Math Testing Lab

Math study guide and test practice from 6th grade to 11th grade.

This page is built as a broad school-math map for students who want lessons, review, and math test practice. It starts with fractions, decimals, percentages, ratios, rates, and geometry basics, then moves through linear equations, systems, exponents, scientific notation, roots, quadratics, logarithms, trigonometry, and early calculus ideas.

Grades 6-11 Curriculum map Reference notes Mini tools
School sequences vary, so this is best read as a strong core roadmap rather than a promise that every school teaches topics in exactly the same order.

Grade map: what 6th-11th graders should learn

The biggest idea is progression: arithmetic becomes proportional reasoning, proportional reasoning becomes algebra, algebra becomes functions, and functions become the bridge to geometry, trigonometry, and calculus.

6th grade essentials

  1. Whole numbers, factors, multiples, primes, and divisibility.
  2. Fractions, decimals, mixed numbers, and percents.
  3. Ratios, unit rates, and basic proportional reasoning.
  4. Expressions, variables, and one-step equations.
  5. Area, surface area, volume, and coordinate-plane basics.
  6. Mean, median, mode, range, and simple data displays.

7th grade essentials

  1. Operations with rational numbers, including negatives.
  2. Percent increase, decrease, tax, discounts, and simple interest.
  3. Proportional relationships and scale drawings.
  4. Multi-step equations and inequalities.
  5. Angles, circles, area, surface area, and volume.
  6. Probability and statistics with samples and simple comparisons.

8th grade essentials

  1. Linear equations, slope, intercepts, and graphing.
  2. Functions as input-output rules.
  3. Systems of linear equations.
  4. Exponents and scientific notation.
  5. Transformations, congruence, and similarity.
  6. Pythagorean theorem, distance, and basic irrational numbers.

9th-11th essentials

  1. Linear, quadratic, exponential, and polynomial functions.
  2. Factoring, radicals, rational expressions, and equations.
  3. Geometry, proofs, circles, similarity, and right-triangle trigonometry.
  4. Logarithms, sequences, series, and modeling.
  5. Function transformations, inverse ideas, and advanced graph reading.
  6. Early calculus ideas like rate of change and derivatives.

6th-8th grade core topics

These topics are the foundation. If these are solid, almost everything in later algebra and geometry becomes easier.

6th grade 7th grade 8th grade

Fractions, decimals, and percents

These are three ways to describe the same amount. A fraction shows parts of a whole, a decimal shows place value, and a percent means "per hundred."

fraction = decimal = percent / 100

Key skills

  1. Simplify fractions.
  2. Convert between forms.
  3. Add, subtract, multiply, and divide fractions.
  4. Use percents in real situations.

Ratios, rates, and percentages

Ratios compare quantities. Rates compare quantities with units. Percentages compare values on a scale of 100.

percent change = (new - old) / old * 100%

Key skills

  1. Find unit rates.
  2. Solve percent increase and decrease problems.
  3. Recognize proportional relationships.
  4. Interpret scale and comparison statements.

Expressions, equations, and inequalities

Variables let you describe patterns and unknown values. Equations say two expressions are equal. Inequalities compare size instead.

3x + 5 = 20 means find the x that makes both sides match

Key skills

  1. Use order of operations.
  2. Combine like terms.
  3. Solve one-step and multi-step equations.
  4. Interpret inequality solutions on a number line.

Geometry basics

Students at this stage learn to measure shapes, compare angles, and use formulas for area, surface area, and volume.

rectangle area = length * width, prism volume = base area * height

Key skills

  1. Find area and perimeter of common polygons.
  2. Find surface area and volume of prisms and cylinders.
  3. Use angle relationships.
  4. Read points on the coordinate plane.

Statistics and probability

This is where students learn how to summarize data and reason about chance.

mean = sum of values / number of values

Key skills

  1. Find mean, median, mode, and range.
  2. Read dot plots, box plots, and histograms.
  3. Describe variability.
  4. Estimate simple probabilities.

Linear equations and slope

This is the door into algebra and graphing. A line shows constant rate of change.

slope = (y2 - y1) / (x2 - x1)

Key skills

  1. Graph lines from tables or equations.
  2. Understand y = mx + b.
  3. Interpret slope and intercepts in context.
  4. Move between words, tables, graphs, and equations.

Systems, Pythagorean theorem, and scientific notation

By the end of middle school, students should be ready for systems of equations, right-triangle relationships, and large or tiny numbers written efficiently.

a^2 + b^2 = c^2 and a * 10^n for scientific notation

Key skills

  1. Solve simple systems by graphing or elimination.
  2. Use the Pythagorean theorem for missing side lengths.
  3. Estimate square roots of non-perfect squares.
  4. Write numbers in scientific notation.

Middle-school checkpoint

If a student can confidently work with fractions, proportional thinking, linear equations, graphs, geometry formulas, and basic statistics, they are in strong shape for Algebra 1 and Geometry.

Weakness in fractions and ratios is one of the biggest reasons later algebra feels harder than it should.

9th-11th grade core topics

High-school math takes the patterns from middle school and organizes them into functions, proof, and modeling.

9th grade 10th grade 11th grade

Functions and graphing

A function gives exactly one output for each input. This idea becomes the center of nearly all high-school math.

f(x) = 2x + 3 means every input x gets turned into 2x + 3

Key skills

  1. Read domain and range.
  2. Compare linear, quadratic, and exponential behavior.
  3. Interpret graphs as stories about change.
  4. Recognize transformations like shifts and stretches.

Polynomials, factoring, and quadratics

These topics dominate Algebra 1 and Algebra 2. Quadratics are especially important because they are the first major nonlinear family students study deeply.

x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a

Key skills

  1. Multiply and factor expressions.
  2. Solve quadratics by factoring, graphing, and the quadratic formula.
  3. Interpret the discriminant.
  4. Connect the equation to the graph of a parabola.

Geometry, congruence, similarity, and proof

Geometry teaches students to justify statements with structure instead of only computation.

If side-angle-side matches, triangles are congruent

Key skills

  1. Work with angle relationships and parallel lines.
  2. Use triangle congruence and similarity.
  3. Understand proof flow and why each step is allowed.
  4. Apply geometry on the coordinate plane.

Circles, area, surface area, and volume

Circle geometry connects algebra and geometry through arc length, area, angles, and coordinate descriptions.

area = pi * r^2, circumference = 2 * pi * r

Key skills

  1. Use circle formulas.
  2. Understand arc length and sector area.
  3. Find surface area and volume for 3D solids.
  4. Connect equations to geometric figures.

Exponential growth, logarithms, and the number e

This is where students start seeing how repeated multiplication models growth and decay.

log base b of x = y means b^y = x

Key skills

  1. Work with integer and rational exponents.
  2. Interpret growth and decay models.
  3. Use logarithms to undo exponents.
  4. Understand why e appears in continuous growth.

Trigonometry and the unit circle

Trigonometry starts in right triangles and grows into a full function system.

sin(theta), cos(theta), tan(theta), unit circle values, and trig graphs

Key skills

  1. Use sine, cosine, and tangent in right triangles.
  2. Understand degree and radian measure.
  3. Read values from the unit circle.
  4. Graph and interpret trig functions.

Sequences, series, and modeling

These topics teach pattern recognition and introduce recursive and explicit thinking.

arithmetic sequences add a constant, geometric sequences multiply by a constant

Key skills

  1. Find terms and formulas for arithmetic sequences.
  2. Find terms and formulas for geometric sequences.
  3. Use sigma notation in simple settings.
  4. Model repeated change over time.

Derivatives and early calculus ideas

Some 11th graders meet derivatives directly, while others only preview rate of change. Either way, slope, change, and function behavior start becoming the main story.

If f(x) = x^n, then f'(x) = n x^(n-1)

Key skills

  1. Interpret average rate of change.
  2. Understand derivative as instant rate of change.
  3. Use the power rule on simple functions.
  4. Connect slope, motion, and graph behavior.

Topic reference notes

These short cards are here so you can jump directly to a rule or concept when you need a refresher.

Square roots

The square root of N is the positive number whose square is N.

sqrt(N) = x means x^2 = N

Estimation matters. Before using a calculator, you should often know the answer is between two nearby whole numbers.

Cube roots

The cube root of N is the number whose cube is N.

cbrt(N) = x means x^3 = N

Cube roots show up less often than square roots in school, but they are a good way to practice reverse operations.

What e means

The number e is about 2.718281828 and is the natural base for continuous growth.

e = limit of (1 + 1/n)^n as n gets large

Why 1 + 1 = 2 can be proved

Formal arithmetic starts with definitions of numbers and rules for addition.

1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2

Logarithms

A logarithm tells you which exponent produces a target value.

log base b of x = y means b^y = x

Trigonometry

Sine, cosine, and tangent begin as side ratios in right triangles and later become functions on the unit circle.

sin = opposite / hypotenuse, cos = adjacent / hypotenuse, tan = opposite / adjacent

Quadratics

Quadratic equations create parabolas and often model maximums, minimums, and curved motion.

x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a

Derivatives

Derivatives describe instant change. If a graph is a road, the derivative is the steepness at one moment.

If f(x) = x^n, then f'(x) = n x^(n-1)

Tool lab

These are not meant to replace full practice. They are here so you can read a rule and instantly try it.

Fraction converter

Percent change

Square root estimate

Cube root estimate

e from a series

Logarithm calculator

Slope from two points

System solver

Scientific notation

Trig ratio and angle

Circle measurements

Quadratic roots

Derivative value

Practice prompts by difficulty

Try a few from each band. The goal is to feel the ladder from arithmetic to algebra to functions.

6th-7th grade practice

  1. Convert 7/8 into a decimal and a percent.
  2. Find the unit rate for 18 miles in 3 hours.
  3. Find the percent increase from 45 to 54.
  4. Solve 3x + 5 = 20.
  5. Find the area of a rectangle with length 12 and width 7.

8th-9th grade practice

  1. Find the slope between (2, 3) and (8, 15).
  2. Solve the system 3x + y = 11 and x - y = 1.
  3. Write 0.00042 in scientific notation.
  4. Use the Pythagorean theorem on a 5-12-13 triangle.
  5. Solve x^2 - 7x + 12 = 0.

10th-11th grade practice

  1. Explain in words why log base 2 of 32 equals 5.
  2. Estimate sqrt(45) and cbrt(70).
  3. For a right triangle with opposite side 5 and adjacent side 12, compute tan(theta) and estimate theta.
  4. If f(x) = x^4, find f'(x) and evaluate it at x = 3.
  5. Use the series definition to approximate e with 6 terms.

Big-picture checkpoint

If you can explain what a ratio is, graph a line, solve a system, factor or solve a quadratic, interpret a log, use basic trig, and talk about derivative as rate of change, then you have a strong school-math core.

The most important skill across all grades is not memorizing every formula. It is learning how to translate words, numbers, graphs, and symbols into each other.