Why students read exponents as quantities, and the two rules that decide whether an answer is finished.
Ask how many times bigger 10⁶ is than 10³ before you slide anything.
A good number will say twice, or three times. It is a thousand, and the strip shows the two sitting only three short steps apart while the readout says 1000.
Steps add; sizes multiply. That sentence is the whole sub-topic, and the strip is the only way I have found to make it land in under a minute.
1 ≤ a < 10, and k an integer. Students check the first and forget that 52 × 10⁶ fails it, because the value is right and it looks like standard form.
5.2E7 scores nothing. It is calculator output, the guide says plainly it is not acceptable, and students copy it off the screen under time pressure. Say it now and say it again before the mock.
| 1. a in 52 × 10⁶ | 5.2, since 52 × 10⁶ = 5.2 × 10⁷. |
| 2. 10⁶ against 10³ | 1000 times. |
| 3. 3 × 10⁵ + 2 × 10⁴ | B, 3.2 × 10⁵. Match the powers, then add the fronts. |
1 markA final answer in proper standard form, with 1 ≤ a < 10.
1 markRewriting calculator notation into mathematics.
1 markMatching powers before adding or subtracting.
| They give | What it means |
|---|---|
| 2 (Q2) | Read 6 against 3 as a ratio of the exponents. This is the misconception the strip exists for; send them back rather than telling them. |
| 3 (Q2) | Gave the difference in the powers. One step from the answer: the ratio is 10 to that difference. |
| 5 × 10⁹ (Q3) | Added the powers, which is the multiplying rule. Worth naming the rule they used, because they have not invented it, they have misapplied it. |
| 5 × 10⁵ (Q3) | Added the fronts without matching powers first. |
| 52 (Q1) | Did not apply the 1 ≤ a < 10 condition at all. |
"Does it matter if I write 52 × 10⁶?" Yes. The value is right and the form is wrong, and the form is what the mark is for. Compare it to writing 6/4 instead of 3/2.
"Why does the calculator use E?" Because a screen cannot superscript. It is a display limitation, not notation, and it never belongs in an answer.
"How do I add them?" Make the powers match, then add the fronts. Writing both over the larger power is the reliable way round.
| What is happening | |
|---|---|
| 1 | The 10⁶ against 10³ question, cold. Then the strip. |
| 2 | The two rules, with the valid-or-not table done as a quick sort. |
| 3 | Multiplying and dividing, which they find easy. |
| 4 | Adding and subtracting, which they do not. Spend the time here. |
| 5 | Quantities from science, so the sizes mean something. |
Do not accept calculator notation in class, even casually. They write what they see you accept.
Do not teach adding before multiplying. Multiplying is where the rule feels natural; adding is the exception that needs the extra step.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.