Teaching Math in Chicago in Grades 8–10

Quick answer: Teach the Pythagorean theorem and linear functions as connected ideas. Start with right triangles and the relationship a^2 + b^2 = c^2, then use slope as a rate of change on the same coordinate grid. The Pythagorean Theorem Applications series builds from finding sides to word and space problems.

In Illinois, math follows the Illinois Learning Standards, which are based on the Common Core. Grade 8 students sit the Illinois Assessment of Readiness (IAR) in spring, and 2025–26 brought new unified performance levels. High school students take the ACT suite, which Illinois adopted in place of the SAT starting in spring 2025.

That puts Grades 8–10 at a hinge point. The same students move from a state test built on grade-level standards to a college readiness test that mixes algebra and geometry. Two topics carry a lot of weight across that shift: the Pythagorean theorem and linear functions. For other levels and subjects, see our Chicago teaching resources page.

What should Grades 8–10 students be able to do with these topics?

For the Pythagorean theorem, students should name the legs and the hypotenuse in any orientation, find a missing side, use the converse to test whether a triangle is right-angled, and apply the theorem to distances on a grid and to problems in three dimensions.

For linear functions, students should move confidently between a rule, a table and a graph. They should read the slope as a constant rate of change and the y-intercept as a starting value, and write an equation from a word problem.

• Vocabulary: leg, hypotenuse, converse, square root, slope, rate of change, y-intercept, function
• Checkable skill: find the distance between (1, 2) and (7, 10) using a right triangle (answer: 10)
• Checkable skill: write the rule for a taxi fare of $3 plus $2 per mile (y = 2x + 3)

How can I sequence the Pythagorean theorem in Grade 8 or 9?

1. Squares on the sides: students draw squares on the sides of a 3-4-5 triangle on grid paper and compare areas.
2. Naming parts: students mark the hypotenuse on ten triangles in different orientations.
3. Finding the hypotenuse: practice with whole-number triples, then with answers that need rounding.
4. Finding a leg: rearrange the relationship and discuss why the answer must be shorter than the hypotenuse.
5. The converse: students test side lengths such as 5, 12, 13 and 6, 8, 11.
6. Distance on a grid: build a right triangle between two points and calculate the distance.
7. Word and space problems: ladders, ramps, screen diagonals and the diagonal of a box.

The Pythagorean Theorem Applications resource follows the same arc with 5 differentiated worksheets, full answer keys, a 7-slide presentation and a step-by-step lesson plan, so you can pick the worksheet level that fits each group.

Where can I find a Chicago connection?

Chicago's architecture is a ready source of right triangles. The Home Insurance Building, completed in 1885, is often called the world's first skyscraper, and tall buildings raise good questions: how long does a cable need to be to reach a point on a facade from a spot on the ground? Students can sketch the situation, label the right angle and solve.

For linear functions, use city life that students know: a transit card balance going down by a fixed fare per ride, or a phone plan with a monthly fee plus a charge per unit. Give real-looking numbers you choose yourself and ask students to explain what the slope and the y-intercept mean in the story.

What errors do students make most often?

• Adding the sides instead of the squares: 3 + 4 = 7, so c = 7. Ask for an area picture every time until the rule sticks.
• Using the theorem on a triangle that has no right angle. Require students to mark the right angle before calculating.
• Treating the longest given side as a leg when finding a missing leg.
• Reading slope as rise over run in the wrong order, or ignoring the sign on a falling line.
• Confusing the y-intercept with the x-intercept on a graph.
• Assuming every straight-looking graph shows a proportional relationship, even when it does not pass through the origin.

How do I support English learners and mixed-skill groups?

Word problems carry most of the language load. Pre-teach context words such as ladder, ramp, diagonal, fee and rate, and let students draw every situation before they write any equation. A shared diagram often does more than a translated text.

Discussion builds precise language. The Linear Functions Teaching Kit with puzzle and Talking Cards gives groups prompts for mathematical argument and practice sheets on three levels, from basic skills to real-life problems. Sentence frames help: The slope is ___, which means ___ for every ___.

How should I plan around the IAR and the ACT?

For Grade 8, the IAR is the main state measure in math, and the 2025–26 performance levels are new, so review how your school reports results. For Grades 9 and 10, the ACT suite is the reference point. Both reward students who can explain their steps and switch between representations, not just calculate.

Keep test practice short and regular. Mix one Pythagorean item and one linear function item into each warm-up, and review errors as a class. For a fuller unit on graphing and word problems, the Linear Functions Unit for Grade 8 covers tables of values, slope, y-intercept and interpretation. You will find more in our Middle School Math collection.

Math in Chicago: at a glance

Frequently asked questions

Should I teach the Pythagorean theorem before linear functions?

Either order works, but teaching the theorem first lets you reuse the coordinate grid. Distance between two points needs a right triangle, and the rise and run of that triangle lead straight into slope. Students then see both topics as parts of one picture.

How do I help students remember which side is the hypotenuse?

Tell them it is always opposite the right angle and always the longest side. Have them mark the right angle first, then draw an arrow across the triangle to the opposite side. Practice with rotated triangles so they do not rely on the hypotenuse being at the bottom.

Which tests do Chicago students in Grades 8 to 10 take?

Grade 8 students take the Illinois Assessment of Readiness in math, which has new unified performance levels from 2025–26. In high school, Illinois uses the ACT suite, which replaced the SAT starting in spring 2025. Your school sets the exact dates and grades tested.

How can I make linear functions less abstract?

Use situations with a starting value and a steady rate, such as a fee plus a charge per unit. Ask students to fill a table, plot the points and then write the rule. When they explain what each number means in the story, the slope and y-intercept become concrete.

Related reading

• Chicago teaching resources: selected units and subjects
• Teaching Art in Chicago in Grades 7–9
• Teaching History in Chicago in Grades 7–9