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Algebra Journey: Generalized Properties

“Distributive Property”

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

January 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

Goals

◊ Examine and further strengthen “relational thinking.”

◊ Explore “generalized properties” as connections between arithmetic and algebra.

Milwaukee Mathematics Partnership, January 2006

Big Ideas of Algebra

EquivalenceAny expression or equation can be represented in equivalent ways.

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


True or False ??

13 9 = 90 + 27

13 9 = 130 – 13

IndividuallyDecide if the statement is true or false.Make notes to keep track of your thinking.

GroupShare your thinking.


13 9 = 90 + 27

13 9 = 130 – 13

What approaches were used? computational strategy relational thinking strategy

Be prepared to share a computational approach and a relational thinking approach for each equation.


8 + 4 = + 5

Computational approach

Compute the “answer” of 12 and reason from there.

Relational thinking approach

Reason using relationships and properties and do not need to find the “answer” of 12.

“5 is 1 more than 4, so the number in the box must be 1 less than 8.”


1. Select a facilitator.

2. Facilitator pulls out an equation strip and shares it with group.

3. Individually decide if the statementis true or false. Keep track of your thinking.

4. Facilitator asks each person to share their decision and the reason behind it.


True or False ???

6 7 = 6 6 + 7

3 8 = 2 8 + 8

8 6 = 8 5 + 6

7 7 + 7 1 = 7 8

6 9 = 5 9 + 1 9

3 8 + 7 8 = 21 8

8 6 = 8 8 – 8 2

13 7 = 70 + 27


1st viewing: Listen for the relational thinking demonstrated by the students.

2nd viewing:Listen and note how the students use the distributive property to explain their relational thinking.

Finally, using the script as a reference, write an expression that corresponds to each student’s reasoning and that highlights the distributive property.


8 6


8 15

The second graders at Clemens School are performing a play for their families. The chairs in the gym are arranged in 8 rows with 15 chairs in each row. How many chairs are there?

1. Cut an array to represent 8 15.

2. Partition the array into two smaller arrays.

3. Label each partial product directly on your array.

4. Write an equation that matches how you partitioned your array.

8 15 =


23 34

The chairs at La Escuela Fratney are set up for the Bring Books Alive program. There are 23 rows of chairs with 34 chairs in each row. How many chairs are set up for the program?

1. Cut an array to represent 23 34.2. Partition into smaller arrays using

benchmarks of 10.3. Label each partial product directly on

your array.4. Write an equation that matches how

you partitioned your array.

23 34 =


32 48The Marcus Center is preparing for a performance by the Milwaukee Symphony Orchestra. There are 32 rows of chairs in the amphitheater with 48 chairs in each row. How many chairs are there all together?

1. Cut an array to represent 32 48.2. Partition array using benchmarks of

10. End up with only 4 smaller arrays.3. Label each partial product directly on

your array.4. Write an equation that matches how

you partitioned your array.32 48 =


Self Assessment

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.


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