Is this in standard form?
A categorical atom from mrbartonmaths.com
What this sequence teaches
One thing: the first number must be one or more and less than ten. Ten itself is out; one itself is in. It also shows that a negative power of ten does not change the answer.
Everything else about standard form (the multiplication sign, the 10, a whole-number power) is held still here and met in the expansion sequence.
The teaching sequence
Example 1 — 10 × 107 (not standard form). Ten itself. Many students accept it because “it’s all tens”.
Example 1 → 2. The verdict clears and the stage holds still for a second. Then the first number fades out, the space sits empty for a moment, and 9.864 fades in. The × 107 never moves.
Example 2 — 9.864 × 107 (standard form). Below ten, with three decimal places, so students see a long decimal accepted straight away.
Example 2 → 3. The 7 slides across a little and a minus sign appears in front of it. Nothing else changes, and the answer stays yes.
Example 3 — 9.864 × 10−7 (standard form). A negative power is fine. Students cannot work this out for themselves, so it is shown.
Example 3 → 4. The first number is replaced by 1.
Example 4 — 1 × 10−7 (standard form). One itself is in, and the first number does not need a decimal part.
Example 4 → 5. The first number is replaced by 0.9.
Example 5 — 0.9 × 10−7 (not standard form). Just below one. The negative power has already been accepted twice, so it cannot be the reason.
What this sequence doesn’t address
Other powers. The power is 7 or −7 throughout. The testing sequence uses powers from −12 to 46.
Mid-range first numbers and other decimals (3.5, 6.48359, 2) are in the testing sequence.
A power that is not a whole number, a plus sign, a base that is not 10, powers of 0 and 1, and a negative first number are in the expansion sequence.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if the first number is one or more and less than ten, the number is in standard form.
- Example 1 — “This is not in standard form. How do I know? Because the first number is one or more — but it isn’t less than ten.”
- Examples 2, 3 and 4 — “This is in standard form. How do I know? Because the first number is one or more and less than ten.”
- Example 5 — “This is not in standard form. How do I know? Because the first number is less than ten — but it isn’t one or more.”
Whether students see the words is your call. Nothing appears unless you click. Show rule puts the rule above the stage until you turn it off. Why? appears after each reveal: it shows the justification and highlights the first number, on every example.
Vocabulary. The page says “the first number” throughout. Once students are fluent, tell them its technical name is the coefficient, and use that from then on.
Running the sequence
Pause before each reveal. The pause is the prediction moment.
Replay appears on Examples 2 to 5 and replays the change from the example before, with the same pauses.
About the testing sequence
Ten items in a new random order each time: five are in standard form, five are not. The rule decides every one. There is no Show rule button here. Why? works as in the teaching sequence, after each answer.
What each item is diagnosing
The first numbers 3.5, 2 and 6.48359. Whether students accept first numbers that look different from the teaching examples: one decimal place, none, and five.
The first numbers 9.999 and 1. The two edges from the inside. One itself is in.
The first number 10, with a negative power. Ten itself is out, whatever the power.
The first numbers 12.5 and 36. Ten or more. The 12.5 looks right at a glance: a decimal and a negative power.
The first numbers 0.999 and 0.62. Below one. These catch “a decimal times a power of ten is always standard form”.
The powers run from −12 to 46. A student who hesitates because the power is not 7 is showing that the teaching sequence left that impression; say plainly that the rule does not mention the power.
Discussion prompts
- “Say the rule. Now use it on the one that starts with 12.5.”
- “On the one that starts with 0.999: which part of the rule does it break?”
- “On the one that starts with 1: is one ‘one or more’?”
Reading the summary
Each cell has two channels: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the number is in standard form.
Saying yes to 10, 12.5 or 36 suggests the top edge is not secure. Saying yes to 0.999 or 0.62 suggests the bottom edge is not. Saying no to 1, 2 or 6.48359 suggests students are judging by how the first number looks.
About the expansion sequence
Six items a little beyond what the rule covers, to see how far students’ understanding transfers. The rule began “For the following examples”; this is where students meet its edge. It works exactly like the testing sequence.
The two kinds of item
The rule still works on an unfamiliar surface. A power of 0 and a power of 1 look odd, but the first number is in range, so both are in standard form.
The rule is not enough on its own. In three items the first number is in range but the number is not in standard form: it is added to the power of ten, not multiplied; the power is not a whole number; the power is on a 5, not a 10. In one item the first number is negative: the rule as worded says no, but it is in standard form, because the minus sign only makes the whole number negative.
Who it’s for
Students who are secure on the testing sequence. The negative first number rarely appears at GCSE; skip it if it would distract.
Discussion prompts
- “On the one with the plus sign: the first number passes the rule. What else is wrong?”
- “On the one with a power of 0: what does the rule say?”
Reading the summary
The same two channels as the testing sequence. Saying yes to the plus sign, the decimal power or the 5 suggests students are checking the first number and nothing else, which is all they have been taught so far. That is the cue for the rest of the definition.