Lesson Plan

Associative and Commutative Properties of Multiplication

Objectives

This lesson focuses on applying the associative and commutative properties to simplify and solve multiplication problems. Students will learn to apply these properties to support them in solving multiplication problems more efficiently and with greater ease. Students will:

  • apply the associative and commutative properties to simplify and solve multiplication problems.

Essential Questions

How are relationships represented mathematically?
How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?
How can mathematics support effective communication?
How is mathematics used to quantify, compare, represent, and model numbers?
  • How is mathematics used to quantify, compare, represent, and model numbers?
  • How can mathematics support effective communication?
  • How are relationships represented mathematically?
  • How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?

Vocabulary

  • Division: The mathematical operation of splitting a quantity into equal groups. (For example, 8 ÷ 2 = 4 because splitting 8 into 2 equal groups results in 2 groups of 4.)
  • Equation: A statement of equality between two mathematical expressions.
  • Factor: A number that is multiplied with another number to form a product.

Duration

60–90 minutes

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

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Related Materials & Resources

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Formative Assessment

Suggested Instructional Supports

  • View
    Scaffolding, Active Engagement, Modeling, Explicit Instruction, Formative Assessment
    W: Students will learn to apply the commutative and associative properties of multiplication to simplify mental computations. 
    H: Color tiles will be used to investigate the commutative property of multiplication for two whole numbers. Miniature whiteboards and calculators will then be used to engage students in investigating the commutative and associative properties, as they relate to the multiplication of three whole numbers. 
    E: Students will experience the commutative property of multiplication visually using color tiles. They will examine and explore the commutative and associative properties using whiteboards and calculators. Students will then focus on making mental computations more quickly using these properties. 
    R: Students will review the commutative and associative properties of multiplication as they complete the Make It Easier practice worksheet. This will be completed in class and requires students to apply the properties to make computations easier. 
    E: Students will be evaluated based on their performance on the Make It Easier practice worksheet. Students will also be evaluated using the Lesson 3 Exit Ticket. 
    T: The lesson may be adapted to meet the needs of students by using the suggestions in the Extension section. Specific suggestions are provided for students who could use extra practice in learning to apply the commutative and associative properties. The Expansion section provides additional challenges for students who have moved beyond the requirements of the standard. 
    O: The focus of the lesson is for students to learn to use the commutative and associative properties of multiplication to simplify computations. The lesson is scaffolded in that students first learn about each of the properties, and then they apply the properties to simplify mental computations.  

Instructional Procedures

  • View

    Commutative Property—Multiplication

    Introduce the problem, “Juanita says 6 × 3 = 18 and 3 × 6 = 18. Juanita’s little brother Fabio doesn’t understand. Let’s help Juanita explain this to Fabio.” Ask students to work in pairs. Be sure each pair has at least 40 color tiles—20 each of two colors.

    “How do we read 6 × 3 = 18?” (6 groups of 3) Use one color of tiles to build 6 groups of 3, as shown.

     

    “How do we read 3 × 6 = 18?” (3 groups of 6) Use the other color of tiles to build 3 groups of 6, as shown.

     

    Rotate the blue rectangle to show that it is congruent (equal) to the green rectangle.

    “Now, we have used color tiles to show Fabio that you can multiply 6 × 3 or 3 × 6  and the product will be the same, 18. But, Fabio now wants to know if you can multiply three numbers in any order and get the same product.”

    Distribute at least one calculator to each pair of students. Distribute miniature whiteboards to five students. Ask two students to write a multiplication sign × on their whiteboards. Ask the other three students to write a number between 1 and 10 on their whiteboards.

    Ask the pairs of students to compute the product using a calculator. For example, if the five miniature whiteboards read 8 × 5 × 6, students should compute this using a calculator. (The product is 240.) Ask one student to write the number sentence or equation on the board, as 8 × 5 × 6 = 240. Now ask students holding the whiteboards with numbers on them to move around to form a new number sentence, such as 6 × 8 × 5. Ask students to again compute the product using calculators. Ask another student to write the number sentence or equation on the board, as 6 × 8 × 5 = 240.

    Ask students holding the whiteboards to move to create another new number sentence, such as 5 × 6 × 8. Ask students to compute the product using calculators, and ask a student to write the number sentence on the board.

    Ask a new group of five students to hold the whiteboards. Again, ask two students to write a multiplication sign × on their whiteboards. Ask the other three students to write a number between 1 and 10 on their whiteboards. Repeat the activity, asking students to find the product of at least two different multiplication number sentences using the three numbers written in different orders.

    Introduce the formal language of commutative property, although you should not require students to know this term. “Multiplication of numbers can be completed in any order and the product will always be the same. Mathematicians say multiplication is commutative, which means it can be done in any order. Why is this important? We will investigate the answer to that question very soon, but first we are going to look at another property of multiplication.”

    Associative Property—Multiplication

    Distribute the miniature whiteboards to five different students. Ask two students to write a multiplication sign  on their whiteboards. Ask the other three students to write a number between 1 and 10 on their whiteboards. Draw one parenthesis on each of the first two whiteboards with numbers on them (e.g., (7 × 5) × 4).

    Ask the pairs of students to compute the product using a calculator. Be sure to help students learn to compute within the parentheses first. In the example (7 × 5) × 4, ask students to compute 7 × 5 first. Ask students holding the whiteboards with (7 × 5) to step aside. Ask one student to write 35 on another whiteboard and hold it in the place of the quantity (7 × 5). Write the simplified version of the number sentence 35 × 4 on the board. Now ask students to compute: 35 × 4 = 140.

    Ask the original five students to return to their positions. Draw parentheses around the last two numbers instead (e.g., 7 × (5 × 4)). Ask students to compute within the parentheses first. Ask one student to hold another whiteboard to represent the product and record the simplified version of the number sentence 7 × 20 on the board. Now ask students to compute: 7 × 20 = 140.

    Ask a new group of five students to hold the whiteboards. Again, ask two students to write a multiplication sign  on their whiteboards. Ask the other three students to write a number between 1 and 10 on their whiteboards. Repeat the activity, asking students to find the product of at least two different multiplication number sentences using the three numbers written and using parentheses to group different sets of numbers.

    Introduce the formal language of associative property, although you should not require students to know this term. “Multiplication of numbers will result in the same product even if the numbers are grouped together in different ways. Mathematicians say multiplication is associative, which means the numbers can be regrouped. Why is this important? We will investigate the answer to that question now!”

    Mental Math—Applying the Properties

    “Juanita and Fabio were both doing their homework. The problem was 6 × 2 × 5. Juanita said 60 quickly, without using a calculator. Fabio said it is equal to 12 × 5, but he was still working. How do you think Juanita multiplied these three numbers so quickly?” Students will likely recognize that Juanita multiplied 2 and 5 first, and then multiplied by 6. These properties allow students to compute more quickly and accurately. This is the reason students should know how to apply these properties, but the standards clearly do not require students to know the names of the properties.

    Present two more examples for class discussion. These examples will prepare students for the Make It Easier practice worksheet.

    Write 4 × 9 × 2 on the board. Ask students, “How can we rewrite this number sentence to make it easier to multiply mentally?” Students may suggest different number sentences, depending on what multiplication facts they find to be the easiest to compute. Ask one student to rewrite the number sentence on the board. Compare the relative ease of the computations for both number sentences.

    Most students will likely recognize that 4 × 2 × 9 is generally easier to compute than 4 × 9 × 2. Multiplying the numbers from left to right, 4 × 2 × 9 becomes 8 × 9 whereas 4 × 9 × 2 becomes 36 × 2. Both products are 72.

    Write (8 × 6) × 5 on the board. “This expression uses parentheses, ( ). In math, parentheses are used to group a certain part of a number sentence together—separate from the other parts. The parentheses also show us which part of the number sentence we are supposed to calculate first. For example, the way the number sentence is written now, we are supposed to multiply  first, and then multiply this product by 5. But we can make it easier! How can we rewrite this number sentence to make it easier to multiply mentally?” Ask one student to rewrite the number sentence on the board. Compare the relative ease of the computations for both number sentences.

    Most students will likely recognize that (8 × 5) × 6 is generally easier to compute than (8 × 6) × 5. Multiplying the numbers within the parentheses first, (8 × 5) × 6  becomes 40 × 6 whereas (8 × 6) × 5 becomes 48 × 5. Both products are 240.

    Distribute a copy of the Make It Easier worksheet (M-3-5-3_Make It Easier and KEY.docx) to all students. Remind students to use the properties to make the multiplication quick and easy to do mentally. Sample solutions are provided.

    To further evaluate student mastery of lesson concepts, use the Commutative and Associative Practice sheet (M-3-5-3_Commutative and Associative Practice and KEY.docx).

    Extension:

    The Routine section provides suggestions for reviewing lesson concepts throughout the year. The Small Group section is intended for students who would benefit from additional instruction or practice. The Expansion section includes ideas for challenging students who are prepared to move beyond the requirements of the standard.

    • Routine: To help students review the use of these operations, continue to emphasize their utility in performing computations mentally. Do not encourage students to reach for a calculator to multiply 5 × 24 × 2 or (5 × 9) × 6. Instead encourage students to find an “easier” way to compute these, such as 5 × 2 × 24 and (5 × 6) × 9. These properties can also be quite helpful as students learn to multiply two-digit and three-digit numbers.
    • Small Group: Students who need additional practice may be pulled into small groups to work on additional problems that utilize these properties. The focus should be on supporting students in identifying how to make the computation easier. Be sure to explicitly discuss the why of each solution. Use the miniature whiteboards to complete these problems, encouraging students to propose how to move the whiteboards to make the computation easier. (Since there is only a small group, lay the mini-whiteboards on the table instead of having students hold them.) Additional problems are provided on the More Examples practice worksheet (M-3-5-3_More Examples and KEY.docx). If the teacher is not available to help the small group, students may find additional instruction on their own at the following Web site.

    http://www.coolmath.com/prealgebra/06-properties/02-properties-commutative-multiplication-02.htm

    • Expansion: Students prepared for a greater challenge should work in groups of two or three to play the following game focused on commutative, associative, distributive, and multiplicative identity properties. A description of each property is provided. http://www.aaamath.com/pro74b-propertiesmult.html

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Final 06/07/2013
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