KS3 · Maths

Substituting values into expressions and formulae

Two expressions. The same letter. The same value. They look almost identical, yet one answer comes out ten times bigger than the other. Can you spot which one, and why?

Substitution · Sort it

Same letter, same value — different answers

Each expression is waiting for a value. Pick one, then drop it into the box showing the value it gives. Before you place it, swap the letter for the number in brackets and read what you've got.

Put (6) where each x was, then read the expression as it now stands. Brackets first, then powers, then multiply or divide, then add or subtract.

Still to sort

31 (0)

35 (0)

360 (0)

3,600 (0)

4 of 4 still to sort.

Watch out: That's exactly why the brackets matter. (−5)² is 25, but without the brackets, −5² would only square the 5 and give −25. The brackets keep the minus sign inside the square.

Written method

Problem

Work out 100 − 3(a + 2)² when a = 4.

Your turn to fill the gaps

Several letters, one formula

(a) A cuboid has length 5, width 3 and height 4. Use V = l × w × h to find its volume. (b) Work out 3ab − 2c when a = −5, b = −2 and c = −6.

  1. Part (a). A cuboid's volume is length × width × height. Put the values in: V = 5 × 3 × 4
  2. missing step
Which line is step 2?

Spot the slip

One line is wrong

A student works out a − b when a = −2 and b = −3, and then 50 − 3n² when n = −5. One line goes wrong. Which?

A student's answer — which line goes wrong?

How the value changes when the letter changes

-3-1.250.52.2540481216y (the value you substitute)y² (the value you get)(0.5, 0.25)

y (the value you substitute): 0.5. y² (the value you get): 0.25

Drag me. What's y² at −2? At 2?

Watch out: y² is never negative: squaring a negative gives a positive. Meanwhile 2y goes up by 2 each time y goes up by 1, and y + 2 is always 2 more than y.

Always, sometimes or never?

Is the perimeter always even?

A square has sides of length L, so its perimeter is P = 4L.

Someone says: "P is always even." Which idea is closest to what you think?
How sure are you?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Substitution looks like the easy bit of algebra. Here's where it quietly goes wrong.

What you need to know

  • To substitute, swap the letter for its number and put the number in brackets exactly where the letter was.
  • Then follow the priority of operations: brackets, powers, multiply or divide, add or subtract.
  • A power only acts on what it directly follows. In 10x² only the x is squared. In (10x)² the 10 and the x are squared together.
  • Squaring a negative number gives a positive, and subtracting a negative number adds.
  • A different value of the letter gives a different value of the expression, so you can try values to test a statement.

The big picture

Substitution means swapping a letter for a number. Put the number in brackets exactly where the letter was, then follow the priority of operations: brackets, then powers, then multiplying and dividing, then adding and subtracting. A power only acts on what it directly follows, so 10x² and (10x)² give different values. Negative values need their brackets so the minus sign stays inside, and trying different values shows how an expression changes and whether a statement is always, sometimes or never true.

Key points

1With x = 6, 5x + 1 = 31 but 5(x + 1) = 35.
2With x = 6, 10x² = 360 but (10x)² = 3,600.
3Write substitution down the page, one step per line, with each value in brackets.
4With several letters, give every letter its own value before you calculate.
5To check whether a statement is always true, try different values. A single value that breaks it shows it is not always true.

Worked example

Problem

Find the value of 4p − q² when p = 3 and q = −2.

⚠ Watch out

Treating look-alike expressions as if they were the same: 5x + 1 and 5(x + 1) give different values, and so do 10x² and (10x)². The other classic slip is thinking a negative stays negative when it is squared. (−5)² is 25, not −25.

🧠

Memory hook

Letter out, number in (in brackets). And a power only grabs what it's touching.

✓

Check yourself

Take y = 4. Work out 3y² and (3y)². Which is bigger, and why? Answer: (3y)² = 144 beats 3y² = 48, because the bracket makes the power act on the 3 too.

Flashcards

(13)
What does it mean to substitute?
Swap each letter for the number it stands for. Put the number in brackets, right where the letter was.
Why put a substituted number in brackets?
The brackets show what belongs together before you start calculating, and they keep a negative sign attached to its number.
What is the priority of operations?
Brackets first, then powers, then multiply or divide, then add or subtract.
What does a power act on?
Only what it directly follows. In 10x² the power acts on the x alone. In (10x)² it acts on everything inside the bracket.
When x = 6, what are 10x² and (10x)²?
10x² = 10 × 36 = 360, but (10x)² = 60² = 3,600. One pair of brackets makes it ten times bigger.
When x = 6, what are 5x + 1 and 5(x + 1)?
5x + 1 = 31, because you multiply first. 5(x + 1) = 35, because the bracket makes the adding go first.
What is (−5)²?
25. Squaring a negative number gives a positive: (−5) × (−5) = 25.
What happens when you subtract a negative number?
It adds. For example, 4 − (−3) = 4 + 3 = 7.
How do you substitute several letters at once?
Give every letter its own value, each in brackets, and then calculate using the priority of operations.
How should substitution working be laid out?
Down the page, one step per line, with each value in brackets.
How does the value of an expression change as the letter changes?
Each new value of the letter gives a new value of the expression. For example y² is never negative, 2y goes up by 2 each time y goes up by 1, and y + 2 is always 2 more than y.
How can you test whether a statement is always, sometimes or never true?
Substitute several different values, including awkward ones such as decimals and negatives. One value that breaks the statement shows it is not always true.
Which numbers can be called odd or even?
Only whole numbers (integers). A decimal such as 5.2 is neither odd nor even.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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