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**NCCTM Leadership Conference Greensboro, NC**

How Many Times Larger? A Progression of Multiplicative Comparisons Grades 3-5 NCCTM Leadership Conference Greensboro, NC March 22,2013 Marta garcia Windy Taylor

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**Bridging Major Work Standards**

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**Think about this problem:**

*How did you approach the problem? *What representations did you use to solve the problem? *What mathematical ideas are embedded in this problem? $120 is shared among 3 persons A, B, and C. If A receives $20 less than B, and B receives 3 times as much money as C, how much money does C receive?

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Grade 3 3.NBT Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. Use base ten blocks to justify why 3 x 50 is ten times larger than 3 x 5.

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**Associative Property: Why ten times larger?**

6 x 5 = 30 3 groups of 10 6 x 50 6 x 50 = 6 x (5 x 10 ) = (6 x 5) x 10= 30 x 10= ( 3 x 10 ) x 10 = 3 x (10 x 10) 3 x 100 3 groups of 100 Modeling: With base ten blocks? On a number line? Model this with the base ten bloc

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**Analyzing Student Work**

Let’s look at some work samples! Paige and Ben each babysat last weekend. Paige babysat three times as many hours as Ben. Ben babysat for four hours. How many hours did Paige babysit?

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**What does it mean to be ten times bigger?**

Student responses to how many times greater is 4 x 100 than 4 x 10? Discuss each response: What understandings are students bringing to this question? What misconceptions are present?

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Student Responses 4 x 100 is 360 times bigger than 4 x 10 4 x 100 is 90 times bigger than 4 x 10 4 x 100 is 100 times bigger than 4 x 10

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**Using Arrays to Compare**

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**Try This ! Build an array which is 3 times larger than a 2 x 4 array.**

Build an array which is 2 times smaller than a 4 x 10 array.

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Fourth Grade 4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1 4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

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Grade 5 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used

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**Decimals: What is happening to the size of the numbers?**

Place the following decimals on a number line. Then discuss with a partner how your number lines are alike or different.

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**What is happening to the products?**

8 x 0.01 = 8 x 0.1 = 8 x 1 = 8 x 10 = 8 x 100 = Just use one set? Copy and insert

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3 x 2.5 = Estimate the product. Between what two whole numbers will the product lie. Use base ten blocks and then the grid paper to model the product of 3 x 2.5. How does the model justify that the product is 3 times larger than 2.5?

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3.5 x 2.5= Estimate the product. Between what two whole numbers will the product lie? Use base ten blocks and then grid paper to find the product. How can you justify that the product is 3 ½ times larger than 2.5?

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**Decimal Multiplication**

How are these ideas building across the grades so that by fifth grade students can make sense of multiplying and dividing with decimals? Looking back at the problem you solved at the beginning of the session: How do these standards support that type of reasoning?

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**Questions? Reflections?**

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Divide with decimals Standard UW.GAP.5.M.NBT.07 Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawing and strategies.

Divide with decimals Standard UW.GAP.5.M.NBT.07 Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawing and strategies.

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