At a glance. One lesson of about 60 minutes. It assumes rearranging from Standard Unit 3 and the sum formulae from lessons 1 and 4. Students rearrange the arithmetic and geometric sum formulae and formulae with squares, and find the multiplier of an output from the multipliers of its inputs, including joint variation.
Calculator: no calculator. No calculator: numbers are chosen to be exact, and the learning is in the structure of the formula.
What students will learn
Know
- Undo operations in reverse order; a root frees a squared letter.
- If y varies as xp, x × k gives y × kp.
- With several inputs, multiply the factors; percentages are converted to factors first.
Understand
- Why the order of undoing matters.
- Why scaling uses the power and not the factor itself.
- Why percentage changes multiply and do not add.
Do
- Rearrange the sum formulae and evaluate.
- Find a constant and use a variation relationship.
- Combine multipliers from several inputs, and convert to a percentage.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: rearranging and variation (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe what the power does to the multiplier.
- Criterion C. Communicating: writing the substitution step for scaling.
- Criterion D. Applying mathematics in real-life contexts: light, a cyclist on a curve and cylinders (hook, questions 5 and 6).
The one thing to do with the figure
Ask for the prediction first. Then set the power to −2 and the factor to 2, and ask what the figure shows. Move the power to 1 and 2 and ask what stays the same.
Ask: “Which has the larger effect when x is multiplied by 3, the power 2 or the power 3?” Then give 9 against 27.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the lamp; collect predictions. |
| 12 min | Figure, key idea and warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose D, ask what (5x) squared is. |
| 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 = (y + 7) ÷ 3; 30; 2. |
|---|---|
| Hook | Light on the page varies inversely as the distance squared, so doubling the distance divides it by 4: a quarter as much. |
| Worked examples | 5; 1; 3/4. |
| Faded example | 4.5. |
| 1. Inverse variation with a power | C. A is direct in x; B is direct in x squared; D forgets the square. |
| 2. Rearranging with a square | 18. 144 ÷ 8. |
| 3. Using a rearranged sum formula | 3. 27 ÷ 9. |
| 4. Using a root relationship | 20. k = 4. |
| 5. Joint variation | 0.5. 4 ÷ 8. |
| 6. Percentage change from variation | 80. 2.25 × 0.8 = 1.8. |
| 7. Rearranging then substituting | 6. (52 − 40) ÷ 2. |
| 8. Scaling backwards | A. B ignores the cube; C moves z the wrong way; D multiplies by the power. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Treated the variation as direct, direct in the square, or inverse in x only. | Write the new value of x squared in the denominator. |
| 1 or 20.5 in question 2 | Skipped the squares, or squared only v. | Write the formula with squares before substituting. |
| 2.7 or 27 in question 3 | Divided by n, or stopped at the top. | Ask: how many differences are there in 10 terms? |
| 33.33 or 4 in question 4 | Treated it as direct proportion, or gave k. | Find k, then substitute the new L. |
| 16 or 4 in question 5 | Multiplied the factors without squaring or dividing, or ignored the radius. | Write the formula as a fraction and change each part. |
| 30 or 20 in question 6 | Added the percentages, or did not square the radius. | Convert each percentage to a factor, and square the radius factor. |
| 12 or 4 in question 7 | Stopped at 2a, or used n differences instead of n − 1. | Check the answer: does 11 ÷ 2 × (2a + 40) give 286? |
| B, C or D in question 8 | Ignored the power, moved z the wrong way, or multiplied the factor by the power. | Write y with x ÷ 2 in place of x and ask what z must become. |
Questions worth asking
- Why does doubling a length multiply an area by 4? Listen for: the length appears twice in an area.
- How do you undo a square when rearranging? Listen for: take the square root of both sides.
- How do you combine a 20% rise and a 10% fall? Listen for: 1.2 × 0.9, a rise of 8%.
- What power gives the same result whatever x is? Listen for: power 0, constant.
Differentiation
A way in for anyone stuck
- Give a table with a row for each input: the factor, its power and the effect on the output.
- Provide a rearranging checklist: write each operation, then undo in reverse order.
- Give the frame: “x is multiplied by …, and the power is …, so y is multiplied by ….”
Stretch
- Make u the subject of v2 = u2 + 2as, and explain why u could be positive or negative.
- T = 2π√(l ÷ g). Make g the subject and say how T changes when l is quadrupled.
Students learning in English as an additional language
“Subject” is the letter on its own on one side of the formula. “Undo” means do the inverse operation. “Factor” here means the number a quantity is multiplied by. “Varies” means changes with.
| Word | Plain meaning | Example |
|---|---|---|
| subject | the letter alone on one side of a formula | the s in s = vt |
| inverse operation | the operation that undoes another | root undoes square |
| varies as | changes with, in a fixed way | y varies as x |
| scale factor | the number by which a quantity is multiplied | ×3 |
| constant | the number that stays the same in a relationship | k |
Sentence starters for explaining aloud or in writing:
- To get … on its own I first undo ….
- x is multiplied by …, so y is multiplied by …
- I converted the percentages to factors first.
Two things not to say
“Whatever you do to x, y does the same.” Say instead: look at the power of x.
“Add the percentages.” Say instead: multiply the factors.
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 = 5x + 2.
x = (y − 2) ÷ 5. - Scaley varies as x2. x is doubled. What happens to y?
Multiplied by 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe area of a screen in Seoul varies as its width squared. The width goes up by 10%. By what percentage does the area go up?
1.12 = 1.21, so 21%. - Earlier work (lesson 5)Find the sum to infinity of 30 + 6 + 1.2 + …
37.5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy does doubling the side of a square not double its area?
The area varies as the side squared, so it is multiplied by 4. - Working backwardsy varies as x3 and y is multiplied by 27. By what was x multiplied?
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 calculator is needed.