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A study was conducted to determine how students reason in various proportional tasks. It was conducted to see if it was developmental or instructional factors that were related to proportional reasoning.

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Great way to organize problems.

Organizes the two variable quantities that are related.

Numbers are compared or used to find a missing number

Connected to seeking the concept of an equivalent ratio rather than only trying to find the missing number

Should be taught prior to the cross product method because this allows the students to better understand what ratios are.

Ratio tables allow uses multiplicative relationship

It is a relationship where two quantities can be expressed as multiples of each other. It can be generalized as y = ax, where y and x are multiples of each other.

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Rate is a quantity that is measured with another measured quantity.

Example: a rate of speed of 60 miles an hour.

Unit rate method: for example- dividing the price of three apples by 3, then multiplying the unit of $0.80 per unit by 10. Will give the answer.

Ex: Tommy bought 3 apples for $2.40. At the same price, how much would 10 apples cost?

Unit rate would be three apples by 3 and $0.80 per unit by 10.

Scale factor is the ratio of the model measurements to the actual measurement in the simplest form.

Buildup strategy is usually done within the scale factor.

Normally done on whole numbers and when the numbers are compatible.

Pizza Party example: Two camps of scouts are having a pizza party. The bear camp ordered enough so that every 3 campers would get 2 pizzas.  The raccoon camp ordered enough so that there would be 3 pizzas for every 5 campers. Which troop will have the most pizza?

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These strategies all students to use simple reasoning, as well as sophisticated strategies.

Most commonly know strategy is cross products

Should only be done after proportional problems with reasoning done using tape or diagram stripes.

Multiple solutions strategies help students solve proportional reasoning problems.

These strategies are:

Rate

Scaling up or down

Scale factors

Ratio Tables

Graphs

Equation

Percent

First three are the most intuitive and the ones that students usually begin with when solving problems with reason.

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5. Recognize that symbolic or mechanical methods (cross-product algorithm) do not develop proportion reasoning and shouldn’t be introduced until students have many experiences with intuitive and conceptual methods.

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4. Students should be engaged in a variety of strategies for solving proportions =. Specifically use ration tables, visuals, question and graphs in order to solve problems. Make sure students apply reasoning strategies.

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3. Ratio and proportion tasks should be provided in a wide range of contexts. This includes situations involving measurements, prices, geometrics and other visual contexts as well as all sorts of rates.

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2. Students should be able to distinguish between proportional and non-proportional comparisons by giving examples of each and discussing differences.

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1. Composed unit and multiplicative comparison should be used to build understanding of rations. A better understanding of multiplicative comparisons leads to a better understanding of rate, a strategy that can then be applied to proportions.

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Traditionally, textbooks taught students to set up equations of two ratios, and proportions b solving for the unknown (x) by cross-multiplying and solving for x.

Students should be encouraged to reason in order to find the missing value rather than just using the cross-multiply algorithm

1. Create a Visual Model: ask students to illustrate a problem that shows what is covarying instead of just setting up proportions. Provide visual cues to set up the proportions in order to support a wide range of learners.

2. Solve the Proportion: unit rate can be found by dividing the amount by number and then multiplying by amount total. Scale factors can be used as well to solve problems. If using the cross-multiply, students need to understand in order to complete problems with more challenging numbers and unit rates/scale factors that are not easy to calculate.

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Percent problems can be solved by using all sorts of strategies that have been talked about in this chapter.

Double number line is a great way to solve percent situations

It can help students figure out the unknown

After double number line is set up, the diagram can be used as a proportion

One line would represent the measures of the problem

The second line would represent the value in terms of percents

Using the double number line not only shows modelling for part-whole, but also increase-decrease situations, as well as comparisons between two distinct quantities

Linear models also do not restrict students from thinking about percents that are larger than 100 since the line can represent more than 100 rather than a circle

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Double number line diagrams are similar to tape or strip diagrams but they might not show proportions. They must however have the two lines labeled, just as tape diagrams should be.

Example: Comparing the height between a zoo animal and the students’ height.

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Tape or strip diagrams can be great visuals that help students visualize multiplicative relationships.

Defined as “drawings that look like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar models or graphs, fraction strips, or length models.”

Example: The ratio of boys to girls in the class is 3 to 4.

Once the ratio is provided students can set up their sketches in different ways.

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STRATEGIES FOR SOLVING PROPORTIONAL SITUATIONS

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STRATEGIES FOR SOLVING PROPORTIONAL SITUATIONS

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C. Algebra

Proportional situations are linear situations. Graphing equivalent rations is a powerful way to illustrate this concept. The ratio or rate is the slope of the graph.

Having two sized shapes start in the same corner is useful for finding the ratio between the two shapes.

Graphing ratios can be challenging for students. Some might have trouble using which points to graph. A good way to help students make sense of unit rates and proportional reasoning is to use real-life contexts and gathering data.

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B. Measurement and geometry

Scale drawings can be a useful tool in seeing connections between multiplicative and proportional reasoning. Slope is the ratio between 2 values. If two figures are proportional then any corresponding linear dimensions will have the same scale factor.

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A. Within and between ratios

Within ratio: two measures in the same setting (ex. mangos to price)

Between ratio: two corresponding measures in different situations (ex. price of 5 mangos compared to price of 10 mangos)

These ratios are popular in 8th grade and students explore similarities with geometric shapes.

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3. Covariation

When two different quantities (ratio) vary together.

Ex. 5 mangos cost $2.00. If there are 10 mangos, the cost will be ____. Once you know the new price or new number, you can figure out the last variable.

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2. Additive and Multiplicative Comparisons in Story Problems
Using either additive or multiplicative reasoning in the wrong situation can lead to wrong answers. An assessment sheet might be useful in the category.

Often, students will notice an answer could be out of place when the wrong comparison is used.

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1. The difference between proportional and nonproportional situations

Is the comparison due to an additive, multiplicative, or constant?

Additive (walks the same amount): Janet and Jeanette walk at the same pace. When Janet is at 6 blocks, Jeanette has started and is at 2 blocks. How far will Jeanette be when Janet is at 12 blocks?

Constant(: Lisa and Linda both plant corn. Lisa plants 4 rows and Linda plants 6 rows. Linda’s corn is ready in 8 weeks, how long is Lisa’s till it’s ready?

Multiplicative: Kendra is making 6 dozen cookies and Kevin is making 3 dozen. Kevin uses 6 ounces of chocolate chips. How many ounces will Kendra use?

For students to distinguish between the different types of questions, they must make distinctions about each.

A ratio is a number that expresses a multiplicative relationship that can be applied to a second situation in which the quantities are measured similarly.

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Uses: investigating interest, taxes, and tips or connecting figures, graphs, and slopes.

Students start using proportional reasoning when they are involved in, one-to-one correspondence, place value, fractional concepts, and multiplicative reasoning.

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Thinking of the ratio as one unit; 5 limes for $1.00 is a unit

Using the base unit, one can think of infinite multiples

Example: 10 limes for $2.00, 15 limes for $3.00, etc.

Composed units can be partitioned, enabling the ability to solve for any amount

Example: 3 limes for $0.30, 1 lime for $0.20

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Introduced in 4th grade (CCSS)

Foundational to understanding ratios

Comparing two different units (ratios as rates) is a 6th grade expectation (i.e., 8 roses per bouquet)

6th graders are also expected to solve real-world problems related to ratios

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Rates are ratios between two measurements with different units (miles per gallon, square yards of wall and gallon of paint, flowers per bouquet) or relationships between two units of measure (centimeters per inch, inches per foot, feet per mile).

They also represent an infinite set of equivalent ratios

Example: A swimmer’s rate of laps swam is 30 laps in 1 day- equivalent ratios would be 60 laps in 2 days, 90 laps in 3 days, etc.

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A ratio that compares a part of a whole, to the whole

Number of girls in an after-school program: 6/11

Can be thought of as a fraction: six-elevenths

Includes percentages and probabilities

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A ratio that relates one part of a whole to another part of the same whole

Students in an after-school program: 6 girls, 5 boys

Not fractions, but can be written using a fraction bar

6/5: A ratio of 6 to 5, not six-fifths

Occurs across curriculum (slope, odds)

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A number that relates to two quantities or measures within a given situation with a multiplicative relationship

One of the four critical areas in 6th grade

Students’ understanding of multiplicative reasoning and multiplicative comparisons is critical

Closely related to fractions, but have distinct differences (i.e., ratios are not part-to-part). However, should be thought of as overlapping concepts (Ratio of cats to dogs in pet store vs. ratio of cats to pets in pet store.

Should be taught as relations that involve multiplicative reasoning rather than simply recording symbols (6:5).

Forming a ratio is a cognitive task, rather than a writing task

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Example: Limes are on sale and you can buy 5 limes for $1.00; The ratio of money for limes is $1.00 for 5 limes, which means the cost per lime is $0.20.