CAS Calculator Hacks for Redesigned SAT Math

Test Preparation SAT Preparation
1

Even with the new no calculator section, there are still several problems on the calculator section that can be hacked with a CAS calculator. I thought I would share a few SAT CAS hacks in this thread.

Hack Number 1: Solving an algebraic equation for one variable in terms of one or more other variables.

Many people know that you can solve algebraic equations with a CAS calculator. Not so many people know that with the proper command line, you can also solve for one variable in terms of other variables. This is something I discovered on my own shortly after I purchased my first CAS, a TI89.

Here is an example:

The average acceleration of a car in miles per hour^2 over a period of h hours is calculated using the formula
a=(f-s)/h , where s is the speed in miles per hour at the beginning of the period and f is the speed in miles per hour at the end of the period. What is s in terms of f, a and h?

To solve this problem with a CAS calculator, just copy the equation and solve for all the variables except the “in terms of” variables (the variables that you want to appear in the answer). That is, you input:

solve(a=(f-s)/h,s) Enter

This returns s=f-ah

2

Similarly, problem 22 of the redesigned PSAT of October 28, 2016 contains the formula for Fahrenheit degrees in terms of Kelvin degrees, and asks the student to solve for Kelvin in terms Fahrenheit.
That is, we are given:

F=9/5 (K-273.15)+32 and asked to solve for K in terms of F.

CAS Solution:

solve(F=9/5 (K-273.15) +32,K) Enter

This returns K=5f/9 + 255.372

3

https://collegereadiness.collegeboard.org/sample-questions/math/calculator-permitted/29

Above is the link to question 29 of the Sample Questions College Board has posted for the new SAT.

If the expression (4x^2)/(2x-1) is written in the equivalent form 1/(2x-1) + A, what is A in terms of x?

CAS Solution:

solve((4x^2)/(2x-1)=1/(2x-1) + A,A) Enter

This returns A=2x+1

4

Hmph…

I’m curious if someone will do or has done a study correlating calculator type with SAT math score.

The AMC 8/10/12 exams banned calculators almost 10 years ago. I think that was a good move.

5

I am pretty sure it correlates…but that does not mean that everyone should run out and get an n-spire or TI89. I have found that students have to be at a certain level of math comfort for those tools to be more helpful than they are intimidating. Students who can handle these calculators are already the stronger students.

But a funny thing about that question #29…it is a classic “back door” problem. I used x= 3 and got 7.2 on the left side of the equation and .2 + A on the right side. So for me, A=7. Then, I put x=3 into each answer choice. Only a) 2x+1 came out to 7.

This was supposed to be “hard”.

I’m not against calculator hacks. But the “back door” play is my favorite SAT hack of all time.

6

Hey in my day we looked up the values for sin, cos, log etc. in a BOOK.

I think Microsoft, Texas Instruments, and College Board are all in it together. Calculator sales are big $$$$.

I am not sure how you define “stronger”. I have students who started around M 480 who can use a TI-Nspire CX CAS with no problem. The operating system is much more user-friendly than was the old TI89 operating system. My students are incredibly fast with buttons and are used to electronics. Of course, I have to show them the commands…

7

Yeah, I am also old enough to have used sine tables and log tables. (But not quite old enough to have needed a slide rule.) Remember when linear interpolation was actually a necessary skill? The idea behind it occasionally pops up on an SAT question. Like: here is a chart for a linear function f but with some missing values…

I was thinking about the TI89. I’ll take a closer look at the n-spire. Thanks.

8

YES! I remember doing linear interpolations in high school. Wow I haven’t thought about slide rules in a long time. My father (a math teacher) used a slide rule all the time so we had one around the house when I was a child, but I never quite understood how to use it.

The Texas Instruments people bent over backwards to make a user-friendly interface. I am going to post some more hacks when I have time.

9

I have seen several questions on the new SAT that ask for the value of x for which a function with a denominator is undefined.

Most students know (or should know) that the function is undefined when the denominator = 0. If you know this, you can use the calculator to do the algebra. You set up an equation
denominator=0 and then solve it with the calculator.

For example, Official Practice Test 1 Section 4 Question 36

h(x)= 1/[(x-5)^2+(x-5)+4] For what value of x is the function undefined?

CAS Solution

solve((x-5)^2+(x-5)+4=0,x) Enter

Returns x=3

10

@Plotinus there’s also the “solve a system of linear equations” method using the reduced row-echelon form.

It’s been a while since I used a graphing calculator (so bear with me!), but if I remember correctly, I think you can define functions whose outputs are boolean (true or false) values. For example, practice test 1, Section 4 (calc. section), #11:

  1. Which of the following numbers is NOT a solution of the inequality 3x - 5 ≥ 4x - 3? A) -1 B) -2 C) -3 D) -5

While it’s not hard at all to solve without a calculator, I feel like 11. could also be solved by cheap calculator use:

f(x) := 3x - 5 ≥ 4x - 3 [Enter]
f(-1), f(-2), f(-3), f(-5) [Enter]

This should return false, true, true, true. There are probably better examples with harder problems that can be solved using the same method.

11

@MIT94er

Definitely a calculator problem, but I think your method is too long.

The calculator solves all inequalities exactly the same way it solves equalities.

CAS Solution

solve(3x - 5 ≥ 4x - 3,x) Enter

returns x<=-2

At that point the student has to understand that -1>-2, so some thought is required. But I usually tell my students that the calculator does not do all the thinking, just the algebra.

12

One of my favorite hacks for the old SAT was for this problem, which some of you may have seen:

         1/(2)(3)  1/(3)(4)  1/(4)(5)

The nth term of the above sequence is given by: 1/[(n)(n+1]) and this is equal to 1/n - 1/(n+1)

What is the sum of the first 80 terms of the sequence?

The TI-Nspire has a summation (sigma) template. You press the picture of the sigma, and the sigma appears with spaces to fill in the upper and lower bounds and the expression for the nth term. I can’t represent it with the sigma sign here so I will just write “sigma”.

CAS solution

80
sigma (1/(n)(n+1)) Enter
n=1

returns 80/81

13

Then there was a similar old SAT problem with a product instead of a sum. So you just use the product (pi) template.

It went something like
What is 1/3 x 2/4 x 3/5 x …x 98/100?

CAS solution

98
pi (n)/(n+2) Enter
n=1

returns 1/4950

14

And what about this one? Redesigned Official Practice Test 3 Section 4 Question 30

3x + b = 5x−7
3y + c = 5y − 7

In the equations above, b and c are constants. If b is c minus 1/2 , which of the following is true?

A) x is y minus 1/4.
B) x is y minus 1
C) x is y minus 1/2
D) x is y plus 1/2

First of all, I just want to say that this question has the worst use of math language I have ever seen.
What is gained by writing “b is c minus 1/2” or “x is y minus 1/4” except to slow down and confuse non-native speakers of English? Or is this how we “narrow the gap”?

That said, if you can translate the words into math, you can put the problem into the calculator. The TI-Nspire has a template for systems of linear equations. You press the template button, choose the number of equations, enter the equations, enter the variables you want to solve for, press enter, and you’re done (almost).

You need to write the third equation: b=c-1/2
Then you need to see what variable you need to solve for.
The answer choices all give x in terms of y. That means you solve for all the variables except y. That is, you solve for a,b,and x.
The calculator will give you answers for all three in terms of y, but you are only interested in the answer for x.

CAS solution

solve(3x + b = 5x−7
3y + c = 5y − 7
b = c - 1/2, a,b,x) Enter

returns x=(4y-1)/4 and b= (4y-15)/2 and c=2y-7

So the answer is x=(4y-1)/4

At this point you have to realize that (4y-1)/4 = y-1/4 and that this corresponds to x is y minus 1/4.

15

…wait what? I don’t think I’ve seen anyone anywhere spell out “minus” in that context.

I think it’s important to note that all of these problems can be solved without a calculator, and can usually be solved in less time than with a calculator (assuming a capable student). Here we are referring to problems that could be solved with a graphing calculator, with very little thought given to the problem itself.

There was also Test 4, Sec. 4, #25 in which @Plotinus and I agreed that it was easily exploitable with a CAS calculator. I don’t think I mentioned this, but it is also exploitable with a non-CAS calculator using a similar method.

16

A lot of calculators can do things that us students don’t even know. My friend knows a bunch of things he can do on a ti-84 that I didn’t even know were possible.

17

@LlenrocPeoh that’s true - you can even do basic (or BASIC, however you interpret it?) programming on the TI-83/84/NSpire.

I have one of the original TI-NSpire (non-CAS) graphing calculators which has the snap-in keyboards. The NSpire seems to have similar functionality to the 84, but with improved resolution and UI.

18

Yes, of course.

Major assumption here. Most of my students in the old SAT M 500-650 were using their CAS’s for several problems during each test. Clearly, the weaker the student is in algebra, the more it helps. I had an athlete come in with around CR 420 M 440 who needed 1010 CR+M to make NCAA eligibility, and I got her to CR 440 M 570 in 6 weeks partly because of the calculator.

In any case, the capable students are not getting more points for solving the problem without CAS.

19

Here’s another one.

Official Practice Test 2 Section 4 Problem 24

x^2 + y^2 + 4x − 2y = −1

The equation of a circle in the xy-plane is shown
above. What is the radius of the circle?

A) 2 B) 3 C) 4 D) 9

You can do this algebraically or graphically. To do it algebraically, use the CompleteSquare command. You enter the expression and then the variables.

CAS Solution

CompleteSquare(x^2 + y^2 + 4x − 2y = −1,x,y) Enter

returns (x-2)^2+(y-1)^2=4

Of course, the student has to know that 4=r^2, otherwise he will pick the trap answer choice C instead of the correct answer choice A.

The graphical solution is slower but safer for students who know nothing except the calculator menus and how to fill in the boxes of the equation template.

Step 1: Open a graphing page.
Step 2: Press Menu, Graph Entry/Edit, Equation, Circle
At this point the calculator offers you a choice of two forms of the circle equation:
(x-h)^2+(y-k)^=r^2
ax^2+ay^2+bx+cy+d=0
In this problem, we have a slightly modified version of the second form.
Step 3: Press ax^2+ay^2+bx+cy+d=0
Step 4: The calculator provides an equation template with fill-in boxes for a,b,c, and d.
Here a small amount of thinking is involved to get the signs right, especially for d. Enter
a: 1
b: 4
c:-2
d:1 Enter
Step 4: The calculator provides a beautiful clear colored graph of the circle from which it is evident that r=2.

20

There are not that many problems you can solve by just copying equations into the calculator with no thinking whatsoever. However, there are some.

See, for example, PSAT October 14 Section 4 Problem 4
(x+2)^2-9=0 What is a possible value of x?

CAS Solution
solve((x+2)^2-9=0,x) Enter
returns x=-5 or x=1

and PSAT Oct. 14 Section 4 Problem 28
3x+2y=9
5x-y=-11 What is y?

CAS Solution
solve(3x+2y=9
5x-y=-11,x,y) Enter

returns x=-1 and y=6