Developing Concepts of Ratio

and Proportion

By David Her

54 ÷ ∫cos2θ dθ = 86 ÷ 317

Big Ideas

1. A ratio is a comparison of any two quantities.

2. Proportions involve multiplicative rather than additive comparison.

3. Proportional thinking is developed through activities involving comparing and determining the equivalence of ratios and solving proportions without recourse to rules.

Proportional Reasoning

1. Provide ratio and proportion tasks in a wide of context. i.e. measurements, prices, geometric or visual context.

2. Encourage discussion and experimentation in predicting and comparing ratios.

3. Relate proportional reasoning to existing process.

4. Recognize that symbolic or mechanical methods, such as cross product, for solving proportions does not develop proportional reasoning.

Question #1)

Two weeks ago, a rose was measured at 8 inches and a sunflower at 12 inches. Today the rose is 11 inches and the sunflower is 15 inches tall. Did the rose or the sunflower grow more?

Informal Activities to Develop Proportional Reasoning

• Equivalent Ratio Selections

• Comparing Ratios

• Scaling with Ratios Tables

• Construction and Measuring Activities

Equivalent Ratio Selections

• Select an equivalent ratio from those that are presented.

• Focus on an intuitive rationale why the pairs are selected.

Question #2)On which cards is the ratio of trucks to boxes the same? Also compare trucks

to trucks and boxes to boxes.

Question #3)Which rectangles are similar?

Question #3)Which rectangles are similar?

Question #3)Which rectangles are similar?

Comparing Ratios

• An understanding of proportional situations includes being able to distinguish between ratios as well as to identify those ratios that are equivalent.

Question #4)Some of the hens in Farmer Brown’s chicken farm lay brown eggs and the others lay white eggs. Farmer Brown noticed that in the large hen house he collected about 4 brown eggs for every 10 white ones. In the smaller hen house the ratio of brown to white was 1 to 3. In which hen house do the hens lay more brown eggs?

Scaling with Ratios Tables

• Ratio table or charts show how two variable quantities are related. This is a good way to organize information.

Question #5)

Fill in the chart then graph it.

Acres 5 10 15 20 25

Pine trees

75 150 225

Question #5)

Fill in the chart then graph it.

Acres 5 10 15 20 25 30 35

Pine trees

75 150 225 350 425 550 600

Construction and Measuring Activities

• In these activities, students make measurement or visual models of equivalent ratios in order to provide tangible example of a proportion as well as look at numeric relationships.

Question #6)

Create a similar object that has 4 times the volume as shown below.

Solving Proportions

• Within and Between Ratios

• An informal approach

• Cross Product Algorithm

Within and Between Ratios• A ratio of two measures in the same setting is a within ratio.• A between ratio is a ratio of two corresponding measures in different

situations.

A a

B b

Between

Between

Within Within

Within BetweenAB

ab

Aa

Bb

= =

An informal approach

• Students find ways to solve proportions using their own ideas first. If you have been exploring proportions informally, students will have a good foundation on which to build their own approaches.

Question #7)

The price of a box of 2 dozen candy bars is $4.80. Bridget wants to buy 5 candy bars. What will she have to pay?

Money money money. Money

1 dozen candy bars cost half of $4.80

So 1 dozen candy bars cost $2.40

There are 12 candy bars in a dozen

$2.40 divided by 12 is $0.20 or 20 cents

So each candy bar cost 20 cents

And 5 candy bars times 20 cents is $1.00

Cross Product Algorithm

• Sketch a simple picture that will determine what parts are related.

Question #8)

Apples are 3 pounds for 89 cents. How much should you pay for 5 pounds? Hint use within and between ratio.

5 pounds3 pounds

n cents89 cents

Within 389

5n

= Between 35

89n

=

Reference

John A. Van De Walle. Elementary and Middle School Mathematics, 5th Edition. Pearson Education Inc., 2004