Then there are the questions of the form, “Which of the following is an equivalent expression?”
Here you have to use the fact that the difference between equivalent expressions=0. So you just use the calculator to subtract each answer choice from the original expression to see which difference= 0. Make sure you put parentheses around the expression in the answer choice to avoid sign mistakes.
PSAT Oct. 14 Section 4 Question 2
Which of the following is equivalent to 3(x+5)-6?
A. 3x-3 B. 3x-1 C. 3x+9 D. 15x-6
CAS Solution
A. 3(x+5)-6 -(3x-3) Enter
returns 12
B. 3(x+5)-6 -(3x-1)Enter
returns 10
C. 3(x+5)-6 -(3x+9) Enter
returns 0
The calculator has copy and edit functions similar to a word-processor, so after the first time you can just copy the expression and change the last number. The whole thing only takes a few seconds.
Ok, I know you’re telling me that question 2 was too easy to get out the calculator. How about question 27? With the calculator, these two have the same difficulty.
PSAT Oct. 14 Section 4 Question 27
(2x-2)^2 -(2x-2)
Which is an equivalent expression?
A. 2x-2 B. 2x^2-6x+6 C. 4x^2-10x+2 D. (2x-3)(2x-2)
CAS Solution
Note: I always tell my students to start from the answer choice that looks the most complicated. So here we should start from D. You also have to remember to put in the multiplication sign between factors.
For a question about whether lines in a system of linear equations are parallel, perpendicular, intersecting, or the same line, graph both lines using the line equation templates. You don’t have to convert any equations, just select the correct template and enter the coefficients in the boxes.
PSAT Oct. 14 Section 4 Question 16
y=x-4
4x-4y=12 Are these lines parallel, perpendicular, intersecting at point blah blah or the same?
CAS Solution
Step 1: open graphing page
Step 2: Menu, Graph Entry, Equation, Line, y=mx+b
Put 1 in the box in front of the x and -4 in the box for b.
Step 3: Menu, Graph Entry, Equation, Line, ax+by=c
Put 4 in the box for a, -4 for b, and 12 for c
The calculator provides a beautiful colored graph showing that the lines are parallel.
For word problems, first you have to be able to translate the words into equations. If you can do that, you can use the calculator to solve the equations.
For example. PSAT Oct. 14 Section 4 Problem 11 had a table with some data, and a cost question about cookies and water. The student has to do a bit of interpreting to get the equations
21w+25c=45.00
w+c =2.00 What is c?
At that point,
solve(21w+25c=45.00
w+c =2.00, w,c)
returns w=1.25, c=0.75
PSAT Oct. 14 Section 4 Problem 29 is a solid geometry word problem. You have a rectangular box with dimensions h, h-3, h+4, the area of the base is 60, and you have to find the volume.
A student who can understand the language and set up the equation will arrive at
(h-3)(h+4)=60 and know that he or she needs the value of h, and that he or she has to multiply h(60) to get the volume.
Many of my students who get to this point will prefer to solve the equation by CAS. It takes less than 20 seconds with the calculator, and no brain power.
Another thing that troubles me about the new test is the order of the no calc and calc sections.
There are quite a few “hard” questions in the no calculator section that can be solved brainlessly with a CAS calculator.
Students will have their CAS calculators in hand during the calculator section. What will stop them from using the calculators (also non-CAS) to solve non-calculator questions from the previous section?
They won’t even have to turn back to the pages of the earlier section (although rumor has it that some students do this, especially if proctoring is lax).
If a student can remember a question and the answer choices, he can use a calculator to solve the problem and then fill in the bubble or grid on the answer sheet during the calculator section.
I admit it would be difficult to remember more than one or two questions without making notes or turning pages. But even one or two questions can make a big difference in some cases
There are many more tricks and useful commands you can exploit. These are just a few obvious ones. I am not even good with a calculator, so I am sure other people have invented better tricks than these, and also special programs.
With the new emphasis on basic algebra 1 and algebra 2 skills and algebraic manipulation even in the calc section, the new test seems more CAS hackable than ever. Data entry skills are crucial.
I understand that there are many new SAT problems that can be exploitable with a graphing calculator.
The questions I probably should’ve posed earlier are:
*How predictive are new SAT math scores going to be in terms of actual ability? From here it seems that scoring 500-600 is pretty doable by just exploiting CAS capabilities.
*What do you think CB or ETS is going to do about it? (either banning/restricting calculator use, or some other method)
If you look at the math test that CB wrote, it is pretty clear that measuring math ABILITY is not a top priority. The top priorities are (1) narrowing the gap between the hot potato groups by dumbing down the test and (2) putting pressure on states to adopt Common Core.
The SAT was forced to align itself to the ACT and test what students are routinely taught in high schools. Calculators are an unneeded crutch that emerged from the schools. Students who have learned reasoning and basic skills have the mental agility to ace the test without a fancy calculator and perhaps faster. For many, the CAS tool is a poisoned gift.
See, for example, PSAT October 14 Section 4 Problem 4
(x+2)^2-9=0 What is a possible value of x?<<<
[/QUOTE]
True, but by the time a student is in the middle of typing the numbers, the pencil tester is already answering the next question. After all, how long does the brain need to process x+2 = 3.
The relative speeds of student brains and fingers vary more than you might imagine.
Further, the correct answer choice to this question is not x+2=3 but x+2=-3.
Just think of all the students who are going to forget to put +/- when they take the square root with their brains.
The calculator won’t forget.
It is going to return both the positive and the negative answer. And of course, only the negative answer is among the answer choices.
The calculator is going to beat many, many brains.
I agree that a CAS calculator has the potential to do a lot of harm. However, it is not harmful to the student if used rigorously in the proper way.
I distinguish between homework practice and testing or testing practice.
During homework practice, the students should solve all problems FIRST without any calculator, and THEN with the calculator. Do not solve problems only with the calculator. For the new test, students should also do all numerical calculations first without a calculator.
During testing, do whatever is easiest, fastest and safest for you. This varies greatly by student. Some students use CAS on easy questions just to conserve brain power for the harder ones, or to avoid sign mistakes.
There are plenty of alternate calculator policies out there. No calculators are allowed on the GMAT. For GMAT, students have to know how to do basic arithmetic, but the numbers are fudged so you don’t have to do any long calculations. The GRE has an online calculator for computations, but personal calculators are banned. National school-leaving exams usually allow calculators but not ones that can be programmed. Texas Instruments is not so influential abroad.
Further, the correct answer choice to this question is not x+2=3 but x+2=-3.<<
[/QUOTE]
You sure about that? My aging brain and archaic method seem to suggest that x+2 = 3 is still x=1 and that THAT is one of the answers! Takes around five seconds!
@Zeldie I think what @Plotinus was saying is that among the four answer choices, x=1 is not an answer choice but x=-5 is. Because -5 is also a solution to the equation.
(x+2)^2-9=0 was a multiple choice question and the answer choices were:
A) -5
B) -2
C) 5
D) 7
x=1 was not among the choices. The pencil student had to do
(x+2)^2-9=0 Add 9 to both sides.
(x+2)^2=9 Take square root of both sides.
x+2=+/- 3 Subtract 2 from both sides.
x=1 or x=-5
Pick x=-5 from among the choices.
Many students (say M 550 or lower) would do what @Zeldie suggested:
(x+2)^2=9
x+2=3
x=1
At this point, the student discovers to his or her chagrin that 1 is not among the choices. Some of these students will realize that they forgot x+2=-3 and then get -5. Some of them will backsolve at this point. Probably many or most students who are over M 450 will figure out the right answer choice eventually. But they are going to lose time. Some may get upset.
By contrast, with the CAS approach, solve((x+2)^2-9=0,x) Enter
returns x=1 or x=-5
So this will be faster for the kind of student who forgets to put +/- when taking square roots. It also will be less disconcerting. Some students get upset and frustrated when they work out a problem but don’t get the right answer, and the loss of psychological composure can affect their performance on other questions. CAS can avoid certain traps, making the testing experience a little more relaxed. Using CAS, an M 450 level student with good data entry skills will solve this problem in virtually the same amount of time and just as calmly as a 700 level pencil student.
@Zeldie I think what @Plotinus was saying is that among the four answer choices, x=1 is not an answer choice but x=-5 is. Because -5 is also a solution to the equation. <<
[/QUOTE]
Unless I missed them, the choices of answer were not listed in the question. I do not think it would change anything to my statement as the line x + 2 = 3 is identical to x + 2 = -3. That leads to x = -5.
All semantics! The reality remains that this problem was easily solved in 5 seconds without a calculator.
Many students (say M 550 or lower) would do what @Zeldie suggested:
(x+2)^2=9
x+2=3
x=1
[/QUOTE]
At this point, the student discovers to his or her chagrin that 1 is not among the choices. Some of these students will realize that they forgot x+2=-3 and then get -5. Some of them will backsolve at this point. Probably many or most students who are over M 450 will figure out the right answer choice eventually. But they are going to lose time. Some may get upset. <<<
To be clear, I only suggested the positive answer because of the incomplete problem statement. I maintain that just about everyone who presents the PSAT or SAT should know that (x+2)^2=9 has two answers being x+2=3 and x+2 = -3.
That is as basic as it gets! Further I happen to think that 450 SAT students will not know much about a graphical calculator.
@Zeldie Consider the following two question statements:
If (x+2)^2 - 9 = 0, then what is a possible value of x?
If (x+2)^2 - 9 = 0, then what is the value of x? (alternatively, "then x = ?")
Suppose the answer choices to both of them were the same (-5, -2, 5, 7).
I claim that the first question is good, while the second question is bad.
The first question is fine, because -5 is a possible value of x, and the other three choices cannot possibly be the value of x. The second question is bad, because it is not logically true that “(x+2)^2 - 9 = 0” implies “x = -5.”
I’m watching this thread and I’m somewhat amazed. You can debate what level student will thrive with whatever calculator. But for my less-mathematical students, my goal is to find the solution that requires the least cognitive load. I look at this problem…
(x+2)^2-9=0 was a multiple choice question and the answer choices were:
A) -5
B) -2
C) 5
D) 7
…and I can’t think of any reason to do anything more than try the answer choices. It takes maybe 30 seconds to find that -5 works. If you don’t trust your arithmetic, you can use any level of calculator. And you don’t ever have to wonder why it works. You don’t have to get lost in the thickets of +/- square roots. You just try answers, find the one that works, move on.
