At a glance. One lesson of about 60 minutes, optional. It is retrieval, so expect some students to need the refresher links. Every question can be done without a calculator; the triangles are chosen so that their sides are whole numbers.
Calculator: no calculator. Nothing here needs one. The square root in question 3 is of a perfect square, and every triangle has whole-number sides, so recalling squares is part of the retrieval.
What students will learn
Know
- The index laws for multiplying, dividing and a power of a power.
- Pythagoras' theorem: the two shorter sides squared add to the longest side squared.
- To solve a linear equation, do the same to both sides.
Understand
- When x² appears on both sides with the same coefficient it cancels, and the equation is linear.
- Year 10 problems often join several Year 9 ideas in one line of working.
- A square root does not split over an addition.
Do
- Use the index laws, including with algebra.
- Find a side with Pythagoras, including when a side is written with x.
- Expand, cancel and solve an equation that looks quadratic but is linear.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: questions 1 to 8 mix the three ideas.
- Criterion B. Investigating patterns: the what-if asks when the triangle has whole-number sides; record k, m and x and look for a rule.
- Criterion C. Communicating: write the theorem first, then each step with its reason, then the three sides as a check.
- Criterion D. Applying mathematics in real-life contexts: the ladder in the hook; ask whether the answer is a believable and safe ladder.
The one thing to do with the figure
Ask for a prediction before sliding: for the first triangle, will the rod fit for a small x or a large x? Then slide until it fits and read the linear equation on the panel.
Then switch triangles and ask students to solve the panel's equation before sliding to check.
A possible order
| 5 min | Warm-up: each item names the unit it comes from. |
|---|---|
| 10 min | Hook and figure: predict, slide, read the panel. |
| 5 min | Key idea and the warning about square roots. |
| 15 min | The three retrieval blocks: read the worked example, do the first-step-given one, compare. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 8. |
The answers
| Warm-up | x² − 8x + 16; 1/100; x = 5. |
|---|---|
| Worked examples | x5; x = 8. |
| First step given | 27; x = 7; x = 24 (sides 24, 10, 26). |
| 1. When x² cancels | C. A, B and D keep an x² term after simplifying. |
| 2. Index laws | 4. 26 ÷ 24 = 2². |
| 3. Pythagoras' theorem | 15. √225. |
| 4. A multi-step equation | 11. 3x − 12 = x + 10. |
| 5. A square that becomes linear | 4. 6x + 9 = 33. |
| 6. Pythagoras with brackets | 84. x = 28: 28 + 21 + 35. |
| 7. Indices in a difference of two squares | 6. x³ × x³ = x6. |
| 8. Expand both sides, then solve | D, x = 7. A is a sign slip; B slips on 1 × (−5); C thinks cancelling means no answer. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Expected every equation with brackets to be linear, or did not expand. | Expand each one and look for x² on both sides. |
| 2 or 32 in question 2 | Added the indices for a power of a power, or squared the index. | Write (2³)² as 2³ × 2³. |
| 21 or 225 in question 3 | Added the sides, or forgot the square root. | Say the theorem in words before using it. |
| 7, −1 or 22 in question 4 | Partial expansion, a sign slip, or stopped early. | The Unit 7 routine: expand, gather, undo, check. |
| 24 or 8 in question 5 | Stopped at 6x, or counted one strip. | Lesson 3's grid for a squared bracket. |
| 28, 140 or 63 in question 6 | Stopped at x, counted one strip, or left out the 21 cm side. | Label all three sides on a sketch before adding. |
| 9 or 3 in question 7 | Multiplied the indices, or kept one power. | x³ × x³ = x × x × x × x × x × x. |
| A, B or C in question 8 | A sign slip, a slip on 1 × (−5), or thinking that cancelling x² leaves no answer. | Check each option in the original equation. |
Questions worth asking
- Why can we take x² from both sides? It is the same amount on both sides, as with any term.
- When would the x² terms not cancel? When their coefficients differ, as in x² + 9 = 4x².
- Is a 13 m ladder with its foot 5 m from the wall believable? Yes: fire service ladders are this long. A common safety guide is 1 m out for every 4 m up, which puts the foot about 3 m out; ask what then changes in the equation.
- Which Year 9 idea did you need most help to bring back? Use the answer to choose a refresher link.
Differentiation
A way in for anyone stuck
- Give a one-page reminder: the three index laws, the theorem with a labelled triangle, the equation routine.
- Let students use the refresher links before the Your turn questions.
- For questions 5 and 6, give the expanded square and ask only for the solving.
Stretch
- The what-if: which k and m give whole-number triangles?
- Solve x² + (x + 7)² = (x + 8)² by trying values. Why does balancing alone not work here?
Students learning in English as an additional language
“Retrieval” means bringing back what you learned before. “Hypotenuse” is the longest side of a right-angled triangle.
| Word | Plain meaning | Example |
|---|---|---|
| hypotenuse | the longest side of a right-angled triangle, opposite the right angle | in a 3, 4, 5 triangle it is 5 |
| index | the small raised number that says how many times to multiply | in 25 the index is 5 |
| linear equation | an equation with x but no x² or higher power | 4x + 1 = 13 |
| cancel | remove the same term from both sides | x² + 7 = x² + x becomes 7 = x |
| retrieval | bringing back something learned earlier | the warm-up questions |
Sentence starters for explaining aloud or in writing:
- By Pythagoras, … = …
- The x² terms cancel, so …
- I check: the sides are …, and … + … = …
Two things not to say
“Square root both sides term by term.” It is the slip in the warning card.
“This is just revision.” Say what it is for: Year 10 joins these ideas together.
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.
- IndicesSimplify x7 ÷ x² × x.
x6. - Earlier workFind the longest side of a right-angled triangle with shorter sides 6 and 8.
10.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- BracketSolve (x + 5)² = x² + 45.
x = 2: 10x + 25 = 45. - From lesson 4Simplify (x + 1)(x + 4) − (x + 2)².
x: x² + 5x + 4 − (x² + 4x + 4).
3Challenge
Reasoning, generalizing and working backwards.
- TriangleShorter sides x and 15, longest side x + 5. Find x.
x = 20: 225 = 10x + 25. Sides 20, 15, 25. - Indices(x4 + 3)(x4 − 3) = xn − 9. Find n.
n = 8.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard and by touch. The rod's fit is also written in words and numbers, so the picture is never the only clue; “Show me the steps” gives the states in words.
No device? Draw the triangle for x = 3, 8 and 12 on squared paper and measure the long side against x + 2.