Teacher guide: Algebra Foundations

The idea in plain words

An expression is a calculation with a letter in it, such as $2x + 3$. It has a value once you know $x$, but it does not say anything is true.

An equation puts an = sign between two expressions, such as $2x + 3 = 11$. It is a statement: "these two things are equal". The solution is the value of the letter that makes the statement true. Here it is $x = 4$, because $2 \times 4 + 3 = 11$.

Solving means finding that value. The unit teaches three ways, in order:

1. Try and check (substitution): put a value in place of the letter, work out both sides, and see if they match.
2. Backtracking: think of the equation as a machine ($x$, then $\times 2$, then $+ 3$, gives 11) and undo each step, last step first.
3. The balance method: do the same thing to both sides until $x$ is on its own. This one also works when $x$ is on both sides.

Whatever the method, the answer is checked by putting it back into the first equation.

Why it matters

• Equations turn word problems into something you can solve: "4 tickets and a $5 booking fee cost $41" is $4x + 5 = 41$.
• Formulas in science and money are equations with one unknown: speed, area, cost per item.
• Checking by substitution is how you know an answer is right without asking anyone.

An everyday picture, and where it breaks

A balance scale. Two bags with the same unknown number of blocks, plus 3 loose blocks, balance 11 blocks. Take 3 blocks from each side and it still balances: two bags balance 8 blocks. Share both sides into 2 equal groups: one bag balances 4 blocks.

Where it breaks: a scale cannot show taking away more than is there, so equations with subtraction ($7x - 3 = 32$) or negative answers need the written method. That is why the balance lesson moves from the board to lines of algebra, and why subtraction comes in a later lesson.

Try it yourself (5 minutes)

1. Solve $3x + 5 = 20$ by trying values: $x = 4$ gives 17, $x = 5$ gives 20. So $x = 5$.
2. Solve it by backtracking: $20 - 5 = 15$, then $15 \div 3 = 5$.
3. Solve it by balancing: take 5 from both sides ($3x = 15$), divide both sides by 3 ($x = 5$).
4. Check in the first equation: $3 \times 5 + 5 = 20$ ✓.

Three methods, one answer. Students meet them in this order because each one explains the next.

Questions students ask

• "Isn't 2x just 2 and x next to each other?" No: $2x$ means $2 \times x$. With $x = 4$ it is 8, not 24. This is the most common slip in the unit.
• "Which step do I undo first?" The last thing that was done to $x$. In $2x + 3$, the 3 was added last, so take it away first.
• "Why do I have to do it to both sides?" Because the equation says both sides are equal. Change only one side and they are not equal any more.
• "Do I have to check?" Yes. A check in the first equation catches almost every mistake, and it takes ten seconds.

How it connects

The Equations unit builds on writing expressions (the Expressions unit) and leads to graphing linear relations in Year 8, where the solution of an equation is where a line crosses a value. It also links to coding: a function machine is a function in Python, and backtracking is running it in reverse.

Mistakes this unit plants on purpose (so students learn to spot them): reading $2x$ as 24, undoing steps in the wrong order, and changing one side of an equation only.

The idea in plain words

The number outside a bracket multiplies every term inside: $3(2x + 4) = 6x + 12$. With algebra tiles, $3(2x + 4)$ is 3 rows, each of two $x$-tiles and four ones, so 6 $x$-tiles and 12 ones. Signs come with the terms: $4(3x - 2) = 12x - 8$, and a negative outside flips every sign: $-2(x - 5) = -2x + 10$.

After expanding, collect like terms: $6x + 12 - x = 5x + 12$.

Factorising goes backwards: share the terms into equal rows. $6x + 9 = 3(2x + 3)$. Take out the highest common factor, letters included ($2x^2 + 4x = 2x(x + 2)$), and check by expanding.

Changing the subject undoes what was done to a letter, last step first, on both sides: $y = 3x + 5$ gives $x = \dfrac{y - 5}{3}$. Check by substituting numbers.

Why it matters

Expanding is how expressions are simplified, how equations with brackets are solved, and, backwards, how expressions are factorised. It is the distributive law students already use in mental arithmetic: $3 \times 24 = 3 \times 20 + 3 \times 4$.

What students get wrong

• Multiplying only the first term: $4(x + 3) = 4x + 3$. Count the tiles: every row has the ones too.
• Signs: $4(3x - 2) = 12x + 8$, or $-2(x - 5) = -2x - 10$. Each row takes away; a negative number of rows flips every tile.
• A common factor that is not the highest: $12x - 8 = 2(6x - 4)$ is equal but not fully factorised; the bracket still shares a 2.
• Dividing only one term when rearranging: $y = 4x - 3$ becomes $\frac{y}{4} = x - 3$. Undo the $- 3$ first.
• Stopping before collecting: $6x + 12 - x$ is expanded but not simplified. The answer checks tell students which.

The words

expand: write without brackets. term: a part of an expression joined by + or −. like terms: terms with the same letters and powers ($6x$ and $-x$). distributive law: $a(b + c) = ab + ac$. factorise: write as a product, usually with a bracket. highest common factor (HCF): the biggest number (and letters) that divides every term. subject: the letter on its own on one side of a formula.