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### Overview

This lesson will allow students to become familiar with the concept of unit rate. Through an open investigation students will develop methods to find unit rate with a table, equivalent ratios, or an equation. This is a lesson to be used as part of a unit with "Painter Problems" and "How Big Should It Be?"

This is a College- and Career-Ready Standards showcase lesson plan.

## UP:MA19.6.1

### Vocabulary

• Ratio
• Ratio Language
• Part-to-Part
• Part-to-Whole
• Attributes
• Quantity
• Measures
• Fraction

### Knowledge

Students know:
• Characteristics of multiplicative situations

### Skills

Students are able to:
• Compare and contrast additive vs. multiplicative contextual situations.
• Identify all ratios and describe them using "For every…, there are…"
• Identify a ratio as a part-to-part or a part-to whole comparison.
• Represent multiplicative comparisons in ratio notation and language (e.g., using words such as "out of" or "to" before using the symbolic notation of the colon and then the fraction bar. for example, 3 out of 7, 3 to 5, 6:7 and then 4/5).

### Understanding

Students understand that:
• In a multiplicative comparison situation one quantity changes at a constant rate with respect to a second related quantity. -Each ratio when expressed in forms: ie 10/5, 10:5 and/or 10 to 5 can be simplified to equivalent ratios, -Explain the relationships and differences between fractions and ratios.

## UP:MA19.6.2

### Vocabulary

• Unit rate
• Ratio
• Rate language
• Per
• Quantity
• Measures
• Attributes

### Knowledge

Students know:
• Characteristics of multiplicative comparison situations.
• Rate and ratio language.
• Techniques for determining unit rates.
• To use reasoning to find unit rates instead of a rule or using algorithms such as cross-products.

### Skills

Students are able to:
• Explain relationships between ratios and the related unit rates.
• Use unit rates to name the amount of either quantity in terms of the other quantity flexibly.
• Represent contextual relationships as ratios.

### Understanding

Students understand that:
• A unit rate is a ratio (a:b) of two measurements in which b is one.
• A unit rate expresses a ratio as part-to-one or one unit of another quantity.

## UP:MA19.6.3

### Vocabulary

• Rate
• Ratio
• Rate reasoning
• Ratio reasoning
• Transform units
• Quantities
• Ratio Tables
• Double Number Line Diagram
• Percents
• Coordinate Plane
• Ordered Pairs
• Tape Diagrams
• Unit Rate
• Constant Speed

### Knowledge

Students know:
• Strategies for representing contexts involving rates and ratios including. tables of equivalent ratios, changing to unit rate, tape diagrams, double number lines, equations, and plots on coordinate planes.
• Strategies for finding equivalent ratios,
• Strategies for using ratio reasoning to convert measurement units.
• Strategies to recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and denominator is the same.
• Strategies for converting between fractions, decimals and percents.
• Strategies for finding the whole when given the part and percent in a mathematical and contextual situation.
• Strategies for finding the part, given the whole and the percent in mathematical and contextual situation.
• Strategies for finding the percent, given the whole and the part in mathematical and contextual situation.

### Skills

Students are able to:
• Represent ratio and rate situations using a variety of strategies (e.g., tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes).
• Use ratio, rates, and multiplicative reasoning to explain connections among representations and justify solutions in various contexts, including measurement, prices and geometry.
• Understand the multiplicative relationship between ratio comparisons in a table by writing an equation.
• Plot ratios as ordered pairs.
• Solve and justify solutions for rate problems including unit pricing, constant speed, measurement conversions, and situations involving percents.
• Solve problems and justify solutions when finding the whole given a part and the percent.
• Model using an equivalent fraction and decimal to percents.
• Use ratio reasoning, multiplication, and division to transform and interpret measurements.

### Understanding

Students understand that:
• A unit rate is a ratio (a:b) of two measurements in which b is one.
• A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.
• When computing with quantities the transformation and interpretation of the resulting unit is dependent on the particular operation performed.

## UP:MA19.7.1

### Vocabulary

• Unit rate
• Ratio
• Unit
• Complex fractions

### Knowledge

Students know:
• What a unit rate is and how to calculate it given a relationship between quantities.
• Quantities compared in ratios are not always whole numbers but can be represented by fractions or decimals.
• A fraction can be used to represent division.

### Skills

Students are able to:
Compute unit rates associated with ratios of fractional
• lengths.
• Areas.
• quantities measured in like or different units.

### Understanding

Students understand that:
• Two measurements that create a unit rate are always different (miles per gallon, dollars per hour)

### Primary Learning Objectives

I CAN identify and develop ratios in real world situations.

I CAN identify equivalent ratios.

I CAN compare ratios in real world situations.

I CAN use equivalent ratios to find the unit rate.

### Procedures/Activities

1. The teacher will instruct the students to search for "fastest car in the world" (if available, students can use individual devices or this can be done in small or whole group). The teacher will ask "How do we know it is the fastest car in the world?" Students will give responses, an ideal response is "because it tells us the speed." The teacher will ask the students, "How is speed displayed?" Ideal student response "Miles per hour." The teacher will introduce speed as a rate and explain that in 1 hour that car can go n miles. The teacher will introduce rate as a ratio where a unit equals 1.

2. The teacher will ask students for other rates we use every day. Ideal student response "dollars per hour". The teacher will show the students a rate/ratio of $40 for five hours. The teacher will ask, "Is this a rate?" Ideal response, "No, because the hours is not equal to one." The teacher will pose an open-ended question "How can we get our hours down to one?" Allow students to give suggestions and strategies. Teacher will drive students to set up a table or use equivalent ratios. 3. The teacher will introduce the activity How far can you leap? The teacher will demonstrate (or have a student demonstrate a leap), start with both feet together and jump with one foot. If needed, teacher will demonstrate the leap progression in the activity (the directions are on the activity guide). Students will begin investigation. As students are working, teacher will act as a facilitator or coach asking questions that drive understanding. 4. Once adequate time (30-45 minutes) is given, the students will share their finding on the document camera. (If a document camera is not available students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently?" Teacher will spawn debate on who has the best rate at jumping tiles. 5. Toward the end of the class students will complete the Rate Exit Slip (found in attachments). 1. The teacher will instruct the students to search for "fastest car in the world" (if available, students can use individual devices or this can be done in small or whole group). The teacher will ask "How do we know it is the fastest car in the world?" Students will give responses, an ideal response is "because it tells us the speed." The teacher will ask the students, "How is speed displayed?" Ideal student response "Miles per hour." The teacher will introduce speed as a rate and explain that in 1 hour that car can go n miles. The teacher will introduce rate as a ratio where a unit equals 1. 2. The teacher will ask students for other rates we use every day. Ideal student response "dollars per hour". The teacher will show the students a rate/ratio of$40 for five hours. The teacher will ask, "Is this a rate?" Ideal response, "No, because the hours is not equal to one." The teacher will pose an open-ended question "How can we get our hours down to one?" Allow students to give suggestions and strategies. Teacher will drive students to set up a table or use equivalent ratios.

3. The teacher will introduce the activity How far can you leap? The teacher will demonstrate (or have a student demonstrate a leap), start with both feet together and jump with one foot. If needed, teacher will demonstrate the leap progression in the activity (the directions are on the activity guide). Students will begin investigation. As students are working, teacher will act as a facilitator or coach asking questions that drive understanding.

4. Once adequate time (30-45 minutes) is given, the students will share their finding on the document camera. (If a document camera is not available students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently?" Teacher will spawn debate on who has the best rate at jumping tiles.

5. Toward the end of the class students will complete the Rate Exit Slip (found in attachments).

1. The teacher will instruct the students to search for "fastest car in the world" (if available, students can use individual devices or this can be done in small or whole group). The teacher will ask "How do we know it is the fastest car in the world?" Students will give responses, an ideal response is "because it tells us the speed." The teacher will ask the students, "How is speed displayed?" Ideal student response "Miles per hour." The teacher will introduce speed as a rate and explain that in 1 hour that car can go n miles. The teacher will introduce rate as a ratio where a unit equals 1.

2. The teacher will ask students for other rates we use every day. Ideal student response "dollars per hour". The teacher will show the students a rate/ratio of $40 for five hours. The teacher will ask, "Is this a rate?" Ideal response, "No, because the hours is not equal to one." The teacher will pose an open-ended question "How can we get our hours down to one?" Allow students to give suggestions and strategies. Teacher will drive students to set up a table or use equivalent ratios. 3. The teacher will introduce the activity How far can you leap? The teacher will demonstrate (or have a student demonstrate a leap), start with both feet together and jump with one foot. If needed, teacher will demonstrate the leap progression in the activity (the directions are on the activity guide). Students will begin investigation. As students are working, teacher will act as a facilitator or coach asking questions that drive understanding. 4. Once adequate time (30-45 minutes) is given, the students will share their finding on the document camera. (If a document camera is not available students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently?" Teacher will spawn debate on who has the best rate at jumping tiles. 5. Toward the end of the class students will complete the Rate Exit Slip (found in attachments). 1. The teacher will instruct the students to search for "fastest car in the world" (if available, students can use individual devices or this can be done in small or whole group). The teacher will ask "How do we know it is the fastest car in the world?" Students will give responses, an ideal response is "because it tells us the speed." The teacher will ask the students, "How is speed displayed?" Ideal student response "Miles per hour." The teacher will introduce speed as a rate and explain that in 1 hour that car can go n miles. The teacher will introduce rate as a ratio where a unit equals 1. 2. The teacher will ask students for other rates we use every day. Ideal student response "dollars per hour". The teacher will show the students a rate/ratio of$40 for five hours. The teacher will ask, "Is this a rate?" Ideal response, "No, because the hours is not equal to one." The teacher will pose an open-ended question "How can we get our hours down to one?" Allow students to give suggestions and strategies. Teacher will drive students to set up a table or use equivalent ratios.

3. The teacher will introduce the activity How far can you leap? The teacher will demonstrate (or have a student demonstrate a leap), start with both feet together and jump with one foot. If needed, teacher will demonstrate the leap progression in the activity (the directions are on the activity guide). Students will begin investigation. As students are working, teacher will act as a facilitator or coach asking questions that drive understanding.

4. Once adequate time (30-45 minutes) is given, the students will share their finding on the document camera. (If a document camera is not available students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently?" Teacher will spawn debate on who has the best rate at jumping tiles.

5. Toward the end of the class students will complete the Rate Exit Slip (found in attachments).

1. The teacher will instruct the students to search for "fastest car in the world" (if available, students can use individual devices or this can be done in small or whole group). The teacher will ask "How do we know it is the fastest car in the world?" Students will give responses, an ideal response is "because it tells us the speed." The teacher will ask the students, "How is speed displayed?" Ideal student response "Miles per hour." The teacher will introduce speed as a rate and explain that in 1 hour that car can go n miles. The teacher will introduce rate as a ratio where a unit equals 1.

2. The teacher will ask students for other rates we use every day. Ideal student response "dollars per hour". The teacher will show the students a rate/ratio of \$40 for five hours. The teacher will ask, "Is this a rate?" Ideal response, "No, because the hours is not equal to one." The teacher will pose an open-ended question "How can we get our hours down to one?" Allow students to give suggestions and strategies. Teacher will drive students to set up a table or use equivalent ratios.

3. The teacher will introduce the activity How far can you leap? The teacher will demonstrate (or have a student demonstrate a leap), start with both feet together and jump with one foot. If needed, teacher will demonstrate the leap progression in the activity (the directions are on the activity guide). Students will begin investigation. As students are working, teacher will act as a facilitator or coach asking questions that drive understanding.

4. Once adequate time (30-45 minutes) is given, the students will share their finding on the document camera. (If a document camera is not available students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently?" Teacher will spawn debate on who has the best rate at jumping tiles.

5. Toward the end of the class students will complete the Rate Exit Slip (found in attachments).

### Assessment Strategies

Formal Formative Assessment: Rate Exit Slip

Formal Assessment: Using the Investigative Activity Rubric (found in attachments) teacher will evaluate students' work.

Informal Formative Assessment: As the students are working, the teacher will act as the facilitator and coach. Teacher will ask questions to evaluate students (i.e. How do you know ______?, What did you do to get that?) Teacher may pull small groups during investigation on a needs basis.

### Acceleration

The investigation has an included extension on the How Far Can You Leap? Activity Sheet (found in attachments).

### Intervention

Because this is part of a unit, teacher may develop small groups based on the Rate Exit Slip or informal questioning as part of the investigative activity.

### Total Duration

61 to 90 Minutes

### Background/Preparation

The teacher must make the appropriate number of copies of the How Far Can You Leap? activity guide (found in attachments). Copies should be made so that students can work collaboratively.

The teacher must make the appropriate number of copies of the Rate Exit Slip (found in attachments). Each student should have one.

Teacher must prepare the appropriate number of Math Toolboxes.

Teacher must mark off a starting point for the leap. In a long hallway with square tiles works best, but this can easily be modified to do an outdoor lesson. Instead of floor tiles, the teacher can mark off feet.

The students must have knowledge of ratios.

### Materials and Resources

How Far Can You Leap? Activity Guide (found in attachments), Investigative Activity Rubric (found in attachments), Rate Exit Slip (found in attachments), Chart paper, painter's tape, Math Toolbox which includes the following: pencil, paper, graph paper, markers, scissors, glue, calculator, sticky notes

### Technology Resources Needed

Interactive whiteboard (Optional) with required software, document camera, projector, access to search engine (individually or whole group)