Math · 8th grade

8th grade math, unit by unit

Q: How do you find the slope of a line?

Divide the change in y by the change in x between two points: m = \frac{y_2 - y_1}{x_2 - x_1}. Between (1, 2) and (3, 8), m = \frac{6}{2} = 3. In y = mx + b, m is the slope and b is the y-intercept.

Q: What is the Pythagorean Theorem?

In a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2. With legs 6 and 8, the hypotenuse is \sqrt{36 + 64} = 10.

Q: How do you write a number in scientific notation?

Write it as a \times 10^n, with a at least 1 and less than 10 and n an integer. For example, 4{,}500{,}000 = 4.5 \times 10^6 and 0.00032 = 3.2 \times 10^{-4}.

Exponents and scientific notation, linear equations and systems, functions, transformations, the Pythagorean Theorem, scatter plots: here is everything 8th grade math covers, with worked problems and the mistakes to avoid. Then practice every unit in the Mathflow app, in three levels.

Start for free See the 15 units

From

A(1, 1)

B(7, 9)

, the legs of the right triangle are

6

and

8

, so

AB = \sqrt{6^2 + 8^2} = \sqrt{100} = 10

The course

What you’ll learn in 8th grade

15 units in five domains. The Common Core names three critical areas for 8th grade: linear equations and systems, including lines that model data; functions; and geometry with distance, angles, similarity and congruence, including the Pythagorean Theorem.

The Number System

2 units

Rational vs Irrational NumbersRecognize numbers that cannot be written as a fraction of two integers, like $\sqrt{2}$ , and turn terminating or repeating decimals into fractions.
Approximating IrrationalsEstimate irrational numbers with rational approximations, compare them and place them on a number line.

Expressions & Equations

4 units

Exponents & Scientific NotationUse the properties of integer exponents, square and cube roots, and compute with very large or very small numbers in scientific notation.
EquationsSolve multi-step linear equations, including equations with the variable on both sides.
Slope & Linear GraphsGraph proportional relationships and read the slope $m$ of a line as a constant rate of change.
Systems of EquationsSolve pairs of linear equations by graphing and by elimination, and explain what the intersection point means.

Functions

3 units

Introduction to FunctionsDefine and evaluate functions, and compare two functions given as graphs, tables, equations or descriptions.
Linear FunctionsRead $y = mx + b$ as a linear function, build one from a situation, and graph it with its slope and $y$ -intercept.
Function BehaviorSpot nonlinear functions, describe where a graph increases, decreases or stays constant, and sketch graphs from a description.

Geometry

4 units

TransformationsDescribe how translations, reflections, rotations and dilations move or resize a figure on the coordinate plane.
Angles & TrianglesUse informal arguments about angles, and the fact that the angles of a triangle add up to $180^\circ$ , to find unknown angles.
Pythagorean TheoremExplain and apply $a^2 + b^2 = c^2$ , use its converse to test for right triangles, and find distances in the coordinate plane.
VolumeUse the volume formulas of cylinders, cones and spheres to solve real-world problems.

Statistics & Probability

2 units

Scatter Plots & Lines of FitBuild scatter plots from two-variable data, and use a line of fit to model the data and make predictions.
Two-Way TablesBuild and read two-way tables, and compute relative frequencies to spot associations between two categories.

Worked problems

Two worked problems

Two problems typical of 8th grade. Try each one on paper before you read the solution.

Problem 1
A linear function from two data points
A plumber charges a fixed fee plus an hourly rate. A $2$ -hour job costs $190 and a $5$ -hour job costs $385. Write the cost $C$ of an $h$ -hour job as a linear function, then find the cost of a $3.5$ -hour job.
Solution
The cost goes up by the same amount every hour, so $C = mh + b$ . The slope is the change in cost divided by the change in hours:
$m = \frac{385 - 190}{5 - 2} = \frac{195}{3} = 65$
Use the $2$ -hour job to find the fixed fee: $190 = 65 \times 2 + b$ , so $b = 190 - 130 = 60$ .
$C = 65h + 60$
$65 \times 3.5 + 60 = 287.5$
A $3.5$ -hour job costs $287.50. In context, $m = 65$ is the hourly rate in dollars and $b = 60$ is the fixed fee.
Check: $65 \times 5 + 60 = 385$ , the cost of the $5$ -hour job.
Problem 2
A system of two equations
A school play sold $120$ tickets for $440 in all. Adult tickets cost $5 and student tickets cost $3. How many tickets of each kind were sold?
Solution
Call $a$ the number of adult tickets and $s$ the number of student tickets. One equation counts tickets, the other counts dollars:
$\begin{cases} a + s = 120 \\ 5a + 3s = 440 \end{cases}$
Multiply the first equation by $3$ to get $3a + 3s = 360$ , then subtract it from the second equation to eliminate $s$ :
$2a = 440 - 360 = 80 \quad\Rightarrow\quad a = 40$
Then $s = 120 - 40 = 80$ : the play sold $40$ adult tickets and $80$ student tickets.
Check: $5 \times 40 + 3 \times 80 = 200 + 240 = 440$ .

In Mathflow

How Mathflow teaches it

Every unit is split into skills, and every skill is a series in three levels. Here are real examples from three 8th grade series.

Level 1, Explore: explore and play with the concepts to build your intuition. Level 2, Explain: turn your intuitions into the right words and uncover hidden mistakes. Level 3, Solve: reason it out, crunch the numbers, and find your way to the answer. If you get stuck, Alphi, the AI math tutor, gives you a hint rather than the answer. When a unit is done, its Chapter Challenge checks it: 8 questions, 3 lifelines and a perfect score to earn the chapter gem.

1Linear Functions · Slope and $y$ -intercept in $y = mx + b$
- Level 1 · Explore
  Drag $P$ along the line $y = 2x - 1$ to the point where $y = 3$ , turn a line until its equation is $y = -3x + 2$ , and change $b$ in $y = \frac{1}{2}x + b$ until the line passes through a given point.
- Level 2 · Explain
  Decide who is right about the $y$ -intercept of $y = 3x$ , tap the first wrong line when Jordan describes the line $y = 5 - 2x$ , and compare the lines $y = -3x + 2$ and $y = 3x + 2$ .
- Level 3 · Solve
  A candle is $h = 12 - 0.5t$ inches tall after burning $t$ hours: find how many hours it takes to lose $3$ inches. The line $y = mx + 4$ passes through $(2, -2)$ : find $m$ .
2Exponents & Scientific Notation · Operations with numbers in scientific notation
- Level 1 · Explore
  Drag the decimal point to write a number in scientific notation, multiply and divide with a machine, and set the exponent $n$ so that $7.3 \times 10^n$ equals a sum.
- Level 2 · Explain
  Sofia and Marcus multiply $(3 \times 10^4)(4 \times 10^5)$ : decide who is right. Then tap the first line where Owen goes wrong in $(4.5 \times 10^3) \div (9 \times 10^{-1})$ .
- Level 3 · Solve
  Work out how many seconds sunlight takes to reach Earth, or how many times heavier than a mouse a blue whale is, and write each answer in scientific notation.
3Pythagorean Theorem · Explain and apply the Pythagorean Theorem
- Level 1 · Explore
  Tap every piece of a tilted square to find its area, drag the foot of a ladder until its top reaches a window sill, and drag the corner of a screen to make a $50$ -inch TV.
- Level 2 · Explain
  A $10$ -foot ladder leans against a wall with its foot $6$ feet away: decide whether Liam or Zoe is right about its height. Then tap the first wrong line of Jordan’s work on a missing leg.
- Level 3 · Solve
  Find the straight distance from home plate to second base, how far from the building a wheelchair ramp starts, and whether a $10$ -inch pencil can lie flat on the bottom of a square box.

Practice 8th grade math

Watch out

Common mistakes

Five slips that cost points in 8th grade, and how to avoid them.

Mixing up the exponent rules
Multiplying powers of the same base adds the exponents: $2^3 \times 2^4 = 2^7$ , not $4^7$ . Raising a power to a power multiplies them: $(x^3)^2 = x^6$ , not $x^5$ .
Leaving a first factor of $10$ or more
In scientific notation, the first factor is at least $1$ and less than $10$ . So $(3 \times 10^4)(4 \times 10^5) = 12 \times 10^9$ must be rewritten as $1.2 \times 10^{10}$ .
Moving a term without changing its sign
In $5x + 3 = 2x + 15$ , subtract $2x$ from both sides: $3x + 3 = 15$ , so $x = 4$ . Moving $2x$ to the left side without changing its sign gives $7x + 3 = 15$ , which is wrong.
Computing the slope as run over rise
The slope is the change in $y$ over the change in $x$ : between $(1, 2)$ and $(3, 8)$ , $m = \frac{8 - 2}{3 - 1} = 3$ , not $\frac{1}{3}$ .
Taking $\sqrt{a^2 + b^2}$ as $a + b$
The square root does not split over a sum: $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$ , not $6 + 8 = 14$ . The hypotenuse is always shorter than the two legs put end to end.

Frequently asked questions

What do you learn in 8th grade math?

Irrational numbers, exponents and scientific notation, linear equations and systems of equations, functions and $y = mx + b$ , transformations, angles in triangles, the Pythagorean Theorem, the volume of cylinders, cones and spheres, scatter plots and two-way tables.

Is 8th grade math the same as Algebra 1?

No. 8th grade math covers linear equations, systems and a first look at functions. Algebra 1 goes further, with exponential functions, polynomials and quadratic functions. Some students take Algebra 1 in 8th grade instead.

How do you find the slope of a line?

Divide the change in $y$ by the change in $x$ between two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Between $(1, 2)$ and $(3, 8)$ , $m = \frac{6}{2} = 3$ . In $y = mx + b$ , $m$ is the slope and $b$ is the $y$ -intercept.

What is the Pythagorean Theorem?

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ , $a^2 + b^2 = c^2$ . With legs $6$ and $8$ , the hypotenuse is $\sqrt{36 + 64} = 10$ .

How do you write a number in scientific notation?

Write it as $a \times 10^n$ , with $a$ at least $1$ and less than $10$ and $n$ an integer. For example, $4{,}500{,}000 = 4.5 \times 10^6$ and $0.00032 = 3.2 \times 10^{-4}$ .

Is there an app for 8th grade math?

Mathflow covers all 15 units of 8th grade math in three levels (Explore, Explain, Solve), with Alphi, an AI math tutor, and a Chapter Challenge for every unit. It is free to start on iOS and Android.

Sources and next steps

Your school’s course may order the units differently: use the list above as a map, and start with the unit you are studying in class. To review proportions, percents and negative numbers, see 7th grade math. Next comes Algebra 1, which builds on linear functions and systems. All courses are on the math hub.

8th grade math, unit by unit

A linear function from two data points

A system of two equations

1Linear Functions · Slope and yyy-intercept in y=mx+by = mx + by=mx+b

2Exponents & Scientific Notation · Operations with numbers in scientific notation

3Pythagorean Theorem · Explain and apply the Pythagorean Theorem

Mixing up the exponent rules

Leaving a first factor of 101010 or more

Moving a term without changing its sign

Computing the slope as run over rise

Taking a2+b2\sqrt{a^2 + b^2}a2+b2​ as a+ba + ba+b

Math is all about flow.

1Linear Functions · Slope and $y$ -intercept in $y = mx + b$

Leaving a first factor of $10$ or more

Taking $\sqrt{a^2 + b^2}$ as $a + b$