No Calculator, No Cry: How to Master Both SAT Math Sections

Students on March 5th were very surprised by the types of questions the College Board considered appropriate for the No-Calculator and Calculator sections of the redesigned SAT. One might think that the questions on the No-Calculator section would not require significant computations, much like the calculator-free Science section of the ACT doesn't require more than the occasional rough estimation. Au contraire, mon frère. Students were surprised to learn that there indeed were calculations to be made and that the College Board is, in fact, actively testing students’ ability to perform computations.

By contrast, they were equally astonished to find that many questions on the Calculator section did not require the calculator. Had the College Board goofed? Had it assigned different labels to the different sections?

Sadly, no. The goal of the No-Calculator section is to require students to solve math problems using approaches, concepts, and formulas that mirror what they are learning in their high school classes. Most high school math teachers require students to show all their work to receive full credit, and the No-Calculator section is the College Board’s way of acting a little more like your Algebra I teacher. You need to know the ins and outs of a problem, and not simply solve it by employing calculator tricks or helpful multiple-choice strategies like working backwards.

By contrast, the Calculator section cares a lot more about concepts that go beyond simple calculations. Variables, tables, study designs, and word problems abound, requiring students to sift through data and interpret information in context, not merely solve for x. Students might be given messy, real world data to work with. Having a calculator means that a student can focus on interpreting the data rather than tedious calculations.

Going Calculator-free

Admittedly, it is going to feel difficult for some students to keep their calculators under their desk. I imagine many hands reached instinctively for empty space during the redesigned SAT’s first administration, a “phantom calculator syndrome” akin to the phantom vibration syndrome that affects the smartphone-touting millennial generation.

While computing a math problem without a calculator may feel as likely as going a day without Internet, it is possible, and the No-Calculator section is your opportunity to show that you can do math with nothing but your pencil, paper, and grey matter. For one, most of the No-Calculator questions allow for multiple pathways to the correct answer. You can still pick numbers or work backwards from answer choices, although the best and quickest processes will involve a deeper awareness of the underlying math concepts.

Also, the No-Calculator section does not mean that each problem will involve ten grueling mathematical operations. On this section, students whose M.O. is immediately to simplify equations, isolate variables, and combine like terms may find themselves stuck in the weeds of the problem. Students will excel when they ask themselves, “What does this question ask? What concepts is it testing? Is the question’s presentation indicative of anything? What’s the method that will take the fewest steps?”

You can overcome calculator anxiety in much the same way that you don’t overcome a fear of sharks: full immersion. The next time you practice on a math section, leave your calculator in your bag. Try working through problems methodically, looking for the “elegant” solution.

What’s the same

Both sections are on the SAT, which means that they’ll share the underlying focus of math fluency and algebra-based questioning. Both sections will feature algebra and modeling as the most commonly-occurring problem types. Both sections will provide real-world scenarios steeped in scientific concepts and equations. Additionally, both sections will test students with multiple choice and grid-in questions, as did the retired SAT.

What are the features that distinguish the two sections?

The No-Calculator Section

Students need to master their process.

With the No-Calculator section, students must recall important rules and equations (such as the quadratic formula, exponent rules, or trigonometric identities), and also work through multiple steps without the calculator as a safety net:

• Understanding and applying laws of distribution to simplify an equation
• Matching a problem’s equation to the equation’s standard form
• Correctly substituting values into an equation
• Distributing negatives, squaring/multiplying values, and simplifying radicals
• Modifying the simplified equation to match one of the answer choices

This process involves time and requires the student to write out all of his or her work. Every step is crucial, and any error will result in lost time or a missed problem.

Picking numbers and working backwards are still potential options for solving problems, but students should not depend on these strategies. They will see questions that either require grid in answers (taking away the working backwards strategy), or values that make picking numbers the less ideal choice (such as answer choices that each offer a complex equation).

Students need to become more flexible in their approach.

The lack of a calculator takes away easy fixes that can mask deficiencies in a student’s math knowledge and fluency. Students are still able, however, to find “elegant” solutions to problems which actually demonstrate their knowledge of the math content. Not every problem involves onerous work. Consider the following example:

EXAMPLE

3x(2x – 3) + 2(4x + 5) = ax² + bx + c

In the equation above, a, b, and c are constants. If the equation is true for all values of x, what is the value of b ?

For our first step, we would distribute and simplify to get the following:

6x² - x + 10 = ax² + bx + c

You then might be tempted to try different values for x (0, 1, and 2)- because the equation is true “for all values of x”- and see if you can get a set of equations to stack and solve for b. You will eventually get the correct answer using that approach, but only after performing three different calculations and solving two equations. There is a simpler approach to the solution.

If you look back at the equation, you see three terms on both sides. What’s more, both sides have an x², an x, and a number. The 6x² corresponds to the ax², the -x corresponds to the bx, and the 10 corresponds to the c. Making this observation helps you to see that b is going to equal -1, since both sides of the equation are equal.

This connection took 5 seconds to make and saved minutes of frantic scrawling and potential arithmetic errors. It didn’t even involve arithmetic, only a few realizations (1. The two equations equal one another, 2. The equations are composed of 3 similar terms, therefore 3. Each matching term must be equal).

Yes, students will need to be able to perform calculations including long division/multiplication, factoring/foiling, exponents/roots, and solving systems of equations without a calculator. But the No-Calculator section is more than simply grinding through laborious equations. On this section you’ll also need to be able to see “the elegant solution” to a problem. If you find yourself grinding through tedious calculations, take a step back. Ask yourself, am I missing something that will get me to the answer more quickly? Can I think of this problem from a different angle? That is the name of the game for the No-Calculator section.

In order to excel on the No-Calculator section, you will need to abandon your dependence on your calculator, but not on your ability to make connections and draw conclusions. Practice on a non-calculator section and see what operations leave you scratching your head. Did you forget how to distribute exponents, or how to solve multiple equations? That feedback will inform your test preparation.

The Calculator section

For the Calculator section, students will need to have a firm grasp of statistics (finding and interpreting the mean and median, understanding basic principles of study design, interpreting a line of best fit, etc.) as well as a strong ability to read charts, tables, and graphs, and draw conclusions from them.

The Calculator section is going to be frustrating for students who finish the previous section and are ready to unleash their mad calculator skills. This section, while allowing students to perform calculations on their calculator, will instead emphasize data analysis, much of which does not require the use of a calculator. Students will work with larger numbers organized into complex tables and charts, and their calculator will be invaluable for computing averages or percentages. However, students with the latest-and-greatest Texas Instrument will have little advantage over the student with a simple scientific calculator.

For this section, students will want to look back over basic statistics concepts (such as the relationship between median and mean). They will also want to review linear, parabolic, and exponential equations and how they can appear in word problems and real-world scenarios. If a cannonball is shot from a cannon and its height is modeled by the equation h = 9.8t^2 + 64t + 5, what does the “5” represent in the equation? If a rainforest’s lizard population, P, follows the equation P = 17t + 23 where t is the number of years since a baseline study, what does the 17 mean? If a bacterial colony on your kitchen sponge grows exponentially, starts with 300 colonies, and doubles every 3 hours, how can we express that as an equation? The sky (and maybe your gag reflex) is the limit in terms of how creative you can get with these types of questions. Just keep in mind that the No-Calculator section will not demand as much from your multi-buttoned, rectangular compadre as you have been used to in the past.

If you are wondering how to excel at the Calculator section, it will be important to get acquainted with its questions. Remember: just because you are permitted to use a calculator doesn’t mean that you need it or would benefit from using it. Try going through a practice section and counting the number of times a calculator helped you answer it, as well as how you used it (simple arithmetic, graphing, etc.). Once you get a sense for how the SAT expects you to use your calculator, you can focus on perfecting your approaches to various problems.

Conclusion

While a bit disorienting for students, the distinction between No-Calculator and Calculator sections will help to test different criteria that are equally necessary for proving math mastery. Students need to show that they can perform multi-step calculations in an organized manner using only their pencils, paper, and math expertise. Additionally, students will need to show that they understand the underlying concepts behind a real world problem, as well as the best path toward the solution.