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Algebra is the branch of maths where we use letters to stand for unknown numbers. It lets you write general rules and solve problems where you don't know all the values yet. This topic is introduced in Year 6.
A variable is a letter that stands for a number we don't know yet (or that can change).
An expression is a combination of numbers, variables, and operations.
Evaluating an expression: replace the variable with a given value and calculate.
A formula shows the relationship between variables. You substitute (plug in) known values to find unknown ones.
Example: The perimeter of a rectangle is: P = 2l + 2w (where l = length, w = width)
Example: Cost formula: C = 5n + 3 (where n = number of items)
A sequence is an ordered list of numbers that follow a pattern. The pattern is called the rule.
Arithmetic sequences increase or decrease by a fixed amount (the common difference).
Finding missing terms:
The nth term: a formula that gives any term in the sequence.
An equation has an equals sign. Solving it means finding the value of the unknown variable.
Use inverse operations to isolate the variable.
Always check your answer by substituting back:
Some equations need two steps to solve. Undo the operations in reverse order.
Function machine view. For an equation like 3n + 5 = 20, picture the operations applied to n. To solve, run the machine in reverse:
flowchart LR
A[Input n] --> B[x 3]
B --> C[+ 5]
C --> D[Output 20]
D -.reverse.-> E[- 5]
E -.reverse.-> F[/ 3]
F -.reverse.-> G[n = 5]
Example: 3n + 5 = 20
Example: x/2 - 3 = 7
Example: 4(y + 2) = 24
Some equations have two unknown variables. You cannot find a unique answer for each — instead you find all the pairs of values that make the equation true.
Example: Find all whole-number pairs for a + b = 10.
| a | b |
|---|---|
| 0 | 10 |
| 1 | 9 |
| 2 | 8 |
| 3 | 7 |
| 4 | 6 |
| 5 | 5 |
| 6 | 4 |
| 7 | 3 |
| 8 | 2 |
| 9 | 1 |
| 10 | 0 |
There are 11 pairs (if a and b must be 0 or positive whole numbers).
Example: Find all positive whole-number pairs where 2x + y = 12.
There are 6 pairs where x and y are non-negative whole numbers.
Enumerating combinations: When a problem asks you to list all possible combinations of two variables, work systematically — start with the smallest value of one variable, find the matching value of the other, then increase.
Algebra makes word problems easier by turning words into equations.
Problem: I think of a number, multiply it by 3, and then add 8. The answer is 29. What is the number?
Problem: Two numbers add to 50. One is 12 more than the other. Find both numbers.
Three children — Aisha, Ben, and Chloe — share sweets between three bags. Aisha's bag has n sweets. Ben's bag has 4 more than Aisha's. Chloe's bag has double Aisha's. Together they have 38 sweets. How many sweets does each child have?
Step 1 — Write each amount in algebra.
Step 2 — Form the equation by adding all three amounts.
n + (n + 4) + 2n = 38
Step 3 — Simplify the left-hand side. Add the n-terms together: n + n + 2n = 4n. The constant is 4.
4n + 4 = 38
Step 4 — Solve using inverse operations. Subtract 4 from both sides:
4n = 38 - 4 = 34
Then divide both sides by 4:
n = 34 / 4 = 8.5
That gives a fractional answer — which is suspicious because you cannot have half a sweet. So the original problem cannot have integer solutions, OR a number must change.
Step 5 — Adjust the problem. Suppose the total is 40 instead of 38. Now:
4n + 4 = 40
4n = 36
n = **9**
So Aisha has 9 sweets.
Step 6 — Find the other children's totals.
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