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Multiplication and Division of Fractions and Decimals Session 3 January 15, 2013

Multiplication and Division of Fractions and Decimals Session 3 January 15, 2013 Huntersville Elementary. Solve………. A number with 3 digits after the decimal point is rounded to 4.8 when rounded to the nearest tenth. What is the smallest this number could be?

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Multiplication and Division of Fractions and Decimals Session 3 January 15, 2013

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  1. Multiplication and Division of Fractions and Decimals Session 3 January 15, 2013 Huntersville Elementary

  2. Solve………. A number with 3 digits after the decimal point is rounded to 4.8 when rounded to the nearest tenth. What is the smallest this number could be? What is the largest this number could be?

  3. Fraction Representations • Pizzas, pies, and candy bars • Number lines/Sentence strips • Fraction bars/Centimeter grid paper • Clocks • Percent strips- percent and decimal equivalents • Arrays (4 x 6 and 5 x 12)

  4. Benchmarks (containers) • Decomposing fractions 7/8 = 3/6 + 1/4 + 1/8 (Fraction Track Game) • Pattern blocks (unit fractions) • Investigations’ Bar diagrams • Paper folding and Open arrays • Equations

  5. Fraction Conjectures • To multiply unit fractions, multiply the denominator times the denominator and • To multiply any two fractions, multiply the numerator times the numerator, and the denominator times the denominator • When you multiply a whole number times a fraction less than 1, the answer is smaller than the whole number • When you multiply a whole number times a fraction greater than 1, the answer is greater than the whole number

  6. Fraction Conjectures continued • When you multiply two fractions that are both less than 1, the product is a fraction smaller than either of the factors • When you divide a whole number by a fraction less than 1, the answer is larger than the whole number • When you divide a fraction by a whole number, the answer is smaller than the whole number and the fraction • 10 ÷ ½ = 10 x 2 • 10 x ½ = 10 ÷ 2

  7. Sharing Submarine Sandwiches The cafeteria made lunches for the fifth graders going on a field trip. They were in four different groups so the number of sandwiches differed. The sandwiches were all the same size. Group One had 4 students sharing 3 subs Group Two had 5 students sharing 4 subs Group Three had 8 students sharing 7 subs Group Four had 5 students sharing 3 subs Did each student get a “fair share?” If not, which group ate the least? Most? How do you know?

  8. Help the Cafeteria Staff Next trip we want to guarantee that each student will receive 2/3 of a sub Using large paper, create a chart for the cafeteria to help them know how many subs to make for up to 15 students • What patterns do you notice? • What strategy could cafeteria workers use for any number of students? • If you knew there were 8 subs made, how could you figure out how many students could each get 2/3 sub? Model this situation using numbers and symbols.

  9. Dividing a Whole Number by a Fraction What does 6 ÷ ½ mean? How does this problem relate to multiplication?

  10. How is multiplication and division of whole numbers connected to multiplication and division of fractions? • What is meant by “there are two wholes when dividing fractions?” Give an example..

  11. Rounding Decimals Using the number line below, label the point 1.68 1.7 1.6 How did you know where to place the number?

  12. Rounding Decimals 1.68

  13. Procedure for Rounding • Underline the rounding place • Circle the digit to the right • If the circled digit is 5 or greater, increase the underlined digit • If the circled digit is less than 5, leave the underlined digit as it is • Drop the digits to the right of the underlined digit 4.923 Round to the nearest hundredth What about 2.97 rounded to the nearest tenth?

  14. Expanded Notation Write the following number using expanded notation: 689,738 Use exponents when possible What number is 5,000 less than this? What number is 200 more? Note the suggestions in the snap-ins!

  15. Decimal Subtraction Problems Mercedes had 1.86 grams of gold. She used 0.73 grams of it in a piece of jewelry. How much gold does she have left? • Use of Hundredths Grids • Subtract in parts • Add up from 0.73 to 1, and then from 1 to 1 and 86 hundredths

  16. Subtract: 0.6 – 0.48 How can equivalent decimals help you to subtract these two numbers?

  17. Multiplying Powers of 10 • Use 4 Hundredths grids to show the following 4 x 0.01 4 x 0.1 How can we use what we know about multiplying fractions to help us solve the two problems above? 4 x 1 = 4 x 0.1 = 4 x 10 = 4 x 0.01 = 4 x 100 = 4 x 0.001 = ?

  18. What relationships do you notice? • 25 x 0.01 = 0.25 • 25 x 0.1 = 2.5 • 25 x 1 = 25 • 25 x 10 = 250 • 25 x 100 = 2,500

  19. Multiplying by “Small” Numbers 2 x 7 = 2 x 0.7 = Use a number line from 0 to 2 to show the answer to the second problem Use the same number line to show the answer to the following: 2 x 0.07 (notice the use of running context p CC110-111) 32 x 0.8 = 2.56 25.6 256

  20. Reasoning with Decimals Problem: 185 x 0.4 = If the answer to 185 x 4 is 740, how can we use reasoning to determine the answer to the above problem?

  21. Writing a Rule for Multiplication of Decimals • Multiply the numbers like they are whole numbers and then think about the size of the factors and place the decimal point so the product is the right size. • Multiply a whole number by a decimal, and the answer has the same number of decimal places as the decimal number being multiplied. Or, if each of the numbers has one decimal place, then the answer has two decimal places. • Therefore solve: 42 x 36 = 1,512 4.2 x 3.6 = ?????

  22. Multiplying Tenths…. 0.2 x 0.4 = Think about one of the conjectures we made about multiplying 2 fractions less than 1 How can this conjecture and what you know about the relationship between decimals and fractions help you solve this problem?

  23. Dividing Powers of 10 2 ÷ 1 = ? 2 ÷ 0.1 = ? 2 ÷ 0.01 = ? What does 2 ÷ 1 mean? Will the answer to each of these be greater or less than 2? How do you know? How can we use 2 hundreds grids to represent these situations?

  24. Comparing Multiplication and Division 25 x 100 = 2,500 25 ÷ 100 = ____ 25 x 10 = 250 25 ÷ 10 = ____ 25 x 1 = 25 25 ÷ 1 = 25 25 x 0.1 = 2.5 25 ÷ 0.1 = 250 25 x 0.01 = 0.25 25 ÷ 0.01 = 2,500 What patterns do you notice? How can you use your understanding of fractions to figure out the answers to the two unsolved problems? Bonus: use the pattern in the division problems to figure out the answer to 25 ÷ 0.001

  25. Closest Estimate 6.8 x 2.3 ≈ 1.4 14 140 74 x 8.1 ≈ 5.6 56 560 166 x 0.08 ≈ 1.66 16.6 166 What is the closest estimate for each? Is the closest estimate greater or less than the actual answer? How do you know?

  26. Dividing Decimals 18 ÷ 6 = 18 ÷ 0.6 = How could you write these problems as missing factors problems? Draw a number line from 0-20 Solve each of these using the number line.

  27. What are some rules for dividing decimals? • Problems can be solved by thinking about both numbers as whole numbers and then reasoning about where the decimal belongs • Problems can be solved by thinking about the division problem as a missing factor problem to solve or to check the reasonableness of the answer

  28. Will the rule of counting decimal places work for division like it does for multiplication? 97.5÷ 6.5 = 21.52 ÷ 0.8 =

  29. Scooping Trail Mix

  30. Math Reasoning Inventory Questions

  31. www.illustrativemathematics.org Resources DPI Fraction Unit Test-item bank CMS Wiki Unpacking Document

  32. Why don’t we just teach them the rule? There are easy rules for multiplying and dividing fractions and decimals. So…..

  33. Reflection…… • Thinking about multiplying and dividing fractions, why do you think it has been said that “fractions are the pathway from arithmetic to algebra? • Is multiplying or dividing fractions more difficult? Explain your reasoning. • Is multiplying or dividing decimals more difficult? Explain your reasoning

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