SBAC Practice Test Questions 25-26 (Day 173)

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

In triangle ABC, AB = 6 and BC = 8. What is the maximum value of the area?

This is yet another one of those problems that straddle the line between Geometry and trig. If we use trig and teach the formula A = (1/2)xy sin(theta), then we know that the maximum value of sin is 1. Thus the max value of the area is:

A = (1/2)xy = (1/2)(6)(8) = (3)(8) = 24

So the desired maximum area is 24 -- and of course, today's date is the 24th.

Now we ask, is there a purely geometric way to solve this problem? Consider the following -- we begin with segment AB of length 6. Now draw a line parallel to line AB that is exactly 8 units away. The locus of all points 8 units from B (that is, the location of all possible points C) is a circle of radius eight. It is tangent to the new line. The maximum area thus appears at the point of tangency -- otherwise C lies in between the two lines and so the height (and thus the area, half the base times the height) is less than that said maximum.

This explanation is a bit off -- it could probably be made more rigorous. The Radius-Tangent Theorem -- that the radius to a point of tangency is perpendicular to the tangent line -- likely comes into play.

This is what I wrote two years ago about today's lesson:

Question 25 of the SBAC Practice Exam is on inequalities:

A student earns $7.50 an hour at her part-time job. She wants to earn at least $200.

Enter an inequality that represents all of the hours (h) the student could work to meet her goal. Enter your response in the first response box.

Enter the least whole number of hours the student needs to work to earn at least $200. Enter your response in the second response box.

This is a first-semester inequality problem.

7.50h > 200

h > 26 2/3

So the inequality is 7.50h > 200 ("at least" = "greater than or equal to") and the smallest whole number value that satisfies it is 27 hours.

Both the girl and the guy from the Pre-Calc class fail to write the correct inequality. Both of them are confused by "at least" and write "7.50h < 200." Actually, the guy writes 200 > 7.50x, which is equivalent to the girl's inequality. The guy doesn't bother solving his inequality. The girl solves it by writing "at least 26.67" hours -- she missed the words whole number. Notice that no inequality needs to be written in the box -- all that's necessary is the single whole number 27.

Question 26 of the SBAC Practice Exam is on comparing statistics:

Michael took 12 tests in his math class. His lowest test score was 78. His highest test score was 98. On the 13th test, he earned a 64. Select whether the value of each statistic for test scores increased, decreased, or could not be determined when the last test score was added.

(The possible stats are standard deviation, median, and mean.)

Stats appears in the last chapter of Glencoe Algebra I, and so this is a second-semester question.

Anyway, the mean must decrease because Michael's last test score is lower than that of any previous test that he has taken. The standard deviation must increase when that low value is added. But we can't be sure about the median. The median of 12 values is the mean of the 6th and 7th value, but the median of 13 values is the 7th value (the old 6th value before the 64 happened). So median could decrease if the original 6th test is less than the 7th -- but the median could stay the same if the original 6th and 7th scores were equal. (It's impossible for the median to increase here!)

The girl from the Pre-Calc class gives the correct responses for the mean and median, but she incorrectly writes that the standard deviation decreased. Then again, most students are less familiar with standard deviation than with the measures of central tendency. But at least she tries -- the guy just leaves it blank.

This lesson was at one point given as an activity. Of course, our activity day this year will be later on.

SBAC Practice Exam Question 25

Common Core Standard:

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

SBAC Practice Exam Question 26

Common Core Standard:

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Commentary: The inequality I wrote for h can be solved as early as Lesson 4-6 of the U of Chicago Algebra I text. Meanwhile, stats isn't covered in the text at all, except for Lesson 1-2 where mean and median appear, but not standard deviation.