# Common Core Standards: Math

### Expressions and Equations 8.EE.B.5

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

When your students have a chance, tell them to take a look at the speedometer in the family car. They'll see the normal range of speeds in miles per hour or mph. Speed is always calculated in terms of the distance you could cover (miles) at that rate if you drove for a certain amount of time (one hour). So if you drove at a speed of 55 mph for one hour, you would cover 55 miles.

If they take a closer look, they'll notice that most speedometers will also have another scale: km/h or kilometers per hour. A speed of 55 mph is approximately equal to a speed of 88.5 km/h. (Since a kilometer is a bit shorter than a mile, you can cover more of kilometers than miles in an hour of driving the same speed.)

Draw your students' attention to this fact: the same speed can, and often is, represented in a number of different ways. Sometimes it's a speedometer and sometimes it's a graph. Sometimes it's an equation and sometimes it's a speeding ticket. Oops.

Students should realize that ultimately, slope is a quotient: the change in something (miles or vertical units) divided by the change in something else (time or horizontal units).

When we graph lines on the x-y plane, we say that the slope is "rise over run," or the number of squares we move up ("rise") for every square we move to the right ("run"). If we let the y-axis represent miles, and the x-axis represent hours, suddenly a slope of 55 miles per hour becomes a slope of .

Students should understand both the concept of slope and how to find the slope when given a verbal description or the graph of a line. You may also want to teach them how to find the slope given two points, as well as slowly introduce them to the concept of a linear equation. Not necessary quite yet, but it'll prove useful soon.

#### Drills

1. The slope of a line is 2. What does that mean?

The line goes up 2 units every time it goes over 1 unit

Slope is the change in y divided by the change in x, or rise over run. We can change 2 to 21, so the rise is 2 and the run is 1. Only (B) makes sense because (C) would mean a slope of ½, (D) would mean a slope of 1 (because 22 = 1), and (A) isn't right because lines have infinite length.

2. Find the slope of a line segment that has endpoints of (1, 1) and (5, 9).

2

We can use the slope formula to find the slope given two points or derive it logically. The difference between the y coordinates represents how much "rise" we have (9 – 1 = 8), while the difference between the x coordinates represents how much "run" we have (5 – 1 = 4). If we put rise over run, we end up with 84 = 2 as our slope.

3. What does the slope of this line segment mean?

It moves up 2 units every time it goes over 1 unit

Slope is the change in the vertical over the change in the horizontal. Since our slope is 2, we know that means it moves 2 units up for every 1 unit to the right. This is the only right answer since (B) would mean a slope of ½ and (C) would require the use of the distance formula. Also, we really hope you didn't pick (D).

4. We find that line a has a larger slope than line b. What does this mean about line a?

Line a is steeper than line b

Since its slope is greater, line a is increasing in height faster than line b. That means it's steeper. If you need proof of this, try graphing two lines with different slopes to understand it visually. By the way, lines have infinite lengths and infinitely small thicknesses, which means that (C) is the only answer that even remotely makes sense.

5. What does it mean if two lines have the same slope?

They have the same rate of change

They could be the same line or they could be parallel. We don't have enough information to answer (A), but we know that slope is another name for rate of change. If the two slopes are equal, the lines have the same rate of change. Perpendicular lines have negative reciprocal slopes, and congruency requires some measurement but lines have infinite length. That's why (C) and (D) are incorrect.

6. In a weeklong free-throw contest out of 10 shots, Julian scored 4 on Monday and 8 by Friday. Connor scored 6 on Monday and all 10 by Friday. Who improved faster in shots per day?

Both showed the same rate of improvement

Since we're looking at shots per day, the top number in our slope should be the shots and the bottom number should be day. If we calculate the difference in shots for Julian (8 – 4 = 4) and Connor (10 – 6 = 4) and calculate shots per day where Monday = day 1 and Friday = day 5, we find that both Julian and Connor had rates of change of 44 = 1 shot per day. While (A) and (C) are right in the calculation, (D) is right because their rates of improvement were the same in terms of shots per day.

7. The Big Cheese, a dairy supermarket, is advertising mac and cheese at 4 boxes for \$5. Cheap'n'Cheesy, their competitor, is advertising a special where buying a box of 6 for \$8 gives you two more boxes for free. Which is the better bargain?

Cheap'n'Cheesy, with a unit price of \$1

The Big Cheese is selling 4 boxes for \$5, or \$54 = \$1.25 each. Cheap'n'Cheesy is selling 8 boxes for \$8, which comes out to \$1 each. It should be clear that Cheap'n'Cheesy has the better deal. Although with the way they taste, you might be better off making your own.

8. On various math tests, Erin scored 58 out of a possible 59 points, Fran scored 81 out of 84 points, Greta scored 30 out of 31, and Henri scored 24 out of 28. Who got the best grade?

Erin

To calculate the grades, we can divide the points received by the total points possible. If we do this for every student, we find that Erin got over 98% correct, while the others scored below 97%. As long as she doesn't rub it in their faces.

9. The population of Happytown increased from 400,000 to 750,000 in the past 5 years. The neighboring town of Smileyville increased its population by 60% over the same time period. Which town showed the greater rate of change per year?

Happytown, with an annual rate of 17.5%

Since all the answers are in percent, we should calculate the population change of Happytown in terms of percentage. That's 750,000 – 400,000 = 350,000 new people, or a change of 350,000 ÷ 400,000 = 87.5% over the course of 5 years. While Smileyville only had an annual rate of 60% ÷ 5 = 12%, Happytown had a change of 87.5% ÷ 5 = 17.5% annually. Answer (B) comes from incorrectly dividing 350,000 people by the new total of 750,000, while (C) forgets to divide by the number of years and (D) seems to forget about Happytown entirely.

10. Why would anyone care about the rate of change of a population?

All of the above