7th grade math: what your child learns this year

Math for 7th graders is the year of proportional relationships and negative numbers. Your child learns to solve rate and percent problems, to add, subtract, multiply and divide negative numbers and to write and solve two-step equations. Geometry adds circles, scale drawings and angle facts, and probability appears for the first time. A child who finishes 7th grade solid in proportional reasoning and negative numbers starts 8th grade ready for slope and linear equations.

The one-minute check

Before the checklist, try this

Ask your child to find 30% of 80 without a calculator, and to say each step out loud.

If they get it

Your child gets 24, for example by finding 10% of 80 (8) and multiplying by 3. That means percents from 6th grade are solid, and 7th grade has something to build on.

If they do not

Your child answers 50 (80 minus 30), or divides 80 by 30, or cannot start without a calculator. That is the gap, and it comes from 6th grade.

Percents come back all through 7th grade: sales tax, tips, discounts, simple interest and percent increase. A child who does not read 30% as 30 out of every 100 loses points on all of them, even when the new idea itself made sense.

The 7th grade skills checklist

Read each line as "my child can…". Confident checks are strengths; hesitations are where to look closer.

Not every topic weighs the same. Proportional relationships, negative numbers, expressions and equations are where most of 7th grade goes, and most state test questions come from them. Geometry, statistics and probability are real topics, but a smaller share of the year.

Where most of the year goes

Proportional relationships and percents

The first big topic of 7th grade. Your child learns to spot when two amounts grow at the same rate, and uses that to solve rate and percent problems from daily life.

Finds a unit rate when the amounts are fractions, such as 1/2 mile every 1/4 hour (2 miles per hour)
Decides whether a table or a graph shows a proportional relationship, with a straight line through (0, 0)
Finds the constant of proportionality and writes an equation such as y = 3x
Explains what a point on the graph means in the story, such as (1, 3) meaning 3 dollars for 1 pound
Solves percent problems: sales tax, tips, discounts, markups, simple interest and percent increase or decrease

Comes from last year: Sixth grade introduced ratios, unit rates and percents, mostly with whole numbers. Seventh grade uses fractions in the same problems, so a child who finds a unit rate only by trial and error gets stuck fast.

7.RP.1 · 7.RP.2 · 7.RP.3

Negative numbers

Your child learns to add, subtract, multiply and divide positive and negative numbers, including fractions and decimals. The sign rules start here, and every later grade uses them.

Adds and subtracts negative numbers and shows the move on a number line
Knows that a number and its opposite add to 0, such as -7 and 7
Explains subtraction as adding the opposite: 3 - (-5) is the same as 3 + 5
Multiplies and divides with negative numbers and gets the sign right
Turns a fraction into a decimal by long division, and knows the decimal either ends or repeats
Solves word problems with negative numbers, such as temperatures, money owed and heights below sea level

Comes from last year: Sixth grade put negative numbers on the number line, for temperatures and money owed, but did not calculate with them. Fraction and decimal arithmetic has to be solid already, because 7th grade adds signs on top of it.

7.NS.1 · 7.NS.2 · 7.NS.3

Expressions

Your child learns to rewrite an expression without changing its value: expanding brackets, collecting like terms and factoring. Two expressions that look different can mean the same amount.

Expands brackets, such as 4(2x - 3) = 8x - 12
Collects like terms, such as 3x + 5 - x + 2 = 2x + 7
Factors out a common number, such as 6x + 9 = 3(2x + 3)
Explains why a new form can help, for example that a + 0.05a and 1.05a both mean "the price plus 5%"

Comes from last year: Sixth grade introduced variables and the distributive property with whole numbers. Seventh grade adds negative numbers and fractions, so the sign rules from this year have to work inside expressions too.

7.EE.1 · 7.EE.2

Equations and inequalities

Your child turns word problems into two-step equations and solves them, then starts on inequalities. Every answer gets checked against a quick estimate.

Solves multi-step word problems that mix whole numbers, fractions, decimals and percents
Checks whether an answer makes sense with a quick estimate
Writes and solves two-step equations such as 3x + 5 = 20 or 2(x + 4) = 18
Writes and solves an inequality such as 3x + 5 > 20, and shows the answers on a number line

Comes from last year: Sixth grade solved one-step equations such as x + 7 = 12 and 4x = 28. A child who solves those by guessing, rather than by undoing the operation, gets stuck as soon as there are two steps.

7.EE.3 · 7.EE.4

Also covered, smaller share of the year

Scale drawings, triangles and cross sections

Your child uses a scale drawing, such as a map or a floor plan, to find real lengths and areas. This area also covers drawing triangles from given measurements and the flat shapes made by slicing a solid.

Finds real lengths and areas from a scale drawing, and redraws it at a different scale
Draws a triangle from three side lengths or three angles, and notices when only one triangle, more than one or none is possible
Describes the cross section made by slicing a prism or a pyramid, such as a rectangle or a triangle

Comes from last year: Sixth grade ratios carry scale drawings, because a scale such as 1 inch to 10 feet is a ratio. Sixth grade nets of 3D shapes help with picturing a slice.

7.G.1 · 7.G.2 · 7.G.3

Circles, angles, area and volume

The two circle formulas and the angle facts used to find a missing angle. Your child also finds area, surface area and volume of shapes built from simpler pieces.

Uses the formulas for the area and the circumference of a circle
Finds a missing angle with an equation, using angle pairs that add to 90° or 180° and vertical angles, which are equal
Finds area, surface area and volume of shapes made from triangles, rectangles, cubes and prisms

Comes from last year: Sixth grade found the area of triangles and of shapes made from rectangles and triangles, plus the volume of boxes. A child who has to look up the area of a triangle will struggle with shapes built from several pieces.

7.G.4 · 7.G.5 · 7.G.6

Random samples and comparing data

How a small random sample can describe a large group, and how to compare two groups by the average and the spread of each one.

Explains why a random sample can stand for a whole group, and why a sample that favors one group cannot
Uses a sample to estimate something about the whole group, and knows that different samples give slightly different estimates
Compares two groups of data using the mean or the median and the spread

Comes from last year: Sixth grade introduced the mean, the median and measures of spread, with dot plots, histograms and box plots. Seventh grade uses all of them to compare two groups.

7.SP.1 · 7.SP.2 · 7.SP.3 · 7.SP.4

Probability

Probability appears for the first time, as a number from 0 to 1. Your child finds probabilities by reasoning, by experiment and with lists, tables and tree diagrams.

Knows that a probability runs from 0 (impossible) to 1 (certain), and that 1/2 means as likely as not
Estimates a probability by repeating an experiment many times, such as flipping a coin or rolling a number cube
Builds a probability model and compares it with what really happens
Finds the probability of compound events, such as two coin flips, with an organized list, a table or a tree diagram

Comes from last year: Nothing directly, because probability is new in 7th grade. It leans on fractions, decimals and percents, since the same probability can be written in all three forms.

7.SP.5 · 7.SP.6 · 7.SP.7 · 7.SP.8

All 24 7th grade math standards, in plain English

These are the Common Core standards for 7th grade math. Each link opens a page that explains the topic in simple words, with examples, the usual sticking points and four practice questions.

Expressions and Equations

Geometry

7.G.A.1 Scale Drawings
7.G.A.2 Constructing Triangles
7.G.A.3 Cross Sections of 3D Figures
7.G.B.4 Area of a Circle
7.G.B.5 Supplementary and Complementary Angles
7.G.B.6 Volume of Prisms

The Number System

7.NS.A.1 Adding and Subtracting Integers
7.NS.A.2 Multiplying Integers
7.NS.A.3 Integer Word Problems

Ratios and Proportional Relationships

Statistics and Probability

What schools teach in 7th grade math, at a glance

Area	What schools teach	Share of the year
Proportional relationships and percents	The first big topic of 7th grade. Your child learns to spot when two amounts grow at the same rate, and uses that to solve rate and percent problems from daily life.	Major
Negative numbers	Your child learns to add, subtract, multiply and divide positive and negative numbers, including fractions and decimals. The sign rules start here, and every later grade uses them.	Major
Expressions	Your child learns to rewrite an expression without changing its value: expanding brackets, collecting like terms and factoring. Two expressions that look different can mean the same amount.	Major
Equations and inequalities	Your child turns word problems into two-step equations and solves them, then starts on inequalities. Every answer gets checked against a quick estimate.	Major
Scale drawings, triangles and cross sections	Your child uses a scale drawing, such as a map or a floor plan, to find real lengths and areas. This area also covers drawing triangles from given measurements and the flat shapes made by slicing a solid.	Smaller
Circles, angles, area and volume	The two circle formulas and the angle facts used to find a missing angle. Your child also finds area, surface area and volume of shapes built from simpler pieces.	Smaller
Random samples and comparing data	How a small random sample can describe a large group, and how to compare two groups by the average and the spread of each one.	Smaller
Probability	Probability appears for the first time, as a number from 0 to 1. Your child finds probabilities by reasoning, by experiment and with lists, tables and tree diagrams.	Smaller

What 7th grade math actually looks like

Five real problems from our 7th grade bank. Work through them with your child, check each answer, and see the steps for any problem.

Problem 1
James simplified the expression below. Which statement best explains the mistake?
Original
$\displaystyle -4x + 3x + 7 + 2$
Step 1
$\displaystyle = 7x + 9$
A
James lost a negative sign when combining the x-terms.
B
James combined unlike terms.
C
James did not combine all like terms.
D
James added the constants incorrectly.
Show the answer and how to get it
The correct explanation is that James lost a negative sign when combining the x-terms, because -4x + 3x = -x, not 7x.
1. Find the like terms in the original expression. The x-terms are -4x and 3x, and the constants are 7 and 2.
2. Check the x-terms. The correct sum is -x, but James wrote 7x. He computed $4 + 3 = 7$ instead of $-4 + 3 = -1$ , so the negative sign on -4x was lost.
3. The constants were combined correctly.
4. The only mistake is in the x-terms: James lost the negative sign on -4x. The best explanation is that James lost a negative sign when combining the x-terms.
Problem 2
The candidate drawing below is supposed to be a scale copy of the original using the scale factor shown. Which statement best describes the candidate drawing?
$\displaystyle \text{Scale factor }= 2$
A
It is a correct scale copy.
B
It is not a correct scale copy because only one dimension was multiplied by the scale factor.
C
It is not a correct scale copy because the same amount was added to each dimension.
D
It is not a correct scale copy because both dimensions were scaled by the same wrong factor.
Show the answer and how to get it
The candidate is not a correct scale copy because the same amount was added to each dimension. A true factor-2 copy of 3 by 5 would be 6 by 10, not 5 by 7.
1. The original rectangle is 3 by 5, the stated scale factor is 2, and the candidate shows 5 by 7.
2. Multiply each original dimension by 2 to find what a true scale copy should measure.
3. Neither dimension matches the scaled value, so the copy is wrong. Check what was done instead.
4. The same amount, 2, was added to each dimension instead of multiplying by 2.

Problem 3

Look at the student's work below. Which mistake did the student make?

Original	$\displaystyle \frac{\frac{1}{3}}{\frac{3}{4}}$
Step 1	$\displaystyle = \frac{1}{3} \times \frac{3}{4}$
Step 2	$\displaystyle = \frac{3}{12}$
Step 3	$\displaystyle = \frac{1}{4}$

The student inverted the first fraction instead of the denominator fraction.

The student changed division to multiplication and left both fractions unchanged.

The student simplified the final fraction incorrectly.

In Step 1, the student swapped the top and bottom fractions of the whole complex fraction.

Show the answer and how to get it

The correct description is that the student changed to multiplication but did not invert the denominator fraction. Dividing by $\frac{3}{4}$ should be rewritten as multiplying by $\frac{4}{3}$ .

Start by checking the move from Original to Step 1. That is where division by a fraction should be rewritten correctly.
When you divide by $\frac{3}{4}$ , you multiply by its reciprocal, $\frac{4}{3}$ .
The student wrote $\frac{1}{3} \times \frac{3}{4}$ . That changes division to multiplication, but it does not flip $\frac{3}{4}$ to $\frac{4}{3}$ .
The simplification at the end is not the issue. $\frac{3}{12}$ really does equal $\frac{1}{4}$ , so the error is not in the final step.
So the matching description is that the student changed to multiplication but did not invert the denominator fraction.

Problem 4

Which student gives both a correct judgment and a valid reason? Use the study summary in the table.

Part	Information
Population	All 7th graders at Riverside Middle School
Sample	40 7th graders chosen at random from the school roster
Result	28 of the 40 said they would bring reusable water bottles to school
Claim	All 7th graders at Riverside Middle School would bring reusable water bottles to school
Jayden	The claim is justified because the sample was random and most students said yes.
Aaliyah	The claim is not justified because students were chosen randomly instead of volunteering.

Aaliyah

Neither student

Jayden

Both students

Show the answer and how to get it

Neither student gives both a correct judgment and a valid reason. Jayden overlooks 'all,' and Aaliyah wrongly treats random selection as a flaw.

Read the Sample, Result, and Claim rows carefully.
Choosing students at random from the roster helps the sample represent the whole grade. Letting students volunteer could leave out students with different views.
70% said yes, not 100%. The claim uses the word 'all,' which would require every 7th grader to agree.
Jayden treats a majority as support for 'all.' Aaliyah correctly rejects the claim but gives a wrong reason: random selection helps rather than harms the study. Neither student gives both a correct judgment and a valid reason.

Problem 5

Look at the student's work. Which statement best describes the work?

Given	$\displaystyle \text{Increase by }0.5\%$
Step 1	$\displaystyle 0.5\% = 0.005$
Step 2	$\displaystyle \text{Multiplier} = 1 + 0.005 = 1.005$

The student used subtraction as if it were a decrease.

No mistake; the work is correct.

The percent was not written as a decimal correctly.

The student forgot to add the original 1 whole.

Show the answer and how to get it

The correct choice is "No mistake; the work is correct." A 0.5% increase uses multiplier $1 + 0.005 = 1.005$ , which the student found correctly.

Start with Step 1: 0.5% = 0.005. Dividing 0.5 by 100 moves the decimal two places left, giving 0.005. That conversion is correct.
For a percent increase, the multiplier must be 1 plus the decimal form of the rate. The correct multiplier for a 0.5% increase is 1.005.
Step 2 in the work shows Multiplier $= 1 + 0.005 = 1.005$ , which matches the correct rule exactly. Because 1.005 > 1, this is properly an increase multiplier.
Both lines check out. The best description is: No mistake; the work is correct.

Where 7th grade math usually goes wrong

These are not careless mistakes. Each one is a rule that worked perfectly well until this year, applied consistently to a place where it no longer holds, which is why the same wrong answer keeps coming back. Recognising the specific rule is far more useful than more practice.

Writes -8 + 3 = -11.
The sign rules were memorized without a picture. On a number line, starting at -8 and moving 3 to the right lands on -5.
Writes 5 - (-2) = 3.
Subtracting looks like it should always make a number smaller. Subtracting -2 is the same as adding 2, so the answer is 7.
Says a $100 jacket that is marked down 20% and then marked up 20% costs $100 again.
Both changes say 20%, so the two seem to cancel. The second 20% is taken of $80, not $100, so the jacket ends at $96.
Mixes up the area and the circumference of a circle, or puts the diameter into πr².
Two formulas with π arrive in the same few weeks. Asking which one measures the fence around a round garden and which one measures the grass inside separates them.

Roughly when each part comes up

Schools vary, but most 7th grade classrooms move in about this order. Useful if you are trying to work out whether something is genuinely behind or simply has not been taught yet.

September to OctoberProportional relationships, unit rates and percent problems
October to DecemberAdding, subtracting, multiplying and dividing negative numbers
December to JanuaryExpressions: expanding brackets, collecting like terms and factoring
January to MarchTwo-step equations, inequalities and multi-step word problems
February to MarchScale drawings, drawing triangles and cross sections
March to AprilRandom samples and comparing two groups of data
March to MayCircles and angle facts; area, surface area and volume
May to JuneProbability

What 8th grade assumes on day one

Next year does not re-teach these. It starts from them, which is why a gap left open over the summer tends to show up in October rather than in June.

Solving two-step equations with negative numbers, without guessing the sign
Proportional relationships and unit rate, which become slope
Expanding brackets and collecting like terms, needed for equations with the variable on both sides
The area of a circle, which comes back in the volume of cylinders, cones and spheres

7th Grade math resources

Free 7th grade math quick check

Ten questions on the year’s main topics, about 10 to 15 minutes, with a worked solution for every one. Take it online or print it with the answer key.

Start the quick check →

Not sure which grade fits?

Pick the quick check for any grade, kindergarten to 8th.

Grade-by-grade readiness check

What should be solid before next September, K through 8.

Common questions

What do 7th graders learn in math?

Proportional relationships and percents, arithmetic with negative numbers, expressions, two-step equations and inequalities, scale drawings, circles and angles, area, surface area and volume, random samples and probability. Proportional relationships, negative numbers, expressions and equations carry the most weight.

Is 7th grade math pre-algebra?

Pre-algebra is a course name, not a Common Core grade. Seventh grade math covers much of what a pre-algebra course teaches: negative numbers, expressions, two-step equations and proportional relationships. Some schools offer a faster course that mixes 7th and 8th grade, for children who are ready early.

How can I check if my child is on track in 7th grade math?

Start with the one-minute percent check on this page. For a broader read, try the free 10-question quick check. The Level Up assessment, about 30 to 40 questions on a computer or tablet, shows where your child stands, including gaps from earlier grades.

Original	$\displaystyle -4x + 3x + 7 + 2$
Step 1	$\displaystyle = 7x + 9$