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This is one A Journey into math and math instruction

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Models by Teachers for Students Often we desire a model to make math easier for students. It is there that there are 2 initial mistakes. 1. Our job is to not make math easier. It is to allow it to make more sense to students. 2. Handing someone a model we have worked to make sense for ourselves is actually adding more for them to learn unless the model is internalized conceptually by the student.

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String Theory Each of you will receive a small length of string Each length may be different This will be the beginning for all the math we do today

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“This is One” Hold up your string so you are displaying its whole length and say, “This is one!” Why is this one? “Because I said so!”

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Using your “one” to show larger numbers Say, “This is one” so this is two” (show what two would be)

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Extend to multiplication You have shown 2 and 3 with your length of 1. Show 2 X 3. Remember, all representations begin with the “one” you set up from the beginning.

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How about division Use your string to show 1÷2 (with result). Do this on your own first Compare your demonstration with a partner

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Second division problem Use your string to show 1÷1/2 (with result). Do this on your own first Compare your demonstration with a partner

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Issue to explore! How can I demonstrate division in one way that works for 1÷2 and 1÷ ½. In other words, where dividend, divisor and quotient are represented in the same consistent manner?

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A Little History We first learn about division through whole numbers We extend that to other rational numbers such as fractions and decimals

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Primary students see division two ways. These two ways are called measurement and partition. Young students do this naturally, but in math instruction the distinction becomes fuzzy.

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Partitive Division (Divisor is number of sets) When dividing an amount by 2 we are taking the amount and separating it into two equal sets. Think of separating what you have into two bags:

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Partitive Division (Divisor is number of sets) Imagine you have 8 dots: When I divide by 2, I split that 8 into two equal groups. Each group has 4: 8÷2 = 4

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Measurement Division (Divisor is size of units to count) Imagine you have 8 dots: This time, you are now counting sets of 2 dots

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Measurement Division (Divisor is size of units to count) Imagine you have 8 dots: 1 This time, you are now counting sets of 2 dots

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Measurement Division (Divisor is size of units to count) Imagine you have 8 dots: 1 2 This time, you are now counting sets of 2 dots

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Measurement Division (Divisor is size of units to count) Imagine you have 8 dots: 1 2 3 This time, you are now counting sets of 2 dots

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Measurement Division (Divisor is size of units to count) Imagine you have 8 dots: 1 2 3 4 This time, you are now counting sets of 2 dots There are 4 sets of 2 in 8. 8÷2=4

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1 ÷ 1/2 1÷ ½“How many one-halves in 1?” Answer: There are two one-halves in 1.

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Dividing a number by 1/2 1÷ 1/2 “How many one-halves in 1?” 1 ÷ ½ = 2 2 ÷ 1/2 “How many one-halves in 2?” 2 ÷ ½ = 4 4 ÷ 1/2 “How many one-halves in 4?” 4 ÷ ½ = 8

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Dividing a number by 1/2 1÷ 1/2 “How many one-halves in 1?” 1 ÷ ½ = 2 2 ÷ 1/2 “How many one-halves in 2?” 2 ÷ ½ = 4 4 ÷ 1/2 “How many one-halves in 4?” 4 ÷ ½ = 8 So, what would 10 ÷ ½ be equal to?

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What is the usual rule? To divide by a fraction, multiply by its reciprocal. “What?” Example --- the reciprocal of ½ is 2/1. Divide 4 by 1/2 : 4÷1/2 = = 8 In short, we multiplied by 2 when dividing by 1/2.

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The common algorithm Divide 4 by 1/2 : 4÷1/2 = = 8 The shortcut algorithm works – why? What is gained from conceptually understanding division by a fraction?

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Common Core and Division Grade 5 Number and Operations- Fractions Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

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Common Core and Division Grade 5 Number and Operations- Fractions Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. What are these “Previous Understandings”?

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Area Model- Whole numbers Use an area model (array) to show 3 X 4. Label factors and product

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Area Model- Whole numbers Use an area model (array) to show 3 X 4. Label factors and product. Use your model to show the relationship between 3x4=12 and division with related facts to that equation.

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Area Model- Where is one? Use your model to show where one is in the factors and in the product.

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Area Model- Product “1” is sq. unit 1 + 1 + 1 + 1 1+1+1 1+1+1 Factors (length) Product (area) 3 x 4 = 12 Factors are dimensions in length. Product is area in square units “1” is a 1x1 unit square. 1

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Area Model- Fractions Use an area model (array) to show 1/2 X 4. Label factors and product. Use your model to show the relationship between ½ x 4 = 2 and division with related facts to that equation.

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Factors are ½ and 4 1 2 4

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4 is 1+1+1+1 in length 1 2 4 1 + 1 + 1 + 1

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Product is measured in area. 1 2 4 1 + 1 + 1 + 1 What is the area of the shaded region?

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This is 4 regions 1 by 1/2 1 2 4 1 + 1 + 1 + 1 What is the area of the shaded region? Each of the 4 regions is ½ x 1. Each has an area of ½ sq. units 1 2

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Total area is the product 1 2 4 1 + 1 + 1 + 1 What is the area of the shaded region? Each of the 4 regions is ½ x 1. Each has an area of ½ sq. units Total area = 2. ½ x 4 = 2 1 2 1 2 1 2 1 2

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Division as inverse of Multiplication 1 2 ? 2 ÷ ½ = ? What times ½ would give the product 2? 2

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Division as inverse of Multiplication 1 2 4 2 ÷ ½ = ? “I need 4 halves to make 2 because 4 X ½ = 2” 2 ÷ ½ = 4 Area (dividend) = 2 1 2 1 2 1 2 1 2

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Making Models Powerful Models for instruction are to provide opportunities for exploring concepts to build understanding. The power of models such as arrays is not for solving problems. The first step to being able to use a model is being able to describe what the parts of the model represent. From there, talking about the mathematics being represented provides a greater window into a student’s mathematical thinking.

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