<p>The main idea of this lesson is to simplify and resolve multiplication problems by using the associative and commutative properties. To&nbsp;help them solve multiplication problems more quickly and easily, students will learn to apply these properties. Students are going to:<br>- Use the commutative and associative qualities to simplify and solve multiplication problems.</p>

<p>- What are the mathematical representations for relationships?&nbsp;<br>- What are some applications for expressions, equations, and inequalities in the quantification, modeling, solving, and/or analysis of mathematical situations?&nbsp;<br>- How does effective communication benefit from mathematics?&nbsp;<br>- How do we represent, compare, quantify, and model numbers using mathematics?</p>

<p>- 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.)&nbsp;<br>- Equation: A statement of equality between two mathematical expressions.&nbsp;<br>- Factor: A number that is multiplied with another number to form a product.</p>

<p>- calculators&nbsp;<br>- miniature whiteboards and markers&nbsp;<br>- one copy of the Make It Easier worksheet (M-3-5-3_Make It Easier and KEY.docx) per student&nbsp;<br>- one copy of the Commutative and Associative Practice sheet (M-3-5-3_Commutative and Associative Practice and KEY.docx) per student&nbsp;<br>- one copy of the Lesson 3 Exit Ticket (M-3-5-3_Lesson 3 Exit Ticket and KEY.docx) per student&nbsp;<br>- More Examples practice worksheet (M-3-5-3_More Examples and KEY.docx)</p>

<p>- Students' ability to apply the commutative and associative properties of multiplication to simplify computations can be assessed using the Make It Easier worksheet (M-3-5-3_Make It Easier and KEY).&nbsp;<br>- Use the M-3-5-3_Commutative and Associative Practice and KEY&nbsp;worksheet to assess students' further comprehension of the lesson's concepts.&nbsp;<br>- Assess students' grasp of applying the commutative and associative properties of multiplication quickly by using the Lesson 3 Exit Ticket (M-3-5-3_Lesson 3 Exit Ticket and KEY).</p>

<p>Explicit instruction, modeling, scaffolding, active engagement, and formative assessment&nbsp;<br>W: They will be able to reduce mental calculations by using the commutative and associative properties of multiplication.&nbsp;<br>H: Using color tiles, we will examine whether the&nbsp;multiplication of two whole numbers is commutative. Next, students will investigate the commutative and associative properties related to the multiplication of three whole numbers using miniature whiteboards and calculators.&nbsp;<br>E: Using color tiles, students will be able to visually experience the commutative property of multiplication. Using calculators and whiteboards, they will investigate and analyze the commutative and associative properties. Subsequently, the students will concentrate on employing these properties to make mental calculations faster.&nbsp;<br>R: As they work through the Make It Easier practice worksheet, students will review the commutative and associative properties of multiplication. Students will apply the properties to simplify computations in order to complete this in class.&nbsp;<br>E: Based on their completion of the Make It Easier practice worksheet, students will receive an evaluation. Lesson 3 Exit Ticket will also be used for student evaluation.&nbsp;<br>T: You can use the ideas in the Extension section to modify the lesson to fit the needs of your students. We offer targeted recommendations to students who might benefit from additional practice in applying the associative and commutative properties. Students who have progressed beyond the standard requirements can find more challenges in the Expansion section.&nbsp;<br>O: The main goal of the lesson is to teach students how to make computations simpler by using the commutative and associative properties of multiplication. The way the lesson is structured is that students learn about each property first, and then they use it to make mental calculations easier.&nbsp;</p>

<p><strong>Commutative Property—Multiplication</strong><br><br>Introduce the problem: <strong>"Juanita claims 6 × 3 = 18 and 3 × 6 = 18. Fabio, Juanita's younger brother, doesn't understand. Let's help Juanita explain this to Fabio."</strong> Let students work in pairs. Make sure every pair has a minimum of 40 color tiles (20 of each of the two colors).<br><br><strong>"How can we calculate 6 × 3 = 18?"</strong> (<i>6 groups of 3</i>). As demonstrated, create six groups of three using a single color of tiles.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_1.jpeg" width="534" height="190"><br><br><strong>"How can we calculate 3 × 6 = 18?"</strong> (<i>3 groups of 6</i>). Build three groups of six using the other color of tiles, as indicated.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_2.jpeg" width="476" height="268"><br><br>Flip the green and blue rectangles over to show that they are congruent (equal).<br><br><strong>"Now, we've used color tiles to show Fabio that you can multiply 6 × 3 or 3 × 6 and the result will be the same, 18. However, Fabio now wants to know if you can multiply three numbers in any combination and get the same result."</strong><br><br>Give a pair of students at least one calculator. Give each of the&nbsp;five students a tiny whiteboard. Request that two students use their whiteboards to write the multiplication symbol ×. Ask the remaining three students to write a number on their whiteboards that ranges from 1 to 10.<br><br>Request that the students work in pairs to calculate the product with a calculator. For instance, students should use a calculator to calculate 8 × 5 × 6 if the five tiny whiteboards display that information. (The quantity is 240.) To write the equation or number sentence 8 × 5 × 6 = 240 on the board, assign one student to do so. Now instruct the students to arrange the numbers on the whiteboards to create a new number sentence, like 6 × 8 × 5. Have students use calculators to calculate the product once more. Ask a different student to write the equation or number sentence, which is 6 × 8 × 5 = 240, on the board.<br><br>To create a new number sentence, like 5 × 6 × 8, ask the students holding the whiteboards to move. Have a student write the number sentence on the board and instruct the other students to use calculators to calculate the result.<br><br>Assign the whiteboards to a fresh group of five students. Once more, instruct two students to write the multiplication symbol × on their whiteboards. Ask the remaining three students to write a number on their whiteboards that ranges from 1 to 10. Once again, use the three numbers written in different orders to have students find the product of at least two different multiplication number sentences.<br><br>Although students don't need to understand this term, introduce them to the formal language of <i>commutative property</i>. <strong>"Number multiplication can be done in any order, and the result will always be the same. Multiplexing numbers in any order can be done because mathematicians say it is </strong><i><strong>commutative</strong></i><strong>. Why does this matter? We're going to look into the answer to that question shortly, but first, let's examine another multiplication property."</strong>&nbsp;<br><br><strong>Associative Property - Multiplication</strong><br><br>Distribute the miniature whiteboards to five students. Ask two students to write the multiplication symbol on their whiteboards. Ask the other three students to write a number from 1 to 10 on their whiteboards. Draw a parenthesis on the first two whiteboards with numbers (e.g., (7 × 5) × 4).&nbsp;<br><br>Request that the students work in pairs to compute the product using a calculator. Make sure to teach students how to compute within parentheses first. In the example (7 × 5) × 4, ask students to calculate 7 × 5 first. Students holding whiteboards (7 × 5) should step aside. Ask one student to write 35 on another whiteboard and hold it in place of the quantity (7 × 5). Write a simpler version of the numerical statement 35 × 4 on the board. Ask students to calculate: 35 × 4 = 140.&nbsp;<br>Request that the original five students return to their respective positions. Use parentheses around the last two numbers (e.g., 7 × (5 × 4)). Ask students to start by computing within parentheses. Assign one student to hold another whiteboard to symbolize the product and write the simplified version of the number statement 7 × 20 on it. Ask students to compute 7 x 20 = 140.&nbsp;<br><br>Assign a new group of five students to hold the whiteboards. Again, have two students write a multiplication symbol on their whiteboards. Ask the other three students to write a number from 1 to 10 on their whiteboards. Repeat the assignment, instructing students to find the product of at least two separate multiplication number sentences using the three numbers listed and using parentheses to arrange different groups of numbers.&nbsp;<br><br>Introduce the formal language of <i>associative property</i>, although you should not require students to know it. <strong>"Multiplication of numbers produces the same result even when the numbers are organized in different ways. According to mathematicians, multiplication is </strong><i><strong>associative</strong></i><strong>, meaning that the numbers can be regrouped. Why is this important? We will examine the answer to your question right now!"</strong>&nbsp;<br><br><strong>Metal Math – Applying the Properties&nbsp;</strong><br><br><strong>"Juanita and Fabio were both doing their schoolwork. The problem was 6 × 2 × 5. Juanita said 60 quickly and without using a calculator. Fabio stated that it is equal to 12 × 5, but he is still working. How did Juanita multiply these three numbers so quickly?"</strong> Students will certainly understand that Juanita multiplied 2 and 5 first, followed by 6. These properties enable students to calculate more quickly and accurately. This is why students should understand how to apply these properties; yet, the standards clearly do not need students to know the names of the properties.&nbsp;<br><br>Provide two additional examples for class discussion. These examples will get students ready for the Make It Easier practice worksheet.&nbsp;<br><br>Write 4 × 9 × 2 on the board. Ask students, <strong>"How can we rewrite this number sentence to make it easier to multiply mentally?"</strong> Students may offer new number sentences based on the multiplication facts that they find easiest to compute. Ask a student to rewrite the number sentence on the board. Compare the relative ease of computation for the two number sentences.&nbsp;<br><br>Most students will notice that 4 × 2 × 9 is typically easier to compute than 4 × 9 × 2. Multiplying from left to right, 4 × 2 × 9 becomes 8 × 9, while 4 × 9 × 2 becomes 36 × 2. Both products are 72.&nbsp;<br><br>"Write (8 × 6) × 5 on the board. <strong>"This expression contains parentheses, (). In math, parentheses are used to group a certain part of a number sentence together to distinguish from the rest. The parentheses also indicate which part of the number sentence we should compute first. For example, the way the number sentence is now written, we are supposed to multiply first, then multiply this product by 5. But we can make things easier! How can we restructure this number sentence so it's easier to multiply mentally?"</strong> Ask a student to rewrite the number sentence on the board. Compare the relative ease of computation for the two number sentences.&nbsp;<br><br>Most students understand that (8 × 5) × 6 is easier to calculate than (8 × 6) × 5. Multiplying the numbers within parentheses first becomes 40 × 6, while (8 × 6) × 5 becomes 48 × 5. Both products are 240.&nbsp;<br><br>Distribute a copy of the Make It Easier worksheet (M-3-5-3_Make It Easier and KEY) to each student. Remind students to use the characteristics to perform multiplication quickly and easily mentally. Sample solutions are provided.&nbsp;<br><br>To assess student knowledge of lesson concepts, use the Commutative and Associative Practice sheet (M-3-5-3_Commutative and Associative Practice and KEY).&nbsp;<br><br><strong>Extension:</strong>&nbsp;<br><br>The Routine section includes strategies for reviewing lesson concepts throughout the year. The Small Group portion is designed for students who could benefit from additional teaching or practice. The Expansion section contains suggestions for challenging students who are willing to go beyond the limits of the standard.&nbsp;<br><br><strong>Routine:</strong> Continue to emphasize the utility of these operations in doing mental computations to help students in reviewing their use. Do not encourage students to use a calculator to multiply 5 × 24 × 2 or 5 × 9 × 6. Instead, ask them to discover a "easier" way to compute these, such as 5 × 2 × 24 and (5 × 6) × 9. These features can also be useful as students learn to multiply two and three-digit numbers.&nbsp;<br><strong>Small Group:</strong> Students that require further practice may be divided into small groups to work on additional problems involving these properties. The emphasis should be on supporting students in identifying ways to make the computation easier. Make sure to explain the why of each solution. Use miniature whiteboards to solve these problems, and encourage students to suggest how to move the whiteboards to make calculation easier. (Since there is only a small group, place the mini-whiteboards on the table rather than having students hold them.) The More Examples practice worksheet (M-3-5-3_More Examples and KEY) includes additional difficulties. If the teacher is not available to assist the small group, students can seek further education on their own at the following website.&nbsp;<br>http://www.coolmath.com/prealgebra/06-properties/02-properties-commutative-multiplication-02.htm&nbsp;<br><br><strong>Expansion:</strong> Students ready for a more difficult task should work in groups of two or three to play the following game, which focuses on commutative, associative, distributive, and multiplicative identity properties. Each property comes with its own description.&nbsp;</p><p>http://www.aaamath.com/pro74b-propertiesmult.html</p>