Rational numbers: the number line, the four operations, and signed numbers
Converting and ordering rational numbers on a number line, combining fractions, decimals and percentages with the four operations, then extending to signed integers, fractions and decimals
About three to four lessons of 45 to 60 minutes
3/4, 0.75 and 75% are the same point on the number line, wearing three different outfits
A fraction, a decimal and a percentage can all describe the exact same amount, and a rational number is simply any number, positive or negative, that can be written as one whole number divided by another. Before you can add, compare or order rational numbers confidently, you need to be able to slide between these forms and place any of them precisely on a number line.
Once that fluency is automatic, the four operations, add, subtract, multiply, divide, work on rational numbers exactly the way they always have, just with fractions and decimals allowed, and eventually negative signs too. This unit builds the number-line fluency first, then the positive-number operations, then extends every rule to signed rational numbers.
- A 75%-off sale sign, a 3/4 measuring cup, a 0.75 batting averagethree forms, the exact same value, 75 hundredths of a whole
- A temperature of -3.5 degrees, or a debt of -3/4 of a dollara rational number can sit anywhere on the number line, not just to the right of 0
- A recipe needing 2/5 of a cup plus 0.3 of a cup of an ingredientcombining a fraction and a decimal in one real calculation
- A diver's depth changing by fractions and decimals of a metre, above and below sea levelsigned rational numbers describing a real, physical quantity
What students will be able to do
Students will find equivalent fraction, decimal and percentage forms of a rational number and locate rational numbers on a number line, use the four operations with positive rational numbers to solve problems, and extend the four operations to integers and signed rational numbers.
- I can convert between a fraction, a decimal and a percentage form of the same rational number.
- I can order a mixed list of fractions and decimals from least to greatest, and locate them on a number line.
- I can add, subtract, multiply and find a percentage of positive fractions, decimals and percentages.
- I can add, subtract and multiply signed integers, fractions and decimals, applying the correct sign rule.
Standards this unit teaches
- AC9M7N04Australian Curriculum v9 (ACARA)Rational numbers on a number line
Find equivalent representations of rational numbers and represent rational numbers on a number line.
- AC9M7N06Australian Curriculum v9 (ACARA)Four operations with rationals
Use the 4 operations with positive rational numbers including fractions, decimals and percentages to solve problems using efficient calculation strategies.
- AC9M8N04Australian Curriculum v9 (ACARA)Operations with integers and rationals
Use the 4 operations with integers and with rational numbers, choosing and using efficient strategies and digital tools where appropriate.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 dividing fractions teaching unitfraction computation this unit's positive-rational operations build directly on
- Grade 6 percentages teaching unitfinding a percentage of an amount, one of the operations combined here
- Grade 6 integers & negative numbers teaching unitnegative numbers and the number line, extended here to signed fractions and decimals
- Fraction in the glossarya refresher on numerator, denominator and what a fraction represents
- Decimal in the glossarya refresher on decimal place value
Words to teach and display
- Rational number
- any number that can be written as one whole number divided by another, including negative fractions and decimals
- Equivalent representation
- a different way of writing the exact same value, such as 3/4, 0.75 and 75%
- Percentage
- a rate out of 100
- Integer
- a whole number, positive, negative or zero, with no fractional part
- Sign rule
- the rule that same-signed numbers multiply or divide to a positive result, and different-signed numbers to a negative result
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Equivalent forms and the rational number line
ConcreteAny rational number can be written as a fraction, a decimal, or (if it fits) a percentage, and converting between these forms is exactly what lets you compare or order rational numbers that are not already written the same way. Once every number in a list is in the same form (decimals are usually easiest), ordering them and placing them on a number line is straightforward.
Order 3/4, 0.65, 7/10 and 0.8 from least to greatest. Convert every value to a decimal first: 3/4 = 0.75, 0.65 stays 0.65, 7/10 = 0.7, and 0.8 stays 0.8. Now that all four are decimals, ordering is direct: 0.65, 0.7, 0.75, 0.8, which is 0.65, 7/10, 3/4, 0.8.
Order 3/4, 0.65, 7/10 and 0.8 from least to greatest.
- Convert every value to a decimal: 3/4 = 0.75, 7/10 = 0.7, and 0.65 and 0.8 are already decimals.
- Compare the decimals: 0.65 < 0.7 < 0.75 < 0.8.
Answer: From least to greatest: 0.65, 7/10, 3/4, 0.8.
- Why does converting every number to the SAME form make ordering them straightforward?
- How would you check that 3/4 and 0.75 really are the same value, without a calculator?
2. The four operations with positive fractions, decimals and percentages
PictorialA single calculation can combine fractions, decimals and percentages together, as long as you convert to a common form (usually decimals) before combining, or work through a multistep problem one operation at a time on the correct base amount.
Calculate 1/4 + 0.35: convert 1/4 to 0.25, then add: 0.25 + 0.35 = 0.6. For a multistep problem, find 20% of 3/4 of 60: first find 3/4 of 60 (60 x 3/4 = 45), then find 20% of that result (45 x 0.20 = 9), working through the phrase from left to right, on the correct amount at each step.
Calculate 1/4 + 0.35 (write as a decimal), then find 20% of 3/4 of 60.
- 1/4 + 0.35: convert 1/4 to 0.25, then add: 0.25 + 0.35 = 0.6.
- 3/4 of 60: 60 x 3/4 = 45.
- 20% of 45: 45 x 0.20 = 9.
Answer: 1/4 + 0.35 = 0.6. 20% of 3/4 of 60 = 9.
- Why is 3/4 of 60 calculated BEFORE 20% of the result, rather than the other way around, in the second problem?
- What is the quickest way to convert 1/4 to a decimal without long division?
3. Extending to signed integers, fractions and decimals
AbstractEvery rule from positive rational numbers still applies once negative signs are allowed, with one extra rule for multiplying and dividing: same signs give a positive result, different signs give a negative result. Adding a negative rational number is still a jump to the LEFT on the number line, exactly like adding a negative integer.
Calculate -3/4 + 1/2: rewrite with a common denominator, -3/4 + 2/4 = -1/4, a jump of +1/2 (to the right) starting from -3/4. For -2.5 x (-4): both factors are negative (same sign), so the result is positive: 2.5 x 4 = 10, giving +10.
Calculate -3/4 + 1/2, then -2.5 x (-4).
- Rewrite with a common denominator: -3/4 + 2/4 = -1/4.
- -2.5 x (-4): both factors are negative, same signs give a positive result, so the answer is 2.5 x 4 = 10.
Answer: -3/4 + 1/2 = -1/4. -2.5 x (-4) = 10.
- Why does adding a POSITIVE rational number to a negative starting value still move you to the RIGHT on the number line?
- If you multiplied three negative rational numbers together, would the result be positive or negative? Why?
Common misconceptions and how to address them
MisconceptionComparing fractions and decimals directly without converting them to the same form first, e.g. assuming 3/4 is smaller than 0.8 because 3 and 4 are smaller digits than 0.8's digits.
Why it happens: Comparing the raw symbols (digits in a fraction versus digits in a decimal) is not the same as comparing the actual values they represent.
How to address it: Always convert every value to the SAME form (decimals are usually quickest) before comparing or ordering. Only once they are in matching form can the digits be compared directly.
MisconceptionIn a multistep phrase like '20% of 3/4 of 60', both percentages are applied to the ORIGINAL number, 60, instead of applying each one to the result of the previous step.
Why it happens: Students see two 'of' operations and treat them as two separate calculations on the same starting amount, rather than a chain where each step feeds the next.
How to address it: Work strictly left to right through the phrase: find 3/4 of 60 first (giving 45), then find 20% of THAT result (45), not of the original 60.
MisconceptionThe sign rule for multiplying and dividing rational numbers only applies to whole numbers, so a fraction or decimal multiplication is assumed to ignore it.
Why it happens: The sign rule is usually first learned with integers, so it can feel like a separate, integer-only rule rather than a general one.
How to address it: The same-signs-positive, different-signs-negative rule applies to EVERY rational number, fractions and decimals included. Work out the sign of the answer first, separately from the size, no matter what kind of numbers are involved.
MisconceptionAdding two rational numbers with a negative sign is treated the same as subtracting, regardless of which one is negative.
Why it happens: Both operations involve a negative sign, so the two get blurred into a single 'do something negative' action.
How to address it: Adding a negative number means moving LEFT on the number line by that amount; subtracting a negative number means moving RIGHT (since it is the same as adding the opposite). Narrate the direction of the jump every time to keep the two apart.
Guided practice (with answers)
1. Order 0.3, 3/8, 0.35 and 2/5 from least to greatest.
Answer: 0.3, 0.35, 3/8, 2/5, because 0.3 = 0.3, 0.35 = 0.35, 3/8 = 0.375, and 2/5 = 0.4.
2. Calculate 2/5 + 0.15.
Answer: 0.55, because 2/5 = 0.4, and 0.4 + 0.15 = 0.55.
3. Find 25% of 2/3 of 90.
Answer: 15, because 2/3 of 90 = 60, and 25% of 60 = 15.
4. Calculate -1/2 + (-1/4).
Answer: -3/4, because -1/2 = -2/4, and -2/4 + (-1/4) = -3/4.
5. Calculate -1.5 x 6.
Answer: -9, because a negative times a positive (different signs) gives a negative result: 1.5 x 6 = 9, so -9.
6. Calculate -3/5 x (-5/6).
Answer: 1/2, because (3x5)/(5x6) = 15/30 = 1/2, and two negatives (same sign) give a positive result.
Independent practice worksheets
Practise converting and ordering rational numbers, the four operations with positive rationals, and signed-number operations, with computed, never-wrong answer keys.
Differentiation
- Give a simple fraction-to-decimal conversion table (halves, quarters, fifths, tenths) as a reference card before requiring mental conversion.
- For multistep 'of' problems, underline each 'of' phrase separately and number them 1, 2, 3 in the order to compute them.
- Practise the sign rule as its own short drill, separate from the size of the calculation: 'same signs positive, different signs negative', with whole numbers first, then fractions and decimals.
- Model every signed addition and subtraction with an actual number-line jump before moving to a rule stated in words.
- Introduce ordering a mixed list including a percentage written awkwardly, such as 137.5%, alongside fractions and decimals greater than 1.
- Pose a three-step percent-of-a-fraction-of-a-decimal problem and have students justify the order of operations.
- Investigate multiplying four negative rational numbers together and generalise the rule for when the product is positive versus negative.
- Explore why a negative fraction's numerator, denominator or the fraction as a whole can carry the negative sign, and show all three are equivalent, e.g. -3/4 = (-3)/4 = 3/(-4).
Assessment: exit ticket
A three-question exit ticket sampling ordering rational numbers, positive four operations, and signed-number operations.
1. Order 0.6, 9/20, 0.5 and 13/20 from least to greatest.
Answer: 9/20, 0.5, 0.6, 13/20, because 9/20 = 0.45, 0.5 = 0.5, 0.6 = 0.6, and 13/20 = 0.65.
2. Find 15% of 1/2 of 80.
Answer: 6, because 1/2 of 80 = 40, and 15% of 40 = 6.
3. Calculate -2/3 x 9, then -1.2 + (-0.8).
Answer: -2/3 x 9 = -6 (different signs give a negative result: 2/3 x 9 = 6). -1.2 + (-0.8) = -2 (both negative, add the magnitudes and keep the sign).
Teacher notes and timings
- Rough timing across three to four lessons: Lesson 1 equivalent forms and the number line (section 1), Lesson 2 the four operations with positive rationals (section 2), Lesson 3 extending to signed numbers (section 3), Lesson 4 mixed review plus the exit ticket.
- This unit assumes comfort with fraction computation, finding a percentage of an amount, and integers on a number line (the three Grade 6 units linked above). Revisit whichever foundation is shakiest before combining all three here.
- Language to keep repeating: convert to the SAME form before comparing; work multistep 'of' phrases strictly left to right; same signs positive, different signs negative, for every kind of rational number.
- The two assessment questions with tied decimal values (9/20 = 0.45 versus 0.55 = 11/20) are deliberately close, so double-check the working shown, not just the final order, when marking.
- Curriculum note: AC9M7N04 and AC9M7N06 (Australian Curriculum v9) are both Year 7 Number descriptors; AC9M8N04 extends the same four operations to integers one year later at Year 8. This unit is deliberately sequenced Year 7 then Year 8 content in one place, since the sign rule is the only genuinely new idea section 3 adds.
- Present mode and print both work: build the number-line jumps live with the class in Present mode for sections 1 and 3, then print the worksheets for independent practice.