DSE Maths - Trapezoidal Rule and Numerical Integration Prep Guide (for tutors) | Four and eight strips, error comparison, convergence

DSE Maths – Trapezoidal Rule and Numerical Integration Prep Guide (for tutors) | Four and eight strips, error comparison, convergence

2026-11-05

Tutor lesson-prep guides · Secondary Maths

Students can quote the formula but cannot say why the two ends count once and the middle counts twice. This lesson runs x squared on 0 to 2 with four and eight strips, giving 2.75 and 2.6875, so the error falls from 0.0833 to 0.0208 in front of them.

Objectives — Students can: (1) apply h over 2 times the two ends plus twice the middle sum to get 2.75 for x squared on 0 to 2 with four strips; (2) double the strips to eight, get 2.6875, compare with the exact 2.6667 and state that the error ratio is 4; (3) use the bound of (b minus a) times h squared over 12 times the largest second derivative, which is 0.08333333333333333 for four strips, to explain why more strips is better.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Function values	What is x squared at x equals 1.5?	2.25, and it must be kept as a decimal
Summation	When adding five values, are the two ends counted with the middle?	No, keep the ends and the middle sum separate
Second derivative	What is the second derivative of x squared?	The constant 2, independent of x

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Prior knowledge check	Ask the three prior questions and write the ordinates on one line	Answer orally and copy any wrong item once
5-17 min	Act one: one formula, two moves	Show four strips: h is 0.5, ends 0 and 4, middle sum 3.5	Copy it and recompute the middle sum
17-30 min	Act two: double the strips	Middle sum 8.75, giving 2.6875	Compute once, then compute the error
30-40 min	Act three: the error has a bound	Use (b-a)h squared over 12 times the second derivative to get 0.08333333333333333	Compute the bound for eight strips
40-43 min	One-sentence summary	Ask for one sentence on why the middle is doubled	Answer orally; tutor corrects only wording
43-45 min	Exit check	Hand out three items and collect them	Work alone with books closed

03 Teaching script (read it out)

Scene 1: Act one: a trapezium is average height times width

Each strip uses the mean of the two end heights times the width h: h times left plus right, divided by 2.
With 4 strips h is 0.5, end heights are 0.0 and 4.0, and the three middle heights are 0.25, 1 and 2.25, summing to 3.5.
Adding the strips up, every interior point is used by the strip on each side, so the middle sum is doubled: 0.5/2 times (0 plus 4 plus 2 times 3.5) is 2.75.
One line to remember: ends once, middle twice, then everything times h over 2.

Ask students to repeat: an interior point belongs to two strips, so it is doubled.

Scene 2: Act two: halving h quarters the error

With eight strips h halves to 0.25, the seven interior values sum to 8.75, and the same formula gives 2.6875.
The exact value of the integral of x squared from 0 to 2 is eight thirds, about 2.6667; the four-strip error is 0.0833 and the eight-strip error is 0.0208.The ratio is 4, so halving h divides the error by four, because the error goes with h squared.
The percentage error falls from 3.13 percent to 0.78 percent, which is what more strips is worth in numbers.

Ask students to repeat: half the width, quarter the error; not half the error.

Scene 3: Act three: the error bound can be computed

The trapezoidal error is at most (b minus a) times h squared over 12, times the largest absolute second derivative on the interval.
The second derivative of x squared is the constant 2, so the four-strip bound is 2 times 0.5 squared over 12 times 2, which is 0.08333333333333333.
The actual error 0.0833 meets the bound because the second derivative is constant; for a cubic the actual error would be far below the bound.
The real use is table data: speeds of 0, 8, 12, 15 and 16 metres per second every 2 seconds give a displacement of 86 metres by the trapezoidal rule, since without an equation there is nothing else to integrate.

Ask students to repeat: the bound reads the second derivative, so a more curved graph errs more.

04 Question types and worked steps

Type	Example	Steps	Answer
Four-strip computation	x squared, 0 to 2, 4 strips	List ordinates and values first, then apply the formula	2.75
Eight-strip computation	Same function, 8 strips	Add every interior value, do not drop the 1.5 point	2.6875
Error comparison	The four and eight strip answers	Subtract the exact value, then take the ratio	Ratio 4, matching h squared
Table application	Speeds 0, 8, 12, 15, 16 with spacing 2 s	Treat it as an area and state the unit	86 metres

Sanity check (teach this): Three read-backs: (1) the middle sum must exceed the two ends, so a negative answer means the ordinates are out of order; (2) the exact value is 2.6667, so 2.75 and 2.6875 must sit close to it, and a wild answer lost the h over 2; (3) the error falls as strips increase, so a larger eight-strip error means the four-strip middle sum was reused.

05 Independent practice (3 questions)

Question	Answer	What to watch
x squared, 0 to 2, two strips: what is the estimate?	3, with an error of 0.3333	Check whether only two points were used
x squared, 0 to 1, four strips: what is the estimate?	0.34375 against an exact 0.3333	Check whether h is 0.25
Why does doubling the strips quarter the error?	The error goes with h squared, so halving h gives a quarter	Check that the squared relation is named

06 Common mistakes and what to say

Mistake	Student writes	Say this
Middle counted once	Four strips computed as 2.0	An interior point belongs to two strips, so multiply by 2.
h over 2 dropped	Getting 11 instead of 2.75	The h over 2 carries the area unit; it is not decoration.
Ordinate count wrong	Listing only four points for four strips	Four strips need five points; strips plus one is the point count.
Error mixed with percentage error	Calling 0.0833 a percentage	Absolute error is a difference; percentage error divides by the exact value first.

07 Homework (paper, section and question numbers)

Redo this chapter — Redo the three items, writing h and the ordinates first
Definite integrals and area — Compare the exact integral with the trapezoidal estimate in three points
Maths chapter index — Pick one more chapter and list the formulas only

08 Exit check (last 5 minutes)

Question	Answer	Decision
x squared, 0 to 2, 4 strips: what is the estimate?	2.75	Error: dropping the 2 on the middle sum
Going from four strips to eight, by what factor does the error change?	One quarter	Error: answering one half
Which quantity sets the error bound?	The largest absolute second derivative	Error: answering the first derivative

09 Tutor FAQ

Which year is this for?

S6 compulsory part, and a preview in S5 term two.

Is 45 minutes enough for three layers?

Yes, because all three share one formula and only one set of numbers is computed in full.

Is there a companion worksheet?

Three practice items with answers on the page, plus the definite integral chapter.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.