Multiplication and Division Arrays and Strategies

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012

Multiplication and Division Arrays and Strategies

Common Core Leadership in Mathematics (CCLM^2)Tuesday, July 10, 2012

This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission.


Learning Intention & Success CriteriaWe are learning to…

– Make connections between the distributive property, the use of arrays, and the area model for multiplication.

– Understand how “applying properties of the operations” leads to fluency with single-digit multiplication and division.

We will be successful when…– We can explain and give examples of CCSSM expectations

for fluency with single-digit multiplication and division.– Extend the application of the distributive property to larger

numbers.


3.OA.5 & 3.OA.7

3.OA.5 Apply properties of operations to multiply and divide.

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of the operations.

What properties will students be asked to use and apply as they multiply and divide?


Scaffolding to CCSSM Expectations for Multiplication and Division Facts

• Jigsaw “Levels in problem representation and solution” p. 25-26 OA Progressions

Level 1, Level 2, Level 3

Be prepared to teach your partners:• Summarize key ideas of your Level and

use 3 x 8 to demonstrate the thinking.


Thinking back to Monday…Breaking Apart a 3x4 Array

• What are the ways a 3 x 4 array can be broken apart? – Split rows or columns, not individual tiles

• (3x3) + (3x1)• (3x2) + (3x2)• (2x4) + (1x4)


(3x3) + (3x1)

(3x2) + (3x2)


Level 3: Distributive Property6x8

• Pretend you don’t know the answer to 6x8• Sketch a 6x8 array on grid paper.• Partition the array into no more than 2

easier problems.• Keep track of the way you partitioned it.

– Use numbers and highlighters to keep track of how you partitioned your arrays.


6 x 8


3.OA.5 & 3.OA.7

3.OA.5 Apply properties of operations to multiply and divide.

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of the operations.

In what ways do we engage students in 3.OA.5 and 3.OA.7 as we work with “breaking apart a hard fact” into smaller, easier to solve facts?


CCSSM Fluency Expectations: Single-Digit Multiplication and

Division


Third Grade Fluency1st viewing:

Listen for the fluency demonstrated by the students.

2nd viewing:Listen and note how the students use the distributive property to fluently get their answer.

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


Fluency Expectations

Read the 1st paragraph on p. 27 of your OA Progressions.

Share with your partner 1-2 ideas you have highlighted that contribute to your thinking around fluency with single-digit multiplication and division.


True or False ??

13 9 = 90 + 2713 9 = 130 – 13

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

Group• Share your thinking.


1.Select a facilitator.2.Facilitator pulls out an equation strip and

shares it with group.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.


Extending to larger numbers3.NBT.3, 4.NBT.5, 5.NBT.6


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?

• Cut an array to represent 8 15.• Partition the array into two smaller arrays.• Label each partial product directly on your array.• Write an equation that matches how you partitioned your

array.

8 15 = _______________________


27 x 34

The chairs in the auditorium are arranged in 27 rows with 34 chairs in each row. How many chairs are in the auditorium.

• Sketch an array to represent 27 34.• Partition into smaller arrays using benchmarks of 10.• Label each partial product directly on your array.• Write an equation that matches how you partitioned

your array.

27 34 = _____________________


7 x 30 7 x 4

Write the partial product for each array and calculate the total.

600 = 20 x 30 (Step 1)80 = 20 x 4 (Step 2)210 = 7 x 30 (Step 3)

28 = 7 x 4 (Step 3)918

20 x 3020 x 4

27 x 34

20

7

30 4This is commonly call the Partial Product Algorithm. Why?


Division Strategies


Standard 3.OA.5Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

What is the relationship between these two standards? Compare and contrast them.


Standard 3.OA.7• Fluently multiply and divide within 100, using

strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.


What Strategies Should Children Know for Division?


What Strategies Should Children Know for Division?Unknown Factor Problems

‘Think Multiplication’ when you know the multiplication fact.

Use Strategies.

35 ÷ 5 = ?5 x ? = 35Answer is 7

Use ‘Count-bys’35÷ 5 = ?Count by 5, 10, 15, 20, 25, 30, 35 … 7 counts

Use easier facts35 ÷ 5 = ?5 x ? = 35 Use easier facts5 x 5 = 25 and 5 x 2 = 1025 + 10 = 35 so 5 + 2 =7


Revisiting Math Practices

As a table group, revisit SMP 3: Construct Viable Argument and Critique the Reasoning of Others and SMP 7: Look for Structure

Divide your slate in half. On one side, show how we were engaged in SMP3 and on the other side show SMP7.


Learning IntentionsWe are learning to…• Understand two types of division situations.• Learn division strategies that develop fluency in

basic division facts.

We will be successful when we can …• Articulate the meaning of third grade standards

that focus on division


Review 3.MD.7c How does 3.MD.7c support this work? Using

your slate, draw a model to explain this standard.


Taking a look back: 2.OA.4• Divide your white board in half.• On one side rephrase the standard in your

own words. On the other side provide an example.

In what way does 2.OA.4 prepare students for Level 2 solution strategies in 3rd grade?

Documents

Multiplication and Division Arrays and Strategies