# ACT and SAT Math: The Good, The Bad, and The Ugly (Part 3)

In our last post, we looked at the way the ACT and SAT address similar topics in two different “easy” level problems. Whether you agree that they were easy or not, you have to know that they get harder. The ACT and SAT both are designed in increasing order of difficulty, meaning generally that the lower number a problem has, the easier it is. The SAT resets difficulty between the Multiple Choice and Grid-In problems, so as soon as you see a new set of directions on your SAT test, the problems will start easy and quickly scale up in difficulty.

Now let’s dive into these “medium” difficulty problems. You’ll notice that these problems are towards the middle of the possible numbers. The ACT problem is 29/60 possible, and the SAT is 22/30. These numbers will give us a hint at how challenging they will be.

Both of our first ‘medium’ difficulty problems are related to median. Some students will look at the ACT problem and immediately know that the answer is B - it is the same as the original median. They will recognize that the median is the middle number and then see that two numbers were added to both sides of the set of data. Without knowing the meaning of median, this problem is likely to be largely guesswork. Reviewing and understanding math content is an absolutely essential part of preparing for the ACT. This ACT problem is relatively simple. It requires the understanding of only a single topic, but it requires more than a surface level understanding, increasing the difficulty level.

ACT Math Medium #1 Answer: B.

The first ‘medium’ difficulty SAT problem requires a deeper level of understanding and has significantly more data to process. In some ways, it is a more traditional problem: calculate the median. Yet, it also requires the student to understand how to read the featured chart (a histogram) and how to use the chart to quickly calculate the median.

Since we are looking for the median or middle number, some students will simply count up and down to the middle of the chart and choose 17 or 20. Students with a little better understanding of median will recall that if there are an even number of options, you have to calculate the average of the two middle numbers. If this were a more difficult problem, 18.5 might be a ‘fool’s gold’ answer choice.

The correct way to solve this problem is to recognize that the column on the right is labelled ‘frequency’. Frequency is roughly how often something occurs. In the context of this chart, it means that there are 4 states with 10 electoral votes, so we have to take this into account as we count to the middle. Those 4 states with 10 electoral votes on the low end of the data set correspond with four states that have 55, 34, 31, and 27 electoral votes. As we count inward, we find that the median number of electoral votes is 15.

Another way to solve this problem is to divide the total number of states by two to get 10.5. We count ten up from the bottom. Because the number is 10.5, we have to take the average of the 10th and 11th number. In this problem, both are the same number.

SAT Math Medium #1 Answer: B.

As we look at the second set of medium problems, we see two different methods for assessing knowledge of quadratic equations. Again, keep in mind that the ACT problem number is relatively low, and notice that the SAT problem has no answer choices. These Grid-In problems don’t provide the guidance of confirming that your answer is right.

The ACT Medium #2 brings in the concept of ‘undefined’. For upper level students, this will be a pretty simple matter of finding out when the denominator is zero. For students that don’t immediately know what it means to find “where the expression is undefined”, they might recognize the pattern in this expression: 9-x2 fits the “difference of squares” pattern.

Difference of squares is one of the ACT’s favorite ways to differentiate between students scoring in the 20s and 30s. During my test prep with students I emphasize it heavily. Students who recognize the pattern can quickly factor 9-x2 into (3-x)(3+x). If you don’t recognize this pattern, it is still possible to solve the denominator for zero by setting 9-x2=0 , moving x2 to the other side and taking the square root of both sides. We would normally get + and - 3 as our solution, but the problem specifically asks for non-negative solutions.

ACT Math Medium Answer #2: J - 3.

SAT Medium #2 is number 17 out of 20 total total problems in the no calculator section. We can expect this problem to be a little harder than the previous one. While you might think this is a very difficult problem based on that fact alone, recall that the increasing level of difficulty resets within math test when the problems type changes to fill in the blank. This is actually a fairly easy problem for the SAT. Part of the challenge associated with this problem comes from the vocabulary thrown in the problem: “x,y-coordinate plane,” “x-axis,” “parabola,” and “line of symmetry.”

As with many problems on the SAT, a thorough understanding of the concepts makes this problem relatively simple. If a student recalls that a parabola’s line of symmetry is halfway between any two points with the same y-value then they can simply find the point halfway between the two points given. If not, this is where drawing diagrams come into play. Sketch the two points and then draw in a parabola. The parabola you drew can be folded across the line x=0. If you don’t know what a parabola is, that’s where preparation comes into play!

SAT Math Medium Answer #2: 0. The parabola’s line of symmetry is half-way between the two points with the same y-value.

The medium problems don’t necessarily take more time than the easy ones. They take more knowledge, more attention to detail, and sometimes more patience to understand what is being asked for. In a word, they take more preparation.