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# The Distance Formula

Lesson 12 of 12

## Objective: SWBAT use the distance formula to solve rate problems

*50 minutes*

#### Introduction

*10 min*

This lesson follows a direct instruction format. I will begin by asking the essential question. In the previous lesson we came up with equations in the form of y = mx. Students should be able to generate similar equations. If students are stuck, I could suggest a sample problem: bottles of OJ are selling of $0.69 each. What equation could be written to find the total, T, of n bottles. I could be even less abstract, if necessary, and ask the total cost of a specific amount of bottles.

I will then relate this equation to the distance formula. Just as our equations multiplied the unit rate times a given amount, the distance formula multiples the unit rate (speed) by a specific amount of time.

Next I will go through 3 examples. We'll find distance, rate and then time. For each example I will substitute the given values into the equation and then solve. Using the equation to solve the problems is an example of **MP4**. Each problem has a mirror problem labeled "You try!"; these are quick checks for understanding. I want students to have a chance to immediately apply the concept to a similar problem type.

#### Resources

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#### Guided Problem Solving

*15 min*

The first 3 problems are identical in structure to the examples. The fourth problem asks students to rewrite the distance equation to solve for rate, r. Students may need a hint. A simple hint could be to give students a multiplication problem and have them find the related division facts. Students could then apply this pattern to the formula to derive r = d/t. This problem then leads into the last problem of the section. Students who are struggling may have difficulty counting the elapsed time. They may see that the time is 2 hours 30 minutes, but they may not realize they need to think of this as 2.5 hours. Some may interpret 2 hours 30 minutes as 230 minutes; others may say 2.3 hours. If this occurs I will ask students to tell me how many minutes there are in an hour.

#### Resources

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#### Independent Problem Solving

*20 min*

This section begins in a similar fashion to the previous section, although the rigor has increased by quite a bit. Problem 1 and 2 involve decimal numbers. Problem 3 requires students to determine a distance before solving. The distance is a range of -70 to 180 for a total of 250. I expect students to see this as 110 feet. They may have ignored the fact that the first value is 70 feet BELOW sea level. Asking students to draw a vertical number line will help students see the distance covered.

Problem 5 is a paper based version of a problem from a sample item in the PARCC assessment. It has 3 parts and requires students to apply what they have learned in a slightly richer way. The final problem ties in what we have learned about the graphs of proportional relationships and also how to determine proportional relationships in a table. Students must then order the speeds from greatest to least. Finding the unit rates of the objects will be the most efficient method here.

#### Resources

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#### Exit Ticket

*5 min*

Before we begin, I will ask students to summarize how to use the distance formula to solve for rate, time, and distance. Answers should hit the following points: 1) substitute the known values into the equation; 2) solve the equation.

Once again, the first 3 questions of the exit ticket are similar to what students have already seen several times during the lesson.

Th final question is a slight variation on problems students have already seen. Students must determine a start time given a rate of travel and a specific end time.

Therefore, if a student is able to answer the first 3 questions, I know they understood the lesson. Students who answer all 4 show mastery beyond the basics of the lesson.

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- LESSON 1: Proportional Relationships of Whole Numbers
- LESSON 2: Proportional Relationships With Decimals
- LESSON 3: Proportional Relationships With Fractions
- LESSON 4: Finding Distances on Maps
- LESSON 5: Scaling a Recipe
- LESSON 6: Determine Equivalent Ratios - Scale Factor Between Ratios
- LESSON 7: Determine Equivalent Ratios - Scale Factor Between Terms
- LESSON 8: Determine The Graph of a Proportional Relationship
- LESSON 9: Determine Equivalent Ratios - Common Unit Rate
- LESSON 10: Writing The Constant of Proportionality Equation
- LESSON 11: Writing Equations for Proportional Relationships
- LESSON 12: The Distance Formula