3.746 to 2 decimal places is 3.75: look at the next digit, 6, and round up. How much to round depends on what the number is for.
A stopwatch says 12.846 seconds. Do you report 12.846, 12.85, 12.8 or 13? It depends who is asking, and what the timer can really measure.
To round a decimal to a number of decimal places, find the two numbers with that many places that it sits between, and pick the nearer one. The shortcut is to look at one digit, the one straight after the place you are rounding to:
5 or more: round up.
4 or less: round down (the digits stay).
So 3.746 to 2 decimal places: the digit after the hundredths is 6, so round up to 3.75. A number exactly halfway rounds up.
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.
Round 3.746
Which two numbers with 2 decimal places is 3.746 between?
TryTry it on the board and compare the result with your prediction.
Where does it sit?
Change the decimal places and watch the line: the two ends are always the nearest numbers with that many places.
Try it yourselfYour edits are saved in this browser.
Set the decimal places to 2. Is 3.746 before or past the halfway mark? What does it round to?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
One digit decides
Look at the digit after the place you are rounding to.
Round 8.372 to 1 decimal place.
Kai rounds 2.349 to 1 decimal place. Click the line where the working goes wrong.
ModifyChange the problem on purpose and see what happens.
How much is enough?
Try it yourselfYour edits are saved in this browser.
A stopwatch gives 12.846 seconds. Round it to the nearest tenth of a second.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Rounding
Round 5.0763 to 3 decimal places.
Round 7.0449 to 2 decimal places.
Three friends share a $50 bill: $50 ÷ 3 = 16.666… Each pays this much, to the nearest cent. (Write it as 16.xx.)
Review: write 3×102+5×100 as a number.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Lands on 4.57
Then find the smallest number you can that still rounds to 4.57.
Try it yourselfYour edits are saved in this browser.
Need a hint?
It must be nearer 4.57 than 4.56 or 4.58.
Another hint
Try 4.568.
Set the number so that, to 2 decimal places, it rounds to 4.57.
Use the board. This ticks when you get there.
For teachers
Students round decimals on a number line to a given number of decimal places, then choose how much to round for a context such as time, money, tins of paint and a ruler's accuracy (AC9M7N05, nodes 1 and 2). About 40 minutes.
I can round a decimal to a number of decimal places.
I can say which digit decides, and why halfway rounds up.
I can choose how much to round for the job, and explain my choice.
Before class
Do the Run board yourself first. Students need decimal place value (the place value lesson and Year 6).
The one idea: rounding picks the nearer of two numbers with the right number of places, and the digit straight after the place shows which. The common mistakes are cutting digits off (3.74), letting later digits push the deciding digit up (Kai's double rounding), and counting places wrongly (rounding to 1 place when 2 was asked).
Node 2 is judgement: rounding to the cent for money, up for tins of paint, to the ruler's accuracy for measurements. Accept any justified answer in class discussion.
Running the lesson
Predict: students who chose 3.74 cut the digits off; 3.7 is one place too few.
Run: slide the decimal places down and ask what the two ends of the line are each time.
Investigate: ask what Kai's 2.35 and 2.4 would mean if 2.349 were a real length.
Modify: for the paint, ask what happens if she buys 3 tins.
Make: collect the smallest and largest numbers that round to 4.57.
The hard part (on purpose)
Deciding how much to round in context. Ask 'what is this number for, and how accurately could we have measured it?' before any digits are cut.
When students get stuck
A student cuts digits off (3.74). Ask: Is 3.746 nearer 3.74 or 3.75? Where does it sit on the line? Fix: Use the number line: the next digit tells which end is nearer.
A student rounds 2.349 to 2.35 and then to 2.4 (Kai's slip). Ask: Which digit is straight after the tenths? Fix: Only that digit decides. The 9 is too far along to matter.
A student reports a ruler measurement to 4 decimal places. Ask: Could the ruler show that? Fix: Match the places to what the instrument can show.
Stretch and support
Stretch: Round at every step of 3.46 + 3.46 to the nearest whole number, then round only at the end. Do you get the same answer? Which should you trust?
Support: Keep the number line in view and say which two numbers it is between before choosing.
Where it leads
Next: Estimating, using rounded numbers to check an answer.
Where it leads: significant figures and accuracy in Year 8 and 9, and error in measurement.
What each step is for
round 3.746: Committing to an answer shows whether they cut the digits off (3.74), look at the right digit (3.75) or round the wrong place (3.7).
where does it sit?: Seeing 3.746 on a line between 3.74 and 3.75, past the halfway mark, shows why the next digit decides: 5 or more is past halfway.
one digit decides: Only the digit straight after the place matters; Kai rounds the 9 into the 4 and then again. Halfway rounds up.
how much is enough?: The right amount of rounding depends on the job: a timer, a purchase and a ruler each suggest a different place, and sometimes 'enough' means always round up.
rounding: Rounding to 3 places, to the cent in a share, and rounding to 2 places where the next digit is small; the review keeps place value fresh.
lands on 4.57: Finding a number that rounds to 4.57 shows a whole range of numbers, 4.565 up to just under 4.575, round to the same value.
Now you can
Round a decimal to a number of decimal places.
Say which digit decides, and why halfway rounds up.
Choose how much to round for the job, and explain my choice.