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Algebraic Thinking Problems: Grades K-2
Solve the problems in each set. You may work alone, with a partner, or in small groups.
Problem Set 1: Understanding Equality
1.1. What is the mathematics underlying the concept of equality? That is, what would you want students
to say if you asked, “What does it mean for two things to be equal?”
1.2. What do we want students to understand about the equal sign (=)?
1.3. Carpenter, Franke, and Levi (2003, p. 9) report data about students’ responses to the question below.
What do you notice in the data? What conclusions can you draw?
What number would you put in the box to make this a true number sentence?
8+4=+5
Response/Percent Responding
Grade
7
12
17
5
58
13
1 and 2
9
49
25
3 and 4
2
76
21
5 and 6
12 and 17
8
10
2
1.4. Write two or three learning targets for equality and the equal sign. Be as precise as possible; that is,
what do you want students to know about the equal sign? Where in the K-8 Mathematics Standards
do these ideas appear?
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K-2 Content Professional Development
Print date: June 10, 2008
page 1
Problem Set 2: Number Relationships
2.1. Which of these number sentences are true, and which are false? Explain your thinking.
a. 8 + 7 = 15
b. 15 = 8 + 7
c. 8 + 7 = 8 + 7
d. 8 + 7 = 7 + 8
e. 8 + 7 = 9 + 6
f. 8 + 7 = 87
g. 8 = 8
2.2. What number(s) could go in the box to make each number sentence true? Explain your thinking.
a. 8 + 7 = 
b. 8 + 7 =  + 7
c.  + 7 = 8 + 7
d.  + 8 = 8 + 7
e. 15 = 
f.  = 7
g. 39 + 57 =  + 59
h.  + 82 = 143 + 89
Continue work on the next page.
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2.3. What number(s) could be substituted for N to make each number sentence true? Explain your
thinking.
a. 8 + 7 = N
b. 8 + 7 = N + 7
c. N + 7 = 8 + 7
d. N + 8 = 8 + 7
e. 15 = N
f. N = 7
g. 39 + 57 = N + 59
h. N + 82 = 143 + 89
2.4. Design a sequence of true/false and/or open number sentences that you might use to engage your
students in thinking about the equal sign. Describe why you selected the problems you did.
(Carpenter, et al., 2003, p. 24)
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Problem Set 3: Making Number Sentences True
3.1. In each number sentence, what number(s) could be substituted for the variable to make that number
sentence true? Explain your thinking.
a. 6 + 9 = 8 + 10 + d
b. 9 - 6 = 8 - 4 + g
c. 10 - 6 = 8 - 4 + a
d. 5 + 8 + d = 6 + 9 + d
e. 5 + 8 - d = 6 + 9 - d
f. 5 + 8 + d + d = 6 + 9 + d
g. 5 + 8 - d - d = 6 + 9 - d
h. 5 + 8 - d = 6 + 9 - d - d
3.2. a. Solve: d + d + d - 20 = 16
b. Look at video 5.1 (Carpenter, et al., 2003). Focus your attention on the student’s strategy. Does the
student’s strategy illustrate algebraic thinking?
Continue work on the next page.
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3.3. a. Solve: k + k + 13 = k + 20
b. Look at video 5.2 (Carpenter, et al., 2003). Focus your attention on the student’s strategy. Does the
student’s strategy illustrate algebraic thinking?
3.4. How are the strategies used in videos 5.1 and 5.2 alike? How are they different?
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Problem Set 4: Relational Thinking
Make an argument that this equation is true,
WITHOUT computing each sum:
25 + 17 = 24 + 18
Make an argument that this equation is false,
WITHOUT computing each difference:
25 - 17 = 24 - 18
4.1. The following number sentences are intended to encourage relational thinking (for adults). The
particular numbers chosen are intended to discourage the use of calculating rather than mathematical
reasoning. Which number sentences are true and which are false? Justify your answers. Try to make
a justification that does not involve computing the sums or differences.
a. 3,765 + 2,987 = 3,565 + 3,187
b. 4,013 – 2,333 = 4,043 – 2,363
c. 8,041 – 3,762 = 8,051 – 3,752
d. 5,328 + 3,933 = 8,328 + 933
e. 6,789 – 6,345 = 789 - 345
4.2. Rank the following problems from easiest to most difficult (for students). Justify your choices.
(Carpenter, et al., 2003, p. 41) Hint: You may want to solve these problems first.
a. 73 + 56 = 71 + d
b. 92 – 57 = g - 56
c. 68 + b = 57 + 69
d. 56 – 23 = f - 25
e. 96 + 67 = 67 + p
f. 87 + 45 = y + 46
g. 74 – 37 = 75 - q
Continue work on the next page.
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4.3. Decide whether each number sentence below is true or false. Justify your choices. How do you think
students would justify the choices? (Carpenter, et al., 2003, p. 42)
a. 56 = 50 + 6
b. 87 = 7 + 80
c. 93 = 9 + 30
d. 94 = 80 + 14
e. 94 = 70 + 24
f. 246 = 24 x 10 + 6
g. 47 + 38 = 40 + 30 + 7 + 8
h. 78 + 24 = 98 + 4
i. 63 – 28 = 60 – 20 – 3 – 8
j. 63 – 28 = 60 – 20 + 3 – 8
4.4. Create a set of problems that might encourage students to use relational thinking. Be ready to explain
the grade-level of your problems and justify your choices of numbers.
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Problem Set 5: Properties of Operations
5.1. Make four groups with each group exploring one operation.
a. Explore the properties of addition. That is, what number sentences can you write using variables
that illustrate important properties of addition? What special numbers are there that are helpful in
understanding these properties?
b. Explore the properties of subtraction. That is, what number sentences can you write using
variables that illustrate important properties of subtraction? What special numbers are there that
are helpful in understanding these properties?
c. Explore the properties of multiplication. That is, what number sentences can you write using
variables that illustrate important properties of multiplication? What special numbers are there
that are helpful in understanding these properties?
d. Explore the properties of division. That is, what number sentences can you write using variables
that illustrate important properties of division? What special numbers are there that are helpful in
understanding these properties?
5.2. Describe the commutative and associative properties for addition. Represent these properties using
symbols. Do the same for multiplication. Can you do the same for subtraction and division? Why or
why not?
Continue work on the next page.
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5.3. Read this equation aloud using words rather than symbols:
a + b = (a + 1) + (b - 1)
What mathematical idea does this equation represent? Is the equation true or false? Explain your
answer.
5.4. Look at video 3.3 (Carpenter, et al., 2003). Focus your attention on the strategies the student uses.
What, if anything, do you think the student learned during this interview? What problem would you
pose to check your hypothesis?
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Problem Set 6: Justification and Proof
6.1. True or false:
a - b - c = a - (b + c). Justify your answer.
6.2. If you have 5 sodas and each person gets half a soda, how many people will get to drink soda?
True or false: N ÷ 1/2 = 2 x N
True or false: N ÷ 1/3 = 3 x N
Justify your answer.
Justify your answer.
6.3. Look at video 7.2. Focus your attention on the student’s explanations. What do you think this
student understands about proof? How do the interviewer’s questions help reveal what the student
knows?
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Problem Set 7: What Happens and Why?
7.1. True or false? Without computing each quotient, explain your choices.
a. 87 ÷ 5 and 7 ÷ 5 have the same remainder
b. 876 ÷ 5 and 6 ÷ 5 have the same remainder
c. 895 ÷ 5 and 5 ÷ 5 have the same remainder
d. If abc represents a 3-digit number, abc ÷ 5 and c ÷ 5 have the same remainder
e. What rule can you state for divisibility by 5? Justify the rule.
7.2. True or false? Without computing each quotient, explain your choices.
a. 65 ÷ 3 and (6 + 5) ÷ 3 have the same remainder
b. 652 ÷ 3 and (6 + 5 + 2) ÷ 3 have the same remainder
c. 651 ÷ 3 and (6 + 5 + 1) ÷ 3 have the same remainder
d. If abc represents a 3-digit number, abc ÷ 3 and (a + b + c) ÷ 3 have the same remainder
e. What rule can you state for divisibility by 3? Justify the rule.
Continue work on the next page.
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7.3. Take any 3 digits (not all the same!!) and make the greatest and least 3-digit numbers. Subtract the
lesser from the greater to make the high-low difference. Repeat this process for that difference. Keep
on repeating the process. What happens? Why? (Driscoll, 1999, p. 77)
7.4. Take a three-digit number (with digits not all the same!!), reverse its digits, and subtract the lesser
from the greater. Reverse the digits of the results and add these two numbers.
132 becomes 231, and 231 - 132 = 99 = 099
099 becomes 990, and 099 + 990 = 1089
Try this process for several numbers. What happens? Why? (Driscoll, 1999, p. 59)
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Problem Set 8: Representations
8.1. Look at video 7.1 and video 8.1. Focus your attention on the students’ representations. How are the
two representations of odd numbers alike? How are they different? Which representation is more
convincing? What assumptions is each student making?
8.2. How could you use variables to represent an even number?
An odd number?
A multiple of 5?
8.3. Suppose that when N is divided by 3 the remainder is 1, and that when P is divided by 3 the
remainder is 2. What is the remainder of N + P when you divide it by 3? What is the remainder
of N - P when you divide it by 3?
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Problem Set 9: Patterns and Conjectures
9.1. What do you notice about each pair of products below? What happens? Why?
6 x 6 and 5 x 7
30 x 30 and 29 x 31
500 x 500 and 499 x 501
N x N and (N - 1) x (N + 1)
9.2. Guess these products. Then check your guesses.
300 x 300 and 298 x 302
300 x 300 and 295 x 305
N x N and (N - a) x (N + a)
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Problem Set 10: Reflecting on Your Thinking
10.1. What did you learn (or re-learn) from working on these problems? How did the videos help you
understand the mathematics ideas?
10.2. Look back over the problems. Which ones could you use directly with children? Which ones could
you adapt for use with children?
10.3. Where do these problems “fit” in the Mathematics Standards? Which problems might you share
with other teachers in your school? Why would those problems be important ones to share?
References
Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and
algebra in elementary school. Portsmouth, NH: Heinemann.
Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers: Grades 6-10. Portsmouth, NH:
Heinemann.
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