Mathematics · Year 13

Active learning ideas

Locating Roots of Equations

How can we prove a solution to an equation exists without actually solving it? This topic introduces a simple but powerful numerical method to confirm the location of a root within a specific interval.

• National Curriculum Attainment Targets
• DfE Subject Content for Mathematics: I1 - Locate roots of f(x) = 0 by considering a change of sign of f(x) in an interval [a, b].

• 15–25 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Collaborative Problem-Solving
• 20 min · Pairs

Root Hunters

Students are given a worksheet with various functions and intervals. In pairs, they must calculate the value of the function at the endpoints of each interval to determine if a change of sign occurs, thereby confirming the presence of a root.

Explain the connection between a change of sign of a continuous function f(x) over an interval [a, b] and the existence of a root in that interval.

Facilitation Tip

Encourage students to write a concluding sentence for each problem, such as 'As there is a change of sign and the function is continuous, a root exists between x=a and x=b'.

What to look for

Use mini-whiteboards. Give students a function and an interval, and have them show their calculations for f(a) and f(b) and a written conclusion about the existence of a root.

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

Activity 02

• Collaborative Problem-Solving
• 25 min · Individual

Graphical Detective

Using graphing software like GeoGebra or Desmos, students first visually identify the roots of several given functions. They then have to find the narrowest integer interval that contains each root and verify it algebraically using the change of sign test.

Analyse a function with multiple roots and identify suitable intervals for each root.

Facilitation Tip

Ask students to use the table feature in the software to quickly check values and narrow down their intervals.

What to look for

Set an exam-style question that requires students to first show that a root lies in a given interval, and then to write a sentence explaining a limitation of the method, perhaps with reference to a provided graph.

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

Activity 03

• Collaborative Problem-Solving
• 15 min · Small Groups

Break the Rule

In small groups, challenge students to sketch graphs of functions that are exceptions to the rule. For example, a function with two roots in an interval but no change of sign, or a discontinuous function with a change of sign but no root.

Justify why a change of sign is a sufficient but not necessary condition for a root to exist within an interval for all functions.

Facilitation Tip

Prompt groups to consider parabolas that touch the x-axis or functions with vertical asymptotes.

What to look for

Provide a checklist where students rate their confidence in skills such as 'I can evaluate a function for a given value', 'I can explain why continuity is important', and 'I can identify a case where no sign change does not mean no root'.

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

A few notes on teaching this unit

Begin by displaying a graph of a continuous function crossing the x-axis and ask students what they notice about the y-coordinates on either side of the root. Use this visual discovery to introduce the algebraic condition f(a)f(b) < 0. Ensure students explicitly state the continuity condition in their conclusions. Use calculators throughout to minimise cognitive load on arithmetic, allowing focus on the underlying logic.

Students will be able to use the change of sign method to locate roots and articulate the conditions under which this method is valid, including the importance of continuity.

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Watch Out for These Misconceptions

• If there is no change of sign in an interval, there cannot be a root in that interval.

  This is not necessarily true. An interval could contain an even number of roots or a repeated root (a turning point on the x-axis), neither of which would cause a change of sign. The rule only confirms a root is present when there is a sign change; it doesn't rule one out when there isn't.

• A change of sign guarantees there is only one root in the interval.

  A change of sign guarantees that there is at least one root. There could be any odd number of roots (e.g., three, five) within the interval that would also produce a single change of sign between the endpoints.

• The change of sign method works for any function.

  The method is only guaranteed to work for continuous functions. A discontinuous function might have a change of sign across an interval containing a vertical asymptote, but the graph never actually crosses the x-axis to form a root.

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Methods used in this brief

• Collaborative Problem-Solving