2008 May 29Algebra: Grades K-2: slide 1 Welcome to Grades K-2 Mathematics Content Sessions The focus is Algebraic Thinking. The goal is to help you understand.

2008 May 29 Algebra: Grades K-2: slide 1

Welcome to Grades K-2 Mathematics Content Sessions

The focus is Algebraic Thinking.

The goal is to help you understand this mathematics better to support your implementation of the K-8 Mathematics Standards.


Importance of Algebraic Thinking

Algebraic thinking includes understanding and use of properties of numbers and relationships among numbers.

Algebraic Thinking was chosen as the content focus because it lays the foundation for the learning of algebra in middle school and high school.

It takes most students a long time to develop algebraic thinking, so it is important to begin this work in the primary grades.


Introduction of Facilitators

• <Insert Facilitator Names Here>


Introduction of Participants

In a minute or two:1. Introduce yourself.

2. Describe an important moment in your life that contributed to your becoming a mathematics educator.

3. Describe a moment in which you hit a “mathematical wall” and had to struggle with learning.


Overview

Some of the problems may be appropriate for students to complete, but other problems are intended ONLY for you as teachers.

As you work the problems, think about how you might adapt them for the students you teach.

Also, think about what Performance Expectations these problems might exemplify.


Problem Set 1

The focus of Problem Set 1 is understanding equality.

You may work alone or with colleagues to solve these problems.

When you are done, share your solutions with others.


Problem 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?”


Problem 1.2

What do we want students to understand about the equal sign (=)?


Problem 1.3

Carpenter, Franke, and Levi (2003, p. 9) report data (shown on the next slide) showing students’ responses to the question below.

What number would you put in the box to

make this a true number sentence?

8 + 4 = + 5

What do you notice in the data? What conclusions can you draw?


Problem 1.3

What number would you put in the box to make this a true number sentence?

8 + 4 = + 5

What do you notice in the data? What conclusions can you draw?

Response/Percent Responding Grade 7 12 17 12 and 17

1 and 2 5 58 13 8 3 and 4 9 49 25 10 5 and 6 2 76 21 2


Problem 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?


Problem Set 2

The focus of Problem Set 2 is number relationships.




Problem 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


Problem 2.2What 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


Problem 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


Problem 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.


Reflection

What did you learn (or re-learn) from solving these problems?


The Standards uses the word, equation, instead of the phrase, number sentence. What is the difference in these two terms?


Problem Set 3

The focus of Problem Set 3 is making number sentences true.

You may work alone or with colleagues to solve the problems in set 3.1.



Problem 3.1In 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


Problem 3.1In each number sentence, what number(s) could be substituted for the variable to make that number sentence true? Explain your thinking.

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


Problem 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?


Problem 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?


Problem 3.4

How are the strategies used in videos 5.1 and 5.2 alike?

How are they different?


Reflection

How might your students solve these or similar problems? What strategies might they use?

How could you adapt these problems for your teaching?



Problem Set 4

The focus of Problem Set 4 is relational thinking.




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


Problem 4.1

Which number sentences are true and which are false? Justify your answers.

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


Problem 4.2

Rank the following problem from easiest to most difficult (for students). Justify your choices.

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


Problem 4.3

Decide whether each number sentence below is true or false. Justify your choices. How do you think students would justify the choices?

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


Problem 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.


Reflection

What is relational thinking?

Why is relational thinking important for students to be able to do?



Problem Set 5

The focus of Problem Set 5 is properties of operations.




Problem 5.1

Make four groups with each group exploring one operation.

a. Explore the properties of addition.

b. Explore the properties of subtraction.

c. Explore the properties of multiplication.

d. Explore the properties of division.


Problem 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?


Problem 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.


Problem 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?


Reflection

What do students need to know about the properties of operations?

How can you help students learn those properties?



Welcome to Grades K-2 Mathematics Content Sessions

The focus is Algebraic Thinking.

The goal is to help you understand this mathematics better to support your implementation of the K-8 Mathematics Standards.


Problem Set 6

The focus of Problem Set 6 is justification and proof.




Problem 6.1

True or false: a - b - c = a - (b + c).

Justify your answer.


Problem 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



Problem 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?


Reflection

What do you look for in a child’s argument about whether something is true or false?

How sophisticated can you expect children’s arguments to be?



Problem Set 7

The focus of Problem Set 7 is what happens and why.




Problem 7.1

True or false? Explain your answers.

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.


Problem 7.2

True or false? Explain your answers.

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.


Problem 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?


Problem 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 result 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?


Reflection

Why is knowledge of divisibility important for students to know?

In K-2, the ideas of odd and even are important for children to learn. How are those ideas related to divisibility rules?



Problem Set 8

The focus of Problem Set 8 is representations.




Problem 8.1

Look at video 7.1 and video 8.1.

Focus your attention on 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?


Problem 8.2

How could you use variables to represent an even number?

An odd number?

A multiple of 5?


Problem 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?


Reflection

How can representations help students reason?

How can we help students learn to use representations to clarify their thinking?



Problem Set 9

The focus of Problem Set 1 is patterns and conjectures.




Problem 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)


Problem 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)


Reflection

What kinds of patterns might you ask K-2 students to explore?



Problem Set 10

The focus of Problem Set 1 is reflection on your thinking.




Problem 10.1

What did you learn (or re-learn) from working on these problems?

How did the videos help you understand the mathematics ideas?


Problem 10.2

Look back over the problems.

Which ones could you use directly with children?

Which ones could you adapt for use with children?


Problem 10.3

Where do these problems “fit” in the K-8 Mathematics Standards?

Which problems might you share with other teachers in your school? Why would those problems be important ones to share?


Closing Comments

Implementing the K-8 Mathematics Standards will require a deep focus of mathematics ideas at each grade.

Personal understanding of these ideas will make the implementation process easier.

2008 May 29Algebra: Grades K-2: slide 1 Welcome to Grades K-2 Mathematics Content Sessions The focus is Algebraic Thinking. The goal is to help you understand.

Documents

d slide

learning of algebra

primary grades

focus of problem

true number sentence

number sentence true

number sentences true

number relationships