Five readings, four strips, and how to make the direction of the error something they can state without calculating.
Ask whether the estimate is too big or too small before pressing anything. Most classes think it is a coin toss, or that the errors cancel. Then run the strips on y = x² and point at the slivers of red sticking out above the curve, every single one on the same side.
Then switch to 8x − x². The slivers are now gaps underneath. Within a minute the class can predict the direction of the error from the shape alone, which is exactly what the question asks for.
| y = x² on [0, 4] | Heights 0, 1, 4, 9, 16. Four strips give 22; the true area is 64⁄3 ≈ 21.33, so it is too big. Eight strips give 21.5, halving the error. |
| y = 8x − x² on [0, 4] | Heights 0, 7, 12, 15, 16. Four strips give 42; the true area is 128⁄3 ≈ 42.67, so it is too small. |
| The lake | h = 10, widths 0, 14, 22, 18, 0. Middles sum to 54, doubled to 108, times 5 gives 540 m². |
| 1. Four-strip estimate | 22. |
| 2. Curve bending down | B, too small, chords below the curve. |
| 3. Lake area | 540 m². |
1 markThe correct h. With five readings there are four strips. This single misreading accounts for more lost marks here than everything else combined.
1 markThe bracket: ends once, middles twice, written out in full.
1 markThe arithmetic and the units.
1 markStating the direction of the error with a reason about the shape. "It is only an estimate" earns nothing.
| They write | What it means |
|---|---|
| 17.6 (Q1) | Five strips. They counted readings instead of gaps, so h became 0.8. The error to watch for. |
| 30 (Q1) | Doubled the end values too. They have the shape of the formula but not its asymmetry. |
| 44 (Q1) | Forgot to multiply by h over 2. Usually a dropped step rather than a misunderstanding. |
| 21.33 (Q1) | Integrated instead. Right mathematics, wrong question, and it shows they can do 5.5. |
| "Errors cancel" (Q2) | The intuition the animation is built to kill. Send them back to look at which side every sliver is on. |
| 1080 (Q3) | Used h rather than h over 2. The commonest slip once the formula is remembered at all. |
"Why not just integrate?" Because there is often no function. The lake has measurements and nothing else, and that is the case the rule exists for. It is worth saying before the function example, not after.
"Does it ever become exact?" Only if the curve is a straight line. More strips always help, and the page shows 4 to 8 halving the error, but a chord is never a curve.
"Can the widths be unequal?" Not for this rule as it is set here; equal intervals are part of it. If data arrives unevenly spaced, that is beyond what is being asked.
| What is happening | |
|---|---|
| 1 | Show a shape with no equation, like a lake or a field. How would you find its area? Let them propose strips themselves. |
| 2 | The animation. Predict the direction of the error, then watch it, then switch curves. |
| 3 | The formula, with ends and middles colour coded on the board. The h question, slowly. |
| 4 | The function example, then the lake. Insist the bracket is written out. |
| 5 | Questions 1 to 3. Question 2 is the one worth taking to the whole class. |
| 6 | Link back to 5.5: when you have a function, integrate; when you have a table, use strips. |
Do not say "n strips" while pointing at a list of n readings. Count the gaps out loud with them the first time. The whole error lives in that sentence.
Do not describe the estimate as "close enough" without the direction. Knowing it is an overestimate is more useful than knowing it is close, and it is the part that is actually assessed.
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.
The Add strips button steps from 1 to 12 rather than sliding, so each new strip is visible as a separate event. No external library, nothing stored, nothing sent.