The shape of the lesson
By the end of the lesson, learners can
- Squares and square roots: In the lab, slide \(n\) and watch the square grow as \(n^{2}\) unit squares; then switch to roots, type a number and see which two squares it lies between.
- Cubes and cube roots: You must recall the cubes of 1, 2, 3, 4, 5 and 10 and their cube roots.
- Other powers and roots, and the calculator: \(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\), so the fifth root of 32 is 2: \(\sqrt[5]{32} = 2\).
- Recall: squares 1 to 15 (up to 225); cubes of 1, 2, 3, 4, 5 and 10.
- Decimals: \(0.3^{2} = 0.09\) (2 d.p.), \(0.2^{3} = 0.008\) (3 d.p.).
- Signs: \(\sqrt[3]{-64} = -4\); \((-3)^{2} = 9\) but \(-3^{2} = -9\).
- Under a root: work out everything underneath first; the root does not split over \(+\) or \(-\).
- Trapping: find the squares (or cubes) either side to locate a root.
One picture every idea returns to
A root undoes a power: \(\sqrt{169} = 13\) because \(13^{2} = 169\), and \(\sqrt[3]{64} = 4\) because \(4^{3} = 64\). A square root of a positive number is taken as positive; a cube root keeps the sign, so \(\sqrt[3]{-125} = -5\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Undo the power.. Squaring and cubing build numbers up; square roots and cube roots take them back down. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Squares and square roots | In the lab, slide \(n\) and watch the square grow as \(n^{2}\) unit squares; then switch to roots, type a number and see which two squares it lies between. Project tab 1 of the lab (Powers and accuracy lab, powers and roots) and let a student drive it while the class predicts. Model the worked example Recall on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Cubes and cube roots | You must recall the cubes of 1, 2, 3, 4, 5 and 10 and their cube roots. Model the worked example Without a calculator on the board, one line at a time. | Pairs |
| 21 to 30 min | Teach and check: Other powers and roots, and the calculator | \(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\), so the fifth root of 32 is 2: \(\sqrt[5]{32} = 2\). Model the worked example Roots of a calculation on the board, one line at a time. Your turn: Work out \(\sqrt{6.4 + 3.6^{2}}\). | Cooperative Concept Mapping |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 15 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Other powers and roots, and the calculator, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Cooperative Concept Mapping
Teams receive the key terms on cards and arrange them on a large sheet, drawing and labelling the links. Compare maps between teams before showing the model map.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| \(-3^{2}\) means \(-(3^{2}) = -9\), but \((-3)^{2} = 9\). | On a calculator, put brackets round a negative number before squaring it. |
| Cubing is not multiplying by 3: \(4^{3} = 4 \times 4 \times 4 = 64\), not 12. | In the same way \(\sqrt[3]{27} = 3\), not 9. |
| \(\sqrt{12.5^{2} - 3.5^{2}}\) is NOT \(12.5 - 3.5 = 9\). | A root does not split over a plus or minus sign; work out the whole expression underneath first. |
Squares, cubes and roots: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the value of (a) \(\sqrt{196}\) (b) \(\sqrt[3]{125}\) (c) \(11^{2}\).
Without a calculator, work out (a) \(2^{5}\) (b) \(\sqrt[3]{-1000}\).
Without a calculator, work out \(\sqrt{0.09} + 0.4^{2}\).
Use your calculator to work out \(\sqrt[3]{\dfrac{9.8^{2} - 6.1}{2.3}}\). Give your answer correct to 3 significant figures.
Without a calculator: (a) Between which two consecutive integers does \(\sqrt{90}\) lie? (b) Show that \(\sqrt{60}\) is larger than \(\sqrt[3]{400}\).
A water tank is a cube with volume 1.728 m3. It stands in the middle of a square concrete slab of area 2.56 m2, leaving a border of equal width all round.
Find the width of the border.
Answer key and marking notes
- (a) 14
- (b) 5
- (c) 121
- (a) \(2 \times 2 \times 2 \times 2 \times 2 = 32\)
- (b) \((-10)^{3} = -1000\), so \(-10\)
- \(\sqrt{0.09} = 0.3\) and \(0.4^{2} = 0.16\)
- \(0.3 + 0.16 = 0.46\)
- \(9.8^{2} - 6.1 = 96.04 - 6.1 = 89.94\)
- \(89.94 \div 2.3 = 39.104\dots\)
- \(\sqrt[3]{39.104\dots} = 3.394\dots = 3.39\)
- (a) \(9^{2} = 81\) and \(10^{2} = 100\), so \(9 < \sqrt{90} < 10\).
- (b) \(7.5^{2} = 56.25 < 60\), so \(\sqrt{60} > 7.5\).
- \(7.5^{3} = 421.875 > 400\), so \(\sqrt[3]{400} < 7.5\). Hence \(\sqrt{60} > \sqrt[3]{400}\).
- Edge of tank \(= \sqrt[3]{1.728} = 1.2\) m (since \(12^{3} = 1728\)).
- Side of slab \(= \sqrt{2.56} = 1.6\) m (since \(16^{2} = 256\)).
- Border \(= (1.6 - 1.2) \div 2 = 0.2\) m.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Find the value of \(\sqrt[3]{-64}\).
- A\(-4\)
- B\(4\)
- C\(-8\)
- DIt has no value
Between which two consecutive whole numbers does \(\sqrt{90}\) lie?
- A9 and 10
- B8 and 9
- C45 and 46
- D30 and 31
Work out \(\sqrt{13^{2} - 5^{2}}\).
- A12
- B8
- C144
- D13.9
Work out \((-5)^{2} - 2^{3}\).
- A17
- B\(-33\)
- C4
- D33
Use a calculator to work out \(\sqrt[3]{\dfrac{6.2^{2} + 3.5}{0.7}}\), giving your answer correct to 3 significant figures.
- A3.91
- B4.96
- C7.74
- D59.9
A cube has a volume of 343 cm3. Find the area of one face of the cube.
- A49 cm2
- B7 cm2
- C294 cm2
- D114.3 cm2
Quiz key
\((-4) \times (-4) \times (-4) = 16 \times (-4) = -64\), so \(\sqrt[3]{-64} = -4\). A cube root keeps the sign, so 4 is wrong (\(4^{3} = +64\)). \(-8\) comes from taking a square root of 64 instead of a cube root. Only SQUARE roots of negative numbers have no value; cube roots of negative numbers always exist.
\(9^{2} = 81\) and \(10^{2} = 100\). Since \(81 < 90 < 100\), \(\sqrt{90}\) lies between 9 and 10 (it is 9.49 to 3 s.f.). 8 and 9 would need 90 to lie between 64 and 81. 45 and 46 halves 90 instead of square rooting, and 30 and 31 divides by 3.
\(13^{2} - 5^{2} = 169 - 25 = 144\), and \(\sqrt{144} = 12\). The tempting 8 is \(13 - 5\): a root does not split over a minus sign. 144 forgets the square root, and 13.9 is \(\sqrt{169 + 25}\), adding instead of subtracting.
\((-5)^{2} = (-5) \times (-5) = 25\) and \(2^{3} = 2 \times 2 \times 2 = 8\), so the answer is \(25 - 8 = 17\). \(-33\) treats \((-5)^{2}\) as \(-25\), but the brackets mean the negative is squared too. 4 is \(10 - 6\), multiplying by the power instead of using it as a power, and 33 adds the 8 instead of subtracting it.
\(6.2^{2} + 3.5 = 38.44 + 3.5 = 41.94\), and \(41.94 \div 0.7 = 59.914\ldots\). Then \(\sqrt[3]{59.914\ldots} = 3.913\ldots = 3.91\) (3 s.f.). The tempting 4.96 takes the cube root of the top only and then divides by 0.7: the root covers the whole fraction. 7.74 is a square root instead of a cube root, and 59.9 forgets the root altogether.
Edge length \(= \sqrt[3]{343} = 7\) cm, because \(7 \times 7 \times 7 = 343\). One face is a 7 cm by 7 cm square, so its area is \(7^{2} = 49\) cm2. 7 is the edge length, not an area. 294 is the total surface area of all six faces (\(6 \times 49\)), and 114.3 is \(343 \div 3\), treating 'cube root' as dividing by 3.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.