Area of triangles and parallelograms: a week of ready-to-teach maths

Five lessons for Year 7 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus shapes to cut and rearrange, area problem cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets: shapes to cut (Days 1 and 2), one copy per student, and area problem cards (Day 4), one copy per pair.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Cut a triangle off the slanted end of a parallelogram and slide it across, and the parallelogram becomes a rectangle with the same base and the same height. So its area is base × height. Two copies of a triangle fit together into a parallelogram, so a triangle’s area is half of base × height. In both formulas the height is the perpendicular height, measured straight up from the base, never the slanted side. Harder shapes split into rectangles and triangles, and every area is in square units that match the lengths.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write area

Write the formula, then the numbers, then the answer with its unit: A = ½ × 10 × 7 = 35 cm². Mark the perpendicular height on every sketch with a dashed line and a small square at its foot. Diagrams in this pack are not drawn to scale unless they are on a grid. Calculators are not needed this week.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 turns a parallelogram into a rectangle; Day 2 halves a parallelogram into two triangles; Day 3 makes sure every base has its own perpendicular height; Day 4 builds harder shapes from pieces; and Day 5 puts units and real problems first. It builds on the area of rectangles (AC9M6M02), and it leads into the volume of prisms (AC9M7M02), where these areas become the ends of solid shapes.

Materials for the week (one trip)

• From the classroom: pencils, rulers, scissors, glue sticks, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per student: two parallelograms and two pairs of triangles (Days 1 and 2).
• Cut-out sheet 2, one per pair: 8 area problem cards (Day 4).
• No calculators and no other equipment to buy.

Words for the word wall

area, square unit, square centimetre (cm²), square metre (m²), square millimetre (mm²), base, perpendicular height, slanted side, right angle, parallelogram, triangle, rearrange, half, composite shape, split, add, take away

Dear families

This week in maths, Year 7 is finding the area of triangles and parallelograms by cutting and rearranging shapes, and using the formulas to solve problems with the right square units.

Try this at home

• Fold a piece of paper corner to corner. How does the area of the triangle compare with the whole sheet? (Half.)
• Find a triangle at home or outside: a shade sail, a roof end, a sandwich. Estimate its base and height and work out its area.
• Ask: would you measure a bathroom floor in square centimetres or square metres? Why?
• Ask: when you lean a rectangle over into a parallelogram, keeping the same height, does its area change? (No.)

Areas this week

Fill one row a day with a shape from class, its base and height, and its area with the unit.

Printed from the free seegongsik Area of triangles and parallelograms teaching pack · seegongsik.com/au/y7/measurement/AC9M7M01/pack

A parallelogram is a rectangle

Cut the slanted end off a parallelogram and slide it across, and it becomes a rectangle with the same base and height. Students cut and rearrange, find areas on grids, and use A = b × h with the perpendicular height.

We are learning to

• explain why a parallelogram has the same area as a rectangle with the same base and height,
• find the area of a parallelogram with A = b × h,
• pick the perpendicular height, not the slanted side.

Success criteria

• I can show by cutting why A = b × h works.
• I can find the area of a parallelogram when the slanted side is given too.

You need

Cut-out sheet 1 (one per student), scissors, glue, the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after cut and slide; start the second with Part A.

Watch for these ideas

• Using the slanted side as the height. The height is straight up from the base.
• Multiplying all three numbers given. Only the base and its height are needed.
• Leaving off the unit, or writing cm for an area. Area is in square units.

Answers

• Cut-outs: Parallelogram A, 24 squares; Parallelogram B, 15 squares.
• Part A: 4 × 3 = 12; 7 × 2 = 14; 2 × 5 = 10 square units.
• Part B: 9 × 4 = 36 cm²; 12 × 6 = 72 m²; 7.5 × 3 = 22.5 cm²; the slanted sides are not used.
• Part C: height = 48 ÷ 8 = 6 cm; base = 35 ÷ 5 = 7 m.
• Part D: no. The height can be 4 cm at most, when the 4 cm side stands straight up, so the largest area is 5 × 4 = 20 cm², and the shape is then a rectangle.

A parallelogram is a rectangle

A. On a grid

Each square is 1 square unit. Find the base and the height, then the area.

B. Which length is the height?

Not drawn to scale. The dashed line is the perpendicular height.

C. A missing length

1. A parallelogram has area 48 cm² and base 8 cm. Find its height.   ____________
2. A parallelogram has area 35 m² and height 5 m. Find its base.   ____________

D. Stretch

A parallelogram has sides 5 cm and 4 cm. Can its area be 30 cm²? What is the largest area it can have, and what shape is it then? Sketch to explain.

A triangle is half

Two copies of any triangle fit together into a parallelogram, so a triangle’s area is half of base × height. Students fit triangle pairs together, find areas on grids, and use A = ½ × b × h.

We are learning to

• explain why a triangle is half a parallelogram,
• find the area of a triangle with A = ½ × b × h,
• find a missing base or height from an area.

Success criteria

• I can fit two copies of a triangle into a parallelogram and say why the area halves.
• I can find a triangle’s area when a slanted side is given too.

You need

Triangle pairs C and D from cut-out sheet 1, scissors, glue, the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the pairs; start the second with Part A.

Watch for these ideas

• Forgetting to halve. The triangle is half of its parallelogram.
• Using the slanted side as the height, as on Day 1.
• Halving twice: ½ × 10 × 7 is 35, not 17.5.

Answers

• Cut-outs: Triangle pair C, parallelogram 24, one triangle 12; Triangle pair D, parallelogram 12, one triangle 6.
• Part A: ½ × 5 × 4 = 10; ½ × 7 × 4 = 14; ½ × 3 × 5 = 7.5 square units.
• Part B: ½ × 10 × 7 = 35 cm²; ½ × 9 × 4 = 18 m²; ½ × 15 × 8 = 60 mm²; the slanted sides are not used.
• Part C: height = 2 × 24 ÷ 8 = 6 cm; base = 2 × 30 ÷ 5 = 12 m.
• Part D: all three are 6 square units: they share the base of 4, and every apex is 3 above it.

A triangle is half

A. On a grid

Each square is 1 square unit. Find the base and the height, then the area.

B. Which length is the height?

Not drawn to scale. The dashed line is the perpendicular height.

C. A missing length

1. A triangle has area 24 cm² and base 8 cm. Find its height.   ____________
2. A triangle has area 30 m² and height 5 m. Find its base.   ____________

D. Stretch

Which height?

Any side can be the base, as long as it is paired with its own height, at right angles to it. Students find one area two or three ways, meet a height that falls outside the shape, and work back from an area to a height.

We are learning to

• pair each base with the height at right angles to it,
• find one area in more than one way,
• find the height of an obtuse triangle outside the triangle.

Success criteria

• I can show that any side and its own height give the same area.
• I can extend a base to meet a height outside a triangle.

You need

The worksheet, a ruler, and a set square or the corner of a sheet of paper for checking right angles.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after three ways round; start the second with Part A.

Watch for these ideas

• Pairing a base with the wrong height. Each base has its own height, at right angles to it.
• Thinking an obtuse triangle has no height. Extend the base until the height meets it.
• Rounding 7.2 to 7 and getting a different area. Keep the decimal.

Answers

• Part A: ½ × 12 × 9 = 54; ½ × 9 × 12 = 54; ½ × 15 × 7.2 = 54 cm².
• Part B: 10 × 4.8 = 48 and 6 × 8 = 48 cm²: the same area either way.
• Part C: the apex is past the end of the base, so the height meets the base line extended; ½ × 5 × 6 = 15 cm².
• Part D: 2 × 30 ÷ 12 = 5 cm; 2 × 30 ÷ 10 = 6 cm.

Which height?

A. One triangle, three ways

B. A parallelogram, two ways

C. A height outside the triangle

D. Stretch

A triangle has area 30 cm². One side is 12 cm: what is the height to that side? Another side is 10 cm: what is the height to it?

Shapes made of pieces

Awkward shapes split into rectangles and triangles. Students find each piece with its own formula and add, or take a piece away from a larger shape, then work through the area problem cards.

We are learning to

• split a shape into rectangles and triangles,
• add the areas of the pieces, or take one away,
• find one area in two different ways as a check.

Success criteria

• I can mark the pieces on a shape and find its total area.
• I can check an area by finding it a second way.

You need

Area problem cards from cut-out sheet 2 (one set per pair) and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the carousel; start the second with Part A.

Watch for these ideas

• Counting a piece twice where two pieces overlap. Mark the pieces before working.
• Using a length that belongs to the whole shape for one piece. Find each piece’s own lengths first.
• Forgetting to halve the triangle pieces.

Answers

• Part A: 10 × 4 + 4 × 4 = 40 + 16 = 56 m²; 6 × 4 + 2 × (½ × 2 × 4) = 24 + 8 = 32 cm²; 6 × 3 + ½ × 6 × 2 = 18 + 6 = 24 m².
• Part B: 10 × 6 − ½ × 4 × 3 = 60 − 6 = 54 m².
• Cards: 1 24, 2 14, 3 18, 4 15, 5 20, 6 19.5, 7 28, 8 12 m².
• Part D: 6 × 4 = 24; and 32 − 2 × 4 = 24 square units.

Shapes made of pieces

A. Split and add

Not drawn to scale. Mark the pieces, then find the total area.

B. Take a piece away

A rectangular lawn is 10 m by 6 m. A triangular garden bed with base 4 m and height 3 m is dug out of it. What area of lawn is left?

C. Card carousel

D. Stretch

Units and real problems

Area answers need square units that match the lengths. Students choose sensible units, solve problems about sails, beds and walls, catch a mix of units, and then play the quiz.

We are learning to

• choose a sensible square unit for an area,
• solve area problems with triangles and parallelograms,
• put every length in the same unit before multiplying.

Success criteria

• I can write every answer with a square unit that matches the lengths.
• I can spot and fix a problem that mixes centimetres and metres.

You need

The worksheet and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Writing cm instead of cm². A length unit for an area is a different thing.
• Multiplying centimetres by metres. Change one length first.
• Thinking 1 m² is 100 cm². It is 100 rows of 100, so 10 000 cm².

Answers with working

1. ½ × 4 × 3 = 6 m².
2. 2.5 × 1.2 = 3 m².
3. a phone screen: cm²; a basketball court: m²; a national park: km².
4. 9 × 4 = 36 m², and 36 ÷ 6 = 6 L.
5. h = 2 × 9 ÷ 4.5 = 4 m.
6. the base is in cm and the height in m; in metres, ½ × 0.3 × 2 = 0.3 m², or in centimetres, ½ × 30 × 200 = 3000 cm².
7. 1 m² is a square 100 cm by 100 cm, so 100 × 100 = 10 000 cm², not 100.
8. any base and height whose product is 24, such as 3 m and 8 m, 4 m and 6 m, 6 m and 4 m.

Units and real problems

Problems (7 and 8 are the stretch)

1. A triangular shade sail has base 4 m and height 3 m. What is its area?
2. A garden bed is a parallelogram with base 2.5 m and perpendicular height 1.2 m. What is its area?
3. Which square unit would you use for the area of a phone screen, a basketball court, and a national park: mm², cm², m² or km²?
4. A feature wall is a parallelogram with base 9 m and height 4 m. One litre of paint covers 6 m². How many litres are needed for one coat?
5. A triangular sail must have an area of 9 m² and a base of 4.5 m. What height does it need?
6. Kim found the area of a triangle with base 30 cm and height 2 m as ½ × 30 × 2 = 30. What went wrong? Find the area in m² and in cm².
7. Stretch. How many square centimetres are in one square metre? Draw a sketch to explain.
8. Stretch. A triangular garden has an area of 12 m². Give three different bases and heights it could have.

Shapes to cut and rearrange

Day 1: cut out each parallelogram, cut along its dashed line, and slide the piece across to make a rectangle. Day 2: cut out both triangles of a pair and fit them together into a parallelogram. Each square is 1 square unit.

Area problem cards

Cut out the 8 cards. Day 4: each square is 1 m by 1 m. Split each shape into rectangles and triangles, and find its area in square metres.

What we know: Area of triangles and parallelograms

Work on your own. Write each formula, the numbers, and the answer with its unit. Diagrams are not to scale.

Answers and marking guide

Answers with working

1. A = 11 × 6 = 66 cm²; the 7 cm side is not used.
2. A = ½ × 14 × 5 = 35 m².
3. Base 3, height 4 (outside the triangle): ½ × 3 × 4 = 6 square units.
4. h = 54 ÷ 9 = 6 cm.
5. 9 × 4 + ½ × 9 × 2 = 36 + 9 = 45 m².
6. 12 × 8 − ½ × 6 × 4 = 96 − 12 = 84 m².
7. 1 m = 100 cm, so ½ × 40 × 100 = 2000 cm².
8. Cutting the slanted end off and sliding it across makes a rectangle as tall as the perpendicular height, not as long as the slanted side.

A quick three-level guide

Eight questions, four ideas. A student at standard finds the area of a triangle or parallelogram with its perpendicular height, splits a shape into pieces, and writes every answer with a square unit.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: parallelograms, triangles, which height, shapes made of pieces, and units.