Transitioning to the Common Core State Standards – Mathematics Pam Hutchison

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AGENDA Dividing Fractions Ratios and Proportional Reasoning Stoplighting the CCSS

Spending Spree David spent of his money on a game. Then he spent of his remaining money on a book. If he has $20 left, how much money did he have at first?

Fractions

The unit is split it into 5 equal pieces so each piece represent of that unit. The point is a distance of 3 of those pieces from zero, so the name of the point is. 01 ●

Fraction Concepts

Multiplying Fractions

x 2 Three groups of two

0 1 / 2 2 / 2 3 / 2 4 / 2 3 x ½ Three groups of one-half 1( 1 / 2 ) 2( 1 / 2 ) 3( 1 / 2 )

Multiplying Fractions Remember our protocol for naming fractions  -size pieces

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Multiplying

Multiplying

Area Model 2 x 3 = 3 2 6

Area Model 1 1

Dividing Fractions

= “How many groups of 2 can I make with 6?”

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/ “How many ¼’s are there in 1?”

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/ “How many ¼’s are there in 2?”

0 1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4 9/4 10/4 11/4 12/ “How many ¼’s are there in 3?”

How Many __ are in __? Solve each problem using pictures and a number sentence involving division. a. How many fives are in 15? b. How many halves are in 3? c. How many sixths are in 4?

How Many __ are in __? d. How many two-thirds are in 2? e. How many three-fourths are in 2? f. How many ’s are in ? g. How many ’s are in ?

= “If I split this into 2 equal parts, now long is each piece?”

0 1/2 1

Dividing Fractions At this point, have you noticed any patterns?

Dividing Fractions Solve the following. 1)2) 3)4)

Dividing Fractions Dan observes that = 6 ÷ 2 He says, I think that if we are dividing a fraction by a fraction with the same denominator, then we can just divide the numerators. Is Dan’s conjecture true for all fractions? Explain how you know.

Running to School 2 The distance between Rosa’s house and her school is mile. She ran mile. What fraction of the way to school did she run?

Running to School 3 Rosa ran of the way from her home to school. She ran mile. How far is it between her home and school?

Dividing Fractions Divide by finding a common denominator. Check using traditional fraction division. 1)2)

= “How many groups of 3 can I make with 5?” 1 whole group with a remainder of 2 or 1 r2 1 whole group and of another group or

“How many groups of ⅔ can I make with 3?” 4 r | | | | | | | | | |

“How many groups of ⅔ can I make with 3?” How many full groups can I make? | | | | | | | | | | How pieces do I have left out of how many I need?

“How many groups of ¾ can I make with 2?”

“How many groups of ¾ can I make with 2½?” | | | | | | | | | | | | | |

Dividing Fractions

So, we typically tell students to Why does it work?

x x x 1

x 1 = 1 =

Ratios and Proportional Reasoning

Games At Recess The students in Mr. Hill’s class played games at recess. 6 boys played soccer 4 girls played soccer 2 boys jumped rope 8 girls jumped rope

Games At Recess Afterward, Mr. Hill asked the students to compare the boys and girls playing different games. Mika said, “Four more girls jumped rope than played soccer.” Chaska said, “For every girl that played soccer, two girls jumped rope.”

Games At Recess Discuss at your tables: How could this activity from Illustrative Mathematics help students develop an understanding of what a ratio is?

Ratios Talk at your tables What is a ratio? How is a ratio similar to a fraction? How is a ratio different from a fraction?

Ratios Talk at your tables What is a ratio? How is a ratio similar to a fraction? How is a ratio different from a fraction?

What is a Ratio? Math is Fun A ratio compares values. A ratio says how much of one thing there is compared to another thing. Purple Math A "ratio" is just a comparison between two different things.

RP 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

Ratios Talk at your tables What is a ratio? How is a ratio similar to a fraction? How is a ratio different from a fraction?

Fractions Is the same as ?

Ratios DEPENDS

Ratios In Ms. Yarrow’s class, there are 3 boys for every 5 girls. Three students wrote ratios to reflect this information. Drew wrote 3 : 5 JoAnn wrote 5 : 3 Kris wrote 5 : 8 Which, if any, of the ratios are correct?

In Ms. Yarrow’s class, there are 3 boys for every 5 girls. Three students wrote the following ratios and explanations to reflect this information. Drew wrote 3 : 5 because there were 3 boys for every 5 girls JoAnn wrote 5 : 3 because there were 5 girls for every 3 boys Kris wrote 5 : 8 because there were 5 girls out of every 8 students in the class Again, which ratios are correct?

Ratios Is the same as ? The ratio of boys to girls The ratio of girls to boys

What is a Ratio? Wikipedia In layman's terms a ratio represents, for every amount of one thing, how much there is of another thing. For example, supposing one has 8 oranges and 6 lemons in a bowl of fruit, the ratio of oranges to lemons would be 4:3 (which is equivalent to 8:6) while the ratio of lemons to oranges would be 3:4. Additionally, the ratio of oranges to the total amount of fruit is 4:7 (equivalent to 8:14). The 4:7 ratio can be further converted to a fraction of 4/7 to represent how much of the fruit is oranges.

Fractions Part to whole Ratios Part to Whole Part to Part Whole to Part Exist in context

Problem #1 The ratio of boys to girls at a party was 2:3. If there were 9 girls, how many boys were at the party?

Ratios Look at Problem #2. How is this problem similar to the previous problem? How is this problem different than the previous problem?

Problem #2 The ratio of boys to girls in the 6 th grade class was 5:7. If there are 108 sixth grade students, how many are boys? How many are girls?

Ratios Now look at Problem #3. How is this problem different than the previous problems? How does this effect the model? What questions or prompts would you give students to help draw their attention to the difference?

Problem #3 The ratio of cheese pizzas to all pizzas sold at Sal’s Pizzeria last Thursday was 2:9. There were 18 cheese pizzas sold that day. How many pizzas were sold in all on Thursday?

Ratios Take a few minutes and solve some more of the problems. How do these types of problems support students’ conceptual understanding of ratio problems?

Proportions What is a proportion? The statement that two ratios are equal in the sense that they both convey the same relationship

Fractions How do we solve 8 SPLITTING

Ratios How do we solve REPLICATING 8

Proportions Solve by replicating.

Replicating Why does replicating make more sense when talking about ratios? For example, consider the following problem: The ratio of boys to girls at the dance was 4:5. If there are 108 students at the dance, how many are boys? How many are girls?

Proportions Solve. What is the most efficient way to solve each problems?

Sharing Gasoline Costs Shell Center Problem Solving Lesson

Analyze Solutions Look at individual solutions, making a list of questions that you might ask the student. Do you see any strategies that you did not think of as you were working on the problem?

Sample Response: Adam What questions might you ask Adam? What errors did Adam make?

Sample Response: Kimberley What questions might you ask Kimberly? What errors did Kimberly make?

Sample Response: Donna P-78 What questions might you ask Donna? What errors did Donna make?

Rates What is a rate? MathSteps: A rate is a special ratio in which the two terms are in different units. For example, if a 12-ounce can of corn costs 69¢, the rate is 69¢ for 12 ounces. The first term of the ratio is measured in cents; the second term in ounces.

RP 2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Expectations for unit rates in this grade are limited to non- complex fractions.

Running at a Constant Speed A runner ran 20 miles in 150 minutes. If she runs at that speed, a. How long would it take her to run 6 miles? b. How far could she run in 15 minutes? c. How fast is she running in miles per hour? d. What is her pace in minutes per mile?

Friends Meeting on Bicycles Taylor and Anya live 63 miles apart. Sometimes on a Saturday, they ride their bikes toward each other's houses and meet somewhere in between. Taylor is a very consistent rider - she finds that her speed is always very close to 12.5 miles per hour. Anya rides more slowly than Taylor, but she is working out and so she is becoming a faster rider as the weeks go by.

RP 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.