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This is one

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.

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This is one

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

  2. 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.

  3. 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

  4. “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!”

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

  6. 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.

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

  8. 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

  9. 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?

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

  11. 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.

  12. 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:

  13. 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

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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 1 ÷ 1/2 • 1÷ ½ “How many one-halves in 1?” • Answer: There are two one-halves in 1.

  20. 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

  21. 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?

  22. 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.

  23. 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?

  24. 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.

  25. 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”?

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

  27. 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.

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

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

  30. 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.

  31. Factors are ½ and 4 4 1 2

  32. 4 is 1+1+1+1 in length 4 1 2 1 + 1 + 1 + 1

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

  34. This is 4 regions 1 by 1/2 4 1 2 1 2 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

  35. Total area is the product 4 1 2 1 2 1 2 1 2 1 2 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

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

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

  38. 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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