Math · 7th grade

7th grade math, unit by unit

Q: What is a proportional relationship?

Two quantities whose ratio \frac{y}{x} is always the same number k. Its equation is y = kx, and its graph is a line through the origin. At $1.50 a pound, the cost of x pounds is y = 1.5x.

Q: How do you subtract negative numbers?

Add the opposite. For example, -4 - (-9) = -4 + 9 = 5.

Q: What are the formulas for the area and circumference of a circle?

With radius r and diameter d: the area is A = \pi r^2 and the circumference is C = 2\pi r = \pi d.

Proportional relationships and percents, negative numbers, two-step equations and inequalities, circles, probability: here is everything 7th 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 16 units

Apples at $1.50 a pound: the cost

y

x

pounds is

y = 1.5x

. The line goes through

(0, 0)

, and

(1, 1.5)

shows the unit rate.

The course

What you’ll learn in 7th grade

16 units in five domains. The Common Core names four critical areas for 7th grade: proportional relationships, rational numbers with expressions and linear equations, scale drawings with area, surface area and volume, and inferences about populations from samples.

Ratios & Proportions

3 units

Rates & ProportionsDecide whether two quantities change at a constant rate, and compute unit rates to compare situations.
Proportional EquationsWrite proportional relationships as $y = kx$ , and explain what a point on their graph means, especially $(0, 0)$ and $(1, r)$ .
Percent ApplicationsFind percent increase, decrease and error, and solve multi-step problems with tax, tips, discounts, markups, commissions and simple interest.

The Number System

3 units

Adding & Subtracting IntegersAdd integers with the same or different signs, subtract by adding the opposite, and see distance on a number line as absolute value.
Multiplying & Dividing IntegersMultiply and divide integers with the rules of signs, and use the associative and distributive properties to compute efficiently.
Rational NumbersAdd, subtract, multiply and divide positive and negative fractions and decimals, and write any fraction as a terminating or repeating decimal.

Expressions & Equations

3 units

Simplifying & FactoringUse the distributive property, combine like terms and factor out the greatest common factor to rewrite linear expressions.
EquationsUse variables for unknown quantities, solve equations like $px + q = r$ , then multi-step equations with fractions and decimals.
InequalitiesSolve inequalities like $px + q > r$ , see when the inequality symbol flips, and graph the solution set on a number line.

Geometry

4 units

Scale DrawingsUse a scale to find actual lengths and areas, and make reduced or enlarged copies that keep the right proportions.
Angle RelationshipsUse supplementary, complementary and vertical angles to find unknown angle measures.
CirclesKnow and explain the formulas for the circumference and the area of a circle, and use them in real problems.
Volume & Surface AreaFind the volume and surface area of prisms, pyramids and cylinders, describe cross-sections of solids, and find the area of composite figures.

Statistics & Probability

3 units

Sampling & InferencesSee why random samples give fair estimates, and use sample data to draw conclusions about a population and compare groups.
Probability BasicsLearn the vocabulary of chance, estimate probabilities from repeated experiments, and build models where every outcome is equally likely.
Compound EventsList sample spaces and use tree diagrams to find the probability of compound events.

Worked problems

Two worked problems

Two problems typical of 7th grade. Try each one on paper before you read the solution.

Problem 1
A discount, then sales tax
A jacket costs $72. It is $25\%$ off, and a $6\%$ sales tax is added to the sale price. How much do you pay?
Solution
Taking $25\%$ off leaves $100\% - 25\% = 75\%$ of the price. Adding a $6\%$ tax gives $106\%$ of the sale price. So multiply by $0.75$ , then by $1.06$ :
$72 \times 0.75 = 54 \qquad 54 \times 1.06 = 57.24$
You pay $57.24: the discount saves $18, and the tax adds $3.24.
Check: the order does not matter, because multiplication is commutative: $72 \times 1.06 \times 0.75$ also gives $57.24$ . But you cannot subtract the percents: the total is not $72 \times 0.81$ , because the tax is a percent of the sale price, not of $72.
Problem 2
A sign-up fee plus a monthly price
A gym charges a $25 sign-up fee plus $18 per month.
1. Maya has paid $151 in all. How many months has she paid for?
2. Her budget is at most $200. What is the greatest number of months she can pay for?
Solution
a. Call $m$ the number of months. The total is the fee plus $18$ dollars per month:
$\begin{aligned} 25 + 18m &= 151 \\ 18m &= 126 \\ m &= 7 \end{aligned}$
Maya has paid for $7$ months. Check: $25 + 18 \times 7 = 25 + 126 = 151$ .
b. The total must be at most $200, so solve an inequality the same way:
$\begin{aligned} 25 + 18m &\le 200 \\ 18m &\le 175 \\ m &\le \tfrac{175}{18} \approx 9.7 \end{aligned}$
Months come in whole numbers, so she can pay for at most $9$ months: $25 + 18 \times 9 = 187$ , while $10$ months would cost $205$ .

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 7th 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.

1Adding & Subtracting Integers · Subtract integers by adding the opposite
- Level 1 · Explore
  Show $2 - (-3)$ with counters by taking $3$ negative counters off the mat, drag a point to show $-1 - 4$ as a jump from $-1$ , and drag a diver to where she stops $13$ feet below the platform.
- Level 2 · Explain
  Find what is wrong with Ethan’s computation of $-8 - 3$ , tap every expression with the same value as $-6 - 5$ , and describe the change from $-7$ °F at 7 a.m. to $-2$ °F at 11 a.m.
- Level 3 · Solve
  Compute $-10 - (-2 - 5)$ , tap the first wrong line of Mason’s computation of $-12 - (-5 - 6)$ , and find how many degrees the temperature rose in Spearfish, South Dakota, when it went from $-4$ °F to $45$ °F in two minutes.
2Equations · Solve two-step equations
- Level 1 · Explore
  Get one $x$ box alone on a balance that stays level, run a machine backward to find $x$ , or slide $x$ on a number line until $-2x + 9 = 15$ .
- Level 2 · Explain
  Tyler and Grace solve $4x + 8 = 20$ in two different orders: decide who is right. Then find what is wrong with Sofia’s check of $x = -3$ in $-4x + 7 = 19$ .
- Level 3 · Solve
  Solve $3(x - 4) = 15$ and $-4x + 7 = -5$ . Jada paid $37 for streaming, a $5 sign-up fee plus $4 per month: find how many months she paid for.
3Percent Applications · Solve multistep percent problems with money
- Level 1 · Explore
  Drag a handle to cut a $30\%$ discount off the price bar, tap the shoes that cost exactly $54 once the $8\%$ sales tax is added, and add years until $300 saved at $4\%$ simple interest becomes $348.
- Level 2 · Explain
  An $80 lamp is marked down $25\%$ , then its price goes up $25\%$ : explain why the new price is not $80. Then find the error in Grace’s markup percent.
- Level 3 · Solve
  Boots cost $80, are $25\%$ off, then get a $5\%$ tax: type both multipliers and the total. Kevin pays $48.15 for a $45 lamp, tax included: find the sales tax rate.

Practice 7th grade math

Watch out

Common mistakes

Five slips that cost points in 7th grade, and how to avoid them.

Writing $5 - (-3) = 2$
Subtracting a number means adding its opposite: $5 - (-3) = 5 + 3 = 8$ . On a number line, subtracting a negative number moves you to the right.
Distributing to the first term only
The factor multiplies every term in the parentheses: $3(x + 4) = 3x + 12$ , not $3x + 4$ . Watch the signs too: $-2(x - 5) = -2x + 10$ .
Forgetting to flip the inequality
When you multiply or divide both sides by a negative number, the symbol flips: $-2x > 6$ gives $x < -3$ . Check with $x = -4$ : $-2 \times (-4) = 8$ , and $8 > 6$ .
Thinking that $25\%$ off, then $25\%$ up, gets you back
The second percent applies to a new amount: $80 becomes $80 \times 0.75 = 60$ , then $60 \times 1.25 = 75$ , not $80.
Using the diameter in $A = \pi r^2$
A circle with a diameter of $10$ cm has a radius of $5$ cm, so its area is $25\pi \approx 78.5$ cm², not $100\pi$ .

Frequently asked questions

What do you learn in 7th grade math?

Proportional relationships and percent problems, operations with negative numbers, fractions and decimals, expressions, two-step equations and inequalities, scale drawings, angles, circles, volume and surface area, sampling and probability.

What is a proportional relationship?

Two quantities whose ratio $\frac{y}{x}$ is always the same number $k$ . Its equation is $y = kx$ , and its graph is a line through the origin. At $1.50 a pound, the cost of $x$ pounds is $y = 1.5x$ .

How do you subtract negative numbers?

Add the opposite. For example, $-4 - (-9) = -4 + 9 = 5$ .

How do you solve a two-step equation?

Undo the addition or subtraction first, then the multiplication or division. For $3x + 5 = 26$ : subtract $5$ to get $3x = 21$ , then divide by $3$ to get $x = 7$ . Check: $3 \times 7 + 5 = 26$ .

What are the formulas for the area and circumference of a circle?

With radius $r$ and diameter $d$ : the area is $A = \pi r^2$ and the circumference is $C = 2\pi r = \pi d$ .

Is there an app for 7th grade math?

Mathflow covers all 16 units of 7th 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 the basics, see 6th grade math. Next year, 8th grade math turns proportional relationships into linear functions. All courses are on the math hub.

7th grade math, unit by unit

A discount, then sales tax

A sign-up fee plus a monthly price

1Adding & Subtracting Integers · Subtract integers by adding the opposite

2Equations · Solve two-step equations

3Percent Applications · Solve multistep percent problems with money

Writing 5−(−3)=25 - (-3) = 25−(−3)=2

Distributing to the first term only

Forgetting to flip the inequality

Thinking that 25%25\%25% off, then 25%25\%25% up, gets you back

Using the diameter in A=πr2A = \pi r^2A=πr2

Math is all about flow.

Writing $5 - (-3) = 2$

Thinking that $25\%$ off, then $25\%$ up, gets you back

Using the diameter in $A = \pi r^2$