Irrational numbers on the line

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min

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

1. Predict: ask what 15² is.
2. Run: ask how they know √5 is nearer 2 than 3.
3. Investigate: ask Ben to check 25 × 25.
4. Modify: ask when 1.41 is good enough.
5. 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

Answers to the checks

1. root thirty: Between which two whole numbers is √40?
   6 and 7: 6² = 36 and 7² = 49, and 40 is between them. (Halving 40 gives 20, but 20² = 400.)
2. place √5: Drag P to where √5 is, to the nearest tenth. Did you use squares, as for √40?
   
   4 < 5 < 9, so 2 < √5 < 3; 2.2² = 4.84 and 2.3² = 5.29, so √5 ≈ 2.2 (2.236…).
3. Squares either side: √50 is between two whole numbers. What is the smaller one?
   7
   7² = 49 and 8² = 64, so √50 is between 7 and 8 (just over 7).
4. Find the mistake: Ben places √50 on a number line. Click the line where his reasoning goes wrong.
   line 2
   Line 2: a square root is the number that multiplies by itself to give 50, not half of it. 7² = 49, so √50 is just over 7.
5. Real numbers: Which is true of the number line?
   Every point is a real number, and there are rational and irrational numbers between any two points
   The real numbers fill the whole line, rationals and irrationals mixed together.
6. Root 10: Place √10 to the nearest tenth.
   
   √10 = 3.162…, so 3.2 to the nearest tenth.
7. π: Place π to the nearest tenth.
   
   π = 3.14159…, so 3.1 to the nearest tenth.
8. In order: Put these in order, smallest first: √3, 1.7, 7/4.
   1.7, √3, 7/4
   1.7 < 1.732… < 1.75.
9. Good enough: A right triangle has legs of 1 m, so its long side is exactly √2 m. A carpenter cuts a brace. What should she use?
   1.41 m: close enough to cut, though √2 m is the exact length
   Exact in the maths (√2), rounded for the job: 1.41 m to the centimetre, or 1.414 m to the millimetre.
10. Bounds: √20 is between two whole numbers. What is the smaller one?
    4
    16 < 20 < 25, so √20 is between 4 and 5.
11. Compare: Which is bigger: √8 or 2.8?
    √8
    2.8² = 7.84 < 8, so √8 > 2.8 (√8 = 2.828…).
12. Round: Write √2 rounded to 2 decimal places.
    1.41
    √2 = 1.41421…, so 1.41.
13. Review: irrational: Review: which number is irrational?
    √12
    12 is not a perfect square.
14. root seven: Find the first 2 decimal digits of √7, without a wrong press: square the ends first.
    
    √7 = 2.6457…: the first two decimal digits are 64, and rounded to 2 decimal places it is 2.65.

Solutions

One possible solution for each task. Other correct programs are fine.

more roots, and π › Root 10

Open task: any program that does what the step asks.

more roots, and π › π

Open task: any program that does what the step asks.

more roots, and π › In order

Open task: any program that does what the step asks.

more roots, and π › Good enough

Open task: any program that does what the step asks.

root seven

Open task: any program that does what the step asks.

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.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Irrational numbers on the line".

The idea: Locating irrational numbers on the real number line: I can Students approximately locate irrational numbers on the real number line and understand the real number system as including both rational and irrational numbers.

What understanding looks like:
- Correctly locates an irrational number between two consecutive integers on the number line.
- Describes the real number system as including both rationals and irrationals.
- Does not treat the decimal approximation as the exact value.

Mistakes students often make here:
- A student halves the number.
- A student cannot list perfect squares.
- A student treats 1.41 as exactly √2.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Irrational numbers as non-terminating, non-recurring decimals), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Irrational numbers on the line".

Exit ticket

Between which two whole numbers is √70? Which is it nearer?

64 < 70 < 81, so between 8 and 9, nearer 8 (√70 ≈ 8.37).

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.

Where it leads

This completes AC9M8N01.

Where it leads: exact answers with Pythagoras, and surds (Year 10).