At a glance. One lesson of about 60 minutes. Students read a power as a count of factors, link squares and cubes to area and volume, and place square and cube roots between whole numbers.
Calculator: no calculator. Nothing needs one. Students should know squares to 15 squared and a few cubes by sight, and a calculator would remove the reason to see the pattern.
What students will learn
Know
- The base, the exponent and what the exponent counts.
- Squared and cubed; the square numbers to 15 squared and the cubes 1, 8, 27, 64, 125.
- The square root and cube root as the inverse of squaring and cubing.
Understand
- A power is repeated multiplication and grows quickly.
- A root is the side of a square (or edge of a cube) of a given area (or volume).
- Most roots are not whole numbers and sit between two whole numbers.
Do
- Evaluate powers with a small whole-number base and exponent.
- Write a repeated multiplication as a power.
- Find square and cube roots of perfect squares and cubes and place other roots between whole numbers.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: evaluating powers and roots in familiar situations (questions 1 to 5).
- Criterion B. Investigating patterns: the power figure. Ask for the pattern in the running products and what happens each time the exponent increases by 1.
- Criterion C. Communicating: explaining why 2 to the power 4 is not 2 times 4, in words and in symbols (questions 1 and 9).
- Criterion D. Applying mathematics in real-life contexts: area of a square and repeated doubling (questions 5 and 7).
The one thing to do with the figure
Ask for a prediction on the first figure. “What is 3 to the power 4?” Take 12 and 81 from the room, then press the exponent up one step at a time.
Then ask what the figure does when the exponent goes from 4 to 5 for base 5 (625 to 3125). Students should say it is multiplied by 5 each step. On the second figure, drag the area from 49 to 64 and ask where the side jumps.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea and roots; the second figure, square then cube. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, return to the figure and count the factors. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 49; 27; 0.04. |
|---|---|
| Worked example | √70 is between 8 and 9. |
| Faded example | ∛40 is between 3 and 4. |
| 1. Value of a power | C, 16. A is adding, B is multiplying base by exponent, D has one factor too many. |
| 2. Evaluating a cube | 64. 4 × 4 × 4. |
| 3. Square root | 12. 12 × 12 = 144. |
| 4. Cube root | 4. 4 × 4 × 4 = 64. |
| 5. Area of a square | 169. 13 × 13 = 130 + 39. |
| 6. Locating a root | B. 7² = 49 and 8² = 64. C and D halve instead of rooting. |
| 7. Doubling | 10 folds. 2⁹ = 512, 2¹⁰ = 1024. |
| 8. Counting perfect squares | 14. 14² = 196 and 15² = 225. |
| 9. Swapped base and exponent | C. A: sees the same digits. B and D: multiply base by exponent, 2 × 5. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 8 or 6 in question 1 | Multiplied base by exponent (2 × 4), or added. | Write the repeated multiplication out in full before calculating. |
| 12 in question 2 | 4 × 3, base times exponent. | Ask how many 4s are being multiplied. |
| 72 in question 3 | Halved instead of finding the root. | Ask what number times itself is 144; use a square on grid paper. |
| 8 in question 4 | Found the square root. | Draw a cube: three equal edges. |
| 52 in question 5 | Perimeter instead of area. | Ask what area means: the number of unit squares. |
| C or D in question 6 | Halved 50 (to 25) instead of finding which two squares surround it. | Square 7 and 8 and see where 50 sits. |
Questions worth asking
- Why is 3 to the power 4 not 3 times 4? Because the exponent counts factors. Listen for “four 3s multiplied”.
- Why do we say squared and cubed? They give the area of a square and the volume of a cube. The names come from the shapes.
- Is the square root of 50 a whole number? How do you know? No: 7 × 7 is 49 and 8 × 8 is 64, so it is between 7 and 8.
- Is there a number whose square is negative? No. A positive times a positive is positive, and in Year 8 you will see that a negative times a negative is positive too, so no square you meet at school is negative.
Differentiation
A way in for anyone stuck
- Give a table of squares from 1 to 15 and cubes 1 to 5 for the whole lesson.
- Draw each power as a grid or a row of boxes before calculating.
- Practise the running product on the first figure before the roots figure.
Stretch
- Question 7, then: how many layers after 30 folds of paper? Why is that impossible in real life? (2³⁰ is over a billion, so a sheet could not be folded 30 times.)
- Which is greater, 2¹⁰ or 10³? How do you know without a calculator? (1024 > 1000.)
Students learning in English as an additional language
“Squared” and “cubed” are used for shapes and for numbers. Point to the shape each time you say the word.
| Word | Plain meaning | Example |
|---|---|---|
| power | a number written with a small raised number, such as 3 with 4 raised | 3 to the power 4 is 81 |
| base | the number that is multiplied by itself | In 5² the base is 5 |
| exponent (index) | the raised number that counts the factors | In 5² the exponent is 2 |
| square number | a whole number multiplied by itself | 36 = 6 × 6 |
| cube number | a whole number used as a factor three times | 27 = 3 × 3 × 3 |
| square root | the number that multiplies by itself to give a number | the square root of 49 is 7 |
| cube root | the number used as a factor three times to give a number | the cube root of 64 is 4 |
Sentence starters for explaining aloud or in writing:
- … to the power … means … used as a factor … times.
- The square root of … is between … and … because …
- … is a perfect square because … × … = …
Two things not to say
“Raise to the power means multiply by the number.” It means use the base as a factor that many times.
“The root is half of the number.” Halving and rooting give different results; use a square on a grid to show it.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Writing a powerWrite 6 × 6 × 6 × 6 as a power and find its value.
6⁴ = 1296. - Earlier workWhat is the value of the 6 in 3.065?
0.06, six hundredths.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- GardenA square garden has area 225 m². How long is its fence?
Side 15 m, so 4 × 15 = 60 m. - BoxA cube-shaped box has edges of 5 cm. How many 1 cm cubes fill it?
125.
3Challenge
Reasoning, generalising and working backwards.
- CountingHow many whole numbers n have 50 < n² < 150?
5: 8, 9, 10, 11 and 12 (64, 81, 100, 121, 144). - GrowthA rumour starts with one person. Each hour, everyone who heard it in the previous hour tells 3 new people. How many people hear it for the first time in hour 5?
3⁵ = 243.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use a printed square-number table and squared paper for the root figure.