At a glance. One lesson of about 60 minutes. It assumes solving multi-step equations from Year 9 and squares and roots. Students make a letter the subject by undoing steps in reverse order, with brackets, fractions, squares and roots, and substitute afterwards.
Calculator: no calculator. The substituted values are chosen so that every step is a mental calculation, and the rearrangement is the skill.
What students will learn
Know
- The subject is the letter on its own on one side.
- Undo operations in the reverse order to how they were applied, using inverses.
- √ means the non-negative root; undoing a square gives ±√, and the context decides whether the negative value is kept.
Understand
- A formula is an equation, so the same balance method applies.
- The order of operations tells which step to undo first.
- A rearranged formula can be checked by substituting numbers.
Do
- Rearrange formulas with one, two and three steps.
- Handle brackets, fractions, squares and roots.
- Substitute into the rearranged formula and check.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: rearranging and substituting (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how the backwards row relates to the forwards row and state the rule.
- Criterion C. Communicating: each step written on its own line with the operation noted.
- Criterion D. Applying mathematics in real-life contexts: temperature, speed and energy formulas (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then move x and ask what the bottom row always ends with. Change a and b and ask which box of the bottom row depends on b and which on a.
Ask: “Why does the bottom row use the operations in the opposite order?” Because the last thing done is the first thing to undo, like taking off a coat and then a jumper that were put on in the opposite order.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the oven temperature; students write their plan. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask what the formula already does to x. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | x = 5; the square root, 7; x = 14. |
|---|---|
| Hook | C = (F − 32) × 5⁄9, so 392 F is (392 − 32) × 5⁄9 = 200 degrees Celsius. |
| Worked examples | x = (y + 7) ÷ 4; r = C ÷ 2π; u = √(v² − 2as). |
| Faded example | t = √(2s ÷ g). |
| 1. Making x the subject | C. A adds 8 again; B undoes the steps in the original order; D divides again. |
| 2. Using a rearranged formula | 12. l = P ÷ 2 − w. |
| 3. Making a letter the subject and substituting | 6. (30 − 6) ÷ 4. |
| 4. A fraction in a formula | 10. b = 2A ÷ h − a. |
| 5. Square root as inverse | 6. v = √(2E ÷ m). |
| 6. Square root after subtracting | 15. b = √(289 − 64). |
| 7. A fraction multiplier | 25. C = (F − 32) × 5⁄9. |
| 8. Rearranging with a square | A. B squares π; C roots only A; D belongs to circumference. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Added 8 again, undid the steps in the original order, or divided by 5 again. | Substitute x = 2 (so y = 2) into each option. |
| 24 or 26 in question 2 | Took away both widths but did not halve, or added w. | Write l + w = 19 as a line before the last step. |
| 24 or 9 in question 3 | Forgot to divide by t, or added u. | Write each undo on its own line. |
| 15 or 2.5 in question 4 | Stopped before taking away a, or forgot to double. | Ask: what is 15? (a + b) |
| 36 or 3 in question 5 | Left it squared, or halved a second time. | Ask: what is the inverse of multiplying by a half? |
| 9 or 225 in question 6 | Subtracted sides, or left it squared. | Draw the triangle and label b. |
| 81 or 45 in question 7 | Multiplied by the same fraction, or stopped after subtracting. | Say “the inverse of nine fifths is five ninths” aloud. |
| B, C or D in question 8 | Squared the pi, rooted only A, or used the circumference formula. | Test r = 2 in each option. |
Questions worth asking
- Why do you undo the last step first? Listen for: it was applied last, so it is on the outside.
- What does it mean for a letter to be the subject? Listen for: alone on one side.
- How can you check your rearranged formula? Listen for: substitute numbers into both versions.
- When do you keep both square roots? Listen for: when the context allows a negative value.
Differentiation
A way in for anyone stuck
- Give a flow diagram for each formula before rearranging, with the operations on arrows.
- Provide the inverse pairs on a card, including square and square root.
- Give the frame: “The formula does … then …. To undo it I … then …”
Stretch
- Make b the subject of A = ½ab sin C where C is given, and use it to find b when A = 15, a = 6 and sin C = 0.5.
- Rearrange the compound interest formula A = P(1 + r)2 to make r the subject.
Students learning in English as an additional language
“Subject” here means the letter on its own on one side. “Rearrange” means to change the form without changing the meaning. “Inverse” means the operation that undoes another. “Substitute” means to replace a letter with a number. “Square root” undoes squaring.
| Word | Plain meaning | Example |
|---|---|---|
| subject | the letter on its own on one side of a formula | c in c = 4d + 1 |
| formula | a rule written with letters that links quantities | area = length × width |
| inverse operation | the operation that undoes another | ÷ undoes × |
| rearrange | to change a formula into another form with the same meaning | from y = x + 5 to x = y − 5 |
| substitute | to replace a letter by a number | put m = 3 into 2m + 1 |
Sentence starters for explaining aloud or in writing:
- The formula does … first, so I undo … last.
- I do the same to both sides: …
- I checked my answer by substituting …
Two things not to say
“Move it to the other side and change the sign.” Say instead: do the same to both sides.
“Square root both sides” before the square is alone. Say instead: isolate the square, then take the root.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- RearrangeMake x the subject of y = x + 9.
x = y − 9. - RearrangeMake n the subject of m = 6n.
n = m ÷ 6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA taxi in Accra charges a fixed 11 plus 3 for each kilometre: C = 3k + 11. How many kilometres are travelled when C = 35?
k = (35 − 11) ÷ 3 = 8. - Earlier work (lesson 5)Write the multiplier for a 12% fall.
0.88.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningRearrange the formula y = 2(x + 3)² to make x the subject, with x + 3 positive.
x = √(y ÷ 2) − 3. - Working backwardsS = 4πr². A sphere has S = 36π. Find r.
r² = 9 so r = 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Strips of paper with the operations written on them can be ordered and then reversed.