Algebra Journey: Generalized Properties

Distributive Property: Part 2

Developed by:Melissa Hedges, Sharonda Harris, DeAnn Huinker, Henry Kepner, Kevin McLeod, & Connie LaughlinUniversity of Wisconsin-Milwaukee

February 2006

This material is based upon work supported by the National Science Foundation under Grant No. 0314898. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Algebra Journey

“Children have a great deal of implicit knowledge about fundamental properties

in mathematics, but it usually is not a regular part of mathematics class to

make that knowledge explicit.”

---Carpenter, Franke, & Levi, 2000, p. 47

Key Questions• Will this work with all numbers?• Will this work with other operations?• Why is this working?

Session Goals

◊ Making “implicit” reasoning “explicit” as part of algebraic thinking.

◊ Generalizing the distributive property as an essential foundation for algebra.

1. Select a facilitator.

2. Facilitator pulls out a card and shares it with the group.

3. Decide if the statements are true or false. Share decisions and reasoning.

4. Reflect on the set of statements,• summarize the underlying idea

that surfaces, and• form a conjecture about the

mathematical principle.

Read p. 47& Skim p. 54-55

in “Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School”

Discuss

“Children do not often have an opportunity to articulate and examine

these (implicit) ideas.” (p. 47)

In what ways might you provide more opportunities for your students?

Algebraic Habit of Mind

Abstracting from Computation

The capacity to think about computations independently or “freed from” the particular numbers that are used.

Guiding Questions

How can I predict what is going to happen without doing all the calculation work? (relational thinking)

What holds true when I do the same thing to different numbers? Is that always true? (making conjectures and generalizations)

What are other ways to write the expression to highlight underlying mathematical principles? (equivalence)

---Fostering Algebraic Thinking (Driscoll, 1999, p. 2)

General Approach for the Next Tasks

1. Work on the problem individually.

2. Discuss reasoning and confusions as a table group.

3. Select a Recorder. This person charts the group’s response.

4. Pass the marker to a new Recorder for the next task.

43 • 52

1. Draw an open array (i.e., rectangle) to represent 43 • 52.

2. Partition the array to show the four partial products for (40 + 3)(50 + 2).

3. Label each partition.

4. Write an equation that shows the use of the distributive property.

43 • 52 =

4 43 • 5 3

2

Do not change to an improper fraction!

1. Draw an open array for 443 • 53

2

2. Partition to show four partial products.

3. Label each partition.

4. Write an equation that shows the use of the distributive property.

4 43 • 5 3

2 =

(4x + 3) • (5x + 2)

1.Represent using an open array. 2. Partition to show four partial products.

3. Label each partition.

4. Write an equation that shows the use of the distributive property.

(4x + 3) • (5x + 2) =

• Select a new Recorder.

• Discuss as a group:

(1) What are you noticing across these tasks?

(2) What might be some relationships among the arrays, the distributive property, and the standard algorithm?

• Record on chart paper, 3 key ideas from the group discussion.

Read pp. 114–116

in “Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School”

Guided reading: New idea

! In agreement with the idea ? Idea you question

Discuss: In what ways can arithmetic work support students’ future work in algebra?

Focus discussion on:

• Distributive property

• Other number properties

• Role of conjectures

True or False?

Use relational thinking to decide if these equations are True or False.

(2x + 5)(7x + 6) = (2x 7x) + (5 6)

(3x + 1)(8x + 2) = (3x + 1)8x + (3x + 1)2

(x + 4) 2 = x 2 + 16

Turn to your neighbor. Explain your reasoning for each statement.

Big Ideas of Algebra

PropertiesFor a given set of numbers, there are relationships that are always true, and these are the rules that govern arithmetic and algebra.

EquivalenceAny number, expression, or equation can be represented in different ways that have the same value.