At a glance. One lesson of about 60 minutes. It assumes the index laws and lesson 3. Students match bases, isolate the power before reading it, write non-matching solutions as logarithms and bracket them, and meet a quadratic in a power.
Calculator: no calculator. No calculator: the solutions are exact. Decimal values for log answers come in lesson 5, once the change of base law is available.
What students will learn
Know
- If bm = bn then m = n, for b above 0 and not 1.
- bx = N has solution x = logb N.
- A power is never zero or negative, so bx = negative has no solution.
Understand
- Why the equation has a solution even when N is not a power of b.
- Why the power must be isolated before it is read.
- Why a root of u2 − 4u − 45 = 0 that is negative must be rejected.
Do
- Solve by matching bases, including brackets in the power.
- Write an exact logarithm answer and bracket it.
- Solve a quadratic in bx.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: matching bases and isolating the power (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say how the crossing moves as the target or the base changes.
- Criterion C. Communicating: writing an exact answer and the whole-number bracket.
- Criterion D. Applying mathematics in real-life contexts: an epidemic and a medicine decaying (hook and question 7).
The one thing to do with the figure
Ask for the prediction first. Then keep the base at 2 and move the target from 8 to 32, and ask where the crossing is and why it is a whole number only for some targets.
Ask: “What happens to the crossing as the base increases but the target stays the same?” It moves left, because a bigger base reaches the target in fewer steps.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the district hospital; collect predictions about the week. |
| 12 min | Figure, key idea and warning card; solve one matching-base equation and one logarithm equation together. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask them to test x = 60 in 5x. |
| 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 | 3; 24; 23x. |
|---|---|
| Hook | 4 × 3n first passes 500 when 3n passes 125. 34 = 81 is short and 35 = 243 passes, so the cases pass 500 in week 5 (4 × 243 = 972), and the crossing is at about 4.4 weeks. |
| Worked examples | 5/3; 6; 3; between 1 and 2. |
| Faded example | log6 100, between 2 and 3. |
| 1. Writing an exact solution | B. A divides; C swaps base and number; D miscounts the powers. |
| 2. Matching bases | 1.5. 2x = 3. |
| 3. Isolating the power | 5. 2x = 32. |
| 4. Matching bases with different forms | 5. 3x − 3 = 2x + 2. |
| 5. Bracketing a solution | 4. 81 < 100 < 243. |
| 6. Quadratic in a power | 2. u = 9, so x = 2. |
| 7. Decay to a target | 18. t ÷ 3 = 6. |
| 8. Matching bases | D. A ignores the + 1; B equates exponents before matching bases; C equates 2x with 8. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Divided, swapped base and number, or miscounted the powers. | Ask them to test x = 60 in 5x against 300, then to label 5, 25, 125, 625 with their powers. |
| 3 or 6 in question 2 | Divided 27 by 9, or doubled instead of halving. | Write both sides as powers of 3 and compare the exponents. |
| 7 or 32 in question 3 | Ignored the 4, or gave the value of the power. | Ask: what is 2x on its own, and what is x? |
| 3 or −1 in question 4 | Did not multiply the whole bracket, or made a sign slip. | Expand 3(x − 1) and 2(x + 1) in a line each before equating. |
| 5 or 33 in question 5 | Gave the larger whole number, or divided 100 by 3. | Ask: which powers of 3 sit either side of 100? |
| 9 or 4.5 in question 6 | Gave u instead of x, or divided by 2. | Ask: what does u stand for, and what is x in 3x = 9? |
| 6 or 192 in question 7 | Gave t ÷ 3, or multiplied by 64. | Check: t = 18 gives 0.56 = 1 ÷ 64 and 640 ÷ 64 = 10. |
| A, B or C in question 8 | Ignored part of the exponent, equated exponents with different bases, or set the exponent equal to the number. | Write both sides of each option as powers of one base, then substitute x = 4. |
Questions worth asking
- Why is the solution of 2x = 20 not a whole number? Listen for: 20 is between the powers 16 and 32.
- Why can we equate the exponents when the bases match? Listen for: each power of the base gives a different number, so equal numbers mean equal exponents.
- Why is 3x = −5 impossible? Listen for: a positive base to any power is positive.
- Is x = log2 20 an answer or a calculation? Listen for: an exact answer that can be approximated later.
Differentiation
A way in for anyone stuck
- Give a list of powers of 2, 3 and 5 to 250 to bracket answers.
- Provide a table with the columns left side, right side as one base, exponents equal, solution.
- Give the frame: “Both sides as powers of …: … = …, so the exponents are equal: ….”
Stretch
- Solve 2x = 3 × 2x − 2 + 1 and say how many solutions it has.
- Show that 4x − 6 × 2x + 8 = 0 has two solutions and find them.
Students learning in English as an additional language
“Exponent” is another word for power. “Isolate” means to get one thing alone on one side. “Match the bases” means to write both sides with the same base number. “Bracket” here means to find two whole numbers that the answer lies between.
| Word | Plain meaning | Example |
|---|---|---|
| exponent | the power a number is raised to | the 4 in 54 |
| isolate | to get a term alone on one side of an equation | divide by 6 to get 3x alone |
| match the bases | to write both sides as powers of the same number | 25 as 52 |
| exact value | an answer left as a logarithm or root, not rounded | log base 7 of 30 |
| reject | to throw out a root that does not fit the problem | a negative value for 2x |
Sentence starters for explaining aloud or in writing:
- Both sides can be written with base …
- The power is isolated when I …
- The solution lies between … and … because …
Two things not to say
“Take logs of both sides” without saying why. Say instead: the logarithm is the power, so it is the unknown.
“Cancel the bases.” Say instead: equal bases give equal exponents, so the exponents are equated.
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.
- FindSolve 2x = 64.
6. - FindBetween which whole numbers does the solution of 5x = 70 lie?
Between 2 and 3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA town of 4000 people grows by a factor of 2 every decade. Write an equation for when it reaches 64 000 and solve it.
4000 × 2t = 64 000, so 2t = 16 and t = 4 decades. - Earlier work (Year 10)Solve 3x = 81.
4.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve 25x = 125x − 2.
52x = 53x − 6, so x = 6. - Working backwardsAn equation 2x = k has a solution between 3 and 4. Give one whole number value of k.
Any of 9 to 15, for example 12.
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.