8/9/16

Teaching Proportional Reasoning: The Role of Schema-Based Instruction Presented at Kansas Multi-Tier System of Supports Symposium Wichita, Kansas September 8, 2016 DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Proportionality •  The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) considers proportionality to be “of such great importance that it merits whatever time and effort must be expended to assure its careful development” (p. 82). •  In the Common Core State Standards (CCSS; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), instructional time focused on proportionality occurs in Grade 7 when students “develop understanding of proportionality to solve single and multistep problems…solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease” (p. 46). DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Adult proficiency with proportionality 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

Incorrect

Correct

Compute Loan Interest

Calculate Miles Per Gallon

Calculate a 10% Tip NMAP (2008)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Proportional Reasoning Proportional reasoning refers to the understanding of “structural relationships among four quantities (say a, b, c, d) in a context simultaneously involving covariance of quantities and invariance of ratios or products; this would consist of the ability to discern a multiplicative relationship between two quantities as well as the ability to extend the same relationship to other pairs of quantities” (Lamon, 2007, p. 638). DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

A problem to get us started If there are 300 calories in 100 g of a certain food, how many calories are there in a 30 g portion of this food? Sorry that I spilled some coffee on part of the problem! Please do your best to solve this problem, even though you can’t see part of one of the numbers, then discuss your strategy and your solution with those around you. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

TIMSS 1999 8th grade problem If there are 300 calories in 100 g of a certain food, how many calories are there in a 30 g portion of this food? A.  B.  C.  D.  E. 

90 100 900 1000 9000

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Different set-ups … 300 cal x cal = 100 g 30 g

300 cal 100 g = x cal 30 g

100 g 30 g = 300 cal x cal

x cal 30 g = 300 cal 100 g

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Different set-ups & strategies

300 cal x cal = 100 g 30 g

300 cal 100 g = x cal 30 g

100 g 30 g = 300 cal x cal

x cal 30 g = 300 cal 100 g

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

TIMSS results International average: 69% US 8th graders: 68% 24 countries scored higher than the US

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Why is this problem hard?

Take a few minutes to talk to those around you.

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Why is this problem hard? 1.  Students are challenged to understand what the problem is about. Because of reading difficulties, unfamiliar words, and unfamiliar units, students have trouble figuring out which parts of the problem they should pay attention to and use in trying to solve it.

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Why is this problem hard? 2. Students are challenged to pull out the mathematics from the words of the problem. In particular...

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Consider these variations Original Problem:

If there are 300 calories in 100 g of a certain food, how many calories are there in a 30 g portion of this food? Variations:

a)  How many calories are in 30 g of a certain food, given that there are 300 calories in 100 g of the same food? b)  A serving size of 100 g of a certain food has 300 calories. How many calories would a smaller serving size of 30 g have? c)  Your dad gives you a 30 ounce Hershey’s dark chocolate mega-kiss in your lunch box. If a 100 ounce brick of Hershey’s chocolate contains 300 micropops of caffeine, how much caffeine is in the mega-kiss? DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

... pull out the mathematics ... Word problems are difficult because there are so many seemingly-different ways to write the same mathematical situation. It is hard for students to look beyond the surface features of a problem to glimpse and recognize the underlying mathematical structure. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Similarly... Look at this page of word problems. How many different kinds of problems do you see?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Mathematical structure Mathematically, there is only ONE kind of problem on this entire page. Yet students have trouble seeing that all of these problems have the same mathematical structure and thus are the same type of word problem. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Why is this problem hard? 3. Even when students are able to figure out the words and the problem type, they might not know what they should do next. Translating from the words to an equation is hard, as is moving fluently between different equivalent forms of the proportion. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

In sum, word problems are hard... ... because students have trouble with the words in the problem, figuring out what type of problem it is, and coming up with strategies for solving the problem. These challenges are especially acute in the domain of ratio and proportion. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Schema-Based Instruction (SBI) •  SBI emphasizes the role of the mathematical structure (semantic) of a problem. –  A schema is an organized structure “consisting of certain elements and relations” specific to a situation (Mayer, 1999, p. 228)

–  Schemata are the appropriate mechanism to “capture both the patterns of relationships as well as their linkages to operations” (Marshall, 1995, p. 67)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Summary of SBI Effects … Goal 2 Studies

Goal 3 Study

Study 11

Study 22

Study 33 4

Study 15 6

Students

148

475 

1,163 (MDRD+MD only: n = 260)

1,999 (MD: n = 806)

Teachers

6

6

15

82

Classrooms

8

21

42

82

Schools Treatment Duration Design

1

3

6

58

2 weeks

6 weeks

6 weeks

6 weeks

Ratio & proportion

Topics ES, Problem Solving

d = 0.45

ES, Retention

d = 0.56

Classrooms randomly assigned to condition Ratio, proportion, Ratio, proportion, & Ratio, proportion, & & percent percent percent Multilevel Multilevel standardized Multilevel standardized standardized ES ES = 0.36 (MDRD+ ES = 0.46 (MD: g = = 0.32 MD only: g = 0.40) 0.32) Multilevel standardized Multilevel standardized ES = 0.32 (MD: g = ES = 0.29 (MDRD+ ns 0.25) MD only: g = 0.42)

1Jitendra,

Star, Starosta, Leh, Sood, Caskie, Hughes, & Mack, 2009; 2Jitendra, Star, Rodriguez, Lindell, & Someki, 2011; 3Jitendra, Star, Dupuis, & Rodriguez, 2013; 4Jitendra, Dupuis, Star, & Rodriguez, 2016; 5Jitendra, Harwell, Dupuis, Karl, Lein, Simonson, & Slater, 2015; 6Jitendra, Harwell, Dupuis, & Karl, in press.

Summary of SBI Effects … Goal 3 Study 15 6

Goal 3 Study 27

Students

1,999 (MD: n = 806)

373 (MD: n = 253)

Teachers

82

20

Classrooms

82

20

Schools

58

Treatment Duration Topics ES, Problem Solving

ES, Retention

10

6 weeks

6 weeks

Ratio, proportion, & percent

Ratio, proportion, & percent

Multilevel standardized ES = 0.46 (MD: g = 0.32) Multilevel standardized ES = 0.32 (MD: g = 0.25)

Multilevel standardized ES = 0.63 (MD: g = 0.51) Multilevel standardized ES = 0.33 (MD: g = 0.35)

5Jitendra, Harwell, Dupuis, Karl, Lein, Simonson, & Slater, 2015; 6Jitendra, Harwell, Dupuis, & Karl, in press; 7Jitendra, Harwell, et al., in review.

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!

SCHEMA'BASED'INSTRUCTION'(SBI)'CURRICULUM'

'

SCOPE'AND'SEQUENCE' LESSON!

! CONTENT!

!

UNIT!1:!RATIOS!AND!PROPORTIONS!

1!&!2! 3!&!4! 5!

SCHEDULE! !

Ratios!

3!days!

Solving!Ratio!Word!Problems:!! PartBtoBPart!and!PartBtoBWhole!Comparisons! Rates!! (Quiz!over!Lessons!1B4)!

3!days! 1!day!

6!&!7!

Solving!Proportion!Problems!

8!&!9!

Scale!Drawing!Problems!! (Quiz!at!end!of!Lesson!9!over!Lessons!1B7)!

3!days!

Unit!1!Review!

1!day!

10!

3!days!

!

!

!

!

UNIT!2:!PERCENTS!

!

11!

Fractions,!Percents,!&!Decimals!

12!

Solving!Percent!Problems:!PartBtoBWhole!Comparisons!

3!days! 13!&!14!

Solving!Percent!Problems:!Percent!of!Change!

15!

Solving!Sales!Tax!and!Tips!Problems:!Percent!of!Change!! (Quiz!over!Lessons!11B14)!

16!

Solving!Markup!&!Discount!Problems:!Percent!of!Change!

17!

Solving!Multistep!Percentage!Adjustment!Problems!

3!days! 3!days!

3!days! 18!

Solving!Simple!Interest!Problems!

19!

Review:!Identifying!and!Categorizing!Problem!types!! (Ratio,!Proportion,!and!Percent)!

20!

Unit!2!Review!

21!

Units!1!&!2!Review!

1!day! 1!day! 1!day!

!

Total! 29'days!

!

Ratio & proportion •  Ratios •  Lessons 1 & 2

•  Proportion Word Problems •  Lessons 6-7

•  Ratio Word Problems •  Lessons 3 & 4

•  Scale Drawing Problems •  Lessons 8-9

•  Rates •  Lesson 5

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

RATIO PROBLEMS DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Consider these problems: The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in Ms. Robinson’s class is 12. How many children are in the class? (Problem 3.6) There are 28 employees at the local bank. Every morning 5 out of 7 employees use Route A to drive to work. How many employees use Route A to drive to work? (Problem 4.3)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Common Ratio Phrases Common%ratio%phrases:%% • a:b$ • a$is$to$b$ $ $ • for$every$b$there$is$an$a$ • n$times$as$ much/many$as$

• a$to$b$ $ th$ • n of$

• a$per$b$ $ • a$out$of$b$$$ (only%for%part3to3whole% comparisons)$

Examples:% • 2$girls$is#to!3$boys$ $ • 7$pizza$orders$for#every!10$ lunch$orders$ $

• 4$cars$to$9$trucks$ $ • 3$times#as#many!nurses$as$ doctors$ •

$

• 24$pencils$per$box$ $ • 1$vacation$day$out#of! 5$work$days$

1 as#many!doctors$as$ 3

nurses$ $

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

“Ratio” problems

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Applying the Schematic Diagram Problem 2: There are 28 employees at the local bank. Every morning 5 out of 7 employees use Route A to drive to work. How many employees use Route A to drive to work? (Problem 4.3)

x Employees using Route A

28 All employees

5 7

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Another Curriculum Feature… DISC Checklist: Step 1. Discover the problem type Step 2: Identify information in the problem to represent in a diagram Step 3: Solve the problem Step 4: Check the solution

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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1. Discover the problem type ü  Read and retell problem to understand it The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in the class is 12. How many children are in the class?

ü  Ask self if this is a ratio problem A ratio describes a multiplicative relationship between two quantities in a single situation.

ü  Ask self if problem is similar or different from others that have been seen before DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Common Ratio Phrases Common%ratio%phrases:%% • a:b$ • a$is$to$b$ $ $ • for$every$b$there$is$an$a$ • n$times$as$ much/many$as$

• a$to$b$ $ th$ • n of$

• a$per$b$ $ • a$out$of$b$$$ (only%for%part3to3whole% comparisons)$

Examples:% • 2$girls$is#to!3$boys$ $ • 7$pizza$orders$for#every!10$ lunch$orders$ $

• 4$cars$to$9$trucks$ $ • 3$times#as#many!nurses$as$ doctors$ •

$

• 24$pencils$per$box$ $ • 1$vacation$day$out#of! 5$work$days$

1 as#many!doctors$as$ 3

nurses$ $

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Identify the information in the problem to represent in a diagram

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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2. Identify the information in the problem to represent in a diagram •  Underline the ratio or comparison sentence •  Write compared and base quantities and units in the diagram •  Write the value of the ratio between the two quantities (the ratio value) •  Write an x for what must be solved

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Identify information in the problem to represent in a diagram ü 

Underline the ratio or comparison sentence.

ü 

Write compared and base quantities and units in the diagram.

ü  ü 

Write the value of the ratio between the two quantities (the ratio value) in the diagram. Write an x for what must be solved.

12 girls x children

2 5

The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in Ms. Robinson’s class is 12. How many children are in the class? (Problem 3.6)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem First, try to come up with an estimate for the answer 12 girls x children

2 5

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Sample Estimate “I know that the number of children is greater than 12 since there are 12 girls and some boys. We can make our estimate slightly more accurate by thinking about 2 . 5 That is, 2 is just a little less than 1 . So this suggests that 5 2 a little less than half the children are girls (and a little more than half of the children are boys). So if there are 12 girls and little more than 12 boys, we can come up with a very rough estimate that there are a little more than 24 children total. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Estimation challenges •  No set strategy for generating estimates – it depends on the problem. •  Some students may not see the value in producing estimates, especially for problems with relatively easy numbers. •  Each student may come up with a different estimate.

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem After an estimate: •  Translate information in the diagram into a math equation •  Plan how to solve the equation •  Which strategy will you use? •  The SBI curriculum uses three strategies…

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Multiple strategies... There are three strategies that are taught:

Unit Rate

We want students to know all three.

Cross Multiply

We want students to be able to use the “best” one for a given problem.

Equivalent Fractions

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem 1)  Try to come up with an estimate for the answer 2)  Translate information in the diagram into a math equation

12 girls 2 = x children 5

3)  Plan how to solve the equation: 4)  Now… Solve the math equation and write the complete answer. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

4. Check the solution •  Look back to see if the estimate in Step 3 is close to the exact answer •  Check to see if the answer makes sense

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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PROPORTION PROBLEMS DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Consider these problems: Amanda has the following banana shake recipe that serves 16 people: Banana$Shake$(16$servings)! 8!bananas!! 1!cup!sugar! 2!quarts!milk! 4!cups!vanilla!ice!cream!

If Amanda wants to make banana shakes for her family of 4 using this recipe, how many bananas does she need? (Review Problem 7.1)

Ming watched TV for 8 hours on Saturday and saw 56 food commercials. How many food commercials did she watch every hour? (Practice Problem 7.3) DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

“Proportion” problems

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Applying the Schematic Diagram Amanda has the following banana shake recipe that serves 16 people: Banana$Shake$(16$servings)! 8!bananas!! 1!cup!sugar! 2!quarts!milk! 4!cups!vanilla!ice!cream!

If Amanda wants to make banana shakes for her family of 4 using this recipe, how many bananas does she need? (Review Problem 7.1)

8 bananas

x bananas

16 people

4 people

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the Schematic Diagram

Ming watched TV for 8 hours on Saturday and saw 56 food commercials. How many food commercials did she watch every hour?

56 food commercials

x food commercials

(Practice Problem 7.3)

8 hours

1 hour

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Let’s Solve These Problems Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing? (Problem 6.4) Carlos is on the school’s track team. He takes 54 minutes to run 6 miles. Assuming that he runs at a constant pace for all 6 miles, how long did it take him to run 2 miles? (Problem 7.2) The Frank family from Arizona, USA, is going to Britain for their summer vacation. They exchanged $50 for £27 (British pounds). At that exchange rate, how many British pounds could they get for $75? (Practice Problem 7.4)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Applying the DISC checklist DISC Checklist: Step 1. Discover the problem type Step 2: Identify information in the problem to represent in a diagram Step 3: Solve the problem Step 4: Check the solution

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

1. Discover the problem type ü  Read and retell problem to understand it Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing?

ü  Ask self if this is a ratio or proportion problem A ratio describes a multiplicative relationship between two quantities in a single situation. A proportion is a statement of equality between two ratios/rates that allows us to think about the ways that two situations are the same.

ü  Ask self if problem is similar or different from others that have been seen before DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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2. Identify the information in the problem to represent in a diagram

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Identify the information in the problem to represent in a diagram •  Underline the two quantities that form a specific ratio/rate •  Write names of the two quantities that form a specific ratio/rate in the diagram •  Write quantities and units for each of the two ratios/rates in the diagram •  Write an x for what must be solved DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Identify information in the problem to represent in a diagram ü 

Underline the two quantities that form a specific ratio/rate

ü 

Write names of the two quantities that form a specific ratio/rate in the diagram

ü  ü 

Write quantities and units for each of the two ratios/rates in the diagram Write an x for what must be solved

cans of icing

cupcakes

2 cans of icing

24 cupcakes

3 cans of icing

x cupcakes

Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing? DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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3. Solve the problem 1)  Try to come up with an estimate for the answer “I know that 2 cans of icing will cover 24 cupcakes. So 3 cans of icing will cover more than 24 cupcakes. I’m not sure exactly how much more (as this is what we will figure out when we solve this problem), but I know my answer needs to be bigger than 24 cupcakes.”

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem 2) Translate information in the diagram into a math equation

2 cans of icing 3 cans of icing = 24 cupcakes x cupcakes 3) Plan how to solve the problem •  Which strategy will you use? DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem 4) Solve the math equation and write the complete answer.

2 cans of icing 3 cans of icing = 24 cupcakes x cupcakes 2 times 12 is 24, so 3 times 12 is 36 Answer: Toshi can ice 36 cupcakes with 3 cans of icing. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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4. Check the solution •  Look back to see if the estimate in Step 3 is close to the exact answer •  Check to see if the answer makes sense

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Estimation Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing? (Problem 6.4)

“I know that 2 cans of icing will cover 24 cupcakes. So 3 cans of icing will cover more than 24 cupcakes. I’m not sure exactly how much more (as this is what we will figure out when we solve this problem), but I know my answer needs to be bigger than 24 cupcakes.”

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Estimation Carlos is on the school’s track team. He takes 54 minutes to run 6 miles. Assuming that he runs at a constant pace for all 6 miles, how long did it take him to run 2 miles? (Problem 7.2)

“Well, I first know that the answer should be less than 54 minutes. If he runs 6 miles in 54 minutes, he should run 2 miles in fewer than 54 minutes. Half of 6 is 3; if he ran 3 miles, this would be half of running 6 miles. But he is running 2 miles, so this is LESS THAN half of 6. Half of 54 is 27, so my estimate is that Carlos will run 2 miles in less than 27 minutes.” “Since 2 miles is around half (or a bit less) than 6, around half (or a bit less) than 54 would be between 20 and 30.” DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Estimation The Frank family from Arizona, USA, is going to Britain for their summer vacation. They exchanged $50 for £27 (British pounds). At that exchange rate, how many British pounds could they get for $75? (Practice Problem 7.4)

“Since $75 is more than $50, I know that the Franks will get more than £27. Double $50 would be $100 (which is more than $75), and double £27 would be around £60, so I estimate that the Franks will get between £27 and £60.”

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Multiple Strategies Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing? (Problem 6.4) Carlos is on the school’s track team. He takes 54 minutes to run 6 miles. Assuming that he runs at a constant pace for all 6 miles, how long did it take him to run 2 miles? (Problem 7.2) The Frank family from Arizona, USA, is going to Britain for their summer vacation. They exchanged $50 for £27 (British pounds). At that exchange rate, how many British pounds could they get for $75? (Practice Problem 7.4)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Using Multiple Solution Strategies… Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing? (Problem 6.4) Unit Rate Cross Multiply Equivalent Fractions

2 times 12 is 24, so 3 times 12 is 36.

2

=

2x = 3 times 24

(not especially easy to use on this problem)

3

24

x

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Using Multiple Solution Strategies… Carlos is on the school’s track team. He takes 54 minutes to run 6 miles. Assuming that he runs at a constant pace for all 6 miles, how long did it take him to run 2 miles? (Problem 7.2) Unit Rate Cross Multiply Equivalent Fractions

6 times 9 is 54, so 2 times 9 is 18.

54

=

6x = 2 times 54

2 times 3 is 6, so what times 3 is 54?

x

6

2

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Using Multiple Solution Strategies… The Frank family from Arizona, USA, is going to Britain for their summer vacation. They exchanged $50 for £27 (British pounds). At that exchange rate, how many British pounds could they get for $75? (Practice Problem 7.4)

Unit Rate

(not especially easy to use on this problem)

Cross Multiply

50x = 27 times 75

Equivalent Fractions

(not especially easy to use on this problem)

50

75

= 27

x

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Nuance #1: Ratios vs. Proportions

Why can’t a ratio problem be solved using the proportion diagram?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Nuance #1, cont. The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in Ms. Robinson’s class is 12. How many children are in the class? (Problem 3.6)

✓ yes 12 girls

✗ no 2 5

12 girls

2

x children

5

x children

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Why not? •  Mathematically OK, but conceptually confusing •  Emphasis in curriculum on identifying the problem type •  Ratio problems do not fit into the IF-THEN structure that is central to the definition of proportion problems (as well as the schema diagram) •  Ratios are fundamentally a different type of problem DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Why not? Ratios are Different! The ratio of the number of girls to the total number of children in Ms. Robinson’s class is 2:5. The number of girls in Ms. Robinson’s class is 12. How many children are in the class? (Problem 3.6)

Ratio: A comparison of any two quantities that expresses a multiplicative relationship between two quantities in a single situation. Proportion: A statement of equality between two ratios/rates that allows us to think about the ways that two situations are the same. Often in a proportion problem, there is an If‐Then statement of equality between two ratios/rates that allows us to think about the ways that two situations are the same. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Nuance #2

Does it matter which way the quantities are arranged in the proportion diagram?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Nuance #2: It matters! Carlos is on the school’s track team. He takes 54 minutes to run 6 miles. Assuming that he runs at a constant pace for all 6 miles, how long did it take him to run 2 miles? (Problem 7.2)

54 minutes

x minutes

6 miles

2 miles

54 minutes

6 miles

6 miles

2 miles

54 minutes

x minutes

x minutes

2 miles

✓ yes

✓ yes

✗ no

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Why does it matter? •  Mathematically OK but conceptually confusing •  Preserve and highlight IF-THEN relationship in proportion problems •  We want to avoid “unitless” units (e.g., dollars/dollar, miles/mile)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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Review Key features of the SBI curriculum: ✓ Recognizing problem types ✓ Generating estimates

✓ Knowing multiple strategies

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Identifying problem types Ratios

Proportions

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Problem Type

Example problem

Ratio Last weekend, Will helped his father plant a garden. For every 3 tomato plants, they planted 2 basil plants. If they planted 6 tomato plants, how many basil plants did they plant? Proportion Construction of a new mall was costing about $3 million a day. At this rate, how long would it take to spend $96 million?

Scale Drawing

Tammy looks on a map and finds that 3 Atlantic City beach is 2 4 cm from the hotel where she is staying. The scale of the map is 1 in. represents 500 ft.. How many feet away is the beach from the hotel?

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Generating estimates •  Estimates should be quick and easy; it is OK to merely identify a rough ballpark range (upper and lower bound). •  Estimates frequently will rely upon benchmark numbers and fractions such as ½ or 50%. •  Estimates are generated for all word problems and then referred back to after the exact answer is computed.

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Using multiple strategies There are three strategies that are taught: We want students to know all three. We want students to be able to use the “best” one for a given problem.

Unit Rate Cross Multiply Equivalent Fractions

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

PERCENT PROBLEMS DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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A problem to get us started If the average salaries of a particular group within a population are 16 percent less than the average salary of the entire population, and one wants to give the individuals in that group a raise to bring them up to parity, what should the raise be – 16 percent, something more, or something less? (Mathematical proficiency for all students : Toward a strategic research and development program in mathematics education / RAND Mathematics Study Panel.) DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Two more problems … 1.  A shop increased its prices by 20%. What is the new price of an item which previously sold for $800? 2.  When a new highway is built, the average time it takes for a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. What is the percent decrease in time taken to travel between the two towns? DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

TIMSS 2003 8th grade problem A shop increased its prices by 20%. What is the new price of an item which previously sold for $800? A.  B.  C.  D. 

$640 $900 $960 $1,000

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

27

8/9/16

TIMSS results International average: 49% US 8th graders: 57% 16 countries scored higher than the US

TIMSS 2003 8th grade problem When a new highway is built, the average time it takes for a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. What is the percent decrease in time taken to travel between the two towns? A.  B.  C.  D. 

4% 5% 20% 25%

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

TIMSS results International average: 31% US 8th graders: 35% 19 countries scored higher than the US

28

8/9/16

Schematic Diagram for “Percent of Change” Problems

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the Schematic Diagram A shop increased its prices by 20%. What is the new price of an item which previously sold for $800?

$?

20 100

$800

$800

$x

$?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the Schematic Diagram A shop increased its prices by 20%. What is the new price of an item which previously sold for $800? $?

$160

$?

$800

20 100

$?

$800

$800

=

=

20 100

20 100

Answer: The new price is $960.

$800

+

$160

$?

$x

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

29

8/9/16

Applying the Schematic Diagram When a new highway is built, the average time it takes for a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. What is the percent decrease in time taken to travel between the two towns?

? min

x 100

25 min

20 min

? min

25 min

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the Schematic Diagram When a new highway is built, the average time it takes for a bus to travel from one town to another is reduced from 25 minutes to 20 minutes. What is the percent decrease in time taken to travel between the two towns?

5 x = !25 100

5 min

? min

x 100

5 times 5 is 25, so 5 times x is 100. x = 20

25 min +

25 min

– 5 min

? min

20 min

5 x = !25 100 25 times 4 is 100, so 5 times 4 is 2 (x = 20)

Answer: 20% DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Percent •  Fractions, Percent, Decimals •  Lesson 11 •  Solving Percent Problems: Part-to-Whole Comparisons •  Lesson 12 •  Solving Percent Problems: Percent of Change •  Lessons 13 & 14

•  Sales Tax & Tips Problems •  Lesson 15 •  Markup & Discount Problems •  Lesson 16 •  Multistep Percentage Adjustment Problems •  Lesson 17 •  Simple Interest Problems •  Lesson 18

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

30

8/9/16

Take a look at this problem Alan’s parents increased his weekly allowance by 10% from last year. If he was getting $20 a week last year how much will he get per week this year? (Problem 13.4)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the DISC checklist DISC Checklist: Step 1. Discover the problem type Step 2: Identify information in the problem to represent in a diagram Step 3: Solve the problem Step 4: Check the solution

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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8/9/16

1. Discover the problem type •  Read and retell problem to understand it •  Ask self if this is a ratio, proportion, or percent problem •  Ask self if problem is similar or different from others that have been seen before Alan’s parents increased his weekly allowance by 10% from last year. If he was getting $20 a week last year how much will he get per week this year?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Identify the information in the problem to represent in a diagram

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Identify the information in the problem to represent in a diagram •  Underline the percent sentence in the problem •  Write information (e.g., part, whole, ratio value; change, original, new, ratio value) in the problem in the diagram •  Write an x for what must be solved

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

32

8/9/16

2. Identify the information in the problem to represent in a diagram Alan’s parents increased his weekly allowance by 10% from last year. If he was getting $20 a week last year how much will he get per week this year? (Problem 13.4) !

!"#$%&'() !2%34')

$?

10 100

!

5&.4.3%0)

,%-.")) /%01')

$20 *%+')

$20 5&.4.3%0)

6)

$?

!

!2%34')

$x 7'8)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem •  Try to come up with an estimate for the answer “It’s hard to come up with something, but I do know a few things about the answer. I know that the answer is more than $20 because Alan received $20 per week from his parents last year, and he got a 10% increase this year. For a rough estimate, I don’t know exactly how much more, but I do not expect the answer will be a big number like $100 or even $50. Just thinking about this situation, I would estimate the amount of increase to be more than $1 and less than $10, since $10 is half of 50% of $20. So, a good estimate of Alan’s weekly allowance this year would be more than $21 but less than $30.” DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem •  Translate information in the diagram into a math equation •  Plan how to solve the equation $? 10 = $20 100

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

33

8/9/16

3. Solve the problem Now… •  Solve the math equation and write the complete answer. $? 10 = $20 100

$? 10 = $20 100

10 times 10 is 100, so ? times 10 is 20. ?=2

20 times 5 is 100, so ? times 5 is 10. ?=2

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Solve the problem !

!"#$%&'() $2

!2%34')

$?

10 100

!

5&.4.3%0)

,%-.")) /%01')

$20 *%+')

$20 5&.4.3%0)

6)

+$2

$?

!

!2%34')

$x 7'8)

Answer: Alan’s allowance this year is $22. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

4. Check the solution •  Look back to see if the estimate in Step 3 is close to the exact answer •  Check to see if the answer makes sense

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

34

8/9/16

Solve these problems: There is a great variation between day and night temperatures in summer in the deserts of Rajasthan, India. Find the percent of change, in degrees Fahrenheit in the deserts of Rajasthan from 120oF in the daytime on May 5, 2006, to 84oF that night. (Problem 14.2) Ricardo took his family out to dinner. The bill was $60. If Ricardo wants to leave the server a 15% tip, how much money should he leave? What was the total cost of the meal? (Problem 15.5) EB Games purchases Wii memory cards for $40 each (the wholesale price). They then sell the memory cards for 25% over (i.e. the markup) the price they paid for them. What is the markup price (the retail price) of each Wii memory card at EB Games? (Problem 16.3) DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

“Percent of change” problems

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the Schematic Diagram There is a great variation between day and night temperatures in summer in the deserts of Rajasthan, India. Find the percent of change, in degrees Fahrenheit in the deserts of Rajasthan from 120oF in the daytime on May 5, 2006, to 84oF that night. (Problem 14.2)

? °F

x 100

120 °F

120 °F

? °F

84 °F

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

35

8/9/16

Applying the Schematic Diagram Ricardo took his family out to dinner. The bill was $60. If Ricardo wants to leave the server a 15% tip, how much money should he leave? What was the total cost of the meal? (Problem 15.5)

$?

15 100 $60

$60

$x

$?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Applying the Schematic Diagram EB Games purchases Wii memory cards for $40 each (the wholesale price). They then sell the memory cards for 25% over the price they paid for them (i.e., the markup or retail price). What is the markup, or retail price of each Wii memory card at EB Games? (Problem 16.3)

$?

25 100

$40

$40

$?

$x

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Shortened DISC Step 1. Fill the information given in the problem into the Ratio and Change diagrams. Step 2: Solve first for the ? in the Ratio diagram or Change diagram. Step 3: Cross out the ? in the Ratio and Change diagrams and fill in with the solution from Step 2. Step 4: Solve for the x (or final answer) in the Ratio or Change diagram DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

36

8/9/16

Step 1. Fill the information given in the problem into the Ratio and Change Diagrams Ricardo took his family out to dinner. The bill was $60. If Ricardo wants to leave the server a 15% tip, how much money should he leave? What was the total cost of the meal? (Problem 15.5)

$?

Ratio Diagram

15 100 $60

Change Diagram

$60

$x

$?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2. Solve for the ? in the Ratio Diagram to find the Change amount

$? 15 = $60 100

$?

15 100 $60

$60

$x

$?

60 times 15 is 900, so ? times 100 is 900. ?=9

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

3. Cross out the ? for the change amount in the Ratio and Change diagrams and fill in with the solution from Step 2 !

!"#$%&'() $9

!2%34')

$?

15 100

!

5&.4.3%0)

,%-.")) /%01')

$60 *%+')

$60 5&.4.3%0)

6)

+$9

$? !2%34')

!

$x 7'8)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

37

8/9/16

4. Solve for x or the final answer Ricardo took his family out to dinner. The bill was $60. If Ricardo wants to leave the server a 15% tip, how much money should he leave? What was the total cost of the meal? (Problem 15.5) !

!"#$%&'() $9

!2%34')

$?

15 100

!

5&.4.3%0)

,%-.")) /%01')

$60 *%+') $9

6)

$60

$?

$x

!

!2%34')

5&.4.3%0)

7'8)

$60 + $9 = $69 Answer: The total cost of the meal was $69. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Summarizing... The three schema diagrams are just about all that are needed to solve any word problem problem involving ratio/rates, proportion, scale drawings, percent, percent of change, mark-up, discount, sales tax, and simple interest. Evidence suggests that this is a strength of our SBI approach. DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Ratio, Proportion, and Percent Schema Diagrams Compared

If

Then

Ra7o Value

Base Compared Change Ra7o Value Original Base

& Original

Change

New

38

8/9/16

Identifying Problem Types Percent – Part-whole comparison

Simple Percent of Change

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Identifying Problem Types Complex Percent of Change (including Simple Interest) Compared Change (Simple Interest)

Ra7o Value

Original (Principal)

Base

Original (Principal)



Change

(Simple Interest)

New

(Balance)

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Problem Type

Example problem

Percent: Part-Whole Comparison On an English quiz that was worth 24 points, Janie got 18 points right. Can you find what percent of questions she got correct? Percent of Change A tree that was 10 feet tall grew by 5 feet. What percent has it grown?

Percent of Change On average the temperatures in the Sonoran desert in Arizona in winter reads 500F; on average in the summer, the thermometer reads 900F. What is the percent of change from winter to summer in degrees Fahrenheit?

39

8/9/16

Problem Type

Example problem

Multistep Percent of Change If a fish tank that costs $80 is decreased in price by 20%, then increased by 20%, will the final cost of the fish tank equal the original price of $80? Explain.

Simple Interest Anna deposits $700 in a savings account at the beginning of the year. The simple annual interest rate for the savings account is 5%. What will be the balance in Anna’s account at the end of the year?

In sum … Our SBI approach addresses the Mathematical Practice Standards in the CCSS and recommendations in the IES Practice Guide on “Improving Mathematical Problem Solving in Grades 4 through 8.”

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Improving Problem Solving

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

40

8/9/16

Common Core State Standards Mathematical Practice Standards 1. Make sense of problems and persevere in solving them. •  Explain the meaning of a problem to oneself •  Analyze givens, relationships, and constraints •  Make conjectures and plan a solution pathway instead of simply jumping in •  Monitor and evaluate progress and change course if necessary •  Continually ask oneself “does this make sense?”

SBI Intervention 1. Discover the Problem Type •  Read and retell problem to understand it •  Ask self the type of problem it is •  Ask self if problem is similar to or different from others that have been seen before 3. Solve the problem •  First, try to come up with an estimate for the answer •  Plan which strategy to use to solve the problem •  Make sure the answer makes sense

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Common Core State Standards Mathematical Practice Standards 4. Model with mathematics. •  Identify important quantities in a situation and map their relationship onto diagrams •  Interpret mathematical results in the context of the situation, reflecting on whether the model served its purpose or not

SBI Intervention 2. Identify information in the problem to represent in a diagram •  Underline the two quantities that form a specific ratio/rate •  Write names of the two quantities that form a specific ratio/rate in the diagram •  Write quantities and units for each of the two ratios/rates in the diagram •  Write an x for what must be solved 3. Solve the problem •  Write the complete answer with units and make sure it makes sense

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Common Core State Standards Mathematical Practice Standards

SBI Intervention

5. Use appropriate tools strategically. •  Make sound decisions about when a tool or model might be helpful

2. Identify information in the problem to represent in a diagram

•  Recognize both the insight to be gained from tools/models as well as their limitations

3. Solve the problem

•  Detect possible errors by strategically using estimation

•  Certain diagrams help to represent specific problem types •  First, try to come up with an estimate for the answer •  Students should generate a “ballpark” estimate for each word problem they solve.

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

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8/9/16

Common Core State Standards Mathematical Practice Standards 7. Look for and make use of structure. •  Look closely at a problem to discern a pattern or a structure •  Step back for an overview and be able to shift perspective See that a ratio value can be viewed as a single quantity and as a comparison between two quantities

SBI Intervention 1. 

Discover the Problem Type

•  Look beyond the surface features of a problem to glimpse and recognize the underlying mathematical structure. •  Ask self if problem is similar or different from others that have been seen before

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

IES Practice Guide R2. Model how to monitor and reflect on the problemsolving process. SBI: Metacognitive Strategy Knowledge Questions/Prompts •  What type of problem is this? •  Is the problem similar to or different from others that have been seen before? •  What is the estimate for the answer? •  Which strategy to use to solve the problem? •  Is the estimate close to the exact answer? •  Does the answer makes sense?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

IES Practice Guide R3. Select visual representations that are appropriate for students and the problems they are solving; Use think-alouds and discussions to   teach students how to represent problems visually. Selecting a Visual Representation This is a proportion problem that describes an equality between two ratios/rates that allows us to think about the ways that two situations are the same. So, use a proportion diagram. Toshi used exactly 2 cans of icing to cover 24 cupcakes. How many cupcakes can she ice with 3 cans of icing?

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

2 cans of icing

3 cans of icing

24 cupcakes

x cupcakes

42

8/9/16

IES Practice Guide R4. Provide instruction in multiple strategies. Multiple Solution Strategies Three strategies are taught.

Unit Rate

Students should know all three.

Cross Multiply

Students learn to use the “best” one for a given problem.

Equivalent Fractions

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Wrap up

Questions & Comments

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

Thank you!

DEPARTMENT OF EDUCATIONAL PSYCHOLOGY

43

Ratio Diagram

Compared

Ratio Value

Base

©2014 Regents of the University of Minnesota. All rights reserved.

 

DISC                RATIO  PROBLEM  CHECKLIST  

 

 

 

   

Step  1:    D iscover  the  problem  type  

 

§ §

Read  and  retell  the  problem  to  understand  it.   Ask  if  the  problem  is  a  …                                                                                                                                                                                                       RATIO  PROBLEM  

 a  :  b   Does  this  problem  have  a  part-­‐to-­‐part  or  part-­‐ to-­‐whole   comparison?   Look   for   symbols,   words,  and  phrases  such  as:  “the  ratio  of  a  to   b,”  “a  :  b,”  “a  per  b,”  “a  for  b,”  “a  for  every  b,”   “for   every   b   there   are   a,”   “n   times   as   th   many/much   as,”   “n of,”   “a   out   of   b,”   to   see   whether   there   is   a   ratio   statement   that   tells   about   a   multiplicative   relationship   between   two  quantities  in  a  single  situation.    

§

Ask  if  this  problem  is  different  from  or  similar  to  another  problem  that  has  already  been  solved.  

Step  2:    I dentify  information  in  the  problem  to  represent  in  a  diagram(s)     §

Underline  the  …  

ratio  or  comparison  statement.        

§

       

 

  Write  …    

                 compared   and   base   quantities   and   units   in  the  diagram.                      value   of   the   ratio   between   the   two   quantities  in  the    diagram  (ratio  value).                  “x”  for  what  must  be  solved.          

 

 

 

Step  3:    S olve  the  problem   § § §

Try  to  come  up  with  an  estimate  for  the  answer.   Translate  the  information  in  the  diagram  into  a  math  equation.   Plan  how  to  solve  the  math  equation.    

§  

 

Solve  the  math  equation,  and  write  the  complete  answer.  

Step  4:    C heck  the  solution   § §

Look  back  to  see  if  the  estimate  in  Step  3  is  close  to  the  exact  answer.         Check  to  see  if  the  answer  makes  sense.    

©2014  Regents  of  the  University  of  Minnesota.  All  rights  reserved.  

DISC  Ratio  Checklist    

Proportion Diagram

If

©2014 Regents of the University of Minnesota. All rights reserved.

Then

DISC RATIO  and  PROPORTION  PROBLEM  CHECKLISTS      

 

           

   

 

 

Step  1:    D iscover  the  problem  type   § §

Read  and  retell  the  problem  to  understand  it.   Ask  if  the  problem  is  a  …                                                                                                                                                                                                                                    PROPORTION  PROBLEM  

RATIO  PROBLEM  

 If…Then  

 a  :  b   Does   this   problem   have   a   part-­‐to-­‐part   or   part-­‐to-­‐whole  comparison?  Look  for  symbols,   words,   and   phrases   such   as:   “the   ratio   of  a   to   b,”  “a  :  b,”  “a  per  b,”  “a  for  b,”  “a  for  every   b,”   “for   every   b   there   are   a,”   “n   times   as   th   many/much  as,”  “n of,”  “a  out  of  b,”  to  see   whether   there   is   a   ratio   statement   that   tells   about   a   multiplicative   relationship   between   two  quantities  in  a  single  situation.    

§

Does  the  problem  describe  an  “If…Then”   statement   of   equality   between   two   ratios/rates   that   allows   us   to   think   about   the   ways   that   two   situations   are   the   same?   That   is,   the   If   statement   describes   a   rate/ratio   between   two   quantities   in   one   situation   and   the   Then   statement   involves  either  an  increase  or  decrease  in   the   two   quantities   in   another   situation,   but  with  the  same  ratio.                                              

Ask  if  this  problem  is  different  from  or  similar  to  another  problem  that  has  already  been  solved.  

Step  2:    I dentify  information  in  the  problem  to  represent  in  a  diagram(s)     §

Underline  the  …  

ratio  or  comparison  statement.        

§

I    

 

  Write     …  

two   quantities   that   form   a   specific   ratio/rate.    

compared  and  base  quantities  and  units   in  the  diagram.     value   of   the   ratio   between   the   two   quantities  in  the    diagram  (ratio  value).   “x”  for  what  must  be  solved.          

names   of   the   two   quantities   that form  a  ratio  in  the  diagram.   quantities   and   units   for   each   of   the    two  ratios/rates  in  the  diagram.       “x”   for   what If   Then   must  be  solved.        

 

 

 

Step  3:    S olve  the  problem   § § §

Try  to  come  up  with  an  estimate  for  the  answer.   Translate  the  information  in  the  diagram  into  a  math  equation.   Plan  how  to  solve  the  math  equation.    

§  

 

Solve  the  math  equation,  and  write  the  complete  answer.  

Step  4:    C heck  the  solution   § §

Look  back  to  see  if  the  estimate  in  Step  3  is  close  to  the  exact  answer.         Check  to  see  if  the  answer  makes  sense.    

©2014  Regents  of  the  University  of  Minnesota.  All  rights  reserved.  

DISC  Ratio  and  Proportion  Checklists  

1 2

Percent of Change Diagram

Compared Change Ratio Value Original Base

&

Original ©2014 Regents of the University of Minnesota. All rights reserved.

Change

New

 

DISC    

 

 

RATIO,  PROPORTION,  and  PERCENT   PROBLEM  CHECKLISTS

Step  1:    D iscover  the  problem  type   § §

Read  and  retell  the  problem  to  understand  it.   Ask  if  the  problem  is  a  …                                                                                                                                                                                                                                    PROPORTION  PROBLEM  

RATIO  PROBLEM  

 a  :  b   Does   this   problem   have   a   part-­‐to-­‐part   or   part-­‐to-­‐whole   comparison?   Look   for   symbols,   words,   and   phrases   such   as:   “the   ratio  of  a  to  b,”  “a  :  b,”  “a  per  b,”  “a  for  b,”   “a   for   every   b,”   “for   every   b   there   are   a,”   “n   th   times  as  many/much  as,”  “n of,”  “a  out  of   b,”   to   see   whether   there   is   a   ratio   statement   that   tells   about   a   multiplicative   relationship   between   two   quantities   in   a   single  situation.    

§

 

                                                   PERCENT,    

 If…Then  

                                   PERCENT  OF  CHANGE,  or    %                                SIMPLE  INTEREST  PROBLEM                  

Does   the   problem   describe   an   “If…Then”   statement   of   equality   between   two   ratios/rates   that   allows   us   to   think   about   the   ways   that   two   situations   are   the   same?   That   is,   the   If   statement   describes   a   rate/ratio   between   two   quantities   in   one   situation   and   the   Then   statement   involves   either   an   increase   or   decrease   in   the   two   quantities   in   another   situation,   but   with   the  same  ratio.                                              

Look   for   symbols   or   words   such   as   “%,”   “percent,”   “percent   of   change,”   or   “simple   interest,”   to   see   whether   there   is   a   percent   or   percent   of   change   statement   that   tells   about   a   multiplicative   relationship   between   two  quantities.                                                                        

Ask  if  this  problem  is  different  from  or  similar  to  another  problem  that  has  already  been  solved.  

Step  2:    I dentify  information  in  the  problem  to  represent  in  a  diagram(s)     §

Underline  the  …  

ratio  or  comparison  statement.        

 

 

two   quantities   that   form   a   specific   percent  or  simple  interest  statement.   ratio/rate.             § Write   …     compared   and   base   quantities   and   names   of   the   two   quantities   that                         information  (part,  whole,  or  ratio       units  in  the  diagram.     form  a  ratio  in  the  diagram.   value;   change,   original,   ratio   value,   or     value   of   the   ratio   between   the   two   quantities   and   units   for   each   of   the     new)   in   the   problem   in   the     quantities  in  the    diagram  (ratio  value).   two  ratios/rates  in  the  diagram.       diagram(s).   “x”  for  what  must  be  solved.   “x”  for  what  must   “x”  for  what   If   Then     be  solved.   must  be  solved.             &                         :             § Try  to  come  up  with  an  estimate  for  the  answer.      

Step  3 § §

S olve  the  problem

Translate  the  information  in  the  diagram  into  a  math  equation.   Plan  how  to  solve  the  math  equation.    

§  

 

Solve  the  math  equation,  and  write  the  complete  answer.  

Step  4:    C heck  the  solution   § §

Look  back  to  see  if  the  estimate  in  Step  3  is  close  to  the  exact  answer.         Check  to  see  if  the  answer  makes  sense.    

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