Speed or velocity is calculated by the basic rate equation: , where r is the rate of speed, d is distance, and t is time.
Students will solve problems using the rate equation for speed (velocity). They will use information from the problem to determine x and y values and create a linear graph to show the relationship of speed to time or distance.
? Math Notebook
? 5 or 6 sheets of graph paper per student
Anticipatory Set (Lead-in):
Speed or velocity is calculated by the basic rate equation: r=d/t, where r is the rate of speed, d is distance, and t is time.
Problems involving rate equations assume an average rate of speed, which is the total distance (d) over the total time (t) of an event. Rates are stated in terms of units of distance divided by units of time, as in miles per hour. Sometimes items or phenomena take the place of distance, as in pages typed per minute, or flowers planted per hour.
A common difficulty for students is failure to make sure that the units for time and distance agree with the units for the rate. For instance, if the rate in the problem is given in feet per second, then all units of time must be in seconds and all distances in feet. Sometimes a problem may use different units. Students will have to convert to matching units before solving the problem.
Students have solved problems involving ratios, rates, and proportions. They have learned formulas for computing rates of distance and time, and graphed the results. In this lesson, they will culminate their work with calculating and graphing rates of speed, distance, and time.
Students will be involved with units of speed, distance, and time throughout their lives. They will build upon the basic skills of calculating and graphing ratios using these units.
Lesson Plan Procedure:
Your bicycle club totaled a distance of 81.9 miles in 3 hours. If you maintain the same speed, how far can you go in 6 hours?
We need a way to figure out the speed of the first leg, and use it for the total distance on that day.
You have worked with comparing ratios and solving proportions before. The goal of this lesson is to use the rate equation, r=d/t, to calculate speed from known distance and time. Then, you will graph the rate of speed to find distances and points of time along the route.
The rate equation is r=d/t. This means that a rate of speed is equal to a distance divided by an amount of time. Rates are calculated as distance per unit of time, as in miles per hour.
What are the distances and times in our bicycle problem?
[81.9 miles, 3 hours, and 6 hours.]
The rate of speed is given in the problem. We know it is 81.9 miles in 3 hours. What is that rate in miles per one hour?
[27.3 miles per hour.]
At 27.3 miles per hour, how far will you go in 6 hours?
Students may multiply 27.3 by 6, or they may realize that the distance in 6 hours will be twice what it was in three hours, and just double 81.9 miles. Either method will give them a distance of 163.8 miles.
To graph the distance relative to the time you cycled, you need to determine the independent and dependent variables. Since the question in the problem asked about unknown miles, make distance the dependent variable. The miles depend on the time you spend cycling. From your knowledge of graphing, which axis is for the dependent variable?
[The y-axis is for the dependent variable.]
What will be plotted on the y-axis?
Which axis is for the independent variable? What will we plot there?
[The x-axis will show the independent variable of time.]
Draw and label your graph and its axes.
As you know, the x-axis is listed first in a coordinate pair. We can plot the first coordinate pair, (1, 27.3), and we know from the equation that the final coordinates are (6, 163.8). How do we determine the rest of the points to plot?
[Each increment of time increases by one hour. Each increment of distance increases by 27.3.]
How do we describe the function of x to get the next y?
[f(x) = y = 27.3x.]
You may remember the function tables we made earlier. Your coordinate pairs will look like a function table with the y values increasing by 27.3. Write down the coordinates you will use on the graph.
[(1, 27.3), (2, 54.6), (3, 81.9), (4, 109.2), (5, 136.5), (6, 163.8).]
Plot these coordinates on your graph.
Students’ graphs should look like this:
Let’s solve and graph another rate problem.
A coyote was spotted running down a city street. It ran 5 miles in 10 minutes. If it ran another 45 minutes, how far into the city did it go? My house is 16 miles from the edge of town. Did it get that far?
What distance did the coyote run in how many minutes?
[5 miles in 10 minutes.]
How many miles per hour is that?
[(5 mi)/(10 min)=(x mi)/(60 min) = 30 miles per hour]
The next part of the problem asks how far it can go in 45 minutes. We know it travels 30 miles per hour. What fraction of an hour is 45 minutes?
[3/4 of an hour.]
Use this fraction to find out how far the coyote went at 30 miles per hour.
[(30 mph)(3/4 hour) = 22.5 miles.]
Did the coyote go more than 16 miles in ¾ of an hour?
[Yes, it went 22.5 miles.]
Next, find the independent and dependent variables for the graph. What are they?
[Independent: hours. Dependent: miles.]
Set up your graphs and label the axes.
How do we describe the function of x to get the next y?
[f(x) = y = 0.5x.]
What coordinate pairs will you plot?
[(10, 5), (20, 10), (30, 15), (40, 20), (55, 27.5).]
With your partner, set up and solve the rate equation to answer the question in each problem. Determine the independent and dependent variables, state the coordinate pairs, and graph the results.
1. The pasta press makes three 18-inch lasagna noodles every minute. How many noodles per hour can it make? If the chef needs 48 noodles for the lunch crowd, how long will it take to make them?
2. The Key Club washed 248 cars during their weekend car wash. If the event lasted from 9:00 until 5:00 both days, how many cars per hour did they wash?
Use the rate equation to solve each problem and answer the questions. Then, determine the independent and dependent variables, list the coordinate pairs, and graph the results.
1. Jeff read the first page of his new 360-page novel and could not put it down. He read non-stop for 2 ¼ hours, and had 90 pages left. How many pages per hour did he read? How long will it take him to finish the book?
2. If a hybrid auto can go 544.5 miles on 11-gallons of fuel, what mileage is it getting? Plot the gasoline use per 100 miles.
Closure (Reflect Anticipatory Set):
Explain how to form coordinate pairs from data calculated from rate equations.
Assessments & notes
Plan for Independent Practice:
Ask your mom how many ounces there are and how many servings she gets from a package of spaghetti.
Create and solve a rate equation from the information, then graph the results.
Explain how you identified the variables and how you calculated the coordinate pairs for your graph.
Assessment Based on Objectives:
Slim had been riding since 6:00 and Old Paint had only walked about 20 miles. If it’s 2:00 now, what was their rate of speed? Graph their progress for each hour of the ride.
Teacher needs to observe as students work. Note any difficulties with solving the rate equation, determining independent and dependent variables, or calculating coordinate pairs.