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Modeling Multiplication of Fractions

Modeling Multiplication of Fractions. MCC4.NF.4; MCC5.NF.4 ; MCC5.NF.5; MCC5.NF.6. Deanna Cross – Hutto Middle School. Fraction by a Whole Number. Multiplying on a Number Line. Fraction by a WHOLE number

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Modeling Multiplication of Fractions

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  1. Modeling Multiplication of Fractions MCC4.NF.4; MCC5.NF.4; MCC5.NF.5; MCC5.NF.6 Deanna Cross – Hutto Middle School

  2. Fraction by a Whole Number

  3. Multiplying on a Number Line • Fraction by a WHOLE number Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over? Suggestions on how to solve?

  4. Number Line Number Lines Start and end with an arrow Divided into equal (equivalent) increments Can start and end at any number Are there any numbers that can be “renamed” or written as an equivalent fraction?

  5. Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over? • Isabel has used only ¾ of the paper. What if she had used ½ of the paper? How much would she have used? • You have to multiply 8 x ¾.

  6. 8 x ¾ • First – Model what you have on a number line – “She had 8 feet of wrapping paper” • Now, she is multiplying by ¾ . What is the denominator? 0 1 2 3 4 5 6 7 8

  7. 8 x ¾ • Now, divide the total amount (8) into 4 pieces. (8 ÷ 4 = 2 – so each piece is equal to 2) • Shade in 3 of the four pieces. • Look to see if this lines up with a number on your number line. 0 1 2 3 4 5 6 7 8

  8. Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over? • So, 8 x ¾ = 6. • Why is the answer smaller than 8? • Because whenever you multiply a whole number by a fraction, you will get a smaller answer.

  9. 0 1 2 3 4 4 x ½ • Will your answer be bigger or smaller than 4? • First – show 4 on the number line.

  10. 0 1 2 3 4 4 x ½ • Now, look at your denominator – 2 • Divide your bar into two EQUAL pieces. • Shade in 1 of the two pieces. • Does this line up with a number on the number line?

  11. 0 1 2 3 4 3 x • Will your answer be bigger or smaller than 3? • First – show 3 on the number line.

  12. 0 1 2 3 4 3 x • Now, look at your denominator – 3 • Divide your bar into three EQUAL pieces. • Shade in 1 of the three pieces. • Does this line up with a number on the number line?

  13. 0 1 2 3 4 5 6 6 x • Will your answer be bigger or smaller than 6? • First – show 6 on the number line.

  14. 0 1 2 3 4 5 6 6 x • Now, look at your denominator – 4 • Divide your bar into four EQUAL pieces. HINT: Divide 6 by 4 and determine the decimal portion to divide this piece into • Shade in 2 of the four pieces. • Does this line up with a number on the number line?

  15. Practice • Optional Practice problems • 8 x • 9 x • 12 x • 10 x • 4 x

  16. Multiplying with an AREA MODEL • Fraction by a WHOLE number Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

  17. Area models • Reminder of area – length x width = area • Area is the amount INSIDE a rectangular shape. • To determine area, you multiply TWO numbers – the length and the width.

  18. Area models • Multiply the length and the width • 2 x 5 = 10 – AREA = 10 2 5

  19. Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over? • Isabel has used only ¾ of the paper. • You have to multiply 8 x ¾. • Suggestions to solve using area model?

  20. 8 x ¾ • Draw a rectangle. • Divide the rectangle into smaller rectangles to represent your WHOLE number. 8

  21. 8 x ¾ • Next, along the vertical side, divide your rectangle into the number of pieces representing your denominator  4 4

  22. 8 x ¾ • Now, shade in 3 rows of the 4 you just created. 8 4

  23. 8 x ¾ • Hard part – This model started out with 8 wholes. How much would 1 box be worth? THINK… 8 4

  24. 8 x ¾ • This is 1 whole… • So how much would 1 box be worth? • 1 box equals ¼ 8 4

  25. 8 x ¾ • Now, count how many ¼’s you have shaded green. • 24 boxes = 8 4 • Can we leave like this, or is there another way to write this improper fraction?

  26. 8 x ¾ • = 24 ÷ 4 = 6 • Proof: If you divided 8 dollars up among 4 people, how much would each get? 8 4

  27. 8 x ¾ • = 24 ÷ 4 = 6 • Proof: If you divided 8 dollars up among 4 people, how much would each get? • TWO • Now, how much would 3 people get? • SIX • So, ¾ of 8 = 6

  28. 4 x ½ • This is one you already know the answer to – if you have ½ of 4 you have 2. Let’s prove that with an area model.

  29. 4 x ½ • First, draw a rectangle divided into your whole number – 4 4

  30. 4 x ½ • Next, divide your rectangle into the number of pieces for your denominator along the vertical edge. 4 2

  31. 4 x ½ • Shade in the number represented by the numerator… 4 2

  32. 4 x ½ • Now THINK – how much is ONE square worth? What is your WHOLE? 4 2

  33. 4 x ½ • One square = ½ • There are 4 “halves” – or 4 2

  34. 4 x ½ • = 4 ÷ 2 = 2 • So – if you have half of 4 you have 2. 4 2

  35. 3 x 1/3 • Try to draw this model on your own – you already know what 1/3 of 3 would be… 3 3

  36. 3 x 1/3 • Now, think about what each square represents… 3 So, each square = 1/3, there are 3 thirds… 3 3 x 1/3 = 1

  37. 6 x 2/4 • Draw the model. 6 4

  38. 6 x 2/4 • What does each square represent? 6 How many fourths? 12 4

  39. Practice • Optional Practice problems • 8 x • 6 x • 12 x • 5 x • 4 x

  40. Multiplying with TAPE DIAGRAM • Fraction by a WHOLE number Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

  41. Tape diagrams • Tape diagrams are like adding strips of paper together to determine lengths. For example, if I had 3 chocolate cupcakes and someone gave me 2 more, I would have five. 3 chocolate 2 more 5 chocolate cupcakes

  42. Multiplying with Tape Diagrams • Fraction by Whole numbers are easy with tape diagrams…it is like repeated addition.

  43. Isabel had 8 feet of wrapping paper to wrap Christmas gifts with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over? • Isabel has used only ¾ of the paper. • You have to multiply 8 x ¾. • Suggestions to solve using tape diagram model?

  44. 8 x 3/4 • Think, how many 3/4ths do you need? • 8 • Make a tape model to represent 3/4. Copy this eight times.

  45. 8 x 3/4 + + + + + + + • Add up how many fourth’s you have… Can you leave the fraction as it is?

  46. 8 x 3/4 How do you change an improper fraction to a mixed number?

  47. 4 x ½ • Draw a diagram to represent ½. • Repeat this 4 times.

  48. 4 x ½ Ahhh…there is a large number on top of a small number! • Add up each piece… + + +

  49. 3 x 1/3 How else can you write a number over itself? • Model • Add • Reduce + +

  50. 6 x 2/4 + + + + + • Model • Add • Reduce Can you simplify this fraction?

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