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← High school

Learn Pre-Algebra unit by unit

Build number sense and the habits needed for algebra.

Start Pre-Algebra practice →See worked Pre-Algebra questionsAsk the tutor
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5Core units
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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 10-week schedule.

Pre-Algebra course map: 5 units
#UnitWhat you can do after itWeeks
01Integers and rational numbersYou can add, subtract, multiply and divide negative numbers, fractions and decimals without a calculator.Weeks 1-2
02Ratios and proportionsYou can compare two amounts as a ratio and scale a recipe, a map or a price up or down.Weeks 3-4
03Expressions and variablesYou can turn a word phrase into an expression with letters, put numbers in for the letters, and collect like terms.Weeks 5-6
04Linear relationshipsYou can read a straight-line graph, measure its slope, and write the rule that links x and y.Weeks 7-8
05Geometry foundationsYou can find perimeter, area and angle sizes for basic shapes and name the angle pairs that must be equal.Weeks 9-10

Pre-Algebra course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

01

Integers and rational numbers

Weeks 1-2

You can add, subtract, multiply and divide negative numbers, fractions and decimals without a calculator. You can also place any of those numbers in order on a number line.

  • Add and subtract negatives, such as -8 + 3, and say which side of zero the answer lands on.
  • Multiply and divide fractions, including rewriting 2 1/2 as 5/2 first.
  • Put a mixed list like -0.75, -2/3 and 0.7 in order from smallest to largest.

Worked example

Work out -8 + 3 x (-2)

  1. Multiply first: 3 x (-2) = -6
  2. Now add: -8 + (-6)

Answer-14

Most common mistakeAdding before multiplying: doing -8 + 3 = -5 first, then -5 x (-2) = 10 instead of -14.

Explain integers and rational numbersPractice integers and rational numbers
02

Ratios and proportions

Weeks 3-4

You can compare two amounts as a ratio and scale a recipe, a map or a price up or down. You can also work out a unit rate to see which package is the better deal.

  • Write a ratio such as 12 boys to 18 girls in lowest terms as 2 to 3.
  • Solve a proportion like 3/4 = x/20 to find a missing amount.
  • Find the price of one ounce and use it to compare two package sizes.

Worked example

A map uses 2 cm for every 15 km. Two towns are 7 cm apart on the map. How far apart are they really?

  1. Set up the proportion: 2/15 = 7/d
  2. Cross multiply: 2d = 105

Answerd = 52.5 km

Most common mistakeFlipping one side of the proportion: writing 2/15 = d/7 gives d = 0.93 km, which would put two towns less than a metre apart.

Explain ratios and proportionsPractice ratios and proportions
03

Expressions and variables

Weeks 5-6

You can turn a word phrase into an expression with letters, put numbers in for the letters, and collect like terms. You can also tell an expression apart from an equation.

  • Write 'seven less than triple a number' as 3n - 7.
  • Evaluate 4a - b when a = 5 and b = -2.
  • Collect like terms so 6x + 3 - 2x + 9 becomes 4x + 12.

Worked example

Simplify 5(2x + 3) - 4x

  1. Distribute the 5: 10x + 15 - 4x
  2. Collect the x terms: 10x - 4x = 6x

Answer6x + 15

Most common mistakeAdding terms that are not alike: turning 6x + 15 into 21x, as if a number of x's and a plain number could be added.

Explain expressions and variablesPractice expressions and variables
04

Linear relationships

Weeks 7-8

You can read a straight-line graph, measure its slope, and write the rule that links x and y. You can also predict a value the table does not show.

  • Find the slope between two points such as (1, 5) and (4, 14).
  • Write y = mx + b straight from a table of values.
  • Use the rule to predict the cost of 20 items when the table stops at 6.

Worked example

Find the slope of the line through (1, 5) and (4, 14)

  1. Rise: 14 - 5 = 9
  2. Run: 4 - 1 = 3

Answerslope = 9/3 = 3

Most common mistakeDividing the run by the rise: reporting 3/9 = 1/3, a gentle slope, when the line actually climbs 3 units for every 1 across.

Explain linear relationshipsPractice linear relationships
05

Geometry foundations

Weeks 9-10

You can find perimeter, area and angle sizes for basic shapes and name the angle pairs that must be equal. You can also work backwards from an area to a missing side.

  • Find the area of a triangle with base 10 cm and height 7 cm.
  • Use the fact that angles on a straight line add to 180 degrees to fill in a missing angle.
  • Find a rectangle's width when its area is 96 and its length is 12.

Worked example

A triangle has base 10 cm and height 7 cm. Find its area.

  1. Area = (1/2) x base x height
  2. Area = (1/2) x 10 x 7

Answer35 square cm

Most common mistakeLeaving out the one-half: reporting 10 x 7 = 70 square cm, which is the whole rectangle rather than the triangle inside it.

Explain geometry foundationsPractice geometry foundations

Prepare for Pre-Algebra practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Integers and rational numbers from a blank page. Then connect it to Ratios and proportions. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Arithmetic calculator can test calculations and representations used in Pre-Algebra. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Pre-Algebra practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesArithmetic explainedConcepts, worked methods and checks behind this course.
PracticeMixed Arithmetic questionsAnswer review after every attempt.
ReferenceFormulas for this courseConditions and a worked example for each one.
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Assess

Take the complete Pre-Algebra test

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Reference

Review essential formulas

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Calculate

Use the Arithmetic calculator

Open tool →
Plan

Prepare around your exam date

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Questions about the Pre-Algebra course

Where should I start?

Start with Integers and rational numbers if you are following the full sequence. If that unit feels automatic, open the Pre-Algebra practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Pre-Algebra test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Arithmetic calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.