Mathematics K–6 · Years 3–4
One straight cut: splitting a trapezium into new shapes
Measurement and space
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
A single straight cut splits a shape into two shapes whose kinds depend on where the cut starts and ends, and the pieces can be joined again into shapes with different features and the same area.
What you need
- 10 paper trapeziums per learner, each with parallel sides of 12 cm and 6 cm and slanted sides of 6 cm (three equilateral triangles of side 6 cm in a row)
- scissors, a ruler, a set square and a pencil
- a recording table: where the cut starts, where it ends, the two shapes made
How to do it
- Describe the trapezium: count its sides and corners, find the pair of parallel sides and the two equal sides.
- Draw one straight line across a trapezium and predict the two shapes it will make. Cut along it and name both pieces.
- Repeat with cuts from corner to corner, from a corner to a side, from a side to the next side and from a side to the opposite side, recording each.
- Cut one trapezium into three equilateral triangles with two cuts and stack them to check that they match.
- Use the set square to cut straight down from one end of the short side to the long side. Flip the triangle cut off over, slide it to the other end and join it to the other slanted side. Name and measure the new shape.
- For each kind of cut, add the numbers of sides of the two pieces and look for a rule.
What you should see
A cut from corner to corner makes two triangles (6 sides between them); a cut from a corner to a point on a side makes a triangle and a four-sided shape (7 sides); a cut from side to side makes either a triangle and a five-sided shape or two four-sided shapes (8 sides). The rule is 6 sides plus one for each end of the cut that lands part way along a side. The two cuts in step 4 give three equilateral triangles with every side 6 cm that stack exactly. The straight-down cut gives a right-angled triangle of 7.8 cm² and a four-sided shape with two right angles; rejoined with the triangle flipped over, they make a rectangle 9 cm by 5.2 cm with the same area as the trapezium, 46.8 cm². Turning the triangle without flipping it leaves its slanted edge running the wrong way, so it does not fit.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Cutting a shape in two always makes two smaller copies of the same shape.
- A shape made by rejoining the pieces has a different area from the original.
Safety card
Hazards
- scissors
Controls
- cut seated at a table and pass scissors handle first
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K-10 Syllabus (2022), NSW Education Standards AuthorityMA2-2DS-01MA2-2DS-02MAO-WM-01
- Australian Curriculum v9AC9M4SP01
Sources
The pages the author read to write this activity.