# This is one A Journey into math and math instruction.

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This is one A Journey into math and math instruction

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.

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

“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!”

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

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.

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

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

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?

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

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.

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:

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

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

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

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

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

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

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

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

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?

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.

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?

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.

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”?

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

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.

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

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

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.

Factors are ½ and 4 1 2 4

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

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

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

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

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

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

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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