The Distance Formula

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.

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.  

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.

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.