Mathematics · Year 13

Active learning ideas

Numerical Integration: The Trapezium Rule

How can we find the area of complex shapes when standard integration isn't possible? This topic introduces the trapezium rule, a powerful numerical tool for approximating the area under any curve.

• National Curriculum Attainment Targets
• DfE Subject Content for Mathematics: I4 - Understand and use numerical integration of functions, including the use of the trapezium rule.

• 15–30 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Collaborative Problem-Solving
• 25 min · Pairs

Graph Paper Approximation

Students are given a known function, such as y = x² + 1, and are asked to plot it on graph paper. They then physically draw a set number of trapezia of equal width underneath the curve and calculate the total area by summing the area of each one, reinforcing the geometric origin of the rule.

Explain how the trapezium rule uses a series of trapezia to approximate the area under a curve.

Facilitation Tip

Provide pre-printed axes and the first trapezium to help students get started quickly.

What to look for

Ask students to use mini-whiteboards to calculate the value of 'h' and list the required x-ordinates for a given integral and number of strips. This checks the initial setup before the main calculation.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 30 min · Individual

Spreadsheet Investigation

Students create a spreadsheet to implement the trapezium rule. This allows them to easily change the number of strips (n) and instantly see the effect on the approximation's accuracy compared to the known analytical solution.

Analyse the concavity of a curve to determine whether the trapezium rule will produce an over-estimate or an under-estimate of the integral.

Facilitation Tip

Provide a template with the basic structure to focus students on the formula rather than spreadsheet formatting.

What to look for

An exam-style question that requires students to use the trapezium rule with a given number of strips, state with a reason whether the answer is an over- or under-estimate, and then use a more accurate estimate to calculate the percentage error.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 15 min · Small Groups

Over or Under? Card Sort

Students work in small groups with a set of cards showing different curves (concave up, concave down, linear, mixed). They must sort them into categories: 'over-estimate', 'under-estimate', or 'exact', and justify their reasoning based on the shape of the curve.

Evaluate the effect of doubling the number of strips on the accuracy of the approximation.

Facilitation Tip

Encourage students to sketch a single, wide trapezium on each graph to make the estimation error obvious.

What to look for

Provide a worksheet with problems that have worked solutions on the back. Students can attempt the problems and then check their own work, identifying where they may have made errors in the process.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start by visually deriving the formula from the sum of individual trapezia under a curve. Use graphing software like GeoGebra or Desmos to demonstrate how increasing the number of strips improves the fit and accuracy. When teaching the formula, explicitly label y₀ as the 'first' ordinate, yₙ as the 'last', and the rest as the 'middle ones' that get doubled to avoid confusion.

Students will be able to apply the trapezium rule to estimate definite integrals and critically evaluate the accuracy of their results based on the shape of the curve.

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Watch Out for These Misconceptions

• The strip width 'h' is one of the y-values or 'heights' of the trapezium.

  'h' is the width of each strip along the x-axis. It is calculated as (b-a)/n, where [a, b] is the integration interval and n is the number of strips. The parallel sides of the trapezia are the y-values (ordinates).

• All the y-values (ordinates) in the formula are treated equally, often by just adding them all up and multiplying by h.

  The formula gives a different weighting to the ordinates. The first (y₀) and the last (yₙ) are used only once, but all the intermediate ordinates (y₁ to yₙ₋₁) are doubled because they form a side for two adjacent trapezia.

• Forgetting the ½ at the start of the formula.

  The formula is derived from the area of a trapezium, which is ½(sum of parallel sides) × height. This ½ factor must be included in the final formula for the sum of all the trapezia.

• Confusing the number of strips with the number of ordinates.

  To create 'n' strips, you need 'n+1' ordinates (or y-values). For example, 4 strips require 5 ordinates (y₀, y₁, y₂, y₃, y₄).

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Methods used in this brief

• Collaborative Problem-Solving