CCSSM Interpretation Guide 3 · DRAFT&DOCUMENT,&UNEDITEDCOPY.This"materialwas"developed"forthe"CommonCore%LeadershipinMathematics"(CCLM) project"at"the"University"of"Wisconsin>Milwaukee."""(07.15.2011)"

DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin-‐Milwaukee. (07.15.2011)

Lori Collenburg CCLM Project 2

July 15, 2011

CCSSM Interpretation Guide

Operations and Algebraic Thinking 3.0A Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) The properties

include the Commutative property of multiplication, Associative property of multiplication and the Distributive property.

In the domain of Operations and Algebraic Thinking, this standard is for third grade and deals with knowing the different properties of multiplication while using them as an approach in seeing how multiplication and division work together.

The Commutative property of multiplication is known by other student friendly terms such as the ‘swap over’, the ‘flip flop’ or the ‘switcheroo’. Commute means to exchange. Changing the order of the factors does not change the product. Therefore, 6 X 4 = 24 is the same as 4 X 6 = 24.

The Associative property of multiplication is about grouping factors. This property states that when multiplying more than two numbers the grouping of the factors does not change the product. Therefore, 3 X 5 X 2 could be grouped as (3 X 5) X 2 or 3 X (5 X 2).

The Distributive property is a special way multiplication is used with the addition of two or more numbers. The numbers being added are usually in parentheses. This property helps students break numbers into more familiar facts or benchmark numbers. Therefore, 8 X 7 could be broken down into 8 X 5 + 8 X 2 this is the same as 8 X (5 + 2).


Points to remember…

• Students do not need use the formal words for each of the properties, they just need to understand how to use each as a strategy.

• Levels of thinking for multiplication o Level 1 – counting all quantities o Level 2 – repeated counting by a given number o Level 3 – using the associative or distributive properties to compose

• Types of models for multiplication: o Arrays – one factor tells the number of rows and one number tells the

number of columns (Three rows of desks and each row has 5 desks) o Equal Groups – how many things are shared equally in how many groups.

(3 people each have 5 cookies) o Compare – how much more (if I have 3 marbles and you have 5 times

more than me…..) o Measurement – using a number line to use repeated addition or

decompose numbers to solve • Unitizing – grouping – the unit you are working with • Subitizing – the quantity

o Perceptual – just seeing the quantity o Conceptual – seeing the parts that make the quantity

• Vocabulary is very important and needs to be consistent. o Informal – what you see o Concept based language – groups of….. o Formal – equations, expressions, discussions

• The strategy of using the properties allows students to understand the meaning of multiplication or division rather just fact recall.

• Students need to know how to read an equation or expression to understand what property is appropriate. What operations are involved?

• Classroom discussion is important to identify different student reasoning using different properties.

• Organize practice so it focuses on understanding of strategies or properties. • These properties help in having fluency with multiplication facts. • Multiplication and Division can be taught at the same time. • Mastery is time consuming. • A variety of unknowns –

o Unknown product o Unknown group size/factor o Unknown number of groups/factor


Scott Foresman Textbooks Development: First Grade:

• Skip counting by 2, 5, 10 • Finding patterns of skip counting on a one hundred chart.

Second Grade: In the last chapter, multiplication and division is addressed.

• Skip counting • Repeated addition • Building Arrays • Multiplication – order of factors (Commutative Property) • Drawing a picture • Division – making equal groups

Third Grade:

• Repeated Addition - Using repeated addition to show using equal groups. • Arrays – rows of objects and columns of the objects (Commutative

Property) • Some real world problems representing a variety of multiplication problems. • Learn to write real world problems • Learn the factors as 2, 5, 10, 9, 3, 4, 6, 7,8 • Use comparison to find the size of a group. • Recognize patterns on a multiplication fact table. • Division as sharing • Repeated Subtraction • Multiplication and division practice using naked numbers • Associative Property for multiplication • Identity of Zero for multiplication • Identity of One for multiplication • Distributive Property (Chapter 11) • Estimating Products • Division is also presented using standard algorithms

Fourth Grade:

• Arrays, equal groups and number lines are being used to solve problems. • Commutative, Associative, Identity of Zero and One and Distributive

Properties.


• Multiplication and Division with both naked numbers and in word problems • Estimate products and multiply money using standard algorithms.

Fifth Grade:

• Associative, Commutative and Distributive Properties. • Identity of Zero and One Properties. • Multiplication and Division practice mostly with naked numbers.

Conclusions:

• In third grade, there is more theory of multiplication and practicing the facts using naked numbers than experiencing the use of arrays, equal groups and number lines. Fluency is dependent on the naked number practice.

• In third grade, there is little practice with using the properties of operations. • In fourth grade, there are more opportunities to use the properties of operations

and but practice is still more heavily weighted using naked numbers rather than real life word problems.

• In fifth grade, it appears to be a simple review for multiplication. There were no apparent opportunities to multiply decimals.

Recommendations:

• Use more word problems to practice the operation and the properties of operations in third grade and beyond.

• Use a variety of situations to facilitate use of different representations (arrays, equal groups, number lines and comparisons).

• Use different problems to represent different unknowns. • In first and second grade use more activities to develop the concepts of

unitizing and subitizing. (quick images, dot plates, etc.) • Specific activities are available by request


Formative Assessment - Commutative Property:

Scott and Lauren are making a brick path in the school garden. Scott places 6 rows of bricks with 2 bricks in each row. Lauren places 4 rows of bricks with 3 bricks in each row. Scott turns to Lauren and says, “Look! This is what we did in math. We’re using the Commutative Property of Multiplication.” Lauren disagrees. Who is right?

1. Draw an array to show how Scott placed his bricks. Write a multiplication equation for the array.

Equation: ___________________________________________________________

2. Draw an array to show how Lauren placed her bricks. Write a multiplication equation for the array.

Equation: ___________________________________________________________

3. Explain who is correct.

____________________________________________________________

____________________________________________________________

_________________________________________________________________



Scott and Lauren are making a brick path in the school garden. Scott places 6 rows of bricks with 2 bricks in each row. Lauren places 4 rows of bricks with 3 bricks in each row. Scott turns to Lauren and says, “Look! This is what we did in math. We’re using the Commutative Property of Multiplication.” Lauren disagrees. Who is right? 1. Draw an array to show how Scott placed his bricks. Write a multiplication equation for the array.

Equation: 6 X 2 = 12




Lauren is correct because even though they both have a product of 12, they used

different factors.



Scott and Lauren are making a brick path in the school garden. Scott places 6 rows of bricks with 2 bricks in each row. Lauren places 4 rows of bricks with 3 bricks in each row. Scott turns to Lauren and says, “Look! This is what we did in math. We’re using the Commutative Property of Multiplication.” Lauren disagrees. Who is right? 1. Draw an array to show how Scott placed his bricks. Write a multiplication equation for the array.





Lauren is correct because even though they both have a product of 12, they used

different factors.

This shows that the student understands how to build arrays to solve problems using the vocabulary of rows.

This shows the student understands the relationship of the array with the equation.

The student demonstrates the understanding of the commutative property as the order of factors not what the product is.


Formative Assessment - Associative Property:

If a bookseller in Chicago wraps every 2 books together in bubble wrap, and then packages them into small boxes that contain 2 bubble-wrap packs each, and then into cartons that will hold 6 small boxes, how many books will the bookseller send out in each carton?

1. Draw a picture to show the books in one carton.

2. Write the multiplication equation for the drawing.

_______________________________________________________________

3. How many books will the bookseller send out in each carton?

_______________________________________________________________



If a bookseller in Chicago wraps every 2 books together in bubble wrap, and then packages them into small boxes that contain 2 bubble-wrap packs each, and then into cartons that will hold 6 small boxes, how many books will the bookseller send out in each carton?



( 2 X 2 ) X 6 = 24


The bookseller sent out 24 books in each carton.



If a bookseller in Chicago wraps every 2 books together in bubble wrap, then packages them into small boxes that contain 2 bubble-wrap packs each, and then into cartons that will hold 6 small boxes, how many books will the bookseller send out in each carton?



( 2 X 2 ) X 6 = 24


The bookseller sent out 24 books in each carton.

The student’s drawing shows that 2 books were wrapped together and 2 sets of wrapped books were boxed. In all there 6 boxes in the carton. The student used grouping to solve the problem.

The equation reveals the student’s knowledge of the associative property and the grouping the problem presents.

The student shows an understanding that the question was answered and had the appropriate label.


Formative Assessment - Distributive Property

Tom and Julie want to find the product of 4 X 27. Tom uses breaking apart to solve the problem and says that 4 X 27 = (4 X 20) + (4 X 7). Julie uses breaking apart, and says that 4 X 27 = (4 X 25) + (4 X 2). Are they both correct, or not?

1. Solve Tom’s breaking apart equation.

2. Solve Julie’s breaking apart equation.


________________________________________________________

________________________________________________________

________________________________________________________





4 X 27 = (4 X 20) + (4 X 7)

4 X 27 = 80 + 28

4 X 27 = 108


4 X 27 = (4 X 25) + (4 X 2)

4 X 27 = 100 + 8

4 X 27 = 108


They are both right in how they solved the problem. Tom broke 27 into 20 + 7 and Julie broke 27 into 25 + 2. They used numbers they were comfortable with.





4 X 27 = (4 X 20) + (4 X 7)

4 X 27 = 80 + 28

4 X 27 = 108


4 X 27 = (4 X 25) + (4 X 2)

4 X 27 = 100 + 8

4 X 27 = 108


They are both right in how they solved the problem. Tom broke 27 into 20 + 7 and Julie broke 27 into 25 + 2. They used numbers they were comfortable with.

The student showed an understanding of using the algorithm to solve these break apart problems to make numbers more friendly

The student showed the understanding that the number 27 was broken down into appropriate addends.

Documents

CCSSM Interpretation Guide 3 · DRAFT&DOCUMENT,&UNEDITEDCOPY.This"materialwas"developed"forthe"CommonCore%LeadershipinMathematics"(CCLM) project"at"the"University"of"Wisconsin>Milwaukee."""(07.15.2011)"