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This project addresses the confusion students have distinguishing between unit rates & ratios.

This project is not meant to give any sure fire ways of addressing the difference between unit rates and ratios, it is meant to provide opportunity for reflection, and some possible lesson ideas. Feel free to make use some, none, or all. Please make it your own. If you have any suggestions feel free to email them to pfossat@alpine.k12.ut.us.

The Problem:

When students are taught about ratios and unit rates together, they often mix up using a unit rate, and a simplified ratio. For example, when asked to write a simplified ratio for "5 boys for every 10 girls", a student who recently learned about unit rates may use a calculator to write 0.5 boys/1 girl. This is not an incorrect statement, but does not reflect understanding of the difference between a unit rate and a reduced ratio (1 boy / 2 girls). How can a teacher address this misunderstanding?

Things To Understand:

According to Math.Com "A rate is a ratio that compares two different kinds of numbers, such as miles per

hour or dollars per pound. A unit rate compares a quantity to its unit of

measure." The key to this is that a unit rate compares to the "unit of measure". For example, a person does not usually say "I am traveling 100 miles / 2 hours."The reason we don't is that we don't measure the unit of 2 hours. The unit of measure is 1 hour, so we say, "I am traveling 50 miles / (1) hour." It becomes apparent that the denominator, will always be one. The reason we do this is to have a uniform measurement by which we can compare things.

Possible Student Challenges or Misunderstandings:

The Problem:

When students are taught about ratios and unit rates together, they often mix up using a unit rate, and a simplified ratio. For example, when asked to write a simplified ratio for "5 boys for every 10 girls", a student who recently learned about unit rates may use a calculator to write 0.5 boys/1 girl. This is not an incorrect statement, but does not reflect understanding of the difference between a unit rate and a reduced ratio (1 boy / 2 girls). How can a teacher address this misunderstanding?

Things To Understand:

According to Math.Com "A rate is a ratio that compares two different kinds of numbers, such as miles per

hour or dollars per pound. A unit rate compares a quantity to its unit of

measure." The key to this is that a unit rate compares to the "unit of measure". For example, a person does not usually say "I am traveling 100 miles / 2 hours."The reason we don't is that we don't measure the unit of 2 hours. The unit of measure is 1 hour, so we say, "I am traveling 50 miles / (1) hour." It becomes apparent that the denominator, will always be one. The reason we do this is to have a uniform measurement by which we can compare things.

Possible Student Challenges or Misunderstandings:

- The student does not see the unit rate as a standardized measurement.
- $150 for 9 hours of work = $16.67 per hour.
- The student does not understand the reason, or value of a unit rate.
- $150 for 9 hours of work = $16.67 per hour. = Better Rate
- $65 for 5 hours of work = $13 per hour. = Lower Rate
- One colleague suggested that students would better understand slope as a

measure of steepness if we had

students express it as a unit rate. - The student does not understand the reason or value of a ratio.
- Unit rates compare different types of data-not same.
- Ratios for part to whole relationships.
- 3 out of 5 boys makes more sense than 0.6 boys per 1 boy.
- 45 out of 50 points-not .9 points for every point.

- Is this a student problem, or a teacher problem?
- Is the student expressing a comparison in a way that makes more sense than what the teacher expects?
- If you wanted to compare which class has a higher "concentration" of boys 0.5 boys per girl or 0.6 boys per girl might make sense.
- Are examples used for unit rates, and ratios that don't allow students to make real life connections?
- 3.5 books / pencil would possibly not be as a good of a unit rate as would 35 pages per hour. Students are more likely to experience measure pp./minute as opposed to books/pencil.
- Following the reflection in step 1, choose some possible uses of ratios and unit rates that make sense.
- Create a need for students to use ratios over unit rates, and unit rates over ratios. When a student encounters a situation that using a ratio is not useful,

or using the unit rate is not useful, they will be more likely to

understand the best times to use one or the other. - Unit Rates over Ratios: Use the link below to obtain some demographic information about states square mileage and population. List population and land area for 5-10 of the states and ask students to order them from least crowded to most crowded.
- Ratios over Unit Rates: Test Scores...
- As a homework assignment ask students to locate examples of unit rates in real life, and examples of ratios in real life.