Math · 6th grade

6th grade math, unit by unit

Q: What do you learn in 6th grade math?

Ratios, rates and percents, dividing fractions, negative numbers and the coordinate plane, expressions, one-step equations and inequalities, area, surface area and volume, and an introduction to statistics: mean, median, mode and data displays.

Q: Is 6th grade math pre-algebra?

It is where algebra starts. In 6th grade you write and evaluate expressions with variables and solve one-step equations like x + p = q and px = q. 7th and 8th grade math build on this before Algebra 1.

Q: How do you divide fractions?

Multiply the first fraction by the reciprocal of the second. For example, \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}.

Q: What is a unit rate?

The amount for one unit of the other quantity, such as a price per ounce or miles per hour. A 15-ounce box for $4.20 has a unit price of 4.20 \div 15 = 0.28, that is $0.28 per ounce.

Q: Is there an app for 6th grade math?

Mathflow covers all 21 units of 6th 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.

Ratios and rates, dividing fractions, negative numbers, your first equations, area and volume, statistics: here is everything 6th 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 21 units

Six jumps of

\frac{2}{3}

reach

4

, and the last

\frac{1}{2}

\frac{3}{4}

of a jump:

4\frac{1}{2} \div \frac{2}{3} = 6\frac{3}{4}

The course

What you’ll learn in 6th grade

21 units in five domains. The Common Core names four critical areas for 6th grade: ratio and rate reasoning, dividing fractions and negative numbers, expressions and equations, and statistical thinking.

Ratios & Proportions

4 units

Introduction to RatiosCompare two quantities with ratio language, written $a:b$ , as a fraction or with the word “to”.
Ratios & TablesBuild tables of equivalent ratios, find missing values and spot the constant that links the two quantities.
Unit RatesFind the price of one item or the distance covered in one hour, and compare unit prices to find the better buy.
Percents & ConversionsFind a percent of a quantity or the whole from a part, convert units, and switch between fractions, decimals and percents.

The Number System

5 units

Dividing FractionsDivide fractions by multiplying by the reciprocal, and solve word problems such as sharing food or cutting material.
Decimals & Long DivisionDivide multi-digit numbers with the standard algorithm, and add, subtract, multiply and divide decimals.
Factors & MultiplesFind the greatest common factor and the least common multiple, factor a sum with the distributive property, and recognize prime numbers.
Integers & Number LineUse negative numbers for temperatures, elevations and debts, place them on a number line, compare them and find absolute values.
The Coordinate PlanePlot points in all four quadrants and find the distance between two points that share a coordinate.

Expressions & Equations

5 units

Exponents & OperationsWrite repeated multiplication with exponents and evaluate expressions with the order of operations.
Variables & ExpressionsTurn words like “add $5$ to $x$ ” into expressions, then evaluate them for given values.
EquationsSolve one-step equations like $x + p = q$ and $px = q$ , and write equations for real situations.
InequalitiesWrite inequalities like $x > 3$ and graph their infinitely many solutions on a number line.
Two-Variable RelationshipsTell the independent variable from the dependent one, and connect tables, graphs and equations.

Geometry

4 units

Area of PolygonsMeasure area in square units, find the area of triangles and special quadrilaterals, and split complex shapes into simpler ones.
Polygons & CoordinatesDraw polygons from the coordinates of their vertices, then find side lengths, perimeters and areas.
VolumePack rectangular prisms with unit cubes and use $V = lwh$ and $V = Bh$ , even with fractional edge lengths.
Surface Area & NetsUnfold prisms and pyramids into nets to find their surface area, and identify solids by their faces, edges and vertices.

Statistics & Probability

3 units

Introduction to StatisticsRecognize statistical questions, organize data in frequency tables, and tell a population from a sample.
Measures of CenterFind the mean, the median and the mode of a data set.
Data DisplaysMake and read dot plots, histograms and box plots, and measure spread with the mean absolute deviation.

Worked problems

Two worked problems

Two problems typical of 6th grade. Try each one on paper before you read the solution.

Problem 1
How many servings?
You have $4\frac{1}{2}$ cups of trail mix. One serving is $\frac{2}{3}$ cup. How many servings can you make?
Solution
The question is “how many $\frac{2}{3}$ fit in $4\frac{1}{2}$ ?”, so it is a division. Write the mixed number as a fraction: $4\frac{1}{2} = \frac{9}{2}$ .
To divide by $\frac{2}{3}$ , multiply by its reciprocal, $\frac{3}{2}$ :
$\frac{9}{2} \div \frac{2}{3} = \frac{9}{2} \times \frac{3}{2} = \frac{27}{4} = 6\frac{3}{4}$
You can make $6$ full servings. What is left is $\frac{3}{4}$ of a serving, that is $\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$ cup.
Check: $6 \times \frac{2}{3} = 4$ cups, plus the $\frac{1}{2}$ cup left over, makes $4\frac{1}{2}$ cups.
Problem 2
Which box is the better buy?
A $15$ -ounce box of cereal costs $4.20. A $24$ -ounce box of the same cereal costs $6.48. Which box is the better buy?
Solution
Compare the unit prices, the price of one ounce: divide each price by the number of ounces.
$4.20 \div 15 = 0.28 \qquad 6.48 \div 24 = 0.27$
The small box costs $0.28 per ounce and the large box $0.27 per ounce. The $24$ -ounce box is the better buy, by $1$ cent per ounce.
Check: buy $120$ ounces both ways. That is $8$ small boxes for $33.60, or $5$ large boxes for $32.40: the large boxes save $1.20.

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 6th 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.

1Dividing Fractions · Divide fractions by fractions
- Level 1 · Explore
  Fill $2$ cups with $\frac{1}{4}$ -cup scoops to find $2 \div \frac{1}{4}$ , cut half a pizza into slices of $\frac{1}{6}$ pizza, or cover a $1\frac{3}{4}$ -yard ribbon with $\frac{1}{2}$ -yard pieces.
- Level 2 · Explain
  Decide who is right about $6 \div \frac{1}{2}$ , find the mistake Jordan made when computing $\frac{5}{6} \div \frac{2}{5}$ , and tell whether $5 \div \frac{2}{3}$ is greater or less than $5$ , with the right reason.
- Level 3 · Solve
  Compute $\frac{7}{8} \div \frac{3}{4}$ in simplest form, then find how many $\frac{3}{8}$ -cup servings a container of $2\frac{1}{4}$ cups of yogurt holds.
2Unit Rates · Find the better buy
- Level 1 · Explore
  Tap the box of cereal that is the better buy, scoop trail mix from the bin until it costs as much as the bag, and drag the price of a big jar until both jars cost the same per ounce.
- Level 2 · Explain
  Limes sell at “ $3$ for $2” or “ $5$ for $3”: decide who is right. Then tap Ethan’s first wrong line when he compares $3$ pounds of rice for $3.90 with $5$ pounds for $7.00.
- Level 3 · Solve
  Use a $1 coupon on a $20$ -ounce jar of honey to pick the better buy, and work out how many dollars the better buy saves on $40$ pounds of dog food.
3Equations · Solve one-step equations
- Level 1 · Explore
  Take cubes off the pans of a balance until $x$ is alone and the balance is level, or drag $x$ so that four jumps of length $x$ land exactly on $10$ .
- Level 2 · Explain
  Tap the question that the equation $4x = 10$ asks, and find the mistake Sam made when he solved $x + 2.5 = 6$ and wrote $x = 8.5$ .
- Level 3 · Solve
  Solve $6x = 27$ and $x + \frac{1}{4} = \frac{3}{2}$ . Sticker packs cost $1.25 each and Rumi spent $7.50: how many packs did she buy?

Practice 6th grade math

Watch out

Common mistakes

Five slips that cost points in 6th grade, and how to avoid them.

Reading $3^4$ as $3 \times 4$
An exponent counts how many times the base is a factor: $3^4 = 3 \times 3 \times 3 \times 3 = 81$ , not $12$ .
Working left to right no matter what
In $8 + 4 \times 3$ , the multiplication comes first: $8 + 12 = 20$ , not $36$ . Multiplication and division are done together, from left to right: $12 \div 3 \times 2 = 8$ .
Thinking that dividing always makes smaller
Dividing by a number less than $1$ gives a bigger result: $5 \div \frac{2}{3} = 5 \times \frac{3}{2} = 7\frac{1}{2}$ . And flip only the divisor, the second number.
Believing that $-7 > -2$ because $7 > 2$
On a number line, $-7$ is to the left of $-2$ , so $-7 < -2$ : $-7$ °F is colder than $-2$ °F. A larger absolute value means farther from $0$ , not greater.
Mixing up the part and the whole in a percent
If $18$ students are $30\%$ of a class, the class is not $30\%$ of $18$ . The whole is $18 \div 0.30 = 60$ students.

Frequently asked questions

What do you learn in 6th grade math?

Ratios, rates and percents, dividing fractions, negative numbers and the coordinate plane, expressions, one-step equations and inequalities, area, surface area and volume, and an introduction to statistics: mean, median, mode and data displays.

Is 6th grade math pre-algebra?

It is where algebra starts. In 6th grade you write and evaluate expressions with variables and solve one-step equations like $x + p = q$ and $px = q$ . 7th and 8th grade math build on this before Algebra 1.

How do you divide fractions?

Multiply the first fraction by the reciprocal of the second. For example, $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$ .

What is a unit rate?

The amount for one unit of the other quantity, such as a price per ounce or miles per hour. A $15$ -ounce box for $4.20 has a unit price of $4.20 \div 15 = 0.28$ , that is $0.28 per ounce.

Is there an app for 6th grade math?

Mathflow covers all 21 units of 6th 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. Next year, 7th grade math builds on ratios, negative numbers and equations. All courses are on the math hub.

6th grade math, unit by unit

How many servings?

Which box is the better buy?

1Dividing Fractions · Divide fractions by fractions

2Unit Rates · Find the better buy

3Equations · Solve one-step equations

Reading 343^434 as 3×43 \times 43×4

Working left to right no matter what

Thinking that dividing always makes smaller

Believing that −7>−2-7 > -2−7>−2 because 7>27 > 27>2

Mixing up the part and the whole in a percent

Math is all about flow.

Reading $3^4$ as $3 \times 4$

Believing that $-7 > -2$ because $7 > 2$