##
* *Reflection: Intervention and Extension
The Distance Formula - Section 2: Guided Problem Solving

Even after several years of teaching under my belt, I am always perplexed at the number of students that can not determine elapsed time. Problem GP5 was troublesome for several students. Many was full of many errors due to students not able to accurately determine the number of hours students said there were 3.5 hours between 8 am and 10:30 am. Problem 5 in the independent problem solving caused even more problems - finding the time between 10:15 am and 4:45 pm.

It makes me think that I should have a distance formula lesson dedicated to these elapsed time problems or at least make include elapsed time problems in a daily review portion of class.

Either way, I generally teach elapsed time problems with finger counting as opposed to using subtraction - to counting the time between 10:15 am and 4:45 pm, I would count - 11:15, 12:15, 1:15, 2:15, 3:15, and 4:15 and note 6 hours. Then I would add the 30 minutes from 4:15 to 4:45. This is easiest for me because I can easily visualize an analog clock. Many of my students cannot. What do you think the best approach is?

*How much times is that again?*

*Intervention and Extension: How much times is that again?*

# 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.

#### Resources

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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