A rational number is a fraction: its decimal stops or repeats. An irrational number cannot be written as a fraction: its decimal never stops and never repeats. √2, √3 and π are irrational; √16 = 4 is not.
The diagonal of a 1 m square tile is exactly √2 metres long. You can measure it, but you can never write it down as a fraction or a decimal that ends.
A rational number can be written as a fraction ba of integers (whole numbers or their negatives, with b=0): 0.25, 71, −3, 0.3. Its decimal terminates or recurs.
An irrational number cannot be written as a fraction. Its decimal goes on forever without repeating. Examples: 2=1.41421356…, 3, 5, and π=3.14159265… The square root of a whole number is irrational unless the number is a perfect square: 16=4 is rational.
Irrational numbers are ordinary numbers you can calculate with; we use the exact form (2, π) or a rounded value (1.414, 3.14).
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.
The digits of √2
TryTry it on the board and compare the result with your prediction.
Zoom in on √2
Press the piece of the line that √2 is in. A wrong piece shows you the squares of its ends.
Try it yourselfYour edits are saved in this browser.
Find the first 3 decimal digits of √2 by zooming in. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Rational or irrational
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
a fraction?Yes: rational.
decimal stops or repeats?Yes: rational (it is a fraction).
n,n a perfect square?√16 = 4: rational. √15: irrational.
π,2,3Never stop, never repeat: irrational.
Read Ana's working. What is √16?
Rational or irrational?
1/7
0.25
√9√9 = 3.
√5
π
0.333…It is 1/3.
22/7A fraction, close to π but not π.
Ana sorts √16. Click the line where her reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
Other numbers
Need a hint?
1.7² = 2.89 and 1.8² = 3.24.
Try it yourselfYour edits are saved in this browser.
Find the first 3 decimal digits of √3.
Use the board. This ticks when you get there.
Zoom in on 1/7 to 6 places. What do its digits do?
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Rational or irrational
Review: what is √49?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Π
Then explain why 3.1416 is not exactly π.
Try it yourselfYour edits are saved in this browser.
Need a hint?
π is a little more than 3.
Another hint
Its first two decimal digits are 14.
Find the first 4 decimal digits of π.
Use the board. This ticks when you get there.
For teachers
Students zoom in on √2, √3, π and 1/7 one digit at a time, and sort numbers into rational (fractions, decimals that stop or repeat) and irrational (AC9M8N01, node 1). About 35 minutes.
I can say what makes a number rational or irrational.
I can sort numbers such as √9, √5, π and 0.333… correctly.
I can explain why a rounded value is not the exact number.
Before class
Do the Run board yourself first. Students need recurring decimals (previous unit) and square roots (Number Foundations).
The one idea: rational numbers are fractions, so their decimals stop or repeat; irrational numbers never stop and never repeat. The common mistake is calling every square root irrational (Ana's mistake).
The board checks each zoom with squares for square roots, and with "too big, too small" for π.
Proving √2 is irrational is beyond Year 8; the board is evidence, not proof (say so). Wider: historical approximations of π (Egyptian, Chinese, Indian, Greek), the golden ratio.
Running the lesson
Predict: ask what a repeating √2 would mean (it would be a fraction).
Run: ask how they know 1.4 is too small.
Investigate: ask Ana to work out √16.
Modify: ask how 1/7's digits differ from √3's.
Make: ask what exact means here.
The hard part (on purpose)
Believing a decimal can go on forever without repeating. Let the zooming carry on past where students expect it to settle.
When students get stuck
A student calls a long calculator decimal irrational. Ask: Does it repeat? Is it a fraction? Fix: Repeating means rational; only endless and non-repeating is irrational.
A student calls every root irrational. Ask: What is √16? Fix: Perfect squares have rational roots.
A student thinks π is 22/7. Ask: What is 22 ÷ 7 to 4 places? Fix: 3.1428… against 3.1415…: close, not equal.
Stretch and support
Stretch: Find how ancient Egyptian, Chinese and Indian mathematicians approximated π. Which approximation was closest?
Support: A card with two columns: 'stops or repeats: rational' and 'never stops, never repeats: irrational'.
Where it leads
This starts AC9M8N01.
Where it leads: placing irrational numbers on the number line (next lesson), Pythagoras' exact answers, and surds (Year 10).
What each step is for
the digits of √2: Committing to whether √2's digits stop or repeat shows whether they expect every number to be a neat fraction.
zoom in on √2: Finding √2 one digit at a time, with the squares as evidence, shows the process never has to end.
rational or irrational: Sorting by the decimal's behaviour; perfect squares are rational (Ana calls every root irrational); irrational numbers are usable numbers.
other numbers: Zooming in on √3 and 1/7 compares an irrational number with a repeating one; π and 22/7 differ; irrationals can combine to a rational.
rational or irrational: Classifying and using roots and decimals, mixed; calling every root irrational is the planted slip; the square root review keeps perfect squares fresh.
π: Finding π to 4 places with only 'too big, too small' feedback applies the zooming method to a number with no squares to help.
Now you can
Say what makes a number rational or irrational.
Sort numbers such as √9, √5, π and 0.333… correctly.
Explain why a rounded value is not the exact number.