Algebra Foundations

In a formula, letters stand for quantities that can change (variables) and numbers stay the same (constants). In an equation, a letter can stand for one unknown number.

1. What changes?
2. Change the rectangle
3. Variables, constants and unknowns › Reading a formula
4. Variables, constants and unknowns › Find the mistake
5. Variables, constants and unknowns › Unknown or variable?
6. Other formulas › A plumber
7. Other formulas › A car trip
8. Other formulas › Area
9. Other formulas › A letter that never changes
10. Practise: read the formula › What the constant means
11. Practise: read the formula › Variables
12. Practise: read the formula › Unknown
13. A formula of your own

To use a formula, replace each letter with its value, write in the × signs, then work it out in the right order.

1. A perimeter
2. Check your value
3. Three steps › Area
4. Three steps › A plumber
5. Three steps › Find the mistake
6. Other formulas › A decimal
7. Other formulas › Three letters
8. Other formulas › Divide
9. Other formulas › Brackets
10. Practise: substitute › Tickets
11. Practise: substitute › Brackets again
12. Practise: substitute › A triangle
13. Practise: substitute › Review
14. Rectangles with the same area

When a formula's answer is known but one letter is not, substitute what you know. That leaves an equation; solve it by undoing, then check.

1. How wide?
2. Find the width
3. Substitute, then solve › Area
4. Substitute, then solve › A plumber
5. Substitute, then solve › Find the mistake
6. Other unknowns › A car trip
7. Other unknowns › A different unknown
8. Other unknowns › Going down
9. Other unknowns › A box
10. Practise: find the unknown › Tickets
11. Practise: find the unknown › Substitute or solve?
12. Practise: find the unknown › A bike ride
13. Practise: find the unknown › Review
14. A garden bed

Words such as "more than", "less than", "twice" and "divided by" each mean an operation. Write the operations in the order the words say them, then check with a number.

1. 5 less than n
2. What do the words give?
3. From words to an expression › Three more than twice n
4. From words to an expression › Find the mistake
5. From words to an expression › Your turn
6. Other words › Half
7. Other words › More than
8. Other words › Less than
9. Other words › Two letters
10. Practise: say it, then write it › Older
11. Practise: say it, then write it › Eggs
12. Practise: say it, then write it › Pizza
13. Practise: say it, then write it › Review
14. Find the rule

Brackets keep a part together so it is worked out first. If a sentence says to do something to a whole result, such as "add 3, then double it", the expression needs brackets.

1. Stickers
2. Check the words
3. Where do the brackets go? › Add 3, then double
4. Where do the brackets go? › Find the mistake
5. Where do the brackets go? › Your turn
6. New sentences › Subtract, then multiply
7. New sentences › No brackets
8. New sentences › Two letters
9. New sentences › Halving
10. Practise: brackets or not? › Books
11. Practise: brackets or not? › A group
12. Practise: brackets or not? › Which is bigger?
13. Practise: brackets or not? › Review
14. Find the rule

Say what the letter stands for, write an expression for the situation, then test it with a number you can check. One expression then gives the answer for every case.

1. What should the letter be?
2. One expression, many trips
3. From a situation to an expression › A hall
4. From a situation to an expression › Find the mistake
5. From a situation to an expression › Your turn
6. New situations › A taxi
7. New situations › A tank
8. New situations › Two groups
9. New situations › A big case
10. Practise: build and use › A plumber
11. Practise: build and use › Using it
12. Practise: build and use › The letter
13. Practise: build and use › Review one
14. Practise: build and use › Review two
15. A situation of your own

An equation says two things are equal. Find the value that makes it true, then check it.

1. Which value balances?
2. Find the balance
3. Check a solution › Check by substitution
4. Check a solution › Find the mistake
5. Check a solution › Expression or equation?
6. New scales › Four bags
7. New scales › $x$ on both sides
8. New scales › A scale that will not balance
9. Practise: check a value › Work it out
10. Practise: check a value › Which value?
11. Practise: check a value › Both sides
12. Your own equation

Undo what was done to x, last step first, to find the number you started with.

1. What number went in?
2. Run the machine backwards
3. Undo in reverse › Opposites
4. Undo in reverse › Backtrack it
5. Undo in reverse › Find the mistake
6. New machines › One step
7. New machines › Taking away
8. New machines › Dividing
9. New machines › Adding first
10. Practise: backtrack these › Two steps
11. Practise: backtrack these › With a fraction
12. Practise: backtrack these › Taking away
13. Your own machine

Do the same to both sides to get x on its own, then check your answer in the first equation.

1. One move on both sides
2. Keep it balanced
3. Write it as algebra › In order
4. Write it as algebra › Find the mistake
5. Write it as algebra › Which step first?
6. Harder scales › Bigger numbers
7. Harder scales › Bags on both sides
8. Harder scales › Bags and blocks on both sides
9. Practise: solve and check › Two steps
10. Practise: solve and check › Bags on both sides
11. Practise: solve and check › Bags and blocks on both sides
12. Your own balance

The number outside a bracket multiplies every term inside it. 3(2x + 4) is 3 rows of 2x + 4, so 6x + 12. Then collect any like terms.

1. 3(x + 2)
2. Build it with tiles
3. Multiply every term › Expand, then collect
4. Multiply every term › Find the mistake
5. Multiply every term › Your working
6. One change at a time › Bigger numbers
7. One change at a time › Subtraction inside
8. One change at a time › Quick check
9. One change at a time › A negative outside
10. One change at a time › Collect afterwards
11. One change at a time › Two brackets
12. Practise: expand and simplify › Positive
13. Practise: expand and simplify › Negative outside
14. Practise: expand and simplify › Full working
15. Practise: expand and simplify › Review
16. Brackets backwards

Factorising is expanding backwards. Share the terms into equal rows: 6x + 9 is 3 rows of 2x + 3, so 3(2x + 3). Take out the highest common factor, then check by expanding.

1. Fully factorised
2. Share into rows
3. The highest common factor › Take out the HCF
4. The highest common factor › Find the mistake
5. The highest common factor › Check it
6. One change at a time › On the board
7. One change at a time › A minus sign
8. One change at a time › Three terms
9. One change at a time › A letter in common
10. One change at a time › Nothing to take out
11. Practise: factorise and check › Positive
12. Practise: factorise and check › A minus sign
13. Practise: factorise and check › Full working
14. Practise: factorise and check › Review: expanding
15. Four rows

A formula gives one letter from the others. To get a different letter, undo the steps in reverse order, doing the same to both sides: y = 3x + 5 becomes x = (y − 5)/3. Check by substituting.

1. Which formula gives w?
2. Check with numbers
3. Undo in reverse order › Undo the last step first
4. Undo in reverse order › Find the mistake
5. Undo in reverse order › Your working
6. One change at a time › Check on the board
7. One change at a time › One step
8. One change at a time › A minus sign
9. One change at a time › Undo a divide
10. One change at a time › A fraction
11. One change at a time › The subject is subtracted
12. Practise: rearrange and check › A formula in context
13. Practise: rearrange and check › Full working
14. Practise: rearrange and check › Use it
15. Practise: rearrange and check › Review: factorising
16. Work it backwards

Get the x terms on one side and the numbers on the other, doing the same to both sides; expand any brackets first. Then check by substituting into the first equation.

1. When do they cost the same?
2. Where both sides are equal
3. Same to both sides › Solve, then check
4. Same to both sides › Find the mistake
5. Same to both sides › Check by substituting
6. One change at a time › A bracket
7. One change at a time › A negative answer
8. One change at a time › A fraction
9. One change at a time › Brackets on both sides
10. One change at a time › No solution
11. Practise: solve and check › x on both sides
12. Practise: solve and check › Negative x
13. Practise: solve and check › Angles
14. Practise: solve and check › Review: expanding
15. Build an equation

Solve an inequality like an equation, doing the same to both sides, but when you multiply or divide by a negative, reverse the inequality sign. Show the answer on a number line: open circle for < and >, closed for ≤ and ≥. Check with a value.

1. A negative in front
2. When is −2x more than 6?
3. Solve, show, check › Solve, show, check
4. Solve, show, check › Find the mistake
5. Solve, show, check › Divide by a negative
6. One change at a time › A closed circle
7. One change at a time › A negative x
8. One change at a time › A fraction
9. One change at a time › A range
10. One change at a time › Why it reverses
11. Practise: inequalities › Positive coefficient
12. Practise: inequalities › Negative coefficient
13. Practise: inequalities › Read the line
14. Practise: inequalities › Review: equations
15. Write one with a negative