7 Foundational SAT®/ACT® Math Skills to Strengthen This Summer
In my last post, 3 Ways Tutors Can Use Summer Sessions to Strengthen Core Academic Skills, I discussed how summer tutoring can offer more space to identify and address any reading, math, and writing skills that need more focused attention.
For math, that starts with determining whether a student’s math struggles come from test strategy, pacing, or knowledge gaps.
A missed math question on the SAT® or ACT® does not necessarily indicate a test-taking problem. What looks like a careless mistake or poor strategy may actually come from a lack of understanding of algebra, percentages, functions, geometry, or another foundational math concept.
A full SAT® or ACT® diagnostic test can reveal broader score patterns, but a shorter math skill check can also show where a student begins to struggle with the problem-solving process. Choose a few questions from several key skill areas and allow your student to work through them without a time limit.
As they work, observe whether they can recognize the type of problem, choose a reasonable problem-solving method, explain the steps, and determine whether the answer makes sense.
A student who understands the math concept but works too slowly may simply need pacing practice or a more efficient problem-solving strategy. A student who cannot recognize the problem or explain the mathematical relationship probably needs more direct instruction.
Sometimes, a student has both a knowledge gap and a test-taking problem. The following seven skill areas provide a practical starting point for identifying the foundational knowledge gaps most likely to interfere with their performance on the SAT® or ACT® Math.
Skill #1: Linear Equations and Systems of Equations
Linear equations are a good place to begin because they reveal a lot about a student’s general understanding of algebra.
Students should be able to distribute, combine like terms, work with variables on both sides of an equation, and rearrange an equation without losing track of signs or operations.
Just as importantly, they need to understand how they arrive at a solution rather than relying on a memorized sequence of steps.
Systems of equations are another essential skill. A student may know how to use substitution or elimination when the problem-solving method is clearly stated but struggle to recognize when a word problem, graph, or table represents a system of equations.
Watch for students who can complete a familiar-looking equation but get stuck when the same relationship appears in a different form. That often indicates that they have memorized the steps without developing a strong understanding of the math concept.
Skill #2: Functions and Graphs
Functions can sometimes expose knowledge gaps that students were able to work around during a school-year math unit.
A student may know how to evaluate basic function notation but struggle to connect an equation to its table or graph. Maybe they can identify slope and intercepts when an equation is written in a familiar form but cannot explain what those values mean in context.
A short skill check might ask whether the student can:
evaluate a function for a given input
identify slope and intercepts
connect an equation, table, and graph
interpret what a point on a graph represents
recognize increasing and decreasing relationships
identify important features of a nonlinear graph
The goal here is to determine whether they understand how different representations describe the same mathematical relationship.
Skill #3: Exponents and Radicals
Exponent and radical questions often reveal whether students understand mathematical rules or only memorized them.
A student may confuse multiplication and power rules, misuse negative exponents, or assume every expression involving a square root can be simplified in the same way. These errors can carry into algebraic expressions, functions, scientific notation, geometry, and other areas.
Instead of reviewing a list of rules, ask your student to explain why an exponent changes, what a negative exponent represents, or how a radical relates to an exponent.
When they grasp those relationships, they’re more likely to apply the rules correctly when an expression looks unfamiliar.
Skill #4: Ratios, Percentages, and Rates
Many students learn ratios, percentages, and rates as separate procedures instead of understanding the relationships between them.
They may know one formula for percent change, another setup for proportions, and a separate method for unit rates without recognizing that all three involve relationships between quantities.
Common trouble areas might include:
identifying the original and new values in a percent-change problem
distinguishing the amount of an increase from the final amount
setting up proportional relationships
keeping track of units
converting between fractions, decimals, and percentages
determining whether a relationship is proportional
interpreting rates in word problems, tables, and graphs
When a student relies on a memorized formula, ask them to explain what each number represents. If they make a mistake, their explanation can help you determine whether they made a calculation error or do not fully understand the relationship between the quantities.
Skill #5: Geometry Basics
Geometry gaps can be difficult to identify because students may solve familiar-looking problems correctly but struggle when the same skills appear in a diagram or unfamiliar context.
A simple geometry check might include:
angle relationships,
triangles,
circles,
area,
volume,
the Pythagorean theorem,
coordinate geometry, and
basic right-triangle relationships
Students should know basic formulas, but formula recall is only one part of the problem-solving process. They also need to recognize which formula fits the problem, identify the correct measurements, and understand what the result means.
This is an important distinction when using a resource such as our review of 21 Important Formulas for the ACT® Math Test. A student may be able to recite an area, distance, or slope formula and still struggle to apply it when the question is presented through a diagram or real-world situation.
A few carefully selected problems can show whether the student can connect the given information to the algebra needed to solve them. Geometry practice should also include figures that are not drawn to scale, so students learn to rely on the stated measurements and relationships instead of assumptions based on how the figure appears.
Skill #6: Data Interpretation
Data questions require more than calculation. Students need to read carefully, understand what the data shows, and decide which values are relevant.
Some students understand averages, percentages, and rates on their own but struggle when those skills appear in a chart, scatterplot, or table. Others begin calculating before checking the labels, units, scale, or what the question is actually asking.
When students begin calculating before deciding which information matters, they are also more likely to choose a trap answer. A wrong answer may be based on a real number from the graph or table, but not the number needed to answer the question. Students need to be able to identify the relevant data before they begin solving the problem.
Useful summer practice can focus on:
reading axes, labels, and units
identifying which data is relevant
comparing values accurately
calculating and interpreting averages
understanding changes over time
identifying patterns and relationships
recognizing trap answers based on irrelevant data
distinguishing conclusions supported by the data from those that are not
Before a student begins solving, ask them to explain what the data shows and which values they plan to use. If they cannot do that, they may need help understanding the information and what the question is actually asking before they can use the information to solve the problem.
Skill #7: Number Sense and Calculator Skills
A calculator is an important test-prep tool, but it can also hide gaps in a student’s number sense and basic math skills.
A student may know how to use a calculator to graph an equation or solve a system of equations without fully understanding the math concepts involved. They may still reach the correct answer, but that does not mean they know how to choose the right strategy, enter the information correctly, or recognize when the result does not make sense or does not answer the question.
During summer review, try including a few problems that students can solve without a calculator. The goal is to see whether the student can estimate, simplify, work with numbers, and complete basic algebraic steps independently.
Students should be able to recognize common fractions, decimals, and percentages, and notice when a result seems unreasonable. These skills will help students catch input errors and avoid using a calculator for steps they could complete more quickly on their own.
Then, allow calculator use on other problems and pay attention to how they use it. Do they use the calculator to save time, or do they enter every basic calculation because they don’t trust their own math skills? Can they recognize an input error? Do they know when graphing will help and when writing a simple equation would be faster?
Effective calculator use depends on knowing when the tool will make a problem easier or faster and when it will only add unnecessary steps. As discussed in my post on improving digital SAT® Math scores, the goal is to help students use calculator tools effectively without becoming so dependent on them that they cannot begin, estimate, or check an answer on their own.
Students should also practice with the calculator they expect to use during test prep and on test day. My calculator buyer’s guide for the ACT®, digital SAT®, and college offers additional guidance on choosing a permitted calculator that can also support future coursework.
Decide Which Math Skills to Strengthen First
The purpose of a summer math skill check is to identify the knowledge gaps that are holding a student back across multiple areas of their math work.
Not every missed question needs the same amount of attention. A student who struggles with linear equations may also have difficulty with systems of equations, functions, and coordinate geometry. Weaknesses with fractions, percentages, or negative numbers can create problems across ratios, data interpretation, algebra, and geometry.
Start with the skill that appears in several types of problems or repeatedly causes the student’s problem-solving process to break down. Addressing that gap first may improve more than one area of their math work.
Once you have identified a priority, keep the practice focused. A summer session might include a short review, direct instruction on one specific concept, and a few problems for the student to complete independently.
At the next session, go over the problems together. Ask them to explain how they approached the problems, where they got stuck, and why they chose a particular solution method. This can help you determine whether the student is becoming more confident with the skill or still needs additional support.
Know When to Add Test-Specific Practice
Skill building and test prep do not necessarily have to happen in completely separate stages. As a student becomes more confident and consistent with a math concept, begin adding test-specific practice.
That might include:
completing problems within a time limit
choosing the fastest reasonable problem-solving method
using estimation or answer choices to check the work
practicing with the student’s test-day calculator
working through mixed sets of SAT® or ACT® Math questions
adjusting to the pacing and format of the chosen test
Pay attention to what happens when those test conditions are added. If a student can solve a problem correctly without a time limit but struggles when the clock is running, the issue may now be pacing or strategy. If they still can’t recognize the problem or explain how to solve it after the time limit is removed, the knowledge gap probably needs a bit more attention.
Use Summer to Build a Stronger Math Foundation
Summer math tutoring is most useful when it helps us understand why a student is missing questions, not just which questions they are missing.
A low SAT® or ACT® Math score may come from pacing, test strategy, calculator use, or unfamiliarity with the test. It may also point to knowledge gaps that have been developing over time. More practice alone will not always solve the problem.
Identify the math skills creating the biggest obstacles, strengthen them through focused instruction and practice, and then connect that work to the student’s SAT® or ACT® prep.
Clear Choice provides custom-branded SAT® and ACT® diagnostic tests, curriculum, score reports, and test-prep software designed to help tutors identify student needs and build more focused prep plans. Contact us today to learn more or request a free demo.