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**Use Properties of Operations to Generate Equivalent Expression**

Common Core: Engage New York 7.EE.A.1 and 7.EE.A.2

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What does 7.EE.A.1 cover? Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

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What does 7.EE.A.2 cover? Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

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**Table of Contents 1/8/14 7.EE.2- Engage NY Lesson 3:**

Date Title Page 12/9/13 Post Investigation 1- Check Up HW Assignment Due THURSDAY 12/12/13 HW Accuracy Grade (not completion only) 12/10/13 7.EE Engage NY Lesson 1: Generating Equivalent Expressions *Attach worksheets* # 12/13/13 7.EE.1- Engage NY Lesson 2: Fresh left 1/8/14 7.EE.2- Engage NY Lesson 3: Writing Products as Sums and Sums as Products *Attach worksheets* Fresh Left

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**Focus 6 Solving Equations Learning Goal**

Students understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

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**Today, my learning target is to…**

Use an area/rectangular array model and distributive property to write products as sums and sums as products. Use the fact that the opposite of a number is the same as multiplying by -1 to write the opposite of a sum in standard form. Recognize that rewriting an expression in a different form can shed light on the problem and how the quantities in it are related.

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**How much prior knowledge do you have regarding that goal?**

MY PROGRESS CHART Before we start the Learning Target Lesson, think about the Learning Target for today…. How much prior knowledge do you have regarding that goal? Chart your prior knowledge using your pre-target score icon.

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**Lesson 3- Math Standard 7. EE. A**

Lesson 3- Math Standard 7.EE.A.2 Writing Products as Sums and Sums as Products OPENING EXERCISE (5 minutes) Purpose: To create tape diagrams to represent the problem and solution. Solve the problem using a tape diagram. A sum of money was shared between George and Brian in a ratio of 3:4. If the sum of money was $56.00, how much did George get? Part B: 3t + 4q + 3t + 4q; 2(3t + 4q), 6t + 8q

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**Opening Exercise- Solutions**

Label one unit by “𝑥" in the diagram.

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**Opening- Solutions What does the rectangle labeled 𝑥 represent?**

8 units, 8 boxes, or 8 rectangles. Draw in 8 smaller rectangles in the unit to represent $8. Is it always necessary to draw in every one unit for all tape diagrams? No, it is unnecessary and tedious most of the time to draw every one unit. Tape diagrams should be representative of problems and should be used as a visual tool to help find unknown quantities.

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**Teacher Directed Example 1 (3 minutes)**

Represent 3+2 using squares for units. Represent 𝑥+2 using the same size square for a unit as above. Draw a rectangular array for 3(3+2). Draw an array for 3(x+2). How many squares are in the shaded rectangle? How many rectangles are in the non-shaded rectangle? Record the total number of squares and rectangles in the center of each rectangle. About how big is 𝑥? Approximately six units. 6 3 Next slide

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Example 1- solutions

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**(for all numbers 𝑎,𝑏, and 𝑐)**

Relevant Vocabulary Distributive Property: The distributive property can be written as the identity: 𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐 (for all numbers 𝑎,𝑏, and 𝑐)

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**Student Practice Exercise 1(3 minutes) Fill in the blanks.**

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**Exercise 1 with Distributive Property- Solution**

Is it necessary to draw in the squares in the diagram to determine the number of square units? Is it easier to just imagine the 176 and 55 square units? NO! YES!

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Teacher Directed Example 2 (5 minutes)- Show students representations of the expression with tape diagrams or arrays 𝑥+𝑦 + 𝑥+𝑦 + 𝑥+𝑦 𝑥+𝑥+𝑥 + 𝑦+𝑦+𝑦 3𝑥+3𝑦 3(𝑥+𝑦) What do you notice about all of these expressions ?

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Exercise 2 (5 minutes)- Observations and discussion about the representations of the expression with tape diagrams or arrays What can we conclude about all of these expressions? How does 3 𝑥+𝑦 =3𝑥+3𝑦? How do you know the three representations of the expressions are equivalent? Under which conditions would each representation be most useful? Which model best represents the distributive property? They are all equivalent. Three groups of (x+y) is the same as multiplying 3 with the x and the y. The arithmetic, algebraic, and graphic representations are equivalent. Problem (c) is the standard form of problems (b) and (d). Problem (a) is the equivalent of problems (b) and (c) before the distributive property is applied. Problem (d) is the expanded form before collecting like terms. Either 3 𝑥+𝑦 𝑜𝑟 3𝑥+3𝑦 because it is clear to see that there are 3 groups of 𝑥+𝑦 which is the product of the sum of 𝑥 𝑎𝑛𝑑 𝑦 or that the 2nd expression is the sum of 3𝑥 𝑎𝑛𝑑 3𝑦. Problem (d)

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**Teacher Directed Exercise 3 (5 minutes)**

Find an equivalent expression by modeling with a rectangular array and applying the distributive property 5(8𝑥+3). Write the array in standard form. Substitute the variable with the value of 2. Solve the expression.

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**Student Practice Exercise 2 (3 minutes)-**

For parts (a) and (b), draw a model for each expression and apply the distributive property to expand each expression. Substitute the given numerical values to demonstrate equivalency.

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**Exercise 2 (3 min)- Independent Student Practice**

For parts (c) and (d), apply the distributive property. Substitute the given numerical values to demonstrate equivalency.

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**Teacher Directed Exercise 4 (3 min)**

Rewrite the expression, (6𝑥+15)÷3, as a sum using the distributive property.

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**Student Practice Exercise 3 (3 min)**

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**Teacher Directed Example 5 (3 min)**

Model the following exercise with the use of rectangular arrays. What is a verbal explanation of 4(𝑥+𝑦+𝑧)? Model the expression 4 𝑥+𝑦+𝑧 . Expand the expression 4 𝑥+𝑦+𝑧 . There are 4 groups of the sum of 𝑥, 𝑦, 𝑎𝑛𝑑 𝑧. The expanded expression is 4𝑥+4𝑦+4𝑧.

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**Student Practice Exercise 4 (3 min)**

Expand the expression from a product to a sum so as to remove grouping symbols using an area model and the repeated use of distributive property: 𝟑(𝒙+𝟐𝒚+𝟓𝒛). Area Model Repeated use of distributive property

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**Teacher Directed Example 6 (5 min)**

Read the problem aloud with the class and begin by using different lengths to represent 𝑠 to come up with expressions with numerical values. A square fountain area with side length 𝑠 is bordered by a single row of square tiles as shown. Express the total number of tiles needed in terms of 𝑠 three different ways.

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**Be prepared to share your expression with the class.**

What if 𝑠=4? How many tiles would you need to border the fountain? What is 𝑠=2? How many tiles would you need to border the fountain? What pattern/generalization do you notice? You have 2 minutes to create as many expressions you can think of to find the total number of tiles in the border in terms of 𝑠. Be prepared to share your expression with the class. There is a need for 20 tiles to border the fountain- 4 for each side and 1 for each corner. There needs to be 12 tiles to border the fountain- 2 for each side and 1 for each corner. Answers may vary. There is one tile for each corner and four times the amount of tiles enough to fit one side length.

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Example 5 Discussion: Of the shared expressions, which one would you use and why? Sample responses of created expressions…. 4 𝑠+1 𝑜𝑟 4𝑠+4 is useful because it is the most simplified, concise and in standard form.

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Closing (3 min) What are some of the methods used to write products as sums? In terms of a rectangular array and equivalent expressions, what does the product form represent, and what does the sum form represent? We used repeated use of the distributive property and rectangular arrays. The total area represents the expression written in sum form, and the length and width represent the expressions written in product form.

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**Exit Ticket (3 min)- Lesson 3 Writing Products as Sums and Sums as Products**

A square fountain area with side length 𝑠 is bordered by two rows of square tiles along its perimeter as shown. Express the total number of grey tiles (only in the second rows) needed in terms of 𝑠 three different ways.

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**Exit Ticket - Lesson 3 Solution- Writing Products as Sums and Sums as Products**

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**Today, I achieved my learning target by…**

Using an area/rectangular array model and distributive property to write products as sums and sums as products. Using the fact that the opposite of a number is the same as multiplying by -1 to write the opposite of a sum in standard form. Recognizing that rewriting an expression in a different form can shed light on the problem and how the quantities in it are related.

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**How much prior knowledge do you have regarding that goal?**

MY PROGRESS CHART Before we start the Learning Target Lesson, think about the Learning Target for today…. How much prior knowledge do you have regarding that goal? Chart your prior knowledge using your pre-target score icon.

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**Problems 1a- 1b Problem 2 Problems 3a- 3b Problems 4a-4𝑙 Problem 5**

Student Homework/Practice: Problem Set The completion of the Problem Set indicates an understanding of the objectives of this lesson. Problems 1a- 1b Problem 2 Problems 3a- 3b Problems 4a-4𝑙 Problem 5

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Problem Set Solutions

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**The End of Lesson 3 Writing Products as Sums and Sums as Products **

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