Growing Algebra Eyes and Ears

Using relational thinking to bring meaning to operations

Connie Laughlin

Kevin McLeod Beth Schefelker

Mary Mooney Melissa Hedges

Session Goals:

  • Building on the understanding of how algebraic reasoning surfaces from computational experiences.
  • Using relational thinking to understand arithmetic

Gr. 1 MPS CABS

True and False Number Sentences

For each number sentence, tell whether it is true or false.

a. 9 = 5 + 4True or False

b. 9 = 9True or False

c. 5+4 = 4+5True or False

d. 5 + 4 = 5 + 3True or False

e. 5 + 4 = 6 + 3True or False

For the letter “e” above, how do you know?

Part 2:

Responses to “What is the mathematics in this CABS?”

  1. Equality is an important concept, an equal sign does not mean problem on left answer on right. It represents balance not an end.
  2. It must align to the 1st grade targets and descriptors
  3. To be sure the student can write a CR as to why an answer is incorrect or correct.
  4. To be sure the student is reading all of the content in the problem.
  5. The CABS is assessing student understanding of equal and of various representations of the same quantity.
  6. It’s a tool to help evaluate students’ thinking about the relationship between numbers.

About the Mathematics

Falkner, K., Levi, L., & Carpenter, T. 1999. Children’s Understanding of Equality: A Foundation for Algebra. Teaching Children Mathematics, pp. 232 – 236.

  1. Read the paragraph.
  1. Highlight one sentence out of this portion that you would like your teachers to understand about the mathematics? Be prepared to share this one sentence with your table group.
  1. Share your one sentence with your table group.
  1. Come to a table consensus around one idea from this paragraph.

Mathematical Ideas

Highlighted by Falkner

  • Equality is the relationship that expresses the idea that two mathematical expressions hold the same value.
  • Understanding equality allows for representationand communicationof arithmetic ideas.
  • Equality is a relationship rather than a signal to do something.

Looking at First Grade Work

How does the student work connect with these three statements?

MKT CABS

Solve these problems in 2 different ways:

72 – 8 = 234 – 96=

Describe in words why your method works.

The strategies are sent under a separate file.

These strategies are sent under a separate file.

Task:

Peter is subtracting 5 from some numbers. Peter says that these are quite easy to do. Do you agree?

37 – 5 = 32

59 – 5 = 54

86 – 5 =81

But Peter says that some others are not so easy, like these:

32 – 5 =

53 – 5 =

84 – 5 =

What might make these harder than the previous problems?

Peter says. “I do these by first adding 5 and then subtracting 10. Working it out this way is easier.”

What is his method? Discuss with your shoulder partner why his method works?

Peter’s Method

32 – 5 = 32 + (10 – 5) – 10

32 – 5 = 32 + 5 – 10

32 – 5 = 37 – 10

32 – 5 = 27

Try Peter’s Method with 84 – 5

84 – 5

84 – 5 = 84 + (10 – 5) – 10

84 – 5 = 84 + 5 – 10

84 – 5 = 89 – 10

84 – 5 = 79

Turn and verbalize Peter’s Method.

Each of you take a turn.

Extending Peter’s Method

Try his method for

73 – 6

Once you have an idea of what he might be doing justify his strategy with these problems.

32 – 6

63 – 6

Extend his method to subtracting 100

Try 173 – 94

Verbalize Peter’s Method to your partner

Each of you take a turn.

Once you and your partner are comfortable…

Make up your own problems and give them a try using Peter’s Method.

Why Does Peter’s Method Work?

Peter recognizes it is easy to subtract 10 or to subtract 100. (Why is this?)

If Peter has a “hard” subtraction problem, he changes it into an easier one by reasoning algebraically:

73 – 6 = 73 + (10 – 6) – 10

This equation is true because of its algebraic form, and not because of the particular numbers involved—it is equally true that

73 – 8 = 73 + (10 – 8) – 10

or that

141 – 4 = 141 + (10 – 4) - 10

Why Does Peter’s Method Work?

A recognition that certain equations are true because of their form, and not because of the specific numbers involved, is sometimes called quasi-variable thinking, and is an important step towards formal algebra.

In algebra, we would express the fact that the equation

73 – 6 = 73 + (10 – 6) – 10

remains true when 6 is replaced by any number by saying that, for any number b,

73 – b = 73 + (10 – b) – 10

We might also replace 73 with any number a:

a – b = a + (10 – b) – 10

Why Does Peter’s Method Work?

Thinking about the extension of Peter’s method to subtracting 2-digit numbers, we might even replace the 10 with any number c:

a – b = a + (c – b) – c

Students in early grades should not be exposed to such an explicit use of variables, but teachers should understand the connections between relational thinking, quasi-variable thinking, and formal algebra.

Session Goal:

Building on the understanding of how algebraic reasoning surfaces from computational experiences.

What does this mean?

This entailslooking at expressions and equations in their entirety rather than as processes to be carried out step by step, and seeing the relational and algebraic properties underlying them.

Some final thoughts….

  • Children need to think about the relationships expressed by number sentences (7+8=7+7+1)
  • By integrating the teaching of arithmetic with the teaching of algebra teachers can help children increase their understanding of arithmetic at the same time that they learn algebraic concepts.

Mathematical Knowledge for Teaching

Computational Fluency

Accuracy

Flexibility

Efficiency

If you are going to value student thinking you have to understand how students think.

NCTM: Communication Standard: Analyzing and evaluate the mathematical thinking and strategies of others

Try a new problem on your CABS

35 – 19

504 – 397

Can you apply Peter’s Method as you work these problems?