Every point on the number line is a real number, rational or irrational. To place √30, find the perfect squares either side: 25 < 30 < 36, so √30 is between 5 and 6, nearer 5. A rounded value like 5.48 is close, not exact.
A carpenter needs a brace across a 1 m by 2 m gate: √5 metres. Where is √5 on a tape measure, and how close is close enough?
The real numbers are every point on the number line: the rational numbers and the irrational ones together. Between any two numbers, however close, there are both.
To place a square root, find the perfect squares either side:
25<30<36⇒5<30<6
30 is a little nearer 25 than 36, so 30 is a bit under 5.5 (check: 5.52=30.25). To get closer, square decimals: 5.42=29.16 and 5.52=30.25; then 5.472=29.92 and 5.482=30.03, so 30≈5.48. A rounded value is an approximation; the exact value is 30.
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.
Root thirty
TryTry it on the board and compare the result with your prediction.
Place √5
Try it yourselfYour edits are saved in this browser.
Drag P to where √5 is, to the nearest tenth. Did you use squares, as for √40?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Squares either side
Read Ben's working. What times itself makes 50?
√50 is between two whole numbers. What is the smaller one?
Ben places √50 on a number line. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
More roots, and π
Need a hint?
9 < 10 < 16.
Another hint
3.1² = 9.61 and 3.2² = 10.24.
Try it yourselfYour edits are saved in this browser.
Place √10 to the nearest tenth.
Use the board. This ticks when you get there.
Place π to the nearest tenth.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Placing roots
√20 is between two whole numbers. What is the smaller one?
Write √2 rounded to 2 decimal places.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Root seven
Then mark √7 on a sketch of the number line between 2 and 3.
Try it yourselfYour edits are saved in this browser.
Need a hint?
4 < 7 < 9.
Another hint
2.6² = 6.76 and 2.7² = 7.29.
Find the first 2 decimal digits of √7, without a wrong press: square the ends first.
Use the board. This ticks when you get there.
For teachers
Students place square roots and π on the number line using the perfect squares either side and decimal squares, and decide when an approximation is enough (AC9M8N01, node 2). About 35 minutes.
I can find the two whole numbers a square root is between.
I can place a root or π to the nearest tenth.
I can tell an exact value from an approximation.
Before class
Do the Run board yourself first. Students need the previous lesson and square numbers to 144.
The one idea: perfect squares either side bound a root (25 < 30 < 36 so 5 < √30 < 6), and decimal squares narrow it. The common mistake is halving (Ben's mistake).
The Run and Modify boards are number lines with a point to drag to the nearest tenth; the Make board zooms and counts wrong presses.
Wider: the real numbers are dense: between any two there are rationals and irrationals.
Running the lesson
Predict: ask what 15² is.
Run: ask how they know √5 is nearer 2 than 3.
Investigate: ask Ben to check 25 × 25.
Modify: ask when 1.41 is good enough.
Make: ask for the squares they checked.
The hard part (on purpose)
Using squares, not halves. Ask students to write the perfect squares either side before they guess.
When students get stuck
A student halves the number. Ask: What times itself gives 50? Fix: Find the perfect squares either side.
A student cannot list perfect squares. Ask: What is 6 × 6? 7 × 7? Fix: Write the squares to 12² down the side of the page.
A student treats 1.41 as exactly √2. Ask: What is 1.41²? Fix: 1.9881: close, not 2. 1.41 is an approximation.
Stretch and support
Stretch: Find a rational number and an irrational number between 1.41 and 1.42.
Support: A list of squares 1 to 144, and a number line from 0 to 12 to mark them on.
Where it leads
This completes AC9M8N01.
Where it leads: exact answers with Pythagoras, and surds (Year 10).
What each step is for
root thirty: Committing to an interval for √40 shows whether they use the squares either side or halve the number.
place √5: Placing √5 to the nearest tenth by dragging uses the squares either side and then a decimal check.
squares either side: Perfect squares bound a root (Ben halves instead); real numbers fill the line.
more roots, and π: Placing √10 and π, ordering a root among decimals, and deciding when an approximation is good enough.
placing roots: Bounding, comparing and rounding roots, mixed; halving instead of rooting is the planted slip; the rational review keeps the classification fresh.
root seven: Finding √7 to 2 places without a wrong press makes them check each square before they choose.
Now you can
Find the two whole numbers a square root is between.