A square number is the area of a square: 6² = 6 × 6 = 36. The square root undoes it: √36 = 6, the side of a square with area 36.
A square patio uses 49 tiles. How long is each side? Squares and square roots go back and forth between a side and an area.
Multiply a number by itself and you get its square: 6×6=36, written 62 ("6 squared"). The name comes from the shape: a square with side 6 covers 36 unit squares. Numbers like 1, 4, 9, 16, 25 and 36 are perfect squares.
The square root goes the other way: 36=6, because 6×6=36. It is the side of a square with that area.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
5 squared
Picture a square with side 5. How many unit squares does it cover?
TryTry it on the board and compare the result with your prediction.
A square of 49
s is the side. The board counts the unit squares: the area.
Try it yourselfYour edits are saved in this browser.
Change the side s until the square's area is 49. What is 49? Was your prediction for 52 right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Squaring and its inverse
The square root asks: what number, times itself, makes this?
Worked exampleBefore each step, say what you think comes next.
62=6×6=36Squaring: multiply the number by itself (a square with side 6).
36=6The square root goes back: 6 × 6 = 36, so the side is 6.
1,4,9,16,25,36,49,64,81,100,121,144The perfect squares from 1² to 12². Learn them.
Now try one: 81.
Read Ava's working. What does the small 2 mean?
Work out 81.
Ava finds the area of a square with side 5 cm. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Try sides of 10, 11 and 12.
Try it yourselfYour edits are saved in this browser.
Make a square with area 144 m².
Use the board. This ticks when you get there.
Work out 121.
A square tile has area 64 cm². How long is each side, in cm?
A square garden has area 49 m². What is its perimeter, in metres?
PractiseA few on your own, to check you have it.
Squares and roots
Work out 92.
Work out 100.
A square rug has area 9 m². How long is each side, in metres?
Review: work out −4×3.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A bigger square
Then write its square root, and check by multiplying.
Try it yourselfYour edits are saved in this browser.
Need a hint?
Try a side of 13 or 14.
Another hint
The area is side × side.
Make a perfect square bigger than 144 (past the list).
Use the board. This ticks when you get there.
For teachers
Students grow a square on a grid to see a square number as an area, learn the perfect squares to 144, and use the square root as the inverse, the side of a square with a given area (AC9M7N01, nodes 1 and 2). About 40 minutes.
I can work out the square of a number and list the perfect squares to 144.
I can find the square root of a perfect square and explain what it means.
I can find the side and the perimeter of a square from its area.
Before class
Do the Run board yourself first. Students need times tables and area as side × side (Year 6).
The one idea: squaring is a number times itself (the area of a square), and the square root goes back (the side). The common mistakes are squaring as doubling (5² = 10) and rooting as halving (√36 = 18); both are planted.
Recall of the perfect squares to 144 is the hinge for the next lesson (roots between whole numbers).
Running the lesson
Predict: take a show of hands. Students who chose 10 doubled.
Run: ask how the board knew the area. Count one row, then the rows.
Investigate: write 6² = 36 and √36 = 6 on the board as a pair that undo each other.
Modify: before the perimeter task, ask what the 49 is: an area or a length?
Make: collect squares past 144 and their roots; look at the last digits.
The hard part (on purpose)
Telling squaring from doubling under time pressure. Ask students to draw the square for 5² before they answer, rather than correcting the 10.
When students get stuck
A student writes 5² = 10 (Ava's mistake). Ask: Draw a square with side 5. How many unit squares? Fix: Squared means times itself: 5 × 5.
A student halves for a root (√36 = 18). Ask: Does 18 × 18 make 36? Fix: The root times itself must give the number back.
A student gives the side when the question asks for a perimeter. Ask: How many sides go round the garden? Fix: Find the side with the root, then multiply by 4.
Stretch and support
Stretch: The last digits of square numbers are 0, 1, 4, 5, 6 or 9. Use this to show that 2023 and 1458 cannot be perfect squares.
Support: Keep the list of squares to 144 visible, and use the grid board for every root.
Where it leads
Next: Between the squares (nodes 3 and 4), then exponent notation (AC9M7N02).
Where it leads: Pythagoras (Year 8) uses squares and roots on every side of a right-angled triangle.
What each step is for
5 squared: Committing to a value for 5² shows whether they read the small 2 as 'times 2' (10) or as 'times itself' (25).
a square of 49: Growing the side until the area is 49 shows the square root as the side of a square with that area.
squaring and its inverse: Squaring multiplies a number by itself, and the square root undoes it; Ava's mistake (5² = 10) is squaring as doubling, and its mirror is halving for a root.
one change at a time: One change at a time: the biggest square in the list, a root, a side from an area in context, a perimeter (two steps), then a number that is not a perfect square.
squares and roots: Squares, roots and a side from an area, mixed; doubling is the planted slip; the review keeps integers fresh.
a bigger square: Choosing a perfect square past 100 and naming its root extends the list beyond 12² by the same reasoning.
Now you can
Work out the square of a number and list the perfect squares to 144.
Find the square root of a perfect square and explain what it means.
Find the side and the perimeter of a square from its area.