EXTENDING NUMBER SENSE TO FRACTIONS 3 RD -5 TH Webinar #3: Multiplying & Dividing Fractions ALASKA MATHEMATICS STANDARDS Content Standards Standards of Mathematical Practice (SMPs) STANDARDS FOR MATHEMATICAL PRACTICE. I can make sense of problems and persevere in solving them. 2. I can use numbers, words, and reasoning habits to help me make sense of problems. 3. I can construct viable arguments and critique the reasoning of others. 4. I can recognize math in everyday life and use math I know to solve problems. 5. I can use appropriate tools strategically.. I can be precise when solving problems and clear when communicating my ideas. 7. I can look for and make use of structure. I can see and understand how numbers are organized and composed as parts and wholes. 8. I can notice patterns and find more general methods and short cuts.
STANDARDS FOR MATHEMATICAL PRACTICE. I can make sense of problems and persevere in solving them. 2. I can use numbers, words, and reasoning habits to help me make sense of problems. 3. I can construct viable arguments and critique the reasoning of others. 4. I can recognize math in everyday life and use math I know to solve problems. 5. I can use appropriate tools strategically.. I can be precise when solving problems and clear when communicating my ideas. 7. I can look for and make use of structure. I can see and understand how numbers are organized and composed as parts and wholes. 8. I can notice patterns and find more general methods and short cuts. REPRESENTATION IN MATH Concrete Pictorial Abstract 5 8 FRACTIONS IN THE NEW STANDARDS 3 rd Grade 4th Grade 3.NF: Develop understanding of fractions as numbers. (limited to fractions w/denominators 2, 3, 4, and 8) 4.NF: Extend understanding of fraction equivalence and ordering. 4.NF: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (Add/Subtract limited to fractions w/denom. 2, 3, 4, 5,, 8, 0, 2 and 00) 5th Grade 5.NF: Use equivalent fractions as a strategy to add and subtract fractions. 5.NF: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 2
SENSE-MAKING VS. ANSWER-GETTING What is the whole? What size pieces do we have? How many of those pieces? Estimate! (Is the answer reasonable?) How can I best represent this problem? 3
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FRACTIONS IN THE NEW STANDARDS 3 rd Grade 4th Grade 3.NF: Develop understanding of fractions as numbers. (limited to fractions w/denominators 2, 3, 4, and 8) 4.NF: Extend understanding of fraction equivalence and ordering. 4.NF: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (multiply a fraction by a whole number) 5th Grade 5.NF: Use equivalent fractions as a strategy to add and subtract fractions. 5.NF: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 5
PREVIOUS UNDERSTANDINGS OF MULTIPLICATION Equal groups/arrays 3x 4 Repeated addition Scaling 4 + 4 + 4 3 times as many Draw two different diagrams that show 5 x 4 = 20. Explain how your diagrams represent 5 x 4 = 20. POSSIBLE MODELS OF 5 X 4 = 20
Which of your representations used for 5 x 4 = 20 could be used to represent 5 x 2/3? POSSIBLE MODELS OF 5 X 2/3 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Understand a fraction a/b as a multiple of /b. For example, use a visual fraction model to represent 5/ as the product 5 x (/), recording the conclusion by the equation 5/ = 5 x (/). 7
4.NF.4a. Understand a fraction a/b as a multiple of /b. 0 0 2 3 4 5 5 = + + + + = 5 x 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. b. Understand a multiple of a/b as a multiple of /b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as x (/5), recognizing this product as /5. Adapted From: 3 2 5 3 8
3 cans x 2/5 cup of sugar in each can can 2 3 x (2/5) can 3 can can can 2 can 2 can 3 can 3 3 x 2 x (/5) 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. c. Solve word problems involving multiplication of a fraction by a whole number (e.g., by using visual fraction models and equations to represent the problem). If each person at a party will eat 3/4 of a rhubarb pie, and there will be 5 people at the party, how many total rhubarb pies will be needed? 5 x ¾ = 5/4 5 x ¾ = 3 ¾ 9
Donovan can hike 5/8 of a mile without stopping. His older brother, Dimitri, can hike three times as far. How far can Dimitri hike? Donovan Dimitri 4th Grade: Connections to Multiplication 5 = + + + + = 5 x 5th Grade: Connections to Multiplication X 5 5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers (e.g., by using visual fraction models or equations to represent the problem). 0
2/3 = 2 3 = 2 x /3 2 candy bars shared equally with 3 people Each person gets 2 pieces that are thirds 2 3 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; 4 X 5 Is part when 5 is partitioned into 4 equal parts. 4 of 5
5 4 4 4 X 5 = 4 of 5 Share to 4 equal groups. 4 X 5 = 4 of 5 Divide each whole into 4 equal parts. Take part from each. 3 of X 2 Is part when ½ is partitioned into 3 equal parts. 3 X 2 = x 3 x 2 2
2 3 X 2 Is 2 parts when ½ is partitioned into 3 equal parts. 2 3 X 2 2 = x 3 x 2 2/3 x /2 2/3 of /2 ½ 2/3 2 parts when you find /3 of 5/2 3
CONTEXT MATTERS! (Sense-making rather than answer getting) I made a pan of brownies last night and I wanted to bring the whole pan to school, but my daughter snuck into the kitchen and only left me ¼ of the pan. I was planning to give you guys ½ of the brownies. How much will you get now? What portion of the whole pan? Two Student Representations: ½ of ¼ is /8 ½ x ¼ = /8 Do they both make sense given the story? Division! -- 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 4
PREVIOUS UNDERSTANDINGS OF DIVISION Equal Sharing: 2 3 Measurement: How many sets of 3 can be partitioned out of 2? 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (/3) 4= /2 because (/2) x 4 = /3. 5
Four friends wish to share /3 of a bar of chocolate. How much of the original bar of chocolate will each friend get? 2 2 3 4 /3 4 = /2 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (/5) = 20 because 20 x (/5) = 4. 4 x 5 = 20
/2 = How many sets of ½ can be partitioned out of? If you share out into /2 of a set, how many are in one whole set? Why flip and multiply? /2 = x 2 = x 2 3 3/4 = How many ¾ s can we partition out of 3? 3x 4 = 3 3 x 4 3 = 2 3 7
Can you estimate the answer? 7 3/4 = About how much? Is the exact answer a little more or a little less? 5.NF.. Solve real world problems involving multiplication of fractions and mixed numbers 5.NF.7.c. Solve real world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions (e.g., by using visual fraction models or equations to represent the problem). a. How many quarter cups does she need for the sour cream? Draw a picture to illustrate your solution, and write an equation that represents the situation. 8
c. Carolina doesn t remember in what order she added the ingredients but the last ingredient added required 2 quarter cups. What was the last ingredient Carolina added? Draw a picture to illustrate your solution, and write an equation that represents the situation. 9
Math Glossary, AK State Math Standards 20
SENSE-MAKING VS. ANSWER-GETTING What is the whole? What size pieces do we have? How many of those pieces? Estimate! (Is the answer reasonable?) How can I best represent this problem? Which is greater? or Why does your answer make sense? 2
MORE QUESTIONS FOR SENSE-MAKING Draw a picture to show that 3 x 2/5 has the same value as 2/5 x 3. Name 2 fractions to multiply that result in: a product that is less than both fractions a product that is greater than both fractions FRACTIONS IN THE NEW STANDARDS 3 rd Grade 4th Grade 3.NF: Develop understanding of fractions as numbers. (limited to fractions w/denominators 2, 3, 4, and 8) 4.NF: Extend understanding of fraction equivalence and ordering. 4.NF: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (limited to fractions w/denominators 2, 3, 4, 5,, 8, 0, 2, & 00) 5th Grade 5.NF: Use equivalent fractions as a strategy to add and subtract fractions. 5.NF: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 22
What are some key takeaways about multiplication & division? Resource for this webinar Multiplying Whole Numbers and Fractions https://www.teachingchannel.org/videos/multiplying-fractions-by-whole-numbers-lesson Agree?, Argue?, Aspire? 23
FRACTION PROBLEM SOLVING Devin spent 2/5 of his money on a computer game that cost $8. How much money did Devin have at first, before he bought the game? FRACTION PROBLEM SOLVING Katrina used 2/5 of a liter of paint. Mai used /0 of a liter less paint than Katrina. How much paint did the two girls use altogether? OR. Katrina used 2/5 of a liter of paint. Mai used /0 of a liter less paint than Katrina. How much paint did the two girls use altogether? 24