43 is not a perfect square, but it sits between 36 and 49, so √43 is between 6 and 7. Closer to 49 or to 36? Square a decimal to find out: 6.6² ≈ 43.6.
A square vegetable patch needs 43 m² of soil. No whole number squares to 43, but a builder still needs to know the side, to the nearest 10 cm.
Most numbers are not perfect squares, but they still have square roots. 43 is the side of a square with area 43: no whole number, but somewhere between two.
The method: find the perfect squares either side. 36<43<49, and 36=6, 49=7, so 43 is between 6 and 7. Then try decimals to get closer.
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.
Where is √43?
Which perfect squares are near 43?
TryTry it on the board and compare the result with your prediction.
As close as you can
The board says how far your square is from 43, over or under.
Try it yourselfYour edits are saved in this browser.
The dashed square has area 43. Change the side (in tenths) to make your square as close to 43 as you can. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
The squares either side
Here is the method for 43.
Worked exampleBefore each step, say what you think comes next.
36<43<49Find the perfect squares either side of 43.
6<43<7Take their roots: √43 is between 6 and 7.
6.52=42.256.62=43.56Square some tenths. 6.5 is under and 6.6 is over.
43≈6.643.56 is closer to 43 (0.56 over) than 42.25 (0.75 under).
Now try one: between which whole numbers is 70?
Read Noah's working. Square his answer to check it.
Your working can be short notes. Then choose the reason for the key step.
70 is between 8 and which whole number?
Noah estimates 50. Click the line where his reasoning goes wrong.
A square garden bed has area 90 m². Between which two whole numbers of metres is its side? Write each step with its reason.
Your working
A model solution
92=81 and 102=100these are the perfect squares either side of 90
81<90<100, so 9<90<10a bigger square has a bigger side
Check your own working
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
4² = 16 and 5² = 25, so start between 4 and 5.
Another hint
Compare how far 4.4² and 4.5² are from 20.
Try it yourselfYour edits are saved in this browser.
The dashed square has area 20 m². Make your square as close to it as you can.
Use the board. This ticks when you get there.
A square has perimeter 36 cm. What is its area, in cm²?
PractiseA few on your own, to check you have it.
Roots between
Estimate 30 to one decimal place. (5² = 25 and 6² = 36.)
A square floor needs 64 square tiles, each 1 m by 1 m. How much skirting board goes round the edge, in metres?
Review: work out 122.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
An area in the thirties
Then find the smallest and the biggest side (in tenths) that work.
Try it yourselfYour edits are saved in this browser.
Need a hint?
5² = 25 is too small and 6² = 36 is inside. Are there others?
Another hint
Step in tenths: 5.5² = 30.25.
Make a square whose area is between 30 and 40.
Use the board. This ticks when you get there.
For teachers
Students bound a square root between two whole numbers using the perfect squares either side, estimate it to one decimal place by squaring tenths on a grid board, and solve area and perimeter problems with roots (AC9M7N01, nodes 3 and 4). About 40 minutes.
I can say which two whole numbers a square root is between, with a reason.
I can estimate a square root to one decimal place by squaring decimals.
I can solve problems with the side, area and perimeter of a square.
Before class
Do the Run board yourself first. Students need the perfect squares to 144 (previous lesson) and multiplying decimals (calculators are fine for squaring tenths).
The one idea: find the perfect squares either side, take their roots, and the answer lies between. The scaffold matters: without it students guess, or halve (Noah's mistake).
The reasoning check never marks the free working; only the reason for the key step is chosen and marked.
Running the lesson
Predict: students who chose 21 and 22 halved. Ask them to square 21.
Run: ask whether 6.6 is a little over or under, and how they decided between 6.5 and 6.6.
Investigate: chant the routine: squares either side, roots, between.
Modify: in the perimeter task, ask which is known first: the side or the area?
Make: collect sides that work and order them; the ends approach √30 and √40.
The hard part (on purpose)
Choosing between two neighbouring decimals (6.5 or 6.6). Ask which square is closer to the target and by how much, and let them compare the two distances.
When students get stuck
A student halves the number (Noah's mistake). Ask: Square your answer. Do you get the number back? Fix: The root times itself gives the number; use the squares either side.
A student finds the interval but cannot get closer. Ask: Square 6.5. Is it over or under 43? Fix: Square tenths and compare the distances.
A student mixes up side and area in a problem. Ask: Is this number a length or an area? Fix: Root for area to side; square for side to area.
Stretch and support
Stretch: Between which two whole numbers is √1000? Estimate it to one decimal place, then check with a calculator.
Support: Keep the list of squares to 144 visible and use a number line marked with 36, 43 and 49.
Where it leads
Next: exponent notation, prime factors and HCF and LCM (AC9M7N02).
Where it leads: Pythagoras (Year 8) gives lengths like √20, and estimating them first checks a calculator answer.
What each step is for
where is √43?: Committing to an interval for √43 shows whether they use the squares either side, or halve, or think of 43's digits.
as close as you can: Stepping the side in tenths until the square is as close to the target as possible estimates a root to one decimal place, and shows it is between the whole numbers found first.
the squares either side: Bounding a root by the perfect squares either side gives an interval with reasons; Noah's halving is the squaring-as-doubling mistake run backwards.
one change at a time: One change at a time: a new target on the board, a bigger root, an area from a perimeter (backwards), then which numbers are perfect squares at all.
roots between: An estimate to one decimal place, an interval, and a two-step problem, mixed; halving is the planted slip; the review keeps squares fresh.
an area in the thirties: Choosing a side whose square lands between 30 and 40 turns the bounding argument round: the side must be between √30 and √40.
Now you can
Say which two whole numbers a square root is between, with a reason.
Estimate a square root to one decimal place by squaring decimals.
Solve problems with the side, area and perimeter of a square.