# Common Core Standards: Math

### Statistics and Probability 8.SP.A.3

3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

As we've already seen, a scatter plot can help us see what data is doing. And if the data follows (or almost follows) the equation of a straight line, we can tell a lot about the data. Not only can we see what it's done so far, we can also anticipate what it will probably do in the future. (Basically, we're fortunetellers minus the crystal balls.)

On one hand, students know about scatter plots and the line of best fit. On the other hand, they know about the y-intercepts and slopes of lines. Time to put those hands together.

Students should apply their knowledge of linear equations to the context of lines of best fit, understanding what the slope and y-intercept mean in terms of the axes of the scatter plot.

For example, let's say we've found that the relationship between Justin Beiber's high notes and capuchin monkey births has a line of best fit.

Students should be able to interpret the slope of  as 1 capuchin monkey birth for every additional 150 Hz in the frequency of the Beib's voice. Reinforcing the concept of slope as "rise over run" and pointing out which variable is on which axis might be helpful for the students. (Remember that x is the independent variable, while y is the dependent variable.)

Additionally, we can interpret the y-intercept of -1 as being the number of monkeys born when Beiber is singing at 0 Hz (or, uh, not singing at all?). But how can we have negative births? Does that mean Beiber is killing baby monkeys whenever he isn't singing? Our guess is probably not.

Before blindly accepting the line of best fit's offerings, students should double-check that they make sense within the experiment, within the scatter plot, and whether or not it's a feasible possibility in general. For example, our graph only goes down to 300 Hz, so we don't have actual data about any lower frequencies. Besides, how can J.B. sing at 50 Hz when the human vocal register doesn't even go below 80 Hz?

Students should remember to interpret the line of best fit within the context of the data provided by the scatter plot. The line of best fit is meant to understand the general trend of data, but it might not be able to explain everything about it, but that's okay. We've got our brains to do that.

#### Drills

1. Kelly conducted a statistics experiment to determine the relationship between her brother's age and how many baby teeth he lost each year. With age (in years) as her independent variable and number of teeth lost as her dependent variable, the line of best fit had a slope of 1. What does this mean?

Kelly's brother lost 1 additional tooth every year

The slope is rise over run, vertical over horizontal, dependent over independent. Since our dependent variable is the number of teeth lost and the independent variable is the number of years, then we know our slope means 1 tooth per year. This only eliminates (B), but what are the differences in the others? If (A) were true, our line would be horizontal, which means a slope of 0. Logically, (D) doesn't make sense because it treats age as the dependent variable (and your age doesn't depend on how many teeth you lose!). The only answer that makes sense is (C).

2. John bragged to all his friends that his average grade in math had gone up by 2% per week for the past 10 weeks. They were unimpressed, since his average in the beginning was 25% and he was still failing. What would be the equation of the line of best fit for this data?

y = 2x + 25

In slope-intercept form y = mx + b, the slope is m and the y-intercept is b. First, we should consider what our variables mean: the independent variable x represents weeks since John first started tracking his grade, and y represents his grade percentage. This makes sense, because his grade depends on time, not the other way around. His grade at the beginning was 25%, so that should be our y-intercept, b. The slope of the line should be 2, because rise over run tells us his grade increases by 2% per week. In other words, we have y = 2x + 25, which is (D).

3. John bragged to all his friends that his average grade in math had gone up by 2% per week for the past 10 weeks. They were unimpressed, since his average in the beginning was 25% and he was still failing. If he follows this trend, what grade will John have in 10 more weeks?

65%

John started tracking his grade when he had 25% in math. If his grade increases by 2% per week, we can represent his average grade using the line of best fit, which has the equation y = 2x + 25, where y is his grade percentage and x is the number of weeks. If 20 weeks have passed, we can plug in x = 20 and solve for his grade: y = 2(20) + 25 = 40 + 25 = 65. We know that (C) is right since we just calculated it, but also because (A) is the grade he started out with and (B) and (D) are increases of 1% or 3% per week, not his current trend of 2% per week.

4. John bragged to all his friends that his average grade in math had gone up by 2% per week for the past 10 weeks. They were unimpressed, since his average in the beginning was 25% and he was still failing. If he follows this trend, how many more weeks will it take him to hit his goal of 85%?

20

John started off with a grade of 25%, but over the past 10 weeks, he's managed to raise it to a 2(10) + 25 = 20 + 25 = 45%. Since we're only looking at how many more weeks it'll take, we can set our new y-intercept to be 45. Since the grade he wants is 85%, we can substitute y = 85 and solve for x using the equation y = 2x + 45 to find the number of additional weeks it'll take to reach that grade. If we do this right, we get 85 = 2x + 45, or x = 20. It'll take John 30 weeks in total, but 20 more weeks to reach his goal.

5. The current commercial for FatAway shows that one of their clients lost 35 pounds in the first month of using the product. The commercial says that these results are not typical.

FatAway, Inc. sets up a scatter plot showing their clients' total weight loss for each month using their product. If the company claims that all their clients lose the exact same amount of weight every month, which of the following would be true?

The line of best fit would go through all data points

This might be a bit confusing, so let's imagine the scatterplot itself. The independent variable (the x-axis) represents months after using the FatAway product, and the dependent variable (y-axis) is the weight lost in total. If all clients lose, say, 5 pounds every month, there should be one data point at (1, 5) for every client. Same goes for (2, 10), (3, 15), and so on. So regardless of how many clients FatAway has, they'll only have one data point per month. We can clearly see that this would form a single straight line with a positive slope, meaning (A) and (B) are wrong. While (D) is tempting, the correct answer is (C).

6. FatAway, Inc. sets up a scatter plot showing their clients' total weight loss for each month using their product. If one of their clients loses weight at a faster rate than the rest, how is this expressed on the scatterplot?

A line with a more positive slope

We can think of each client as having a line that tracks his or her weight, with points for each month and amount of pounds lost per month. That means one client corresponds to a line, or a set of points rather than just one. So basically, (A) is out. The more weight the client loses per month, the higher the data points will be, but "faster rate" means more weight per month, and not overall.

7. Malcolm has decided to conduct an experiment comparing the number of hours spent studying per week with GPA. Assuming that his results follow a linear relationship, what would you expect to find?

A positive slope

It's no secret that overall, the more studying you do, the higher your GPA is likely to be. You know it. We know it. It's no big mystery. A negative slope would mean that the more studying you do, the lower your GPA will be, which doesn't make sense (unless "studying" really means playing World of Warcraft). An undefined slope is a perfectly vertical line and a slope of zero means a horizontal line. So either all GPAs are the same or all students study the exact same number of hours? Doubtful. The answer that makes the most sense is (A).

8. Malcolm has decided to conduct an experiment comparing the number of hours spent studying per week with GPA. If the line of best fit in his scatterplot has a y-intercept of 1.8, what does this mean?

A student who doesn't study at all has an average GPA of 1.8

The y-intercept is the value of the dependent variable (y) when the independent variable (x) equals zero. So…what are the independent and dependent variables, anyway? Well, if the GPA depends on the hours of studying, then GPA is the y value and time studying is the x value. The y-intercept means that when x = 0 (in other words, you study for 0 hours), your GPA is given by y = 1.8. Slope isn't the issue here, and we wouldn't need a scatter plot to find the average student's GPA.

9. Casey is doing research on whether or not peanut allergies in elementary school kids are due to the amount of peanut-laden snacks served in school. After plotting her data, she found a line of best fit with the equation . What conclusion could be drawn from this?

Schools that serve more peanut snacks are likely to have fewer students with peanut allergies

The key aspect to focus on here is the equation: . The slope is , which means that x gets bigger, y gets smaller. Since x, the independent variable, represents the number of peanut snacks served in school and y, the dependent variable, represents the number of kids with peanut allergies, we know that (B) is the right answer. If we look closely, (C) and (D) could mean the same thing, but this would mean a line of best fit with slope 0 or no line at all. If you thought it was (A), you may not have seen the negative sign.

10. Casey is doing research on whether or not peanut allergies in elementary school kids are due to the amount of peanut-laden snacks served in school. After plotting her data, she found a line of best fit with the equation . What is the reason for this trend?