Could You Pass New York’s Algebra I Exam?

Only 63% of NY Students Did This Year

Published in

Math Hacks

5 min readDec 1, 2015

Yesterday there was an interesting article in the New York Times discussing New York’s Regents Algebra I exam. This year the exam, which aligned with new standards, had a passing rate of only 63%, down from 72% last year. This left parents and educators wondering,

“Is the exam too difficult or are student’s underperforming?”

Let’s find out.

The New York Times posted 5 questions from the exam in an interactive feature. Give them a shot and then I’ll walk you through the solutions.

Question 1

The relationship presented in both scenarios is linear. Therefore we have two parts to consider: the initial amount (y-intercept) and the rate of increase/decrease (slope).

Recall the form of a linear equation:

Rowan begins with $50 in his savings jar, so b = 50. He adds $5 per week, so the rate of increase is $5 per week, i.e. m = 5.

Jonah starts with $10 and adds $15 per week. So Jonah’s equation will have a slope of 15 and y-intercept of 10.

Now compare the slopes. Since Jonah’s slope is a larger value, it’s graph will be steeper than Rowan’s graph.

Question 2

Julia has $22 to spend. Therefore her total expenditure on juice and gum must be less than or equal to $22. So you can immediately eliminate the second and fourth options.

She buys 7 packs of gum at $0.75 per pack, therefore she spends 0.75(7) on gum. We have an unknown quantity of juice, b, that cost $1.25 each. So the total spending on juice is 1.25 times b.

Combine and set less than or equal to 22.

Question 3

Since we are working with two unknown values, assign them variables.

We’ll need two equations to solve this problem. The first equation deals with the quantity of small and large candles, which must total to 20 candles.

equation expressing quantity of small and large candles

The second equation deals with candle sales.

Calculate the sales from each type of candle by multiplying the quantity sold with the price per candle.

Small candles sell for $10.98 each so their total is 10.98 times x
Large candles sell for $27.98 each so their total is 27.98 times y

The grand total is $355.60. So our second equation is:

equation representing profits from each type of candle

This is a system of equations because we must use both equations to solve for x and y. The substitution or elimination methods are both viable options for solving. I think the substitution method is easiest here, so that’s what I’ll demonstrate.

Begin by isolating one variable in one of the equations. I isolated x in the first equation.

Then substitute (20-y) for x in the other equation, yielding an equation with only one type of variable, y.

Distribute the 10.98, combine like terms and solve for y.

Therefore 8 candles are large (and 12 are small).

Question 4

The above graph is in the shape of a cubic equation. First check to make sure all the equations are 3rd degree.

Since they are, next try matching them up by their roots, i.e. places they intersect the horizontal x-axis. The graph intersects the x-axis at the following coordinates: (-4,0), (-2,0) and (1,0).

The question then becomes, “which equation yields y = 0 when I plug in each of these values for x?”

It’s easy to see that the first and third equations equal zero when x = -2 because -2 + 2 = 0. Next plug x = 1 into the first equation to see if it equals 0 as well.

Now plug x = -4 into the equation to make sure it also equals zero. Since it does, the first equation is likely the answer. To be certain plug in x = -4 and 1 into the other equation to make sure it doesn’t yield y = 0.

Question 5

The easiest way to solve this problem is to list out the payout values for each company until you get to the month where Company B’s offer is larger than Company A’s.