SBAC Practice Test Questions 23-24 (Day 172)

This is what I wrote last year about today's lesson:

Question 23 of the SBAC Practice Exam is on comparing rates:

Nina has some money saved for a vacation she has planned.

• The vacation will cost a total of $1600.
• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation? Enter the result in the response box.

First of all, let's ignore the given value $1600 completely, since it has nothing to do with the solution of this problem. What we want to know is, how long will it take Nina to save the same amount of money at $200/wk. as she does at $150/wk. for eight weeks? Thus this is the equation:

200w = 150(8)

200w = 1200

w = 6

So it takes her six weeks to save the money as opposed to eight weeks. Therefore it will take her two fewer weeks (that is, 8 - 6) to save the money -- students should enter the number 2.

Sometimes students might forget that even though w = 6, they must enter the number 2, since the question is not how many weeks will she save, but how many fewer weeks. Sometimes I avoid this problem by letting x = 8 - w, so that as soon as I find x, I have the number I need to enter:

200(8 - x) = 150(8)

1600 - 200x = 1200

-200x = -400

x = 2

I often solve Pappas problems this way -- using the variable x only for the final value that I need to find rather than any intermediate values. Also, if there's a system of equation and Pappas asks us to solve for x, I might eliminate y even if x is easier to eliminate, so that I find the actual asked-for value more directly.

Both the girl and the guy from the Pre-Calc class have trouble with this question. In fact, the guy has only written:

150x = 1600

on his paper, with no final answer. The girl, on the other hand, has written a little more:

1600 = 150w

150(8) = 1600

200

and then she gives 6 weeks as her final answer. Yes, it does take Nina six weeks to save the money -- but that answer isn't justified by anything on the girl's paper. Actually, I'm starting to wonder whether I've attempted to help her during class, perhaps by telling either her only or the entire class that it takes eight weeks to save the money at the slower rate or six weeks at the faster rate. So she tries to include the numbers 8 and 6 somewhere on her paper.

Once again, I wonder whether there's anything to do to make this question easier for the students to understand and answer correctly. Earlier, I wrote that 1600 has nothing to do with the solution -- so let's restate the problem with the 1600 line left out:

Nina has some money saved for a vacation she has planned.

• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation? Enter the result in the response box.

Now look at those first two lines again. The lady will save $150 per week for eight weeks -- that is, she will save a total of $1200. The problem is less confusing because we now know that the target amount is $1200, not $1600. No student would write an equation with 1600 at all, and so more students are likely to get the right answer.

But what is a teacher to do, since we can't make that 1600 line disappear from the computer we're using to administer the SBAC? Well, we can ask the students to look for extraneous information -- for example, we read the following to them out loud:

Nina has some money saved for a vacation she has planned.

• The vacation will cost a total of $1600.
• She will put $150 every week into her account to help pay for the vacation.
• She will have enough money for the vacation in 8 weeks.

Question for the class: how much money will she have saved in eight weeks?

The hope is that that point, the students see that 150 * 8 = 1200, so it's $1200. At this point, the teacher tells the students that so far, 1600 has nothing to do with the problem. The phrase "enough money for the vacation" is another way of saying $1200. This is not the same as the total cost of the vacation, $1600, since presumably Nina already has $400 in the bank before saving begins.

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save enough money for the vacation?

And since the phrase "enough money for the vacation" is another way of saying $1200, we can rewrite the question as:

If Nina was able to save $200 a week instead of $150 a week, how many fewer weeks would it take her to save $1200?

And now it becomes obvious that at $200 a week, it takes six weeks to save $1200. It now remains for the teacher only to emphasize fewer -- it's not how many weeks, but how many fewer weeks (than eight) it takes for her to earn the money. Therefore we must subtract 8 - 6 = 2.

Some people criticize the Common Core over wording in questions such as these. Indeed, yesterday and today we have two consecutive Questions 22-23 with the phrases "average rate of change" and "enough money for the vacation" -- and both are likely to trip students up.

But I'm not sure how this question can be improved. We might really be in a situation where we want to save money for a trip, but already have money in the account. Thus we must realize to consider that we already have some money, and so it's the remainder that we need to save.

Question 24 of the SBAC Practice Exam is on quadrilaterals:

Consider parallelogram ABCD with point X at the intersection of diagonal segments AC and BD.

Evelyn claims that ABCD is a square. Select all the statements that must be true for Evelyn's claim to be true.

• AB = BD
• AD = AB
• AC = BX
• Angle ABC isn't 90
• Angle AXB = 90

Here's some Geometry -- yes, it's been some time since I posted Geometry on this Geometry blog. I'll look at the three length equations first. AD = AB is obviously true, since the sides of a square must indeed be congruent. On the other hand, AB = BD is definitely false -- the diagonal of a square is sqrt(2) times the length of a side, not the same length as a side. And AC = BX is false as well -- the diagonals of a square (or any rectangle) are congruent, and so AC can't be half of BD (which is what BX is, as the diagonals of a parallelogram bisect each other).

Now let's look at the angle statements. The statement that ABC isn't 90 is obviously false -- the angles of a square must be exactly 90. The other statement that AXB = 90 is true, since the diagonals of a square are also perpendicular. So students must select two correct answers, AB = AD and AXB = 90.

Notice that if all Evelyn knew about her parallelogram is AB = AD and AXB = 90, that would not be sufficient for her to conclude that it's a square. A rhombus, after all, also has congruent sides and perpendicular diagonals. (In fact, given that ABCD is a parallelogram, just one of these two is sufficient for her to conclude that ABCD is a rhombus.) On the other hand, adding the falsity of the claim that ABC isn't 90 (that is, the truth that ABC = 90) to either of the two selected statements is enough for Evelyn to conclude that ABCD is a square.

Both the girl and the guy from the Pre-Calc class correctly answer this question.

SBAC Practice Exam Question 23

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 24

Common Core Standard:

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Commentary: The first equation I wrote for w can be solved as early as Lesson 4-4 of the U of Chicago Algebra I text, although the distributive property in the x equation must wait for Lesson 6-8 a little later on. Meanwhile, the properties of rectangles, rhombuses, and squares are covered in Lesson 5-4 of the Geometry text.