Fractions and decimals on the line

Classroom resources

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Students will be able to

• I can place positive and negative fractions, decimals and mixed numerals on a number line.
• I can explain why −1.5 is between −2 and −1, not between −1 and 0.
• I can put a mixed set of fractions and decimals in order.

Timing

Predict and Run 6 min · Investigate 12 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board and the Make task yourself first. Students need Comparing integers (further left is less) and converting between fractions and decimals, which this lesson assumes was taught in class (node 2 of the descriptor); the Investigate step opens with a quick check of it.

The one idea: split each unit into equal parts (the denominator) and count from zero, to the left for a negative. The common mistake is counting the part of a negative number back towards zero (−1.5 at −0.5); it is planted after the method, in Jo's working.

On the board, a point's readout gives the fraction and the decimal (−1 1/4 (−1.25)).

Running the lesson

1. Predict: take a show of hands. Students who chose 'between −1 and 0' are reading −1.5 as −1 then +0.5. Do not correct it yet.
2. Run: ask students to say where −1.5 will land before they let go of P.
3. Investigate: if many students miss the quick check, do the quarters (0.25, 0.5, 0.75) together first. Then model the worked example live: draw a line from −2 to 1, mark the quarters, and count aloud from zero for −3/4 and −1 1/4. Reveal the prepared lines afterwards to compare.
4. Modify: before the line from −2 to 5, ask: where is zero on this line?
5. Practise: in the ordering question, students can write every number as a decimal first.
6. Make: the board lands on −0.25; ask for other answers on paper (−0.21, −0.275). Then ask: is there a number between −0.3 and −0.25 too?

When students get stuck

Answers to the checks

1. where is −1.5?: Where does $-1.5$ go on a number line?
   Halfway between −2 and −1. −1.5 is one and a half units below zero: count 1 unit left from 0, then half a unit further left.
2. place it on the line: Put $P$ at $-1.5$. Was your prediction right?
   
3. Quick check: Quick check: write $\tfrac{3}{4}$ as a decimal.
   0.75
   One quarter is 0.25, so three quarters is 0.75.
4. Placing −3/4 and −1 1/4: Which is less, $-\tfrac{3}{4}$ or $-\tfrac{1}{4}$?
   $-\tfrac{3}{4}$
   −3/4 is three quarters left of 0, further left than −1/4, so it is less.
5. Find the mistake: Jo places $-1.5$. Click the line where her working goes wrong.
   line 4
   Line 4: −1.5 is 1.5 below zero, so the 0.5 is further below too. From −1, move half a unit left, to halfway between −2 and −1.
6. A negative fraction: Put $P$ at $-\tfrac{1}{4}$.
   
   One quarter left of zero.
7. A negative mixed numeral: Put $P$ at $-1\tfrac{3}{4}$.
   
   One unit left of zero, then three quarters further: a quarter short of −2.
8. Tenths: Put $P$ at $-0.6$.
   
   Six tenths left of zero, a bit past halfway to −1.
9. Zero is not in the middle: This line runs from $-2$ to $5$. Put $P$ at $-1\tfrac{1}{2}$ and $Q$ at $2\tfrac{1}{2}$.
   
   Find zero first, then count: P is 1½ left of 0, Q is 2½ right.
10. Two names, one place: Edge case: where do $-\tfrac{2}{4}$ and $-0.5$ go?
    At the same place: halfway between $-1$ and $0$
    −2/4 = −1/2 = −0.5: two names for one place on the line.
11. In order: Which list is in order from least to greatest?
    $-1\tfrac{1}{2},\; -1.2,\; -\tfrac{3}{4},\; 0.5$
    As decimals: −1.5, −1.2, −0.75, 0.5. Read left to right.
12. Which is greater?: Which is greater, $-0.3$ or $-0.25$?
    $-0.25$
    −0.25 is closer to zero, so it is greater: −0.3 < −0.25.
13. Halfway: What number is halfway between $-3$ and $-2$?
    -2.5
    −2.5, or −2 1/2.
14. Review: Review: which is less, $-6$ or $-9$?
    $-9$
    −9 < −6: −9 is further left.
15. always another number: Put $P$ at a number between $-0.3$ and $-0.2$.
    
    −0.25 is halfway: between any two numbers on the line, there is always another.

Solutions

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

one change at a time › A negative fraction

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

one change at a time › A negative mixed numeral

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

one change at a time › Tenths

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

one change at a time › Zero is not in the middle

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

one change at a time › Two names, one place

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

always another number

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

Stretch and support

Stretch: Find a number between −1/3 and −1/4. Then explain why there is always another number between any two.

Support: Use the board for every question: the readout gives the fraction and the decimal. A strip of paper folded into quarters helps for the fractions.

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 7 student in Australia. Help me understand ONE idea only, from the lesson "Fractions and decimals on the line".

The idea: Rational numbers on the number line: I can Students place positive and negative fractions, decimals and mixed numerals accurately on a number line, including intervals other than 0–1.

What understanding looks like:
- Correctly places a given set of rational numbers (including negatives and mixed numerals) on a number line.
- Selects and labels appropriate scale for the interval.
- Correctly orders the set from least to greatest.

Mistakes students often make here:
- A student puts −1.5 between −1 and 0 (Jo's mistake).
- A student cannot place −3/4.
- A student thinks −0.3 is greater than −0.25 because 3 is less than 25.

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 (Converting between fraction, decimal and percentage), 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 "Fractions and decimals on the line".

Exit ticket

Put in order from least to greatest: 0.4, −1 1/4, −0.8, −1/2.

−1 1/4, −0.8, −1/2, 0.4.

What each step is for

• where is −1.5?: Committing to a place for −1.5 before using the board tests whether they count the half towards zero or away from it.
• place it on the line: Dragging P to −1.5 and reading where it lands settles the prediction: the half is counted further from zero, to the left.
• count the parts: Split each unit into the denominator's number of parts and count from zero, to the left for a negative. Jo's mistake counts the 0.5 back towards zero.
• one change at a time: One change at a time: a negative fraction, a negative mixed numeral, tenths, then a line that does not have zero in the middle, then the edge case of two names for one place.
• place and order: Placing, ordering and comparing a mix of fractions and decimals; the review keeps integer ordering fresh.
• always another number: Finding a number between two close numbers reverses the lesson: from the places back to a number, and shows there is always another one in between.

Where it leads

Next in Number: the four operations with positive fractions, decimals and percentages (AC9M7N06), then negative fractions and decimals, with the same moves as integers.

Where it leads: plotting points with fractional coordinates, reading scales in science (thermometers, measuring cylinders), and the density of the rational numbers.