Triangle Area Investigators

Overview

Students review how to calculate the area of triangles by connecting the formula to the area of a rectangle. They use folding, cutting, diagrams and worked examples to recognise that a right-angled triangle is half of a rectangle with the same base and perpendicular height, then apply the idea to practical problems in cm² and m².

Learning intentions

• WALT connect the area of a triangle to the area of a rectangle.
• WALT identify the base and perpendicular height of a triangle.
• WALT use ( \text{area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height} ).
• WALT explain our mathematical thinking using correct units.

Success criteria

• I can identify the base and the perpendicular height.
• I can halve the area of a matching rectangle to find the area of a right-angled triangle.
• I can calculate triangle area and record the answer in cm² or m².
• I can explain why the formula includes one-half.

Curriculum links

• Te Mātaiaho Mathematics and Statistics — Phase 2 Measurement: area and the relationship between rectangles and right-angled triangles.
• Te Mātaiaho Mathematics and Statistics — Mathsteasers: higher-order thinking and challenge for advanced learners.
• Mathematical communication, reasoning and problem-solving through explaining, representing and checking solutions.

Lesson structure (60 minutes)

1. 0–7 min · Hook and prior knowledge. Display a rectangle divided corner-to-corner on the opening comparison slide. Ask, “What do you notice? How could we find the area of one triangle without measuring every square?” Students turn and talk, then recall the rectangle area formula and estimate which half is larger.

2. 7–17 min · Explore the connection. Give pairs paper rectangles, scissors and rulers. Students draw a diagonal, cut or fold along it, and compare the two triangles. Model that each triangle is half the rectangle, so (A=\frac{1}{2}\times b\times h). Emphasise that the height must meet the base at a right angle; label the base and perpendicular height on the rectangle-to-triangle visual. Students record the relationship in words and symbols.

3. 17–27 min · Teacher modelling. Model two examples, thinking aloud and using the steps on the worked-example slides:

• Base 8 cm, perpendicular height 5 cm: rectangle area (8\times5=40\text{ cm}^2); triangle area (40\div2=20\text{ cm}^2).
• Base 6 m, perpendicular height 4 m: rectangle area (6\times4=24\text{ m}^2); triangle area (24\div2=12\text{ m}^2).

Model the alternative calculation, (\frac{1}{2}\times8\times5=20), and ask students to estimate before calculating. Explicitly address misconceptions: multiplying base by the sloping side, using a non-perpendicular measurement as height, forgetting to halve, and writing linear units instead of square units.

4. 27–38 min · Guided practice. Solve three examples together using mini-whiteboards and the guided-practice questions. Students first highlight or point to the base and perpendicular height, estimate, calculate, then show units. Pause after each question for “show me”, partner checking and correction. Include a triangle with a clearly marked perpendicular height and a right-angled triangle whose sloping side could distract students.

5. 38–53 min · Independent differentiated practice. Distribute the differentiated triangle-area worksheet. Students complete the section that matches their readiness, moving on when secure:

• Support: labelled right-angled triangles, rectangle halves, multiplication facts, grid paper and a four-step frame: identify, multiply, halve, label units.
• Core: triangles in cm and m, missing diagrams, and short practical problems.
• Challenge: compare triangles with the same area, find a missing base or height, and explain whether two different-looking triangles can have equal area.

Confer with individuals and ask, “Where is the perpendicular height?” and “How do you know your answer is reasonable?”

6. 53–60 min · Plenary and exit ticket. Revisit the opening question using the final reflection slide. Students complete an exit ticket: “A triangle has base 10 cm and perpendicular height 7 cm. Find its area and explain why you halve the rectangle area.” Add: “Circle the measurement that is the perpendicular height” on a small diagram. Collect responses as students leave.

Resources

• the triangle-area teaching deck
• the differentiated triangle-area worksheet
• Paper rectangles for cutting or folding
• Scissors, rulers and pencils
• Mini-whiteboards and pens
• Grid paper
• Coloured pencils or highlighters
• Exit-ticket slips

Assessment

• Check prior knowledge during the hook and use mini-whiteboard responses to identify errors with multiplication, halving and units.
• During guided and independent work, observe whether students select the perpendicular height rather than the sloping side; question students and record names for follow-up.
• Use the exit ticket to group students for the next lesson: secure, needs a reminder about the formula, or needs further work linking triangles to rectangles.

Differentiation

• Provide dyslexia-friendly access: read all instructions aloud, use a clear sans-serif font, generous spacing, short numbered steps, uncluttered diagrams and colour-coded base and height. Allow students to listen to instructions, work with a partner or explain answers orally.
• Support diverse learners with concrete paper rectangles, pre-labelled diagrams, multiplication charts, grid paper, a calculator for checking and repeated teacher modelling. Accept drawings, verbal explanations or equations as evidence.
• Pair students strategically and provide sentence starters: “The base is…”, “The perpendicular height is…”, and “I halve because…”.
• Advanced learners complete the challenge section and justify generalisations, such as why triangles with equal base and perpendicular height have equal area even when their sloping sides differ.

Extension

• Design two different triangles with an area of 24 cm² and label possible base and perpendicular-height pairs.
• Find the missing height of a triangle with area 30 m² and base 10 m, then explain the inverse calculation.
• Investigate whether doubling the base, height or both changes the area in the same way.