Operations with rational numbers: a week of ready-to-teach maths
Five lessons for Year 7 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a fraction wall to cut into strips, 16 decimal loop cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets: the fraction wall, one per pair (Days 1 to 3), and the loop cards, one set per group of up to 16 (Day 4).
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
Fractions add only when their pieces are the same size, so 1/3 + 1/4 is rewritten as 4/12 + 3/12 = 7/12. Multiplying takes a part of a part: 2/3 of 3/4 is 6/12 = 1/2, smaller than both. Dividing asks how many fit: 2 ÷ 1/3 = 6, because six thirds fit in 2, which is why dividing by a fraction is the same as multiplying by its reciprocal. Decimals and percentages are other names for the same numbers, and the efficient strategy is often to switch names: 25% of 48 is a quarter of 48, and 0.25 × 48 is the same thing.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.
How we write the working
Fractions are written with a slash, 3/4, and mixed numbers with a space, 2 1/4. Answers are given in simplest form, as mixed numbers when they are over 1. Show the common denominator or the reciprocal step every time. Calculators are kept for checking on Day 5; the week is about choosing a quick way and doing it by hand.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Adding and subtracting fractions | Rewrite fractions over a common denominator to add and subtract them, including mixed numbers | A common denominator |
| 2 | Multiplying fractions | Multiply fractions as a part of a part, simplify first where it helps, and find a fraction of an amount | Multiplying as a part of a part |
| 3 | Dividing fractions | Divide by a fraction as how many fit, then by multiplying by the reciprocal, and solve sharing and cutting problems | Dividing by a unit fraction |
| 4 | Working with decimals | Choose efficient strategies for the four operations with decimals and check with an estimate | Sliding between forms |
| 5 | Percentages at work | Find percentages of amounts, discounts and GST with efficient strategies, and compare offers | A percentage of an amount; self-check quiz |
How the week builds
Days 1 to 3 take the four operations through fractions: common pieces to add and subtract, a part of a part to multiply, and how many fit to divide. Day 4 carries efficient strategies to decimals, and Day 5 to percentages, where switching between forms is often the quickest route. It builds on equivalent fractions, decimals and percentages, and on rounding and estimating to check, from the last two units, and it prepares the ratio and financial problems later in Year 7.
Materials for the week (one trip)
- From the classroom: pencils, coloured pencils, scissors, rulers, this pack printed.
- Cut-out sheet 1, one per pair: a fraction wall, cut into strips (Days 1 to 3).
- Cut-out sheet 2, one set per group of up to 16: decimal loop cards (Day 4).
- Calculators for checking on Day 5 only. No other equipment to buy.
Words for the word wall
common denominator, lowest common multiple, equivalent, mixed number, improper fraction, product, of, reciprocal, quotient, simplify, decimal, percentage, discount, GST, efficient, estimate
Dear families
This week in maths, Year 7 is adding, subtracting, multiplying and dividing fractions, decimals and percentages, and choosing quick ways to do it. For example, 25% of $48 is a quarter of $48, or $12; and 2 ÷ 1/3 = 6, because six thirds fit into 2.
Try this at home
- Halve or double a recipe together. What is half of 3/4 cup? (3/8 cup.)
- At a sale, ask: what is 30% off $60? (Take off $18, so $42.)
- Ask: how many quarter-hour episodes fit in 2 1/2 hours? (10.)
- Ask: is 10% of 70 the same as 70% of 10? (Yes, both are 7.)
Calculations this week
Fill one row a day with a real calculation, the quick way you used, and the answer.
| Day | The calculation | My quick way | Answer |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Operations with rational numbers teaching pack · seegongsik.com/au/y7/number/AC9M7N06/pack
Adding and subtracting fractions
Fractions add only when their pieces are the same size. Students rewrite fractions over a common denominator, ideally the lowest one, to add and subtract them, including mixed numbers.
We are learning to
- rewrite two fractions over a common denominator,
- add and subtract fractions and mixed numbers, giving answers in simplest form,
- choose the lowest common denominator to keep the numbers small.
Success criteria
- I can work out 2/5 + 1/3 = 11/15 and show the common denominator.
- I can say why 1/3 + 1/4 is not 2/7.
You need
The fraction wall from cut-out sheet 1, cut into strips (one per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Same-size pieces Show the board (see the box). Then line up the 1/3 and 1/4 strips against the twelfths strip: 4/12 and 3/12. Ask: “Why can we not just add the tops and the bottoms to get 2/7?” |
| 15 min | Find the common pieces Model 2/5 + 1/3 = 6/15 + 5/15 = 11/15, then a subtraction with mixed numbers, 4 1/3 − 1 1/2, by renaming as improper fractions: 13/3 − 3/2 = 26/6 − 9/6 = 17/6 = 2 5/6. Ask: “Which denominator could you use for 1/6 + 3/4: 24 or 12? Which is quicker?” |
| 35 min | Add and subtract Students complete Parts A to C, using the strips to check one answer in each part; early finishers do Part D. |
Two shorter sessions instead? Split after Part A; start the second with the mixed number model.
Watch for these ideas
- Adding the tops and the bottoms: 1/3 + 1/4 is not 2/7, which is less than 1/3.
- Changing the denominator but not the numerator: 2/5 = 6/15, not 2/15.
- Subtracting part by part when the second fraction is bigger: rename as improper fractions first.
Answers
- Part A: 1/2 + 2/5 = 5/10 + 4/10 = 9/10; 3/4 − 1/6 = 9/12 − 2/12 = 7/12; 2/3 + 1/4 = 8/12 + 3/12 = 11/12; 5/6 − 3/8 = 20/24 − 9/24 = 11/24; 3/10 + 4/15 = 9/30 + 8/30 = 17/30; 7/8 − 2/3 = 21/24 − 16/24 = 5/24.
- Part B: 2 1/3 + 1 3/4: 7/3 + 7/4 = 28/12 + 21/12 = 49/12 = 4 1/12; 3 1/2 − 1 2/3: 7/2 − 5/3 = 21/6 − 10/6 = 11/6 = 1 5/6; 4 2/5 + 2 7/10: 22/5 + 27/10 = 44/10 + 27/10 = 71/10 = 7 1/10; 5 1/4 − 2 5/6: 21/4 − 17/6 = 63/12 − 34/12 = 29/12 = 2 5/12.
- Part C: 3/4 + 2/3 = 17/12 = 1 5/12 cups; 2 1/2 − 1 3/5 = 9/10 m.
- Part D: 18, the lowest common multiple of 6 and 9; 54 works but makes bigger numbers. 5/6 + 4/9 = 15/18 + 8/18 = 23/18 = 1 5/18.
Adding and subtracting fractions
A. Find a common denominator, then add or subtract
B. Mixed numbers
Rename as improper fractions, or work with the wholes and the parts. Give answers as mixed numbers.
C. Problems
A recipe uses 3/4 cup of milk and 2/3 cup of water. How much liquid is that altogether? ________________________
A rope is 2 1/2 m long. 1 3/5 m is cut off. How much is left? ________________________
D. Stretch
To add 5/6 + 4/9, would you use 54 or 18 as the common denominator? Why? Work it out. ____________________________
Multiplying fractions
Multiplying fractions takes a part of a part: tops multiply and bottoms multiply. Students multiply and simplify, find a fraction of an amount, and multiply mixed numbers by renaming them first.
We are learning to
- multiply two fractions, and explain the product with an area picture,
- simplify before multiplying when a top and a bottom share a factor,
- find a fraction of an amount and multiply mixed numbers.
Success criteria
- I can work out 3/5 × 2/3 = 2/5 and draw it as part of a square.
- I can say why 3/5 × 2/3 is smaller than both fractions.
You need
Grid paper or scrap paper for area pictures, the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | A part of a part Show the board (see the box). Then draw 3/5 × 2/3 together: shade 3 of 5 columns and 2 of 3 rows; the overlap is 6 of 15 cells = 2/5. Ask: “Why is the answer smaller than both 3/5 and 2/3?” |
| 15 min | Simplify first Show 4/9 × 3/8: cancel 4 with 8 and 3 with 9 before multiplying, giving 1/3 × 1/2 = 1/6. Then 2 1/2 × 1 1/5 = 5/2 × 6/5 = 3. Ask: “Why can we not multiply 2 × 1 and 1/2 × 1/5 and add them?” |
| 35 min | Products and amounts Students complete Parts A to C, drawing an area picture for one product; early finishers do Part D. |
Two shorter sessions instead? Split after Part A; start the second with the mixed number model.
Watch for these ideas
- Expecting multiplying to make bigger. Taking a part of a part makes smaller when both are less than 1.
- Finding a common denominator first. It is not needed for multiplying.
- Multiplying the wholes and the fractions of mixed numbers separately: rename as improper fractions.
Answers
- Part A: 2/3 × 4/5 = 8/15; 3/4 × 2/9 = 1/6; 5/6 × 3/10 = 1/4; 4/7 × 7/8 = 1/2; 3/8 × 4 = 3/2 = 1 1/2; 2/5 × 15 = 6.
- Part B: 3/4 of 28 kg = 28 ÷ 4 × 3 = 21 kg; 2/3 of $45 = 45 ÷ 3 × 2 = $30; 5/8 of 120 L = 120 ÷ 8 × 5 = 75 L.
- Part C: 1 1/2 × 2 2/3 = 3/2 × 8/3 = 4; 2 1/4 × 1 1/3 = 9/4 × 4/3 = 3; 3 1/3 × 3/5 = 10/3 × 3/5 = 2.
- Part D: for example 1/2 × 2/3 = 1/3 or 3/4 × 4/9 = 1/3; any pair whose product simplifies to 1/3. A part of a part is smaller when both parts are less than 1.
Multiplying fractions
A. Multiply, then simplify
Draw an area picture for the first product.
B. A fraction of an amount
3/4 of 28 kg = ____________
2/3 of $45 = ____________
5/8 of 120 L = ____________
C. Mixed numbers
D. Stretch
Find two fractions, neither of them 1/3, whose product is 1/3. Why is a part of a part smaller than both parts when they are less than 1? ______________________________
Dividing fractions
Dividing asks how many fit. Students count how many pieces of a fraction fit into a whole number, see that dividing by a fraction is multiplying by its reciprocal, and solve cutting and sharing problems.
We are learning to
- divide a whole number by a fraction by counting how many pieces fit,
- divide by a fraction by multiplying by its reciprocal,
- solve problems that ask how many pieces fit into an amount.
Success criteria
- I can say why 3 ÷ 1/4 = 12 using the fraction strips.
- I can work out 5/8 ÷ 1/4 = 5/8 × 4 = 2 1/2.
You need
The fraction wall strips from cut-out sheet 1 (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | How many fit? Show the board (see the box). Then lay quarter strips along three whole strips: 12 quarters fit, so 3 ÷ 1/4 = 12. Ask: “We divided and the answer got bigger. Why does that make sense here?” |
| 15 min | Turn it over Build a table: 3 ÷ 1/4 = 3 × 4; 3 ÷ 2/4 = 6 = 3 × 4/2. Then model 5/8 ÷ 1/4 = 5/8 × 4/1 = 20/8 = 2 1/2. Ask: “Which fraction gets turned over, and why that one?” |
| 35 min | Divide and fit Students complete Parts A to C, checking one Part A answer with strips; early finishers do Part D. |
Two shorter sessions instead? Split after Part A; start the second with the reciprocal table.
Watch for these ideas
- Expecting division to make smaller. Dividing by a number less than 1 makes bigger.
- Turning over the first fraction instead of the one you divide by.
- Dividing a mixed number without renaming it: 1 1/2 is 3/2 first.
Answers
- Part A: 4 ÷ 1/3 = 4 × 3 = 12; 5 ÷ 1/2 = 5 × 2 = 10; 6 ÷ 2/3 = 6 × 3/2 = 9; 3 ÷ 3/4 = 3 × 4/3 = 4.
- Part B: 3/4 ÷ 1/8 = 3/4 × 8 = 6; 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2 = 1 1/2; 5/6 ÷ 5/12 = 5/6 × 12/5 = 2; 7/10 ÷ 2/5 = 7/10 × 5/2 = 7/4 = 1 3/4; 1 1/2 ÷ 3/4 = 3/2 × 4/3 = 2.
- Part C: 3 1/2 kg ÷ 1/4 kg = 14 bags of rice; 4 1/2 m ÷ 3/4 m = 6 lengths of ribbon.
- Part D: dividing by 1/2 counts the halves, and every whole holds two, so the count doubles. A story for 2 ÷ 2/5: how many 2/5 m pieces from 2 m? 5 pieces, as 2 m = 5 × 2/5 m.
Dividing fractions
A. How many fit?
B. Multiply by the reciprocal
Turn over the fraction you divide by, then multiply. Simplify the answer.
C. Problems
How many 1/4 kg bags of rice can be made from 3 1/2 kg? Show the division. ____________________
How many 3/4 m lengths of ribbon can be made from 4 1/2 m? Show the division. ____________________
D. Stretch
Explain why dividing by 1/2 doubles a number. ________________________________________
Write a story that matches 2 ÷ 2/5, and solve it. ____________________________________
Working with decimals
The four operations with decimals, done efficiently: switch a decimal to a fraction it equals, scale a division so the divisor is whole, and check every written answer with an estimate.
We are learning to
- choose an efficient route for a decimal calculation,
- divide by a decimal by scaling both numbers by 10 or 100,
- check a decimal answer with an estimate.
Success criteria
- I can work out 0.5 × 36 as half of 36, and 6.3 ÷ 0.7 as 63 ÷ 7.
- I can say why 0.3 × 0.2 is 0.06, not 0.6.
You need
The loop cards from cut-out sheet 2 (one set per group of up to 16), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Quick routes Show the board (see the box). Then model 0.5 × 36 = 18 (half), 6.3 ÷ 0.7 = 63 ÷ 7 = 9 (scale both by 10), and 0.3 × 0.2 = 0.06 (3 tenths of 2 tenths). Ask: “Why does scaling both numbers by 10 leave a division answer unchanged?” |
| 15 min | Loop game Hand out the 16 loop cards (doubled up if the group is small). The holder of 1.2 reads the first question; whoever has the answer reads theirs, until the loop comes back to 1.2. Ask: “Which question needed the quickest thinking? What route did you use?” |
| 35 min | Routes and checks Students complete Parts A to C; early finishers do Part D. Share Part C: which estimate caught each mistake? |
Two shorter sessions instead? Split after the loop game; start the second with Part A.
Watch for these ideas
- Counting decimal places wrongly: 0.3 × 0.2 has two places in all, so 0.06.
- Scaling only one number in a division: 7.2 ÷ 0.08 is 720 ÷ 8, not 72 ÷ 8.
- Lining up the last digits, not the decimal points, when adding: 3.47 + 12.8.
Answers
- Part A: 0.25 × 48 = 12 (a quarter of 48); 3.6 × 5 = 18 (3.6 × 10 ÷ 2); 4.8 ÷ 0.4 = 12 (48 ÷ 4, scaling both by 10); 0.15 × 400 = 60 (15 × 4); 7.2 ÷ 0.08 = 90 (720 ÷ 8, scaling both by 100); 2.5 × 3.2 = 8 (2.5 × 4 × 0.8 = 10 × 0.8).
- Part B: 3.47 + 12.8 = 16.27; 20 − 6.35 = 13.65; 4.6 × 2.3 = 10.58; 9.36 ÷ 1.2 = 7.8.
- Part C: 0.6 × 0.4 = 2.4 wrong, it is 0.24; 5.4 ÷ 0.9 = 6 right; 12.5 × 0.8 = 10 right; 3.2 ÷ 0.04 = 8 wrong, it is 80; 1.5 + 0.75 = 0.9 wrong, it is 2.25.
- Part D: no. × 0.5 halves a number and ÷ 0.5 doubles it, so for any positive number, ÷ 0.5 gives the bigger answer (for 8: 4 and 16).
Working with decimals
A. Find a quick route
| Calculation | My route | Answer |
|---|---|---|
| 0.25 × 48 | ||
| 3.6 × 5 | ||
| 4.8 ÷ 0.4 | ||
| 0.15 × 400 | ||
| 7.2 ÷ 0.08 | ||
| 2.5 × 3.2 |
B. Written methods
C. Right or wrong?
Estimate first. Tick the right ones; correct the others.
0.6 × 0.4 = 2.4 Estimate: ________ Right or wrong? ________
5.4 ÷ 0.9 = 6 Estimate: ________ Right or wrong? ________
12.5 × 0.8 = 10 Estimate: ________ Right or wrong? ________
3.2 ÷ 0.04 = 8 Estimate: ________ Right or wrong? ________
1.5 + 0.75 = 0.9 Estimate: ________ Right or wrong? ________
D. Stretch
Is a number × 0.5 ever bigger than the same number ÷ 0.5? Explain with an example. ________________________________
Percentages at work
Percentages of amounts, discounts and GST the efficient way: build from 10%, 5% and 1%, or switch to the fraction a percentage equals.
We are learning to
- find a percentage of an amount by building from 10%, 5% and 1%, or by using a fraction,
- work out a discount or an increase such as GST,
- compare offers written as fractions and as percentages.
Success criteria
- I can find 35% of $80 as 10% + 10% + 10% + 5%: $28.
- I can find the price of a $120 item with 25% off: $90.
You need
The worksheet, calculators for checking, and the unit’s self-check quiz on the board.
Lesson flow (about 60 minutes)
| 10 min | Build it from 10% Show the board (see the box). Then find 35% of $80 as 3 × $8 + $4 = $28, and $120 with 25% off as three quarters of $120 = $90. Ask: “When is a fraction quicker than building from 10%?” |
| 30 min | Problems Students work through problems 1 to 6, writing each step and checking with a calculator; 7 and 8 are the stretch. |
| 20 min | Class quiz game Run the self-check quiz as a class game, then wrap up. Ask: “Which operation this week surprised you most, and why?” |
Two shorter sessions instead? Split after problem 3; start the second with problem 4.
Watch for these ideas
- Taking the percentage off the percentage: 20% off $45 is not $25. Find 20% of $45 first.
- Thinking a rise and an equal fall cancel. The fall is a percentage of a bigger amount.
- Adding GST to the wrong amount. GST is 10% of the price before GST.
Answers with working
- 10% of $60 = $6, 5% = $3, so 15% = $9.
- 20% of $45 = $9; $45 − $9 = $36 (or 80% of $45).
- 10% of 640 = 64, so 30% = 192, 5% = 32, and 35% = 224 students.
- 10% of $350 = $35; $350 + $35 = $385.
- 3/8 of $64 = $24, leaving $40; 40% of $64 = $25.60, leaving $38.40; 40% off is better.
- 12.5% = 1/8, so 96 ÷ 8 = 12.
- +10% gives $110; −10% of that is $11, leaving $99: not back, as the fall is of a bigger amount.
- both are 21: 30/100 × 70 and 70/100 × 30 multiply the same numbers in a different order.
Percentages at work
Write each step. Check with a calculator at the end.
Problems (7 and 8 are the stretch)
- Find 15% of $60 using 10% and 5%.
- A $45 jumper is reduced by 20%. What is the sale price?
- A school has 640 students and 35% of them walk to school. How many walk?
- A bike costs $350 before GST. GST of 10% is added. What is the price with GST?
- An item costs $64. Which is the better offer: 3/8 off or 40% off?
- Find 12.5% of 96 the quick way. Which fraction is 12.5%?
- Stretch. A $100 price rises by 10%, then the new price falls by 10%. Is it back to $100? Explain.
- Stretch. Which is more: 30% of 70 or 70% of 30? Explain why.
Fraction wall
Cut along each row to make strips, one per fraction family. Lay strips under each other to compare, add and subtract (Day 1), and count how many pieces fit (Day 3).
Keep one wall uncut for checking. The twelfths strip lines up with halves, thirds, quarters and sixths.
Decimal loop cards
The student holding 1.2 starts. Read your question; whoever has the answer reads theirs next. The loop ends back at 1.2.
Teacher note: the loop runs 1.2 → 6 → 1.5 → 0.3 → 3.05 → 30.5 → 30 → 50 → 2 → 0.25 → 4.5 → 3.15 → 6.3 → 7 → 7.7 → 8 → 1.2.
What we know: Operations with rational numbers
Work on your own, without a calculator. Show your working and give fractions in simplest form.
- 2/3 + 3/5 = ______________________
- 4 1/4 − 1 5/6 = ______________________
- 5/6 × 9/10 = ______________________
- 2 ÷ 2/5 = ______________________
- 3/4 ÷ 3/8 = ______________________
- 0.36 × 25 = ____________ Show a quick route.
- 4.2 ÷ 0.07 = ____________
- Find 15% of $120. What is the price of a $120 item after a 15% discount? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- 10/15 + 9/15 = 19/15 = 1 4/15.
- 17/4 − 11/6 = 29/12 = 2 5/12.
- 45/60 = 3/4.
- 2 × 5/2 = 5: five lots of 2/5 make 2.
- 3/4 × 8/3 = 2.
- 9: 0.36 × 100 ÷ 4 = 36 ÷ 4.
- 60: scale both by 100, 420 ÷ 7.
- 15% of $120 = $18 (10% = $12, 5% = $6); after the discount the price is $102.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Adding and subtracting (Q1, Q2) | adds the tops and the bottoms | adds and subtracts fractions and mixed numbers over a common denominator | uses the lowest common denominator by choice |
| Multiplying and dividing (Q3, Q4, Q5) | finds a common denominator to multiply, or turns over the wrong fraction | multiplies as a part of a part and divides by multiplying by the reciprocal | simplifies before multiplying and explains why dividing can make bigger |
| Decimals (Q6, Q7) | miscounts decimal places, or scales only one number in a division | chooses an efficient route and checks with an estimate | switches to a fraction where it is quicker, as 0.25 = 1/4 |
| Percentages (Q8) | takes the percentage off the percentage | finds a percentage of an amount and the price after a discount | compares offers and explains why a rise and a fall do not cancel |
Eight questions, four ideas. A student at standard answers most questions with working and chooses an efficient route rather than the longest one.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Add and subtract | Multiply | Divide | Decimals | Percentages |
|---|---|---|---|---|---|
The five columns are the five days: adding and subtracting, multiplying, dividing, decimals, and percentages.