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A6.1 Stage 6 Algebra

Expressions, equations & the nth term

Use letters for unknowns & variables; write, read & evaluate expressions; substitute into formulae; simplify (collect like terms, expand, factor); solve linear equations; write & solve inequalities on a number line; generate linear sequences & the nth term; dependent vs independent variables; represent with graphs.

Algebra is arithmetic with the numbers left blank. A letter stands in for a number we don’t know yet, or one that is allowed to change — and once we can write those blanks down, we can reason about whole families of numbers at once.

What it means

An expression is a recipe made of numbers, letters and operations, like 3n + 2. It has no equals sign — it just tells you what to do with n. To evaluate it, you substitute a value: if n = 4, then 3n + 2 means 3 × 4 + 2 = 14.

A letter can play two roles. As an unknown it is one fixed but hidden number we want to find. As a variable it is free to take many values, and the expression changes with it.

An equation puts an equals sign between two expressions, e.g. 3n + 2 = 14. The = is a balance: whatever is true on the left is also true on the right. To solve a linear equation you undo the operations, keeping the balance by doing the same thing to both sides. Subtract 2 from each side, then divide each side by 3, and n = 4.

Simplifying makes an expression tidier without changing its value. You collect like terms — 2n + 5n = 7n, because n and n are the same kind of thing (you cannot merge 2n with 5). You expand brackets — 3(n + 2) = 3n + 6 — and you factor back the other way — 3n + 6 = 3(n + 2).

An inequality uses ≤, ≥, > or < instead of =: n ≤ 5 means every value up to and including 5. On a number line you shade that whole range, with a filled dot at 5 (included) or an open dot (not included).

Sometimes an equation links two unknowns at once, like a + b = 10. A single equation like this does not pin down one answer — it has a whole family of solutions. You can enumerate the combinations systematically: (0,10), (1,9), (2,8), …, (10,0). Listing them in order (rather than at random) is what guarantees you find them all and miss none — the same “be systematic” habit that powers combinatorics.

A sequence is a list following a rule: 5, 8, 11, 14, … Here you add 3 each step. The nth term is a single expression that gives any term directly: term n is 3n + 2. Put in n = 1 and get 5; n = 100 and get 302 — no counting needed.

Finally, when one quantity depends on another we call the input the independent variable and the output the dependent variable. Plotting them as points gives a graph — for a linear rule, a straight line.

Worked examples

Evaluate and simplify:

Substitute n = 3 into 4n - 1:   4×3 - 1 = 11
Collect like terms:             2a + 3a + 4 = 5a + 4
Expand:                         2(x + 5)  = 2x + 10

Solve 5n - 3 = 12:

5n - 3 = 12
   +3       +3     →  5n = 15
   ÷5       ÷5     →   n = 3

Find the nth term of 7, 10, 13, 16, …:

term (n)1234
value7101316

The step is 3, so start from 3n (which gives 3, 6, 9, …) and add 4 to line it up: nth term = 3n + 4.

The generative-art connection

An nth-term rule is a machine for placing things, and that is exactly what generative art needs. Feed n = 1, 2, 3, … into a rule and let each output become a position, size or colour. In dot-multiplier (dot-multiplier.nico.art) the number of arms is a variable — change one number and the whole starburst regenerates, so the count you set is the picture. On a circle, drawing a line from point n to point 2n for every n — a pure linear rule — sweeps out a heart-shaped cardioid (see The Coding Train’s times-tables challenge). And on Desmos, typing an expression in n and dragging a slider turns algebra into motion you can watch: the graph of y = 3n + 2 is a straight line whose steepness is the “+3 each step”.

Common misconceptions

  • 3n does not mean “3 and then n”. It means 3 × n — three lots of n. Substituting n = 4 gives 12, not 34.
  • The equals sign is not “here comes the answer”. It means both sides are equal. That is why solving works by doing the same thing to both sides.
  • You cannot collect unlike terms. 2n + 5 stays as 2n + 5; only matching letters combine.

What a learner should be able to solve

The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.

  1. Simplify `4x + 7 + 2x - 3`, then evaluate your simplified expression when `x = 5`.

    Answer

    Collect like terms: `4x + 2x = 6x` and `7 - 3 = 4`, so it simplifies to `6x + 4`. Substituting `x = 5`: `6 × 5 + 4 = 34`.

    Art hook Draw a horizontal number line 0-40. Place a dot at the value of `6x+4` for `x = 0..6` (4, 10, 16, ..., 40). The dots land evenly spaced by 6 — colour each dot by its `x` so the constant `+4` shift shows as the whole row sliding 4 to the right of the pure `6x` row drawn beneath it.

  2. Solve the equation `3n + 5 = 26`. Show the step where you keep the balance by doing the same thing to both sides.

    Answer

    Subtract 5 from both sides: `3n = 21`. Divide both sides by 3: `n = 7`. Check: `3 × 7 + 5 = 26`.

    Art hook A two-pan balance in Canvas: left pan shows 3 identical `n`-blocks plus 5 unit dots, right pan shows 26 dots. Animate removing 5 dots from each side, then splitting each side into 3 equal groups to reveal one block = 7 dots, the pans staying level throughout.

  3. A pattern of dots grows like this: term 1 has `4` dots, term 2 has `7`, term 3 has `10`, term 4 has `13`. Write the nth-term rule, then use it to find term `20`.

    Answer

    It goes up by 3 each time, so start from `3n` and adjust: `3n` gives 3, 6, 9, 12 which is 1 too few each time, so the rule is `3n + 1`. Term 20: `3 × 20 + 1 = 61`.

    Art hook Render each term as a row of dots and stack rows down the canvas. The `+3` growth shows as a straight staircase edge on the right; the `+1` is the constant single-dot overhang on the left. A slider for `n` rebuilds the whole staircase live.

  4. Show the inequality `n > 3` on a number line. Describe exactly what you draw, including whether the dot at 3 is filled or open. Then give the SMALLEST whole number that is a solution, and one more solution.

    Answer

    Draw an open (unfilled) circle at 3, because 3 itself is NOT included, then shade the line to the right for every value bigger than 3. The smallest whole-number solution is `4` (not 3, precisely because the circle is open); another solution is any whole number above it, e.g. `7`.

    Art hook Interactive number line: click a value and the dot lights green if it satisfies `n > 3`, red if not — 3 itself flashes red, driving home the open circle. Drag a toggle to switch between `>` and `≥` and watch the boundary dot at 3 fill in and turn green.

  5. Two numbers add up to 6: `a + b = 6`. List every pair of whole numbers (including 0) that works, in order, so you are sure you have missed none.

    Answer

    Fix `a` from 0 upward and read off `b`: `(0,6), (1,5), (2,4), (3,3), (4,2), (5,1), (6,0)` — 7 pairs. Listing `a` in increasing order guarantees completeness.

    Art hook Plot each pair as a point `(a, b)` on a first-quadrant grid. All 7 points sit on one straight diagonal line — the graph of `a + b = 6`. Connect them to reveal the line, showing that enumerating solutions and drawing the graph are the same picture.

Training exercises

Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.

  1. The expression `5n` means five lots of `n`. What does `5n` equal when `n = 3`? (Careful: it is not 53!)

    Answer

    `5n` means `5 × n`, so `5 × 3 = 15`.

    Art hook Show 5 rows of `n` dots. Drag a slider for `n` and the dot-rectangle grows; a label reads the total `5 × n` live, making `5n = 15` concrete as a 5-by-3 grid of dots.

  2. Substitute `x = 6` into the expression `2x + 3`.

    Answer

    `2 × 6 + 3 = 12 + 3 = 15`.

    Art hook A little 'machine' animation: `x = 6` enters a box labelled '×2 then +3' and 15 comes out. Feed different `x` values and watch outputs stream onto a number line.

  3. Which of these are 'like terms' that can be collected together: `3a`, `5b`, `2a`, `7`? Collect the ones that match and write the tidy expression.

    Answer

    `3a` and `2a` are like terms (both are lots of `a`): `3a + 2a = 5a`. `5b` and `7` cannot join them or each other, so the tidy form is `5a + 5b + 7`.

    Art hook Colour-code terms: all `a`-terms one hue, `b`-terms another, plain numbers a third. Dots of the same hue slide together and merge into a single labelled stack; unlike hues bump apart and refuse to merge.

  4. Expand the brackets: `4(n + 2)`.

    Answer

    Multiply each part inside by 4: `4 × n + 4 × 2 = 4n + 8`.

    Art hook Draw a rectangle of height 4 split into two parts of widths `n` and 2. The two sub-areas `4n` and `8` tile to fill the whole rectangle, showing expansion as area: `4(n+2) = 4n + 8`.

  5. Solve `n - 4 = 9`, then solve `2n = 16`. Check each answer by substituting back.

    Answer

    First: add 4 to both sides, `n = 13` (check `13 - 4 = 9`). Second: divide both sides by 2, `n = 8` (check `2 × 8 = 16`).

    Art hook Balance-scale canvas: each equation is a see-saw. Do the same operation to both pans and watch it stay level until a single `n`-block is isolated on one side.

  6. Spot the mistake. A friend solved `2n + 1 = 9` like this: 'Take away 1 from the left to get `2n = 9`, so `n = 4.5`.' What went wrong, and what is the correct answer?

    Answer

    They only subtracted 1 from the left, breaking the balance. You must subtract 1 from BOTH sides: `2n = 8`, so `n = 4`. Check: `2 × 4 + 1 = 9`.

    Art hook Split-screen balance: the 'wrong' scale tips over because only one pan changed; the 'right' scale stays level. Toggle between them to see tip vs balance.

  7. A sequence starts 6, 8, 10, 12, … What is the rule to get the next term, and what is the 10th term?

    Answer

    Add 2 each time. Continuing: 14, 16, 18, 20, 22, 24 — the 10th term is 24. (Or use the nth term `2n + 4`: `2 × 10 + 4 = 24`.)

    Art hook Place term `n` as a dot at height 'value' along an x-axis of `n`. The dots march up a straight ramp of slope 2; extend the ramp to read term 10 off the line.

  8. Find the nth-term rule for the sequence 5, 9, 13, 17, … and use it to find the 6th term.

    Answer

    It goes up by 4, so start from `4n` (4, 8, 12, 16) and add 1 to match: nth term `4n + 1`. 6th term: `4 × 6 + 1 = 25`.

    Art hook Two overlaid dot-rows: the `4n` ladder and the actual sequence, offset by a constant `+1` gap. The steady gap of 1 dot makes the constant term visible along the whole row.

  9. Show `n ≤ 5` on a number line. Is the dot at 5 filled or open, and which side do you shade? Name the largest whole-number solution.

    Answer

    The dot at 5 is FILLED (because `≤` includes 5), and you shade to the LEFT for all values up to 5. The largest whole-number solution is 5.

    Art hook Number line where clicking toggles `≤` vs `<`: the boundary dot at 5 fills or hollows and the shaded region snaps left. Solutions glow as you sweep the mouse along the line.

  10. In the rule `y = 2x`, `x` is the input (independent) and `y` is the output (dependent). Make a table for `x = 0, 1, 2, 3` and say what happens to `y` as `x` grows.

    Answer

    `x=0→y=0`, `x=1→y=2`, `x=2→y=4`, `x=3→y=6`. As `x` grows by 1, `y` grows by 2 each time.

    Art hook Plot the points (0,0), (1,2), (2,4), (3,6) on a grid; they line up straight. A slider drags `x` and a moving dot climbs the line — the steepness IS the '×2'.

  11. Two whole numbers multiply to give 12: `a × b = 12`. List every pair of whole numbers (with `a` and `b` at least 1) in order, starting from `a = 1`.

    Answer

    `(1,12), (2,6), (3,4), (4,3), (6,2), (12,1)` — the factor pairs of 12. Working `a` upward in order (skipping 5, 7, 8, 9, 10, 11 which don't divide 12) ensures none are missed.

    Art hook For each pair draw a rectangle `a` wide by `b` tall made of unit squares. Step through the pairs and the rectangle reshapes — tall-and-thin to short-and-wide — but always holds exactly 12 squares, showing `a × b = 12` as a constant area.

  12. Challenge: two numbers add to 10, `a + b = 10`, with both whole numbers from 0 to 10. List all the pairs, then plot them as points `(a, b)`. What shape do the points make?

    Answer

    Pairs: `(0,10),(1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1),(10,0)` — 11 pairs. Plotted, they fall on a straight diagonal line (the graph of `a + b = 10`).

    Art hook The keystone hook: enumerate then plot `(a, b)` on a first-quadrant grid — 11 dots on a perfect descending diagonal. Animate connecting them into the line `a + b = 10`, showing that listing solutions and drawing the graph are two views of one idea.

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