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Quantitative Reasoning Overview and Strategies

The Quantitative Reasoning section is the second part of the ISEE, and the first of two math sections.

You will have 35 minutes to answer 37 questions, and you are not allowed to use a calculator.
Approximately half of the section is made up of Word Problems, while the other half is made up of Quantitative Comparisons.

How is it different from Mathematics Achievement?

Aside from length and timing, this section is very similar to Mathematics Achievement, except that it includes Quantitative Comparison questions - more on those below. Additionally, where Mathematics Achievement involves a greater focus on calculations and knowledge of math terminology, Quantitative Reasoning questions are designed to be solved primarily through logic and reasoning (on top of a healthy knowledge of math concepts).

Piqosity’s math lessons are applicable to both sections; mastering them will ensure that you are well-prepared to tackle all of the math you will see on test day.

Because of the differences in the format for each section, however, it is important to practice questions from both sections on a regular basis.

Steps to Success on the Quantitative Reasoning Test

Learn and review concepts - Use Piqosity’s lessons and associated practice problems to learn new math concepts, and to revisit those you haven’t encountered recently
Practice - Complete timed, strategic practice on a regular basis, and always make sure you understand your mistakes
Revisit and reinforce - As you practice, continue to review the lessons and other supplementary materials to fortify your weak areas

Learn and Review Concepts

Your Piqosity account includes access to curated lessons on every math topic that appears on the ISEE. Reviewing these lessons and topics, especially as you review your first practice test, will give you a better idea of:

What you know
What you need to review
What you still need to learn

Practice

As with all sections of the ISEE, your practice for the Quantitative Reasoning section should be frequent, thoughtful, and reflective:

Frequent:

Complete at least 1-2 hours of practice per week, across all sections of the ISEE
Break up your practice time as needed to establish a schedule you can stick to

Thoughtful:

Answer every question, and always try your best
Pay attention to your pacing
Utilize the strategies you have learned

Reflective:

Review your mistakes and re-work questions you missed
Mark hard questions so you can return to them later
Focus your practice on your weakest areas

Systematic Approach to Word Problems

First, we’ll walk through a systematic approach to the Quantitative Reasoning section’s Word Problems, which comprise approximately half of the section.

Here is a sample Word Problem from the creators of the ISEE:

Now, here is a systematic process you can follow as you tackle any Word Problem on the Quantitative Reasoning section of the ISEE:

This systematic approach to answering any World Problem on the Quantitative Reasoning section involves asking yourself three key questions, which we’ll break down further below.

This process should take you less than a minute, and substantially less time for questions you choose to guess on and skip. This is why practice is important: you must master not only the content being tested, but also your strategy and pacing.

Three Key Questions

Let’s break down the three key questions from the systematic process illustrated above in a little more detail:

Do you understand what the question is asking? Underline any key information. In your head, summarize what the question is asking you to do.
Can you recall the tools, formulas, or strategies needed to solve? Think about what you’ve learned in school and reviewed on Piqosity. Will you be able to solve confidently using what you know?
Can you solve the problem relatively quickly? You only get 51 seconds per question on average, so every second counts!

Educated Guesses

Now, let’s break down the steps to follow when you decide to make an educated guess on a Quantitative Reasoning Word Problem:

Applying the Systematic Approach

Let’s break down what this systematic approach might look like when applied to the sample question cited earlier:

Educated Guess

Eliminate answer choices. Since Mrs. Grange is adding 4 points to all the test scores, I don’t think the range would decrease, so I can eliminate choice A. I also think 43 is too high, because I don’t see why the range would increase by 8. I’ll eliminate D as well.
Guess. Between B and C, I’m tempted to choose C, because it makes sense that the range would also increase by 4. Then again, that seems a little too obvious. I think range has to do with the difference between the highest and lowest values, so I think I’ll go with B; if all the test scores go up by the same amount, I don’t see why that difference would change at all.
Mark the question. If I have time, I’d like to come back to this question so I can think about it some more. I’ll circle the question in my booklet so I can find it easily.
Move on!

Practice Problem Solution

Range is found by subtracting the minimum value in a data set from the maximum value.
If 4 points are added to each student’s score, both the minimum and maximum value will increase by 4
Since the minimum and maximum will change by the same amount, the range will not change.
We can verify this by considering an example:
1. Let’s say the original minimum was 60, and the original maximum was 95. The range would be 95 − 60 = 35.
2. If all the test scores were increased by 4, the new minimum would be 64 and the new maximum would be 99.
3. The new range would be 99 − 64 = 35.
The correct answer is B!

Systematic Approach to Quantitative Comparisons

Because of their unique structure, the Quantitative Reasoning section’s Quantitative Comparison questions will require you to develop a somewhat different systematic approach from the one we’ve reviewed for Word Problems.

Understanding how Quantitative Comparisons are designed and having a specific plan of attack for them will allow you to maximize your accuracy and efficiency on the second half of the Quantitative Reasoning section on test day.

Here is a sample Quantitative Comparison from the creators of the ISEE:

At first glance, this problem might look a little unusual - where are the answer choices? For Quantitative Comparisons, the answer choices are always the same. Here they are, in the form of the reminder that the ISEE will provide on every page that contains Quantitative Comparisons:

In essence, your task is to compare the values in Columns A and B. If A is greater, you’ll pick A. If B is greater, you’ll pick B. If the two quantities are equal, you’ll pick C. And if the relationship between A and B cannot be determined, you’ll pick D (more on this below).

Here is our systematic approach for answering these questions:

Applying the Systematic Approach

Let’s see what this process looks like when applied to the sample question above. Remember: this process should take you less than a minute, and substantially less for questions you choose to guess on and skip. Again, you will need to practice your systematic approach regularly in order to implement it efficiently.

Note that, in this case, we were easily able to compare Columns A and B once we simplified them. This won’t always be the case, however. After our discussion of educated guesses, keep scrolling for an example of a problem where we need to follow some additional steps to compare Columns A and B.

Educated Guess:

Compare the columns to the best of your ability. I remember that negative exponents mean that you take the reciprocal of the base, which means that Column A is 17 squared and Column B is 17 to the one-half power. I know that 17 squared is 289, but I’m not sure about Column B.
Guess (do NOT choose D just because you don’t know). Since the exponents are different while the bases are the same, I think it’s a safe bet that Columns A and B are not equal. I also don’t think it’s D – both columns are constants and will always have the same value. Even though I can’t recall what happens with a fractional exponent, I know that larger exponents generally result in larger numbers, at least when the base is greater than one. I’ll go with “A.”
Mark the question. If I have time, I’d like to come back to this question so I can think about it some more. I’ll circle the question in my booklet so I can find it easily.

Inputting Values

What if Column A and Column B do not simplify into constants that can be easily compared? Take this Quantitative Comparison, for example:

In this case, we cannot do anything to simplify either column. Since there are no constants given – just variables – we will need to use our own numbers.

Solution to Practice Problem

Unlike with the previous problem, we cannot do anything here to simplify either column. We will also need a different strategy for comparing them, as there are no constants given here – just variables.
You may be tempted to pick B, because 3 is a larger exponent than 2. Don’t fall into that trap! Since it’s a variable that we’re raising to an exponent, we need to consider what values we’ll get when we substitute in different possible values for x.
Simple values such as -1, 0, 1, and 2 are usually great choices to plug in as a way of comparing the two columns.
1. First, let’s try 2:
  1. Column A: (2)² = (2)(2) = 4
  2. Column B: (2⁾³= (2)(2)(2) = 8
  3. Here, Column B is greater than Column A, as you might have assumed it would be upon seeing the problem.
2. Now, let’s try 1:
  1. Column A: (1)²= (1)(1) = 1
  2. Column B: (1)³= (1)(1)(1) = 1
  3. Here, Columns A and B are equal.
3. The same is true for 0:
  1. Column A: (0)²= (0)(0) = 0
  2. Column B: (0)³= (0)(0)(0) = 0
  3. Columns A and B are equal once again.
4. Finally, let’s try -1:
  1. Column A: (-1)²= (-1)(-1) = 1
  2. Column B: (-1)²= (-1)(-1)(-1) = -1
  3. Now, Column A is greater!
When you input values and do not get consistent results, D is the correct answer.

This particular problem is relatively simple, but it illustrates an important point. You must consider how the relationship between the columns will change – or not change – depending on the values you input. Never assume the answer is D without testing some values!

Your knowledge of the math concepts involved in each question is very important here. If you know how negative numbers behave when raised to even and odd exponents, or if you can recognize right away that 0 and 1 will stay the same when squared or cubed, you can answer questions like this one rather quickly.

Key Takeaway - If the problem contains variables, input values to help you determine whether one of the columns is always greater, or if the two columns are always equal. If different values produce different results, choose D.

Revisit and Reinforce

Preparing for the ISEE’s math sections is a cyclical process. It’s not enough to review every math topic once, take a practice test, and then call it a day. On the contrary, seeing your Quantitative Reasoning score improve will require you to continually revisit concepts as you work through practice material. After all, you’re highly unlikely to master a topic after practicing it just once, and the large number of topics that can appear on the ISEE means that you’ll need to review topics repeatedly as you get rusty on them over time.

In other words, your preparations should end up looking like this: