Kim Malacara
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
CCSSM Analysis of Standard 3.OA.5 Part 1: Standard 3.OA.5 Apply 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 of multiplication.) 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 of multiplication.) 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.) Part 2: Overall Explanation Standard 3.OA.5 refers to the properties that apply to multiplication and division. These properties are important to third grade multiplication and division development, as well as subsequent grade levels, because they explain the reasoning behind “why” certain strategies work mathematically. The more teachers understand these three properties themselves, the more they can guide their students in using the properties when navigating through strategies when working with number. Explanation and Example of Each Property Explanation: Examples and Ideas for Classroom Practice: The commutative property states that when multiplying a series of Students may use the commutative property when they are asked numbers, the order of the numbers being multiplied doesn’t matter. to find the product in a multiplication problem such as 5x7 (5 groups (3x4=4x3) of 7). They may be more familiar with finding 7x5 (7 groups of 5), so they will switch the problem around to solve it. Teachers should encourage using the commutative property when students are learning their facts and they are using skip counting as a strategy. Students are usually more familiar with skip counting by certain numbers such as 2s, 5s, and 10s, so should be encouraged to switch the problem around to work with these easier numbers. The associative property states that when multiplying, certain Students may utilize the associative property when they are numbers can be associated with other numbers to make multiplying multiplying numbers they are more familiar with first to make an easier, resulting in the same answer. ( 3x4x5=12x5) easier problem. Teachers should encourage the associative property when students are ready for level 3 strategies that involve converting to an easier, equivalent problem. For example, when a student encounters 3x5x2, they may multiply 5x2 to get 10 and then multiply 10x3 to get 30 for an answer. The student may be more familiar with multiplying by 10 and this may make the problem easier for students. However, teachers need to encourage this type of flexible thinking when having mathematical discussions in the classrooms, whether explicitly or implicitly. The distributive property, which relates to both multiplication and division, states that numbers can be broken down into smaller, more familiar numbers (whose sum equals the original numbers) to make it easier to arrive at an answer. {7x6=7(5+1)=35+7} Students may use the distributive property when working with larger numbers in multiplication and division. For example, a student may break up a larger number into smaller, easier numbers in order to solve them problem. (36x4 could be broken down into the following: 4(10+10+10+6)=40+40+40+24. Part 3: Alignment to Expressions 2nd Grade Unit 13 Lesson 4 Introduction of Arrays-uses the language of switching arrays
(commutative property) 3x4=4x3
Activity 2 “My Garden”-Students work with rows and columns
and which number represents which in a multiplication sentence.
Lays groundwork for the properties in third grade and beyond. Unit 13 Lesson 5 Students work with arrays. Commutative property is explicitly taught within lesson. 3rd Grade Unit 7 Lesson 3 Commutative property explicitly taught in an array lesson-array
model is used to show that multiplication is commutative.
Activity 4 directs students to switch the order of the factors and
points out that the product won’t change (shows with arrays).
Students explore the commutative property and write equations
that symbolize the property. Students are also asked to draw the
arrays that match the commutative property. The TE tells
teachers to encourage students to use this relationship to make
more difficult problems easier for them to solve (Ex. 5x7 (5
groups of 7) to 7x5 (7 groups of 5-because I can count by 5s more
efficiently). P.484
Multiplication and Area
Students are finding smaller areas within a larger area model.
(Distributive Property)
For example, 3x7 is the same as 3x5 and 3x2.
Unit 7 Lesson 10 Unit 7 Lesson 12 Not explicitly in lesson, but lists as an alternative approach-to use
doubles (ex. 4x6 is the same as 2x6 and 2x6 or 2(2x6), which is
the distributive property.
Also, lists the using 5’s strategy for multiplication. 7x4 is the
same as 5x4+4+4, which is 28.
Explicitly introduces the Associative Property related to arrayssuggests having students come up to the board and show answers.
Suggests asking students which numbers should be multiplied
first, second, etc. Also, relates addition, multiplication, and
division.
Class Activity 7.14 clearly outlines the commutative and
associative properties and gives examples and tells teachers to
discuss whether they are true for division, also.
Worksheet asks students to use the properties and then asks
students to write an equation to match the property (including the
zero property for multiplication).
Unit 7 Lesson 14 4th Grade Unit 1 Lesson 2 Activity 4 relates to the Distributive Property (have them see 9’s
as 10’s minus 1 group)-1x9 is 1 less than 1x10, so 2x9 is 2 less
than 2x10. Also, in Activity 5 they relate this strategy to division
(63/9 and 63=70-7. So, using the 10s minus 1s patterns, we know
that 63=7x9, or equivalently 63/9=7).
Arrays and Commutative Property
Students are looking at groups of objects commutatively and then
relating to an array that is arranged in both directions showing the
commutative property.
Student Activity Book P. 13 uses the term commutative and asks
students to explain the term in relation to arrays and equations.
Unit 1 Lesson 3 Unit 1 Lesson 5 Activity 2 asks students to use the distributive property indirectly
by asking them to see 5x3 in 7x3. If you know 5x3=15, then you
can count on two more groups of three to get 15+3=18 and
18+3=21, so 7x3=21. They refer to this as the “close count-by
shortcut.” On p. 46 they ask for whole class discussions on what
familiar number facts are close to unknown facts.
Students are asked to find related equations using a factor
triangle. They are encouraged to use the term commutative
property when discussing relationships.
P. 76&77 they are encouraged to write fast arrays using the
commutative property and then they are asked to write about the
relationships of multiplication and division in an open response
activity on p. 77.
Teacher discusses 6’s and has students see the relationship of 5’s
to 6’s (distributive property). They are asked to describe how the
6’s relate to the 5’s shortcut idea.
Activity 2 p. 90 describes three multiplication strategies using the
distributive property.
Students use 7’s and 8’s to find relationships of unknown factors
to known factors (distributive property).
Explicitly teaches the commutative, associative, and distributive
properties. The terms are used when working with the students,
as well as being explained and practiced in relation to
multiplication strategies. Students are asked to explain their
thinking, as well as use various models (arrays, drawings).
Unit 1 Lesson 9 Unit 1 Lesson 11 Unit 1 Lesson 12 Unit 1 Lesson 16 Unit 5 Lesson 1 Multiplication Array-reviews commutative property and utilizes
distributive property related to large arrays broken down into
smaller arrays.
Unit 5 Lesson 2 Reviews multiplication with tens-utilizes the associative and
distributive property (2x3, 2x1x3x1, 6x1)
Unit 7 Lesson 2 The Area Model for Multiplication
Teaches students to see larger numbers broken into place value
using the area model to make easier problems that are added
together when multiplied separately (distributive property).
Multiplying Multi-digit Numbers
Students use the area model to multiply two-digit numbers by
breaking them up into place value making them easier numbers to
multiply (distributive property).
Multiply with Larger Numbers-relates prior distributive property
methods to larger numbers.
Applying strategies to division problems-students can utilize the
break-apart methods (distributive) to find divisors that are a part
of a dividend.
Relate Fractional Relationships
Teachers discuss the idea of multiplying fractions and the
commutative property and show with area models that it doesn’t
matter which order you multiply fractions.
Division as Reverse Multiplication when multiplying fractionsthe lesson shows the idea of the associative property when
dividing fractions-divide the numerator and denominator
separately and you will get the same answer (axc/bxd divided by
c/d is the same as axc/c divided by bxd/d, which is a/b).
Reviewing operations with fractions- utilize the associative
property when reviewing the flexibility of working with fractions.
Unit 7 Lesson 3 Unit 7 Lesson 4 Unit 7 Lesson 12 Unit 9 Lesson 6 Unit 9 Lesson 11 Unit 9 Lesson 14 Conclusions and Suggestions: All three properties from Standard 3.OA.5 are taught within the 3rd grade Expressions curriculum. Only the commutative property is brought up in 2nd grade, and all three properties are extended more fully in 4th grade. Although all three properties are introduced and utilized in 3rd grade, students should experience multiple opportunities applying the properties within contextual situations. Contextual situations refer to providing a context for students to “think” about the numbers. This can be done by continually relating numbers to a situation, whether it is days of the week, groups of objects, or another applicable situation. In addition to offering multiple experiences and contextual situations, students should connect to the properties through the use of visual models. Expressions offers multiple experiences using arrays. However, when students work with arrays, they should strategically be led from more concrete models to more abstract models. When working with arrays, students should use tiles or blocks to form arrays, as well as utilize base-­‐10 blocks to work with arrays utilizing tens. After students have had ample experiences constructing arrays through the use of blocks or tiles, they can move on to forming arrays on grid paper. Array grid paper is available in larger and smaller squares, and in a more structured 100x100 square design to a more open grid paper format. Again, students should move strategically through the use of a more structured grid paper to eventually using open arrays, which are basically drawing a rectangular array divided up into sections according to equivalent, easier to work with numbers. This open array is not drawn to proportion, but is merely a quick model to assist students through their organization of the numbers. Another activity that would encourage the use of the properties of multiplication is “Quick Images.” This activity involves flashing a shape or group of objects up and having students describe what they see. Students utilize the properties through their formation of how they “see the objects. For example, an image of three groups of seven objects could be seen as 3(6+1), which is the distributive property. Quick image activities should be integrated with drawing images on paper and connecting mathematical equations or expressions to these images. Teachers should explicitly pull in the properties and connect to students’ strategies when discussing Quick Images. In summary, in order for students in 3rd grade Expressions to have ample opportunities experiencing the three properties of multiplication, the 3rd grade lessons could be extended to include some of the ideas above or some of the 4th grade lessons could be utilized within 3rd grade.