amount as –$100, and your account balance will be reduced by $100. If you make a $100 purchase using a credit card, your credit card statement may show the amount as $100, and your account balance will be increased by $100.

Bank statements represent the amount of money that one has stored in the bank, whereas credit card

statements represent the amount of money that is owed to the credit card company. As a result, different notational conventions are used to record transactions that the purchaser may regard as essentially the same. Mathematically literate adults are able to make sense of the different ways that numbers are used to represent real-world situations. Mathematics instruction that supports students’ understanding of these different nota-tional conventions will help students become more mathematically literate and see mathematics as relevant to everyday life.

VALUING DIFFERENT INTERPRETATIONSWhen using contexts to help students make sense of integers, it is useful to be open and to share with stu-dents that more than one way makes sense. Otherwise, many students may get the message, either explicitly or implicitly, that the sense they make is incorrect. If we expect the use of contexts to support students’ learning about integers, we need to be aware of the different ways that students reason about relationships between numbers and contexts. As teachers, we need to have the ability to recog-nize students’ perspectives, so that we can be aware of differences between our own reasoning and that of our students. If we do not recognize these differences, we may view our students as “not getting it” or “being con-fused,” rather than recognizing that these students are thinking in consis-tent and reasonable ways that simply do not match a convention used in the textbook.

What can we, as teachers, do with the ideas and findings presented in this article? How can this information be useful? We hope that teachers will find different ways of incorporating the information into their teaching. We offer a few suggestions:

Initial Equation Student Wrote 8 + 5 = 13

Student Wrote an Equation Involving Signed Numbers

Number of students

32 (80%) 8 (20%)

Explanation Explanation addressed magnitude only.

Explanation addressed both magnitude and sign.

Interpretation Negatives as Debt(Negative numbers represent money

owed)

Negatives as Loss (Negative numbers represent money

lost)

No Interpretation(Negative numbers have nothing to do

with the story)

Number of students

19 (47.5%) 19 (47.5%) 2 (5%)

Table 1 Students’ equations and explanations

Table 2 Interpretations of equations involving negative numbers


1. Introduce a story about borrowing money.

2. Discuss the relationship between the context and different equations.

3. Ask students to write an equation and explain how it relates to the story or ask students to write more than one equation that they see fitting the story.

When given a task presented as having multiple solutions and given opportunities to share what they think, students will likely produce a variety of equations. In discussing their equations, students will likely ex-press both the Negatives as Debt and Negatives as Loss perspectives. If we as teachers recognize these different ways of reasoning, we can highlight the distinctions to help students be more precise in relating equations to contexts. Students could suggest names for different ways of reason-ing (e.g., “Negative means losing, positive means gaining”), or the class could refer to them by students’ names (e.g., “Anh’s way” or “Chrissy’s way”). It is important for students to have opportunities to explain how they are making sense of their reasoning, espe-cially when they are grappling with a challenging topic. Therefore, encour-age students to express how they are thinking about the task.

The relationships of integers to contexts are not simple matters. Dis-cussing these relationships may raise ideas that are strange or counter- intuitive for students. We want to be open and honest with students about the challenges associated with connecting symbolic notations to contexts and acknowledge that dif-ferent ways of reasoning make sense. Using examples from real life (like bank and credit card statements) can show students the relevance of differ-ent reasoning. If we wish to see only one correct answer, we may want to BIS

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include specific parameters regarding what part of the situation the solution is meant to represent. These details would be great topics for discussion.

UNDERSTANDING INTEGERS DEEPLYIn mathematics, understanding integers deeply includes being able to think about ways in which integers can be related to contexts. Under-standing involves being familiar with and making sense of conventions and recognizing that conventions are merely conventional; in other words, they could be otherwise. Students have distinct, reasonable ways of relat-ing integers to contexts; these may or may not match a particular conven-tion. We suggest that, as teachers, we recognize when we adopt a conven-tion and keep in mind that it is a way, not the way. We plan to engage students in discussions in which they consider different ways of relating integers to contexts. These discus-sions can serve to clarify and enhance students’ understandings of integers and help them see mathematics as sensible, relevant, and interesting.

ACKNOWLEDGMENTSThis manuscript is based on work supported by the National Science Foundation (NSF) under Grant No. DRL-0918780. Any opinions, findings, conclusions, and recommendations ex-pressed in this material are those of the authors and do not necessarily reflect the views of NSF. We thank the stu-dents for their participation and the co-operating teachers and school staff for their help arranging the interviews. We also thank Spencer Bagley, Michelle Kendrick, Rob Nanna, Sara-Lynn Gopalkrishna, and the anonymous reviewers for their helpful feedback on drafts of this article.

Any thoughts on this article? Send an e-mail to [email protected].—Ed.

REFERENCESCommon Core State Standards Initia-

tive (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards .org/wp-content/uploads/Math _Standards.pdf

Whitacre, Ian, Jessica Pierson Bishop, Lisa L. Lamb, Randolph A. Philipp, Bonnie P. Schappelle, and Mindy Lewis. 2011. “Integers: History, Textbook Approaches, and Children’s Productive Mathematical Intuitions.” In Proceedings of the 33rd Annual Con-ference of the North American Chapter of the International Group for the Psychol-ogy of Mathematics Education, edited by Linda R. Weist and Teruni D. Lamberg, pp. 913–20. Reno, NV: Uni-versity of Nevada. http://www.pmena .org/2011/PMENA_Proc_2011.pdf

Ian Whitacre, [email protected], is a faculty member in the School of Teacher Education at Florida State University in Tallahassee. Jessica Pierson Bishop, [email protected], is at the University of Georgia in Athens. Randolph A. Philipp, rphilipp@mail .sdsu.edu, Lisa L. Lamb, [email protected], and Bonnie P. Schappelle, [email protected] .edu, work at the Center for Research in Mathemat-ics and Science Education at San Diego State Uni-versity. This group studies children’s mathematical thinking. In the past few years, they have focused

on how students reason about integers and integer-related tasks.

C H A R L E S. A. (A N D Y) R E E V E S A N D D A R C Y W E B B

HAVE YOU EVER HAD STUDENTS ASK you for “more of those kinds of prob-lems?” We have, and the topic was inte-ger arithmetic—a difficult topic for manymiddle grades students.

A typical introduction to integer arithmetic—walking on a number line, studying temperatures ris-ing and falling, or using two-color chips—makes theideas reasonable to students, but because of a lack ofan inherent problem-solving focus, these ideas usu-ally do not pique their curiosity. Kieran and Chalouh(1992) report that research on integer arithmetic issparse but does reveal a lack of conceptual knowl-edge on students’ part with typical introductions.

As NCTM states, “Problem solving is central toinquiry and application and should be interwoventhroughout the mathematics curriculum to providea context for learning and applying mathematicalideas” (NCTM 2000, p. 256). Kent goes even fur-ther with the importance of context by stating that

“. . . contextualized situations,whether real or imaginary, help

students make visual andsometimes physical links be-

tween their informal knowl-edge and formal math-

ematical ideas” (Kent2000, p. 62). As im-portant as integersare to a student’s fu-

ture success in math-ematics, we should in-

troduce the topic throughintriguing, real-world situa-tions, with problem solvingbeing the central theme.

We designed an intro-duction to integers for anaverage class of fifthgraders who had not

seen negative numbers previously in their school-work. We used helium balloons—the type com-monly bought in grocery stores—as a “light and hu-morous” setting for the problem-solving exploration.We were interested in whether these students couldgain a conceptual understanding of integers and alsoin how far we might go with integer arithmetic. Ourseven-day unit was to be spread over several months,from February to May, to see if students retainedtheir knowledge and their interest. The exploratoryunit exceeded our expectations, and we would like toshare our experiences with you.

Introduction

VALENTINE’S DAY WAS RAPIDLY APPROACHING, SOstores were filled with heart-shaped helium balloons.One such balloon—a very large one—was broughtto class but kept out of sight until we were ready forthe “unveiling.” We started class that day by asking,“Is there an adult you want to give a Valentine to whosometimes complains that they’d like to lose weight?Is there a Valentine you could give them that helpsthem weigh less?” The students were hooked andguessed all sorts of magic potions. After much dis-cussion, the helium balloon was revealed and passedaround the room so the students could feel its“pulling power.” The students wanted to hold theballoon to determine if they could feel it lift theirarms. We asked, “If you held this balloon in yourhand, would you weigh less than before?” The stu-dents discussed the idea and eventually decided that

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ANDY REEVES, [email protected], teaches at theUniversity of South Florida St. Petersburg. He is interestedin algebraic thinking and problem solving. DARCY WEBB,[email protected], is currently a mathcoach/grant coordinator at Melrose Elementary School inSt. Petersburg, Florida. She is charged with implementingthe Everyday Mathematics textbook series at that school.

476 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

A Problem-Solving Introduction to Integers

Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

holding the balloon really would make someone a lit-tle lighter. We then asked, “Would it be possible totie enough of these around your waist to actually liftyou off the ground?” Their imaginations were begin-ning to take over.

The Larry Walters story was introduced with anews clipping from People magazine dated Decem-ber 13, 1993 (see fig. 1). In the late 1980s, Walterstied 42 helium balloons to a lawn chair and as-cended over the Los Angeles International Airport(LAX), creating total confusion in the area for aboutforty-five minutes. He gradually descended byusing a pellet gun to pop the balloons one by one.His “fifteen minutes of fame” included an appear-ance on The Tonight Show. Walters came to beknown to the students as “Crazy Larry” for obviousreasons. They were hooked and did not have muchtrouble determining the “pulling power” of eachballoon as 4 pounds each, assuming that Larry andhis baggage weighed 168 pounds. The studentswho did not realize how to calculate that figuremathematically agreed when other students ex-plained their thinking.

The class had recently studied gravity in sci-ence, so a discussion of the tug-of-war betweengravity and helium balloons made sense to them.What happened if gravity won the tug-of-war, andwhat happened if helium won? Everyone agreedthat they personally wanted gravity to win most ofthe time, because otherwise they might become“Crazy Larry.” However, they could also see theprospects of using the balloons advantageously if

VOL. 9, NO. 9 . MAY 2004 477

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Fig. 1 Larry Walters preparing for his flight


their flight could be controlled, to help do some“heavy lifting.”

The students were asked to sketch themselvesor an object they owned, standing on a scale withthe weight displayed. Then they drew themselves

or their object on another scale but with enough ofWalters’s balloons so that they floated, just barelytouching the scale, with the scale showing 0. Somestudents weighed themselves or their objects, thenworked in pairs to help each other decide howmany balloons were needed. Some students evenbecame creative and decided to use “partial bal-loons.” (See fig. 2.)

The Valentine’s Day balloon was left in class for aweek to stimulate informal dialogue as studentstried to lift various objects off the floor. The stu-dents tried pencils, markers, scissors, chalk, andother things to see if the balloon would win the tug-of-war with gravity for that item. Informal discus-sions related to the balloon’s pulling power oc-curred throughout the week, revealing students’continued interest in the idea.

Continuing the Adventure

WE DISCUSSED HELIUM BALLOONS ONE DAY Aweek for the next three weeks. We held whole-classdiscussions, then asked students to work in pairs tosolve subsequent problems. On the second day, webrought in two more balloons, one medium and onesmall. We assumed that the Valentine’s Day balloonwas similar to one of Walters’s and could lift 4pounds, so we assigned the other two balloons’pulling powers of 2 and 1 pounds based on theirsizes. Students often complain about lifting theirheavy backpacks, so we discussed the balloons’ actu-ally doing some work by lightening their loads. Thestudents weighed their backpacks, sketched thebackpack on a scale with the weight displayed, thensketched the backpack with a correct combination ofballoons attached so that the scale would read 0.After students had solved the problem of lifting theirown backpacks, they worked as a group to lift all thebackpacks at their tables. Each student was then as-signed to write a story using balloons to help themdo a household chore (see fig. 3). At this point, wehad not labeled the balloons with numbers—theywere simply labeled B (big), M (medium), and S(small). But the students did know the pulling powerof B as 4 pounds, M as 2 pounds, and S as 1 pound asthey used this knowledge to help with their back-pack problems and in their stories.

The third day we discussed showing the bal-loons’ pulling power with numbers rather than let-ters. We asked what they knew about negative num-bers and briefly discussed temperatures below zeroand net worth if more money was owed than was onhand. We suggested using negative numbers to de-scribe a balloon’s pulling power, since it had the op-posite effect of gravity, and students had alwayswritten weight using positive numbers. The stu-Fig. 3 Amy S. uses the balloons to help her take out the garbage.

Fig. 2 Phuc shows his weight and that it takes 19 3/4 balloons to balancethe tug of gravity.

VOL. 9, NO. 9 . MAY 2004 479

dents then completed several problems in whichthey attached balloons to lift objects and this timelabeled the balloons –4, –2, and –1. Students thenwrote addition number sentences under their draw-ings to show how a scale might read 0.

Eventually, we added to our supply of balloons sothat students could consider scenarios that used dif-ferent negative numbers other than –4, –2, and –1.Students solved several similar problems and wrotecorresponding number sentences—see figure 4.We also gave them some problems involving famil-iar objects in which they had to determine the nega-tive number for the balloons. For example, in figure5 a dog that weighed 27 pounds was shown on theleft, and the same dog with two balloons attachedwas shown weighing 13 pounds. The students hadto label the balloons, then label both the balloonsand the weight of the dog with three balloons at-tached. After completing a set of problems such asthis, the students were ready to put their newlydemonstrated knowledge to work. We announcedthe upcoming “Great Cartoon Contest” in which he-lium balloons would play a featured role.

The Great Cartoon Contest

THE FOLLOWING WEEK, THE CARTOON IN FIGURE6 was chosen to introduce using balloons in a humor-ous setting. The students could see in the left sketch

Fig. 4 Phuc and Sam attach balloons and write number sentences for twoscenarios with a bunny.

Fig. 5 An example of a problem setting in which students had to label the balloonsfirst, then solve another problem with the object and the same balloons

Fig. 6 A Close to Home cartoon used to stimulate interest inthe Great Cartoon Contest

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of figure 7 that the toddler weighed 18 pounds—theyhad to label the balloon on the right so that heliumwon the tug-of-war by –2 and show –2 on the scale.“Raising the American flag,” a familiar task for stu-

dents, was chosen as the subject of future cartoons.Students entered the contest in pairs and had a weekto design their entry. They had to show a flag beingweighed on a scale, then show the flag being raisedby a proper combination of balloons. They also had tocreate a funny caption. The results revealed their nu-merical understandings but also that many studentshad well-developed senses of humor. The cartooncontest was a huge success and the winning team,Monica and Stacey, became the proud owners of theValentine’s Day balloon (see fig. 8 for their cartoon).

Time Passes

WE PLANNED A FINAL DAY ON HELIUM BALLOONSin May to see what the students had retained.Memorial Day was approaching, so holiday bal-loons were easy to acquire. Three such balloonswere brought to class, and students reviewed whatthey had learned about negative numbers. We as-signed the balloons’ pulling powers of –6, –4, and –1based on their estimated sizes. We planned anothercartoon contest, this time allowing students tochoose any subject they wanted, but they had to de-cide to lift someone either slowly or quickly off theground. For example, Santa Claus might want to goup the chimney quickly after delivering toys to ahouse; on the other hand, a ghost at Halloweenmight want to hover over the ground, rising gradu-ally. We agreed that if you wanted to get awayquickly, then helium should beat gravity by –5, andif you wanted to rise slowly, then helium should winby –1. The students were again to work in pairs andtheir entry was due in one week, complete with cor-rect number sentences. A completed cartoon isshown in figure 9.

One unexpected event occurred as students triedto lift objects that required a large number of the –6balloons. For example, Mishal and Melody wantedto lift a 270-pound Santa slowly, requiring forty-five–6 balloons and a –1 balloon. We casually mentionedthat they could use multiplication rather than writeout all of the –6s in their number sentence. Inchecking the papers, two student pairs used themultiplication sign correctly even though it wasonly mentioned in passing. Mishal and Melodywrote this number sentence for lifting Santa slowlyup the chimney: 270 + 45 × –6 + –1 = –1.

The wrap-up class discussion and second car-toon contest were successful in that students re-membered what we had done several months ear-lier and were still very much interested in the topic.The use of negative numbers to describe their car-toons seemed to come naturally.

Their last assignment was to respond to twoquestions. The questions and one student pair’s re-

Fig. 7 The child from the Close to Home cartoon was assigned a weight of 18,and students solved a balloon problem with helium winning by –2.

Fig. 8 Monica and Stacey’s winning cartoon entry

VOL. 9, NO. 9 . MAY 2004 481

sponses, with their spelling and language intact, areincluded below:

• What have you learned about numbers with neg-ative signs in front of them? How are they likeregular numbers, and how are they different?“They are like regular numbers because theyhave the same numbers its meaning is just thesame its just that there is a mines sing in front.They are different because the negative signs areunder the regular number zero and that’s whythey have a mines sign in front.”

• What do you think about what we have done withhelium balloons? What have you learned? “WhatI learned is about negative and positive things. Ireally like doing this because we learned, wonthings, and had fun at the same time.”

Extensions for the Future

WE DID NOT GO FURTHER THAN ADDING INTEGERSwith this class, except for the casual mention ofmultiplication described above. However, it is inter-esting to contemplate the next steps provided bythis model. Subtracting negative numbers wouldcome from the original story of Larry Walters anddiscussing what happened when he popped the bal-loons one-by-one with his pellet gun. This is thestandard interpretation of subtraction with wholenumbers—remove them or “take them away”—soremoving negative numbers has a positive effect(i.e., Larry’s total package floating around the skyhad an increased weight of +4 when each balloonwas popped, so subtracting –4 has the same effectas adding +4).

Multiplication of a positive times a negative wasbriefly mentioned, and several members of theclass learned it quite easily—+45 × –6 simply meansadding forty-five helium balloons of –6 each, produc-

ing an overall change of –270. Multiplication of anegative times a negative would be repeatedly re-moving several balloons with the same weight, aswhen Walters had popped ten of his balloons withhis pellet gun to descend to the ground (i.e., –10 × –4= – –4 – –4 – –4 – –4 – –4 – –4 – –4 – –4 – –4 – –4 = +4 ++4 + +4 + +4 + +4 + +4 + +4 + +4 + +4 + +4 = +40). The ef-fect on the tug-of-war would be for gravity to win by+40, so a negative times a negative is positive.

Multiplication of a negative times a positivewould be interpreted as removing several objectsWalters was carrying on his flight that all had thesame positive weight. In the old days, hot-air bal-loonists would carry sandbags so that if theywanted to ascend even higher, they could dropsome weight. If Larry Walters had carried tenbooks for “light reading” during his trip, and eachbook weighed 1/2 pound, what would happen if hedropped them all at some point? Mathematically,the effect would be for helium to win the tug-of-warby –10 × 1/2, or –5, and he would ascend. A negativetimes a positive is therefore negative.

Fig. 9 An example of a cartoon written and drawn by students that wasentered in the final contest

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TS R

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VED


Conclusion

OUR MAIN OBJECTIVE WAS ACCOMPLISHED—introducing integer arithmetic in a problem-solvingexploration that piqued the interest of students. Thenumber sentences came naturally to students in ex-pressing the relationship between gravity and he-lium, and they could explain to us exactly what thenumbers represented in the real world. Over time,of course, we would move them from this single in-terpretation of negative numbers into a more gen-eral context, such as temperature, financial worth,and so forth.

As teachers, we should not be satisfied to simplyshow that integer arithmetic is reasonable. The ex-ploration described here was conceptual in nature,and problem solving was at its core. Students un-derstood, explained, applied, and retained whatthey had learned. A difficult-to-teach topic had beentransformed into one that was interesting to teach,

one that students could not get enough of. Weagree with the Standards and with Kent—contextu-alized, problem-based situations do enable studentsto make sense of mathematical ideas. We shouldemploy them whenever possible in our curriculum.

References

Kent, Laura Brinker. “Innovations in Curriculum: Con-necting Integers to Meaningful Contexts.” Mathemat-ics Teaching in the Middle School 6 (September 2000):62–66.

Kieran, Carolyn, and Louise Chalouh. “Prealgebra: TheTransition from Arithmetic to Algebra.” In ResearchIdeas for the Classroom: Middle Grades Mathematics,edited by Douglas T. Owens, pp. 179–98. Reston, Va.:National Council of Teachers of Mathematics, 1992.

National Council of Teachers of Mathematics (NCTM).Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. �

HandoutBush et al., 2014Handout6EEAlgebraMagic3CarlaBenAlicePicturesInstructions38Think of a number.21914Add 6.42Multiply by 2.26Subtract 2.2013Divide by 2.5Subtract your original number.FranEliDejPicturesInstructions7Think of a number.18Multiply by 6.5026Add 8.2530910Divide by 3.IanHiroshiGiamPicturesInstructionsThink…AbbreviationDescription of PicturesPicturesInstructionsa bucketThink of a number.b + 6a bucket and 6Add 6.2b + 12Multiply by 2.Subtract 2.Divide by 2.Subtract your original number.AbbreviationDescription of PicturesPicturesInstructionsa bucketThink of a number.b + 2Multiply by 3.2b + 6Divide by 2.3

Handout6EECoverUp3Handout6EEStripDiagrams3HandoutWhitacre, et al., 2014Reeves & Webb, 2004

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