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Ratio, Proportion, and Proportional Reasoning.

A professional development program to help teachers better understand the mathematical reasoning needed for

ratio, proportion, and proportional reasoning.

To understand the importance and significance of ratio, proportion, and proportional reasoning in the middle grades mathematics curriculum.

To examine some common ways of operating with ratios.

To identify possible misconceptions students might have when solving ratio problems.

Session # 1 Goals

Connected to elementary topics. * Multiplication

Connected to Algebra 1 and Geometry. * Similarity * Slope * Direct Variation * Percents

Why is topic so important?

Framework for studying various middle school topics/standards

*algebra *geometry *measurement *probability and statistics

Lanius, C.S. & Williams, S.A. (2003). Proportionality: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School, 8(8), 392 – 396.

Why is topic so important (con’t)

A ratio is a comparison between two or more quantities, which are either numbers or measurement

Chapin, S & Johnson, A. (2006). Math Matters: Understanding the Math You Teach, Grades 6 – 8, 2nd Ed. Math Solutions Publications, Sausilito, CA.

Defining ratio.

5 pencils for $1.45 2 degrees per hour 1 point added to final grade for every 10

completed homework assignments ½ cup lemonade concentrate for every 4

cups of water 12 red jellybeans compared with 55

jellybeans total. Any other examples?

Examples of Ratios.

What are some benefits students gain from learning about ratios

Inside the classroom? outside the classroom?

Meet Stephen and Sarah!

Sssssstephen Sssssarah

Stephen, who will ultimately be 12 ft. long, has only grown to 6 ft. Sarah, who will grow to 9 ft., has only grown to 5 ft.

1.) What are some possible questions we can ask from this information?

2.) How might students respond?

Who has grown more?

Additive thinking is needed for comparing quantities in one variable (e.g. heights)

Multiplicative thinking is needed when comparing the fraction of one quantity to another quantity (e.g. current height is what fraction/percent of full grown height).

Two main types of thinking.

How does the relationship between two quantities in a ratio convey different multiplicative info? (Lamon, 1999)

A question to consider…

Suppose there are 40 people in a classroom.

Suppose there are 40 people in a cafeteria.

Suppose there are 40 people in an auditorium.

When we ask which situation is the most crowded, we think multiplicatively and consider both quantities at the same time. (In this case, a quantity of interest might be maximum capacity.) Namely we use multiplication and division in our solution.


Which setting is the most crowded?

Additive reasoning not always sufficient.

Can you relate the values in these ratios multiplicatively?

Pizza, anyone?

Which ratio of pizza slices to girls gives each girl the most pizza? The least pizza?

Number of Pizza Slices Number of Girls

A key idea!

Ratios can be meaningfully reinterpreted as quotients.

Some other interpretations.

Wait a minute! How can we have 1.5 girls per pizza?

Perhaps we should say that not all quotients are meaningful when working with ratios.

Let’s Take a Walk!

A man walks 14 feet in 6 seconds. How many feet can he walk in

12 seconds? 3 seconds? 9 seconds? 8 seconds?

How do you think a student would answer the above four questions?

Consider the following problem.

Another way to form a ratio is by joining (composing) two quantities to create a new unit. Many times, students create composed units when they iterate (repeat) a quantity additively or partitioning (break into equal-sized sections) (Lobato et al, 2010).

Ratio as a composed unit.

Repeated iteration of 7 ft./3 sec.

Distance 7 ft 14 ft 21 ft 28 ft 35 ft.

Time 3 sec. 6 sec. 9 sec. 12 sec. 15 sec.

For the walking man, the easiest partition would probably be 7 ft. in 3 seconds.

How might we partition into six parts to consider distance traveled in 1 second?

The pizza/girls problem explored earlier is another example of partitioning, or sharing equally among each girl.

Partitioning Ratios.

Juice, anyone?

Complete the handout “Which Tastes More Juicy?” You are encouraged to use previous strategies learned such as ratios as a composed unit, the multiplicative nature of a ratio, or meaningful interpretations of quotients.

Pay close attention to the students’ thought processes described on the handout. In addition to determine whether their answers are correct or not, think about their reasoning.

Using some of the ideas discussed today, how might you correct students’ misconceptions that simply adding more water and more juice does not guarantee more juiciness?

Activity: Which Tastes More Juicy

Time to wrap up!

Reasoning with Ratios involves attending to and coordinating two quantities.

A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit.


Three Essential Ideas About Ratios

Session # 2

Reasoning with ratios involves attending to and coordinating two quantities



Let’s recap the three main understandings from session # 1.

To compare and contrast fractions and ratios.

To compare and contrast different methods used to solve proportion problems.

To view proportional reasoning as conceptual rather than procedural.

To examine and evaluate student work and reasoning for the depth of understanding of proportional reasoning.

Session # 2 Goals

Fractions vs. Ratios

Same? Or Different?

Fractions vs.Ratios

Fractions Ratios

Fractions vs. Ratios

Fraction Ratio

A ratio of two integers, where the denominator is nonzero

Sometimes called rational numbers.

Fractions are real numbers.

Fractions only express a part-whole relationship.

Fractions

Ratios can often be meaningfully interpreted as fractions.

Ratios can be compared to zero.

Ratios can compare numbers that are not necessarily whole or rational (e.g. the Golden Ratio)

Ratios can exhibit part-whole relationships and can exhibit part-part relationships.

Ratios

Complete the handout “What is equal?” You are encouraged to use previous strategies learned such as ratios as a composed unit, the multiplicative nature of a ratio, or meaningful interpretations of quotients.

Discuss what methods you are using to answer the question. See if you can come up with “What is Equal?”

Activity: What is Equal?

A proportion is a mathematical statement of equality between two ratios. Stated another way, proportions tell us about the equivalence of ratios.


Proportion

A proportion includes multiplicative relationships


Proportions (continued)

Multiplying within each ratio. Notice that the multiplier is the same in both ratios.

Within the Ratios…

Multiplying across each ratio. Notice that each multiplier is the same, once again.

Between the ratios…

Why is it important for us as teachers to realize the two multiplicative natures in a proportion?

What benefits might our knowledge provide for our students as they engage in solving problems involving proportions?

Questions for Discussion

Proportional reasoning describes the thinking that has been applied to the solution of problems that involve multiplicative relationships.

Proportional reasoning requires examining two quantities in relation to one another.


What is proportional reasoning

equivalent ratios can be created by iterating a composed unit

Proportional reasoning is complex and involves understanding that

Distance 8 ft 16 ft 24 ft

Time 4 sec 8 sec 12 sec

Equivalent ratios can be created by partitioning a composed unit.

If we split the original 8 feet into 4 parts, each 2 ft. part would get 1 second.


In short, we are reducing to a smaller unit and iterating the smaller unit.

Partitioning and Iterating a composed unit.

Distance 2 ft. 4 ft. 6 ft. 8 ft.

Time 1 sec. 2 sec. 3 sec. 4 sec.

if one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship


the two types of ratios – composed units and multiplicative comparisons – are related.


“For every 4 batches of cookies, 6 eggs are required. How many eggs are required for 14 batches of cookies.”

Can you reason through this problem using some of the ideas we have discussed (e.g. multiplicative nature of proportions, partitioning, iteration, etc.).

How would you help a student reason through this type of a problem?

Think and Discuss

“Several ways of reasoning, all grounded in sense

making, can be generalized into algorithms for solving

proportion problems.”Lobato et al (2010)

Complete the handout “A Special Property of Proportions.” Discuss your observations with one or two other people in your group.

See if you can make a conjecture but more importantly, justify it.

A Special Property of Proportions

Two ratios that form a proportion, have equal cross products.

A Special Property of Proportions

Using lowest common denominators.

Why Is the Cross-Product Property True?

Rewrite as

What can you say about ?

Think about what equal.

Notice the appearance of the cross-products in the numerators?

Using Factor of Change

You will be provided with some student work (Canada et al, 2008). With one or two other people, see if you can not only determine whether the answer is correct but also see if you can describe the types of thinking the students were doing (e.g. cross products, composed units, etc.). Which student(s) method do you feel expresses a deep conceptual understanding of proportional reasoning and why?

Understanding Student Thinking

A number of mathematical connections link ratios and fractions.

A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change.

Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

Summary: What are some big ideas to take away from this session?


Equivalent ratios can be created by iterating and/or

partitioning a composed unit If one quantity in a ratio is multiplied or divided by a

particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship

The two types of ratios – composed units and

multiplicative comparisons – are related

Big Ideas (continued)

Session # 3

Welcome back class!

To isolate attributes of a situation needed for ratio formation.

To compare and contrast rates and ratios. To evaluate and reflect on a teacher’s

pedagogical strategies on a ratio and proportion lesson as presented in a case study.

To recognize that not all situations, despite certain key words, will use direct proportional reasoning.

Session # 3 Goals


Proportional reasoning is complex and requires understanding of many important ideas.


What have we learned so far?

How do we calculate and maintain the steepness of a ramp?

Forming a ratio as a measure of a real-world attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest.

How did you coordinate the changing of the base and height of the ramp to maintain the ramp’s steepness?

When calculating steepness…

The steepness of a ramp, sometimes referred to as the slope or the grade, is the ratio of the rise (vertical measure) to the run (horizontal measure)

Steepness can also be viewed as a rate of change…

For every y vertical units measured, you will measure x horizontal units.

Steepness of a ramp wrap-up.

Take the cards with different definitions of the term rate. A short one page reading will also be handed out on ratios and rates.

What definitions from the cards and reading do you feel will generate understanding of rate for students?

Which definitions help you understand how a rate and ratio are the same? Which definitions help you understand how rates and ratios are different?

Let’s now consider rates vs. ratios.

When comparing two different types of measures, the ratio is usually called a rate.

When a rate is simplified so that a quantity is compared with 1, it is called a unit rate.

Some unit rates are constant while others vary.


Defining Rate

Express the following ratios as two different unit rates. Try and use some of the methods discussed in the previous two sessions.

120 miles in 2 hours 7 pizzas for 4 teenagers $24.75 for 25 songs on iTunes 225 Euros for 300 U.S. Dollars 20 candies for $2.50


Understanding the dual nature of unit rates.

60 miles in 1 hour or 1 mile in of an hour.

pizzas for each teenager or 1 pizza for teenager.

$0.99 for 1 song or $1.00 for song.

0.75 Euros for 1 U.S. Dollar or 1 Euro for 1.33 U.S. Dollars.

1 candy for $0.125 or $1.00 for 8 candies.

Two different unit rates are possible for each of the previous ratios.

60

1

7

31 7

4

01.1

Which ones make sense and why?

Which ones don’t make sense and why?

Some unit rates make more sense than others.

As you read the case, you are encouraged to make a list of aspects of Marie Hanson’s pedagogy that appear to support her students’ learning throughout the lesson.

Think about some specific actions that Marie Hanson carries out (e.g. eliciting incorrect additive solution first, returning to two students’ ratio table later in the lesson to show multiplicative relationships).

What intuitive proportional strategies might students you work with use?

Might you encourage students to try other methods as discussed throughout this workshop? What methods might you encourage them to try and why?

Schwan-Smith, M., Silver, E.A., & Stein, M.K. (2005). Improving Instruction in Rational Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and Learning, Vol. 1. New York, Teachers’ College Press.

The Case of Marie Hanson

Solve each of the problems on the next slide.

Think about how students might answer some of these questions as you work through these questions.

How as teachers can we help students recognize that not everything is necessarily proportional?

Not Everything Is Proportional

A train travels 160 miles in 3 hours. At the same rate, how many miles does the train travel in 12 hours?

Three people can paint the exterior of a house in 8 days. At the same rate how many days will it take for seven people to paint the same house?

Three children weigh a total of 195 pounds. How many pounds will 7 children weigh?

A student earned 19 out of 25 points on a quiz. What percent did the student earn?

Make a table.

Actively model the situation.

Use common sense.

Ask questions *What do we know? *What else do we need to know?

Some teaching strategies to consider.

Time to wrap up!

Reasoning with Ratios involves attending to and coordinating two quantities.



10 key ideas behind ratio, proportion, and proportional reasoning.

A number of mathematical connections link ratios and fractions.


Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

10 key ideas (con’t)

Proportional reasoning is complex and involves understanding that…

Equivalent ratios can be created by iterating and/or

partitioning a composed unit If one quantity in a ratio is multiplied or divided by a

particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship

The two types of ratios – composed units and

multiplicative comparisons – are related


Forming a ratio as a measure of a real-world attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest.

A rate is a set of infinitely many equivalent ratios

Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities.


Final questionnaire/evaluation

Thank you all for attending

I hope that many of you have found some good ideas for your teaching!

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