The sign rules for integers work for every fraction and decimal: work out the sign, then the size. Choose an efficient order: multiplying can be done in any order (25 × 4 first), and a hard product can be split (6 × 2.5 = 6 × 2 + 6 × 0.5).
A temperature falls half a degree an hour for three hours. A share loses 2.4 dollars over 6 days. Negative fractions and decimals are everywhere money and weather are measured.
You know the sign rules for integers: the same signs give a positive, different signs a negative, for multiplying and dividing. They work just the same for fractions and decimals: −0.5×(−4)=2, and −43×52=−103.
This lesson also looks for efficient ways to calculate: choosing the order, or splitting a number, so the arithmetic is easy.
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.
A negative half
Which sign rule applies? Then: what is 3×21?
TryTry it on the board and compare the result with your prediction.
Jumps of a half
Then try −3×21 and 3×(−21). Which sign rule does each follow?
Try it yourselfYour edits are saved in this browser.
Set the board to −3×(−21). Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Sign, size, strategy
Here is how to work with negative fractions and decimals, and two shortcuts.
Worked exampleBefore each step, say what you think comes next.
−43×52Sign first: different signs, so negative.
=−206=−103Then size: multiply tops and bottoms, then simplify.
25×(−3.7)×4=100×(−3.7)Any order: pair 25 and 4 first (commutative and associative).
=−370× 100 is easy.
6×(−2.5)=6×(−2)+6×(−0.5)Split a hard number (distributive): −12+(−3).
=−15Then add.
Try one: −43×52.
Read Leo's working. Which rule is he using?
Work out −43×52. Simplify.
Leo works out −0.4×(−0.5). Click the line where his working goes wrong.
Work out 25×(−3.7)×4 the easy way. Write your steps your own way.
Your working
A model solution
25×4=100multiplying can be done in any order, so pair the friendly numbers first
100×(−3.7)=−370different signs give a negative; × 100 moves the digits two places
Check your own working
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Start at −1 1/4 and move 2 1/2 right.
Try it yourselfYour edits are saved in this browser.
Set the board to −141+221. Where do you land?
Use the board. This ticks when you get there.
Work out −2.4÷0.6.
Work out −32÷(−94). Simplify.
Work out 8×(−1.25) by splitting −1.25. Write your steps.
Your working
A model solution
8×(−1)=−8 and 8×(−0.25)=−2split −1.25 into −1 and −0.25 (distributive law)
−8+(−2)=−10add the two parts
Check your own working
PractiseA few on your own, to check you have it.
Mixed rational calculations
Work out −1.5+(−2.75).
Work out −0.3×40.
Work out 65−(−31). Simplify.
Review: write 3515 in its simplest form.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Two negatives that make −1
Then: two negative fractions whose product is 41?
Try it yourselfYour edits are saved in this browser.
Need a hint?
Both must be between −1 and 0.
Another hint
Try −1/4 and −3/4.
Set a and b to two negative fractions (not whole numbers) that add to −1.
Use the board. This ticks when you get there.
For teachers
Students carry the integer sign rules to negative fractions and decimals (sign first, then size) and choose efficient strategies, reordering products and splitting numbers with the distributive law (AC9M8N04, node 2). About 45 minutes.
I can add, subtract, multiply and divide negative fractions and decimals.
I can work out the sign first, then the size.
I can choose an efficient order or split a number to make a calculation easier, and say why it works.
Before class
Do the reasoning task yourself first. Students need Multiplying and dividing integers, Adding and subtracting integers, and Equivalent fractions.
The one idea: the sign rules do not change for fractions and decimals; and the order of multiplying, or a split, can make the size easy. The common mistake is the addition rule for multiplying (two negatives make a negative); it is planted in Leo's working and again in Practise.
The strategy tasks use the reasoning check: students write their own steps (not marked); the answer and the reason for the key step are marked, then they compare with a model.
Running the lesson
Predict: take a show of hands. Students who chose −1 1/2 used the addition rule.
Run: ask whether it matters that the jump is a half, not a whole number.
Investigate: collect different ways to do 25 × (−3.7) × 4 and ask which is least work.
Modify: for the fraction division, recall multiplying by the reciprocal if it is new.
Make: ask for the product question too.
When students get stuck
A student uses the addition rule for multiplying (Leo's mistake). Ask: What does −3 × (−2) give? Does a decimal change that? Fix: Same signs give a positive for multiplying and dividing, whatever the numbers.
A student gets the size wrong with decimals. Ask: Can you work with whole numbers first (0.3 × 40 = 3 × 4)? Fix: Count the decimal places, or scale: 2.4 ÷ 0.6 = 24 ÷ 6.
A student works strictly left to right. Ask: Which two numbers are friendly together? Fix: Multiplication can be done in any order: pair 25 and 4.
Stretch and support
Stretch: Work out (−1/2)⁵ and (−0.2)³. When is a power of a negative number positive?
Support: Use the number line board for every addition and for multiplying by a whole number of jumps.
Where it leads
This completes AC9M8N04 (both nodes).
Where it leads: substituting negative fractions into formulas, gradient from two points, and financial calculations (Year 8 and 9).
What each step is for
a negative half: Committing to −3 × (−1/2) shows whether the integer sign rules carry over to fractions, before the board shows the jumps.
jumps of a half: Setting −3 jumps of −1/2 on the line shows the same picture as with integers: the sign rules do not care whether the jump is a whole number.
sign, size, strategy: Sign first, then size; and choose an order or a split that makes the size easy. Leo's mistake uses 'two negatives make a negative' from adding.
one change at a time: One change at a time: adding negative fractions on the line, dividing decimals, dividing fractions, splitting with the distributive law, then the edge case of a power of a negative decimal.
mixed rational calculations: Adding, multiplying and subtracting with negative fractions and decimals, mixed; the addition-rule slip planted again; the review keeps simplifying fresh.
two negatives that make −1: Choosing two negative fractions that add to −1 reverses the lesson: from the answer back to the parts, using quarters.
Now you can
Add, subtract, multiply and divide negative fractions and decimals.
Work out the sign first, then the size.
Choose an efficient order or split a number to make a calculation easier, and say why it works.