Algebra Journey: Generalized Properties with a focus on the Distributive Property
Math Teacher Leader Seminar
January 2006
Prepared by Melissa Hedges, Sharonda Harris, & DeAnn Huinker
Big Idea #1: How is the distributive property related to Algebra or Algebraic Thinking?
Represent and analyze mathematical situations and structures using algebraic symbols (PSSM, p.160)
Elements of algebraic reasoning as generalized number properties:
 Commutative Property
 Associative Property
 Distributive Property
 Properties of 1
 Properties of 0
Activity 1a: Review of Relational Thinking
13 x 9 = 90 + 27 and 13 x 9 = 130 – 13
1. Individually review each equation above and decide if each is true or false. Make notes to help you
keep track of your thinking.
2. Share your thinking as a table group.
3. You might remember from the last time we met that we spent some time thinking about relational
thinking versus computational thinking. For example, last time we looked at ways you might
approach 8 + 4 =  + 5 from a computational approach or a relational thinking approach. (Gather
examples from the group for both approaches.)
4. Now, at your tables decide what approaches were used in reasoning about the equations with 13 x 9.
Did individuals in your group use a computational strategy or a relational thinking strategy? If you
discover that everyone at your table used the computational approach, how could you approach this
using relational thinking?
Overhead for this activity:
13 x 9 = 90 + 27 and 13 x 9 = 130 – 13
1. Individually review each equation above and decide if each is true or false. Make notes
to help you keep track of your thinking.
2. Share your thinking as a table group.
3. Now, at your tables decide what approaches were used; either a computational strategy
or a relational thinking strategy. If you discover that everyone at your table used the
computational approach, how could you approach this using relational thinking?
Highlight ways the distributive property was used for the two equations.
13 x 9 = (10 + 3) x 9 = 10 x 9 + 3 x 9 = 90 + 27
13 x 9 = 13 x (10 – 1) = (13 x 10) – (13 x 1) = 130 – 13
Milwaukee Mathematics Partnership (MMP), January 2006
Activity 1b: True/False Statements
As a table group, pull each equation out one by one, and explain why the statement is either true or
false using relational thinking – not computing to get the answer. Make note to keep track of your
thinking. You may use computational thinking to check your work but this should not be your first
approach.
1. Select a facilitator
2. Pull out an equation.
3. Individually decide if the statement is true or false Keep track of your thinking.
4. Facilitator asks each person to share their decision and the reason behind it.
Debrief by highlighting one or two problems – either discussing one or two of the false problems or the
true problems.
Problems in the envelope include:
3x8=2x8+8
8 x 6 + 8 x 5 + 6 False
7x7+7x1 = 7 x 8
6 x 9 = 5 x 9+1 x 9
3 x 8 +7 x 8 = 21 x 8 False
8 x 6 = 8 x 8-8 x 2
13 x 7 = 70 + 27
6x7=6x6+7
Big Idea #2: Finding relations among numbers that make learning number facts easier and “set the
foundation for algebra” and the “algebraic character of working with mathematics.”
Activity 2a. Examining Student Reasoning
♦
1st Viewing: Watch and listen for the relational thinking demonstrated by the students
♦
2nd Viewing: Listen for a 2nd time and highlight how the students used the distributive property
to explain their relational thinking
♦
Finally, using the script as a reference, write equivalent expressions that highlight the
distributive property.
What might this look like? Let’s think about 6 x 9. If one of Meghan’s students solved 6 x 9 like this:
5 x 9 = 45 and one more 9 gives me 54. How would this look using the distributive property? 6 x 9 =
(5 + 1) x 9 = 5 x 9 + 1 x 9. The last one is what we want to see in this portion. Keep in mind the
meaning of the equal sign.
Give participants time to work.
Then review the equations written by selecting one or two to highlight.
Milwaukee Mathematics Partnership (MMP), January 2006
Activity 2b (Segue to Activity 3): Partitioning Arrays for 6 x 8
Let’s take a look at 6 x 8. How might we visually represent 6 x 8 so we can “see” the distributive
property in 5 x 8 + 1 x 8? An array will help us represent this very clearly.
Demonstrate partitioning the 6 x 8 array on the document camera or chart paper. Put up an array that
represents 6 x 8. Partition 6 x 8 into two partial products. Write out the equation for the partial
products in a way that highlights the distributive property. Repeat with another way to partition 6 x 8.
Big Idea #3: Making the implicit use of the distributive property in solving multiplication problems
explicit by building and partitioning arrays and writing related equations. Thus setting the
foundation for algebra by moving toward a generalized understanding of the distributive property.
Materials: Grid paper & scissors
Purpose: To get participants thinking about how to use arrays or area models to illustrate the
distributive property and to guide them into choosing benchmarks of 10 for partitioning the arrays.
Activity 3a: Partitioning an 8 x 15 Array into Two Partial Products
We’ve now spent some time looking at single-digit multiplication but we really see the distributive
property highlighted when working with larger numbers.
Start with 8 15. Use overhead to introduce 8 x 15. Using grid paper, demonstrate how we can
decompose 8 x 15 into 2 smaller problems that demonstrate use of the distributive property. Work on
this individually for a few minutes. After some individual work time share your thinking with the
group. An important question to ask might be: How did you know when you had the answer? (This
alludes to the idea of partial products.) See overhead for specific instructions.
Activity 3b: Partitioning a 23 x 34 Array Using Benchmarks of Ten
Move on to 23  34 (as a table group). Use grid paper to show how you may use benchmarks of 10 to
solve this problem by finding partial products. See overhead for specific instructions.
Activity 3c: Partitioning a 32 x 48 Array into Four Partial Products
Move on to 32 x 48 (as a table group). Use grid paper to show how you may use benchmarks of 10 to
partition the array into four partial products. See overhead for specific instructions.
Activity 4. Closing: True or False?
Decide if the following statement is true or false. Turn to your neighbor and explain why you believe
what you do.
Is this statement True or False?
Use relational thinking to reason about these equations.
25  46 = (20  40) + (5  6)
Turn to your neighbor. Explain your reasoning.
Milwaukee Mathematics Partnership (MMP), January 2006