Composite objects: a week of ready-to-teach maths

Five lessons for Year 10 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 twelve formula cards to match, twelve surface area cards to sort, 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, one copy per pair. Cut the 12 formula cards before Day 1 and the 12 surface area cards before Day 4, and keep each set in an envelope.
4. Calculators are used every day. The formulas for cones and spheres are printed on the sheets; students use them, not recall them.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A composite object is built from simple solids, by joining them or by cutting pieces away. Volume is the easy measure: the volumes of the parts simply add, or subtract for a hole. Surface area takes care. Where two parts touch, a face of each is pressed against the other and no longer lies on the outside, so the contact area is left out twice; and cutting a piece away can lose some faces and expose new ones. Units matter too: area is in square units, volume in cubic units, and a litre is 1000 cubic centimetres.

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 these

Answers keep π exactly first, such as 26π m³, then give a decimal, 81.68 m³. Area is in square units (cm², m²) and volume in cubic units (cm³, m³) or litres: 1 L = 1000 cm³ and 1 m³ = 1000 L. The sketches show each object from the front; they are not to scale.

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 refreshes the simple solids; Day 2 adds volumes; Day 3 deals with the faces hidden by joining; Day 4 with the faces lost and gained by cutting; and Day 5 puts it all to work with real units. It builds on Year 9, where students found the volume and surface area of right prisms and cylinders, and it prepares Proportion and Scaling later in Year 10, where doubling the lengths of a solid multiplies its surface area by 4 and its volume by 8.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, pencils, this pack printed.
• Cut-out sheet 1, one per pair: 12 formula cards (Day 1).
• Cut-out sheet 2, one per pair: 12 surface area cards (Day 4).
• If you have them: a few real objects to measure on Day 5, such as a tin, a ball and a box. No other equipment to buy.

Words for the word wall

composite object, prism, cylinder, cone, sphere, hemisphere, pyramid, radius, slant height, volume, capacity, surface area, contact face, hollow, square units, cubic units, litre, kilolitre

Dear families

This week in maths, Year 10 is finding the volume and surface area of objects built from simpler shapes: a grain silo, an ice-cream cone, a block with a hole through it, a water tank and a swimming pool.

Try this at home

• Ask: how many litres are in a cubic metre? (1000.)
• Ask: two dice are glued face to face. Does the surface area go up or down? (Down: two faces are hidden.)
• Ask: what two shapes make an ice-cream cone with a scoop on top? (A cone and a hemisphere.)
• Ask: does slicing a loaf of bread add surface? (Yes: every cut makes two new faces.)

My composite objects week

Fill one row a day. Write an object you saw, the simple solids it is made of, and whether any faces are hidden.

Simple solids

Every composite object is made of simple solids, so the week starts with them. Students find the volume and surface area of a box, a cylinder, a cone and a sphere, keep π exact, and match six solids to their volume formulas.

We are learning to

• use formulas to find the volume and surface area of prisms, cylinders, cones and spheres,
• give answers exactly in terms of π and as decimals, with the right units,
• recognise which volume formula belongs to which solid.

Success criteria

• I can find the volume of a cylinder of radius 3 cm and height 10 cm as 90π ≈ 282.74 cm³.
• I can explain how a cone’s volume compares with the cylinder on the same base and height.

You need

Calculators; the formula cards (cut-out sheet 1), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Using the diameter as the radius: halve it first.
• Using the height of a cone for its slant: the curved surface uses s = √(r² + h²).
• Square units for volume: volume is in cubic units, cm³.

Answers

• Part A: Box 6 by 4 by 3 cm: V = 72 cm³, SA = 108 cm²; Cylinder, radius 3 cm, height 10 cm: V = 90π ≈ 282.74 cm³, SA = 78π ≈ 245.04 cm²; Cone, radius 3 cm, height 4 cm: V = 12π ≈ 37.70 cm³, SA = 24π ≈ 75.40 cm²; Sphere, radius 5 cm: V = 500π/3 ≈ 523.60 cm³, SA = 100π ≈ 314.16 cm² (slant height 5 cm).
• Part B: the cylinder on the same base and height holds 36π ≈ 113.10 cm³, three times the cone’s 12π ≈ 37.70 cm³.
• Part C: the card key is with the mini-check answers.
• Part D: the cube is 10 cm wide, so 1000 cm³; the sphere fills 523.60 cm³ of it, about 52% (exactly π/6 of the cube, whatever the size).

Simple solids

A. Volume and surface area

Give each answer exactly (in terms of π) and to two decimal places. The box is done for you.

B. Cone against cylinder

C. Match the formulas

Match each solid card to its volume formula card. Write the letters.

D. Stretch: a ball in a box

Volume adds up

The volume of a composite object is the sum of its parts’ volumes, or a difference when a piece is missing. Students find the volumes of a silo, an ice-cream cone, a house and an L-shaped prism, and check the L-shape two ways.

We are learning to

• split a composite object into simple solids,
• add or subtract their volumes,
• give answers exactly and to two decimal places, in cubic units.

Success criteria

• I can find the silo’s volume as 26π ≈ 81.68 m³.
• I can find the L-shaped prism’s volume by adding parts and by subtracting one, and get the same answer.

You need

Calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Forgetting the third in a cone’s volume: (1/3)πr²h.
• Using a whole sphere for a hemisphere: half of (4/3)πr³ is (2/3)πr³.
• Counting a piece twice: in the L-shape each part of the cross-section belongs to one piece only.

Answers

• Part A: silo 24π ≈ 75.40 + 2π ≈ 6.28 = 26π ≈ 81.68 m³; ice-cream cone 30π ≈ 94.25 + 18π ≈ 56.55 = 48π ≈ 150.80 cm³.
• Part B: house 144 + 48 = 192 m³; L-shape: cross-section 24 − 6 = 12 + 6 = 18 cm², times 5 = 90 cm³.
• Part C: the box holds 83.2 m³ and the silo 81.68 m³, so the box holds 1.52 m³ more.
• Part D: a cylinder 4π ≈ 12.57 plus two hemispheres, one sphere, 4π/3 ≈ 4.19: 16π/3 ≈ 16.76 cm³.

Volume adds up

Give each answer exactly (in terms of π where it appears) and to two decimal places. The sketches are not to scale.

A. Curved parts

B. Flat faces

C. Which holds more?

D. Stretch: a capsule

Hidden faces

Joining two solids hides the faces that touch. Students find the surface areas of stacked boxes, a cylinder on a box and a dome, leaving out each contact area twice, once from each solid, and check one answer by listing the outside faces.

We are learning to

• add the surface areas of the parts of a composite object,
• take away each contact area twice, once for each solid it hides,
• check a surface area by listing the outside faces directly.

Success criteria

• I can find the surface area of the stacked boxes as 136 cm².
• I can explain why the contact area is taken away twice.

You need

Calculators; a few dice or blocks to glue in imagination; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Taking the contact off once: it is hidden on both solids.
• Counting the flat circle under the dome’s hemisphere: it is inside the object.
• Forgetting the floor: read whether the face an object stands on counts.

Answers

• Part A: stacked boxes 120 + 24 − 2 × 4 = 136 cm²; cylinder on a box 360 + 48π ≈ 150.80 − 2 × 9π ≈ 28.27 = 360 + 30π ≈ 454.25 cm².
• Part B: floor 4π ≈ 12.57 + wall 12π ≈ 37.70 + dome 8π ≈ 25.13 = 24π ≈ 75.40 m².
• Part C: base top 36 − 4, base bottom 36, base sides 48, cube five faces 20: total 136 cm², the same.
• Part D: 24 + 6 − 2 × 1 = 28. Gluing only ever hides faces, so it can never add surface.

Hidden faces

Count every face on the outside, including the face the object stands on. The sketches are not to scale.

A. Joined solids

B. A dome

C. Check by listing

D. Stretch: can gluing add surface?

Holes and hollows

Cutting a piece away changes volume and surface area differently. Students bore a hole through a block, find the material in a pipe, cut a corner from a cube, and sort twelve changes by what they do to surface area.

We are learning to

• subtract the volume of a removed piece,
• track the faces lost and the faces gained when a piece is removed,
• decide whether a change raises, lowers or keeps the surface area, and say why.

Success criteria

• I can find the volume and the surface area of a block with a hole through it.
• I can explain why cutting a cube from a corner leaves the surface area unchanged.

You need

Calculators; the surface area cards (cut-out sheet 2), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Treating a hole like a join: a hole adds the inner wall as well as removing two circles.
• The pipe’s material as πR²h: take the hollow away, π(R² − r²)h.
• “Less volume means less surface.” The corner cut takes volume but keeps the surface the same.

Answers

• Part A: V = 240 − 5π ≈ 15.71 = 240 − 5π ≈ 224.29 cm³; surface 236 − 2 × π ≈ 3.14 + 10π ≈ 31.42 = 236 + 8π ≈ 261.13 cm².
• Part B: pipe π(5² − 4²) × 100 = 900π ≈ 2827.43 cm³; corner cut V = 216 − 8 = 208 cm³, surface still 216 cm²: three faces of 4 cm² are lost and three equal faces appear.
• Part C: the card key, with a reason for each card, is with the mini-check answers.
• Part D: the hole adds 2πrh and removes 2πr², a change of 2πr(h − r): more surface when the block is thicker than the radius, less when it is thinner, and the same when they are equal.

Holes and hollows

A. A block with a hole

B. Hollows and corners

C. Up, down or the same?

Sort the twelve cards by what each change does to the total surface area, then write the letters.

D. Stretch: when does a hole add surface?

Units and real jobs

Real jobs need the right units: square units to paint, cubic units or litres to fill. Students convert between cubic centimetres, litres and cubic metres, paint a water tank, fill a pool, and finish with the self-check quiz.

We are learning to

• convert between cm³, mL, L and m³, and between cm² and m²,
• choose square or cubic units to suit the job,
• solve a practical problem and answer with a sensible amount to buy or a time to wait.

Success criteria

• I can find how many litres the water tank holds.
• I can work out how much paint to buy for the tank, and say why I rounded up.

You need

Calculators; the worksheet, one per student; a tin or a box to measure if you have one; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; do Parts B to D and the quiz next time.

Watch for these ideas

• “1 m³ = 100 cm³.” It is 100 × 100 × 100 = 1 000 000 cm³.
• Painting the base of a tank that stands on the ground: read which surfaces the job includes.
• Buying exactly 3.27 L: paint comes in whole tins, so round up.

Answers

• Part A: 2.5 L = 2500 cm³; 750 mL = 0.75 L; 3.2 m³ = 3200 L; 1 m² = 10 000 cm²; 4500 cm² = 0.45 m²; 1 m³ = 1 000 000 cm³.
• Part B: V = 2.88π ≈ 9.048 m³ = about 9,048 L. Paint the wall and the lid: 6.24π ≈ 19.60 m², so 39.21 m² for 2 coats, and 39.21 ÷ 12 ≈ 3.27 L: buy 4 L.
• Part C: 10 × 4 × 1.2 = 48 m³ = 48,000 L = 48 kL; at 20 L a minute that is 2,400 minutes, or 40 hours.
• Part D: surface 22 → 88 cm² (× 4); volume 6 → 48 cm³ (× 8).

Units and real jobs

A. Convert

1 mL = 1 cm³, 1 L = 1000 cm³ and 1 m³ = 1000 L. The first is done for you.

B. A water tank

A cylindrical rainwater tank has a radius of 1.2 m and a height of 2 m. It stands on the ground.

C. A pool

D. Stretch: twice as big

Solids and their volume formulas

For Day 1, one set per pair. Six cards name a solid; six give a volume formula. Match each solid to its formula.

Teacher note: the key is with the mini-check answers.

Up, down or the same?

For Day 4, one set per pair. Each card describes a change to one or more solids. Sort the cards by what the change does to the total surface area: it goes down, it goes up, or it stays the same.

Teacher note: the key is with the mini-check answers.

What we know: Composite objects

Work on your own. A calculator is allowed. Give answers exactly and to two decimal places, with units.

1. Find the volume of a cylinder of radius 4 cm and height 10 cm. ______________________
2. A tower is a cube 3 m wide with a square pyramid 2 m high on top, on the same base. Find its volume (pyramid V = (1/3)a²h). ______________________
3. Two cubes of side 4 cm are glued face to face. Find the surface area of the joined object. ______________________
4. A hole of radius 2 cm is drilled through a block 10 by 5 by 4 cm, through the 4 cm height. Find the volume left. ______________________
5. How many litres are in 2.5 m³? ______________________
6. When two solids are joined, the total surface area will, circle one:   stay the same  ·  go down  ·  go up
7. A bowl is a hemisphere of radius 9 cm. How many litres does it hold when full? ______________________
8. Every length of a box is multiplied by 3. What happens to its surface area and its volume? ______________________

Answers and marking guide

Answers with working

1. π × 4² × 10 = 160π ≈ 502.65 cm³.
2. 27 + (1/3) × 9 × 2 = 33 m³.
3. 2 × 96 − 2 × 16 = 160 cm² (ten faces of 16 cm²).
4. 200 − π × 2² × 4 = 200 − 16π ≈ 149.73 cm³.
5. 2.5 × 1000 = 2500 L.
6. go down: the faces that touch are hidden.
7. (2/3)π × 9³ = 486π ≈ 1526.81 cm³ ≈ 1.53 L.
8. Surface area × 9, volume × 27: areas grow with two lengths, volumes with three.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and units, and keeps π exact before giving a decimal.

Card keys

Formula cards (Day 1): Rectangular prism: G and A; Cylinder: D and J; Cone: E and L; Sphere: B and H; Hemisphere: K and F; Square pyramid: I and C. Surface area cards (Day 4): goes down M (two faces are hidden), P (two circles are hidden), R (two circles are hidden), U (the touching covers are hidden), X (two circles are hidden); goes up N (the inner wall is bigger than the two circles lost), Q (the cut makes two new faces), T (every cut makes two new faces), W (the tunnel wall is far bigger than its two openings); stays the same O (three faces are lost and three equal faces appear), S (the faces removed match the faces exposed), V (it is the same box).

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: simple solids, adding volumes, hidden faces, holes and hollows, and units in real jobs.