Beyond the Basics: Advanced Math Concepts That Can Give You an Edge on the CPS HSAT

To do well on the HSAT Math section, you must have a rock-solid foundation in the core topics of pre-algebra, algebra, and geometry. But to earn a top-tier score that truly stands out on the 900-Point Matrix, it pays to understand a few "advanced" concepts.

These are the types of problems designed to separate a good score from a great one. As you continue your studies this summer, mastering these topics can give you the competitive edge you’re looking for. This guide will introduce a few key areas that go just beyond the basics.

Advanced Concept 1: Multi-Step Geometry

The HSAT won’t just ask for the area of a single square. It will present you with complex, composite shapes and ask you to find an area, perimeter, or shaded region.

  • What it is: These problems require you to use multiple geometric principles in a single question. You might need to use the Pythagorean theorem to find the side of a triangle, and then use that length to calculate its area, or find the area of a large rectangle and subtract the area of a smaller circle within it.

  • The Strategy: Break the complex shape down into simpler shapes you recognize (rectangles, triangles, circles). Calculate what you need for each simple shape and then add or subtract them as required by the question. A firm grasp of the formulas covered in our "Conquering the HSAT Math Section" guide is the essential first step.

Advanced Concept 2: Basic Probability

Probability measures the likelihood of an event occurring. While you might not see many of these questions, they can be easy points if you know the fundamental formula.

  • What it is: Probability is expressed as a fraction: Probability=Total Number of Possible Outcomes/Number of Favorable Outcomes​

  • Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly drawing a blue marble?

    • Favorable Outcomes (blue marbles): 3

    • Total Possible Outcomes (total marbles): 5 + 3 + 2 = 10

    • Probability: 3/10

  • The Strategy: Carefully read the question to identify the "favorable" outcome and the "total" number of outcomes. Always simplify your fraction if possible.

Advanced Concept 3: Introduction to Quadratic Equations

While you won’t need to master the entire Algebra II curriculum, the HSAT can touch on the most basic concepts of quadratic equations (equations with an x2 term).

  • What it is: Recognizing a quadratic equation and solving it in its simplest forms. Often, this involves factoring.

  • Example: If x2−9=0, what could be the value of x?

    • You can solve this by isolating x2: x2=9

    • Then, take the square root of both sides: x=3 or x=−3

  • The Strategy: When you see an x2 term, think about solving for the variable by taking the square root or by simple factoring. Remember that there are often two possible solutions.

How to Master These Edge Concepts

Knowing these topics is one thing, but identifying them and solving them quickly on test day is another skill entirely. This is where your practice method becomes crucial. The best way to see how these advanced ideas are actually tested is by working through a wide range of problems. Using comprehensive CPS HSAT Practice Tests will expose you to these less-common question types, ensuring you aren't surprised on test day. It allows you to practice applying these principles so they become second nature.

By going beyond the basics and putting in the effort to learn these distinguishing concepts, you position yourself to achieve the highest possible score and open the door to your top-choice high schools.

Frequently Asked Questions about Advanced HSAT Math

Here are answers to some common questions students have when they begin to tackle more advanced math concepts in their test prep.

1. Are these "advanced" topics guaranteed to be on every HSAT?

While core algebra and geometry are always heavily featured, the inclusion and frequency of specific advanced topics like probability or quadratics can vary from test to test. Think of them as "low frequency, high impact." There may only be a few such questions, but correctly answering them can be what separates a good score from a truly great one.

2. If I'm still struggling with basic algebra, should I spend time on these topics?

No. Your first and most important priority must be to achieve 100% mastery of the fundamentals covered in our main Conquering the Math Section guide. A solid understanding of pre-algebra and linear equations will earn you far more points than a shaky understanding of advanced concepts. Only move on to these topics once your foundation is rock-solid.

3. Are there other advanced topics that might appear besides the ones listed?

The HSAT is designed to test concepts generally learned through an 8th-grade curriculum. The topics in this article—multi-step geometry, basic probability, and simple quadratics—are the most common "edge" topics. While you might see other concepts like functions or basic statistics, the test will always prioritize strong mathematical reasoning over the memorization of dozens of obscure, high-level formulas.

4. How do I know which advanced topics I need to work on?

The only way to know for sure is by analyzing your performance data. When you take a full-length CPS HSAT Practice Test, categorize every question you get wrong. If you see a pattern—for instance, you miss every probability question you encounter—then you have identified a high-priority area to focus on in your studies.

5. Where can I find more practice problems for these specific concepts?

Our practice tests are the best source for problems that mimic the style and difficulty of the actual HSAT. Once you've used a practice test to identify a weakness, you can then use free online resources like Khan Academy, which has excellent, in-depth modules on topics like probability and quadratics, to drill that specific skill with hundreds of problems.