Area and perimeter: a week of ready-to-teach maths

Five lessons for Year 8 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 every shape drawn to scale, shape cards and area cards, square tiles, a tracing grid, 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. Every shape is drawn on the worksheets.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 3: the shape and area cards are for Day 3, the tiles for Day 4.
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

A composite shape is made of simpler shapes, so its area can be found two ways: split it and add the pieces, or start from a bigger rectangle round it and subtract what is missing. The pieces may be rectangles, triangles or parallelograms, using the area formulas from Year 7. Missing lengths come from opposite sides: across the top adds to the same total as across the bottom, and up the left side to the same total as up the right. Perimeter is a different measure, the distance round the outside, so a split line inside the shape never counts, and two shapes with the same area can have very different perimeters. An irregular shape such as a leaf or a lake has no formula, so we lay a grid over it and count: every full square, plus about half of the part squares.

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.

Units in this pack

Lengths are in mm, cm, m or km and areas in square units: mm², cm², m² and km², with the hectare for land (1 ha = 10 000 m², a square 100 m by 100 m). Before multiplying, both lengths go in the same unit, and every answer carries its unit. The shapes are drawn to scale on the sheets, but students work from the lengths printed, not by measuring the drawing.

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 reads a shape and finds every length; Days 2 and 3 give the two ways to an area, adding and subtracting; Day 4 turns to the distance round the outside and shows that area and perimeter are different measures; and Day 5 meets shapes with no straight edges at all. It builds on the area of rectangles, triangles and parallelograms in Years 6 and 7, and prepares the area of a circle later in Year 8 and surface area in Year 9.

Materials for the week (one trip)

• From the classroom: rulers, sharp pencils, two coloured pencils each, scissors, this pack printed.
• Cut-out sheet 1, one per pair: 6 shape cards and 8 area cards (Day 3).
• Cut-out sheet 2, one per pair: 12 square tiles (Day 4) and a tracing grid (Day 5).
• For Day 5: a leaf from the school grounds and about 50 cm of string for each pair. No other equipment to buy.

Words for the word wall

composite shape, rectilinear, missing length, opposite sides, split, piece, add, subtract, overlap, area, square centimetre, square metre, hectare, square kilometre, perimeter, outside edge, irregular shape, full square, part square, estimate

Dear families

This week in maths, Year 8 is finding the area and perimeter of shapes made of rectangles and triangles, like L-shaped rooms, garden beds with a path round them and walls with a pointed roof. They learn two ways to an area (adding the pieces, or taking a piece away from a bigger rectangle), how perimeter differs from area, and how to estimate the area of shapes with curved edges, such as a leaf or a lake on a map.

Try this at home

• Find an L-shaped room, rug or garden bed. Measure it, split it into two rectangles and find its area.
• Ask: a rug is 3 m by 2 m. What is its area, and its perimeter? (6 m² and 10 m.)
• Ask: a 2 m by 2 m square is dug out of a lawn 6 m by 5 m. How much lawn is left? (26 m².)
• Ask: how many square metres are in a hectare? (10 000: a square 100 m by 100 m.)

My shapes this week

Fill one row a day. Sketch a shape you find, or one from class, and give its area and perimeter with units.

Printed from the free seegongsik Area and perimeter teaching pack · seegongsik.com/au/y8/measurement/AC9M8M01/pack

Splitting a composite shape

Most shapes in a house plan are rectangles joined together. Students find every missing length from the opposite side, split a shape into rectangles two ways, and catch the slip of counting an overlap twice.

We are learning to

• find a missing length from the lengths along the opposite side,
• split a composite shape into rectangles and add their areas, in square units,
• check that two different splits give the same area.

Success criteria

• I can find the two missing lengths of an L-shape from its other four sides.
• I can split an L-shape two ways and show that both give the same area.

You need

The worksheet, a ruler and two coloured pencils each.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Taking a missing length from the wrong side, or adding where the lengths should be subtracted.
• Two rectangles that overlap, so a corner is counted twice.
• Writing cm for an area: an area is in square units, cm² or m².

Answers

• Part A: A: 6 cm and 4 cm; B: 2 cm and 3 cm; C: 4 m and 2 m.
• Part B: A = 10 × 3 + 4 × 4 = 30 + 16 = 46, or 4 × 7 + 6 × 3 = 28 + 18 = 46 cm²; B = 9 × 2 + 3 × 5 = 33 cm²; C = 8 × 2 + 2 × 4 + 2 × 4 = 32 m².
• Part C: the 4 by 3 corner is in both of Mia’s rectangles, so 30 + 28 − 12 = 46 cm².
• Part D: many answers, for example an 8 m by 4 m room with a 2 m by 4 m nook: 32 + 8 = 40 m².

Splitting a composite shape

A. Missing lengths

Every corner is a right angle. Find each length marked ?, and write which side told you.

B. Split and add

Draw a line to split shape A into two rectangles and find its area. Then split the second copy a different way and find the area again.

C. Find the mistake

Mia split shape A into a 10 cm by 3 cm rectangle and a 4 cm by 7 cm rectangle, and wrote 30 + 28 = 58 cm². What went wrong, and what is the right area?

D. Stretch: design a room

Draw an L-shaped room with an area of 40 m² and label every side. Swap with a partner: can they find the area from your lengths?

Adding the pieces

A shape with a sloping edge splits into a rectangle and a triangle. Students add the pieces with the Year 7 area formulas, find one area two ways, and work out how many tins of paint a wall needs.

We are learning to

• split a shape into rectangles and triangles and add their areas,
• find the height of a triangle piece from the lengths we are given,
• round up when a real job needs whole tins or packets.

Success criteria

• I can find the area of a rectangle with a triangle on top, using ½ × base × height for the triangle.
• I can say how many tins of paint to buy and why I rounded up.

You need

The worksheet and a ruler each; calculators for Part C.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Forgetting to halve the triangle: base × height is the rectangle round it.
• Using a sloping edge as the triangle’s height: the height is square to the base.
• Rounding the tins down, or to the nearest whole number.

Answers

• Part A: A = 4 × 4 + ½ × 6 × 4 = 16 + 12 = 28 cm²; B = 8 × 5 + ½ × 8 × 3 = 40 + 12 = 52 m²; C = 12 × 4 + ½ × 12 × 3 = 48 + 18 = 66 cm².
• Part B: rows 9 × 2 + 6 × 2 + 3 × 2 = 18 + 12 + 6 = 36 m²; columns 3 × 6 + 3 × 4 + 3 × 2 = 18 + 12 + 6 = 36 m².
• Part C: 52 ÷ 16 = 3.25, so buy 4 tins; two coats: 104 ÷ 16 = 6.5, so 7 tins.
• Part D: the diagonal makes two triangles, ½ × 12 × 7 + ½ × 4 × 12 = 42 + 24 = 66 cm², the same area.

Adding the pieces

A. Rectangles and triangles

Split each shape into a rectangle and a triangle. Mark the triangle’s height, then find the area.

B. One shape, two ways

Find the area of shape D by splitting it into rows, then again by splitting it into columns.

C. Paint for a barn wall

Shape B is the end wall of a barn. One tin of paint covers 16 m². How many tins are needed for one coat? How many for two coats?

D. Stretch: two triangles

Draw a diagonal on shape C from its top left corner to its bottom right corner. It makes two triangles. Find the area of each and add them. Do you get the same area as in Part A?

Subtracting a piece

Some shapes are quicker to see as a big rectangle with a piece missing. Students subtract to find the areas of frames, paths and cut corners, then match shape cards to area cards and catch the two traps.

We are learning to

• find an area as the whole rectangle minus the missing piece,
• choose between adding and subtracting for a shape, and check one way with the other,
• explain a wrong answer by the slip that made it.

Success criteria

• I can find the area of a frame or a path as the outer rectangle minus the inner one.
• I can say which way, adding or subtracting, is quicker for a shape, and why.

You need

The worksheet and a calculator each; cut-out sheet 1 (shape and area cards), one set per pair, cut out.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the shape cards.

Watch for these ideas

• Adding a path’s width to one side only, so the outer rectangle is too small.
• Taking away a triangle as if it were the whole rectangle round it.
• Thinking the two ways give different areas: if they disagree, one has a slip.

Answers

• Part A: 30 × 24 − 22 × 16 = 720 − 352 = 368 cm²; 12 × 6 − 10 × 4 = 72 − 40 = 32 m²; 15 × 9 − 5 × 3 = 135 − 15 = 120 m².
• Part B: the cut triangle is 4 by 3, so 10 × 6 − ½ × 4 × 3 = 60 − 6 = 54 cm².
• Part C: card 1 18, card 2 33, card 3 40, card 4 36, card 5 30, card 6 29 cm²; the traps are 56 (card 3 with the notch added on instead of taken away, 48 + 8) and 42 (card 2 with the triangle not halved, 24 + 6 × 3).
• Part D: Sam’s four strips leave out the four corner squares, each 1 m by 1 m: 28 + 4 = 32 m².

Subtracting a piece

A. Frames, paths and lawns

Find the area of the whole rectangle, then subtract the piece that is not there. A quick sketch in the middle column helps.

B. A cut corner

C. Shape cards

For each card, choose a way, find the area and find its area card. Two area cards are traps: each is the answer to a slip.

The traps are ______ and ______. The slip that made each: ____________________________________

D. Stretch: Sam’s path

Sam found the area of the pool path in Part A by adding four strips: 2 × 10 × 1 + 2 × 4 × 1 = 28 m². Why is Sam’s answer too small? Sketch the path to show it.

Perimeter: the outside edge only

Perimeter is the distance round the outside, so a line inside the shape never counts. Students find perimeters with missing lengths, cost a fence, and build shapes of the same area with different perimeters.

We are learning to

• add the outside edges only to find a perimeter, with a length unit,
• find the missing lengths first, or see when we do not need them,
• explain why shapes with the same area can have different perimeters.

Success criteria

• I can find the perimeter of an L-shape from its width and its height alone.
• I can build two shapes from the same tiles that have different perimeters.

You need

The worksheet and a calculator each; cut-out sheet 2 (12 square tiles), one set per pair, cut out.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the tiles.

Watch for these ideas

• Counting a split line, or an edge twice.
• Mixing up the units: metres for a perimeter, square metres for an area.
• Thinking that the same area means the same perimeter.

Answers

• Part A: A = 2 × (9 + 6) = 30 m, the rectangle round it; B = 10 + 7 + 3 + 3 + 4 + 3 + 3 + 7 = 40 m (missing 3 m and 3 m).
• Part B: A: 30 × $12.50 = $375.00; B: (40 − 2) × $12.50 = $475.00.
• Part C: smallest 14 (a 3 by 4 rectangle), largest 26 (any shape with 11 shared edges); perimeter = 48 − 2 × shared edges, as each shared edge hides two tile sides.
• Part D: 1 by 24: 50 m, 2 by 12: 28 m, 3 by 8: 22 m, 4 by 6: 20 m; the 4 by 6 needs least; for 36 m², the 6 by 6 square, 24 m.

Perimeter: the outside edge only

A. Perimeters

Every corner is a right angle. Find the perimeter of each garden. Find any missing lengths first, if you need them.

B. Fencing

Fencing costs $12.50 a metre. Garden A is fenced all the way round. Garden B leaves a 2 m gap for a gate. Find the cost of each fence.

C. Twelve tiles

Use all 12 tiles each time, joined so that whole edges touch. Sketch each shape and count its perimeter in tile edges.

Smallest perimeter: ________   Largest perimeter: ________   What rule links the two right-hand columns?

D. Stretch: same area, least fence

List every rectangle with whole-metre sides and an area of 24 m², with its perimeter. Which needs the least fencing? Which rectangle of area 36 m² would need the least?

Estimating irregular areas

A leaf or a lake has no formula, so we lay a grid over it. Students count full squares and about half the part squares, scale a map count up to hectares, choose sensible units, and play the self-check quiz as a class game.

We are learning to

• estimate an irregular area as the full squares plus about half the part squares,
• scale a count of map squares to a real area, in m² and hectares,
• choose a sensible unit for an area, from mm² to km².

Success criteria

• I can estimate the area of a shape on a centimetre grid and say why my answer is an estimate.
• I can change an area in square metres to hectares.

You need

The worksheet, two coloured pencils and a calculator each; the tracing grid, a leaf and string for Part D.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.

Watch for these ideas

• Counting only the full squares, or counting every part square as a whole one.
• Scaling a map square by its side, 50, instead of its area, 50 × 50.
• Giving an estimate as if it were exact: different counters get answers a little apart.

Answers

• Part A: 17 full and 28 part squares, so about 17 + 14 = 31 cm² (the drawn leaf is 30.3 cm²; answers within a square or two are fine).
• Part B: 10 full and 24 part, about 22 squares; 22 × 2500 = 55 000 m² = 5.5 ha.
• Part C: the screen of a phone, cm²; the head of a drawing pin, mm²; a wheat farm, ha; the floor of a classroom, m²; the state of Tasmania, km².
• Part D: answers vary with the leaf; check that full and part squares were counted separately.

Estimating irregular areas

A. A leaf on a centimetre grid

Each square is 1 cm by 1 cm. Colour the full squares in one colour and the part squares in another.

B. A lake on a map

Each map square is 50 m by 50 m. Estimate the number of squares the lake covers, then its area.

1 hectare (ha) = 10 000 m², a square 100 m by 100 m.

C. Which unit?

Choose mm², cm², m², ha or km² for the area of each.

D. Stretch: your own leaf

On the tracing grid (cut-out sheet 2), trace round a leaf and estimate its area. Then lay string along its edge, straighten it and measure it: that is the leaf’s perimeter.

Shape cards and area cards

For Day 3, one set per pair: six shapes, every length in cm

Eight area cards

Six of these are the areas of the shapes. Two are traps: each is the answer someone gets from a common slip.

Teacher note: card 1 is 18 cm² (adding: a 6 by 2 and a 2 by 3); card 2 is 33 cm² (adding: a 6 by 4 rectangle and a triangle, base 6, height 3); card 3 is 40 cm² (subtracting: 8 by 6 take away the 4 by 2 notch); card 4 is 36 cm² (either way: 4 by 3 plus 8 by 3, or 8 by 6 take away 4 by 3); card 5 is 30 cm² (subtracting: 8 by 4 take away a triangle 2 by 2); card 6 is 29 cm² (subtracting: 7 by 5 take away the 3 by 2 notch, the only way with the lengths given). The traps: 56 is card 3 with the notch added on instead of taken away, 48 + 8; 42 is card 2 with the triangle not halved, 24 + 6 × 3.

Square tiles and a tracing grid

For Day 4, one set per pair: 12 square tiles

Cut along the dashed lines. Every tile is the same size, so every shape made from all 12 has the same area.

For Day 5: a tracing grid (do not cut)

Lay a leaf on the grid and trace round it. If the sheet printed at full size, each square is 1 cm by 1 cm: check one with a ruler, and if it is not, find the area of one square first. The darker lines mark every fifth square.

What we know: Area and perimeter

Work on your own. Use a calculator only for Questions 3, 4 and 6. Show your working and give units.

1. Every corner of shape P is a right angle. Find the two missing lengths. ______________________
2. Find the area of shape P. ______________________
3. A lawn 12 m by 8 m has a shed 3 m by 2 m standing on it. What area of lawn is left? ______________________
4. Find the area of shape Q, a wall with a pointed roof. ______________________
5. Find the perimeter of shape P. ______________________
6. A park is 250 m by 120 m. Give its area in square metres and in hectares. ______________________
7. Each square is 1 cm by 1 cm. Estimate the area of the shape. ______________________
8. Jai says, “Two shapes with the same area always have the same perimeter.” Is Jai right? Give an example. ______________________

Answers and marking guide

Answers with working

1. 9 − 5 = 4 cm and 7 − 4 = 3 cm.
2. 9 × 4 + 5 × 3 = 36 + 15 = 51 cm² (or 5 × 7 + 4 × 4 = 35 + 16 = 51).
3. 12 × 8 − 3 × 2 = 96 − 6 = 90 m².
4. 10 × 4 + ½ × 10 × 3 = 40 + 15 = 55 m².
5. 2 × (9 + 7) = 32 cm: pushing the corner out gives the rectangle round it.
6. 250 × 120 = 30 000 m² = 3 ha.
7. 7 full and 18 part squares: about 7 + 9 = 16 cm² (accept 14 to 18; the shape is 14.8 cm²).
8. No: a 1 by 12 and a 3 by 4 rectangle both have area 12, but perimeters 26 and 14. Any correct example earns the mark.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and the right unit, and gives Question 7 as an estimate.

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: missing lengths, adding pieces, subtracting a piece, perimeter and grid estimates.