Split each unit into equal parts and count from zero. A negative fraction or decimal counts to the left, so −1.5 sits halfway between −2 and −1.
A tide of −0.6 m, a share price down 1¼ dollars, a recipe with ¾ of a cup: numbers between the whole numbers, above and below zero.
Between any two whole numbers on a number line there are fractions and decimals: 21, 0.3, 143. Below zero there are negative ones too: −21, −0.3, −143.
In this lesson you find out how to place them: positive and negative, as fractions, decimals and mixed numerals.
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 −1.5?
Think of a temperature of −1.5 °C. Is it colder or warmer than −1 °C?
TryTry it on the board and compare the result with your prediction.
Place it on the line
Drag P along the line, or pick it under Move and use the arrows. It lands on the halves. The board says where it is under the line.
Try it yourselfYour edits are saved in this browser.
Put P at −1.5. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Count the parts
Converting fractions and decimals, from class.
Here is how to place a negative fraction and a negative mixed numeral.
Worked exampleBefore each step, say what you think comes next.
−43The denominator is 4, so split each unit into 4 equal parts (quarters).
count 3 quarters left from 0Negative means left of zero.
−43 is between −1 and 0It is 3 of the 4 quarters from 0 to −1.
−141=−1−41A negative mixed numeral is a whole number and a part, both below zero.
count 1 unit left, then 1 quarter further leftThe quarter is further from zero, not back towards it.
−141 is between −2 and −1Just past −1, a quarter of the way to −2.
Then: which is less, −43 or −41?
Read Jo's working one line at a time. Which line moves the wrong way?
Quick check: write 43 as a decimal.
Jo places −1.5. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing. First: quarters below zero.
Need a hint?
Each unit is split into 4. Count 1 quarter left from 0.
Need a hint?
Count 1 whole unit left from 0.
Another hint
Then 3 more quarters further left.
Need a hint?
This line is split into tenths. Count 6 tenths left from 0.
Need a hint?
Zero is not in the middle of this line. Find it first.
Try it yourselfYour edits are saved in this browser.
Put P at −41.
Use the board. This ticks when you get there.
Put P at −143.
Use the board. This ticks when you get there.
Put P at −0.6.
Use the board. This ticks when you get there.
This line runs from −2 to 5. Put P at −121 and Q at 221.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Place and order
What number is halfway between −3 and −2?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Always another number
The line now shows only −1 to 0, in twentieths. Find a number between −0.3 and −0.2.
Stretch: find a number between −31 and −41. Can you always find another between any two?
Try it yourselfYour edits are saved in this browser.
Need a hint?
This line is from −1 to 0, in steps of 0.05. Find −0.3 and −0.2.
Another hint
What number is halfway between them?
Put P at a number between −0.3 and −0.2.
Use the board. This ticks when you get there.
For teachers
Students place positive and negative fractions, decimals and mixed numerals on number lines split into halves, quarters and tenths, including lines without zero in the middle, and order them (AC9M7N04, node 3). About 40 minutes.
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.
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
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.
Run: ask students to say where −1.5 will land before they let go of P.
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.
Modify: before the line from −2 to 5, ask: where is zero on this line?
Practise: in the ordering question, students can write every number as a decimal first.
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
A student puts −1.5 between −1 and 0 (Jo's mistake). Ask: Is −1.5 °C colder or warmer than −1 °C? Fix: Colder, so further left. Count 1 left, then half more left: −1.5 is further from zero than −1.
A student cannot place −3/4. Ask: How many equal parts is each unit split into? Which way is negative? Fix: Count the quarters aloud from zero: −1/4, −2/4, −3/4.
A student thinks −0.3 is greater than −0.25 because 3 is less than 25. Ask: Write both with two decimal places. Which is further left? Fix: −0.30 and −0.25: −0.30 is further from zero, so it is less.
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.
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.
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.
Now you can
Place positive and negative fractions, decimals and mixed numerals on a number line.
Explain why −1.5 is between −2 and −1, not between −1 and 0.
Put a mixed set of fractions and decimals in order.