Ratios: 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 ratio family cards, mix cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: ratio family cards (Day 2) and mix cards (Day 5). Collect 30 small counters or paper clips per pair for Day 3.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A ratio compares amounts by how many parts each has. A jug of 2 cups of cordial to 8 cups of water is 2 : 8, and because both parts divide by 2, it is the same mix as 1 : 4. Order matters: cordial to water is not water to cordial. A ratio compares parts with each other; the whole is the sum of the parts, so in a 1 : 4 mix the cordial is 1/5 of the drink. Multiplying both parts by the same number keeps the ratio, which is how we scale a recipe, share a total and find a missing amount.

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 ratios

We write a ratio with a colon and a space on each side, 3 : 5, and read it as “3 to 5”. The order follows the words: cordial to water puts cordial first. Simplest form uses whole numbers with no common factor except 1. Before simplifying, write both amounts in the same unit: 45 minutes to 2 hours is 45 : 120, which is 3 : 8. Calculators are not needed this week.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 reads and writes ratios; Day 2 makes equivalent ratios and simplest forms; Day 3 shares a total in a ratio; Day 4 scales a ratio to find a missing amount; and Day 5 compares mixes and puts everything to work in problems. It builds on the fractions, decimals and percentages of the last units, and it prepares the rates, percentages and proportional reasoning of Year 8.

Materials for the week (one trip)

• From the classroom: pencils, scissors, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 24 ratio family cards (Day 2).
• Cut-out sheet 2, one per pair: 12 mix cards (Day 5).
• Day 3: 30 small counters, paper clips or blocks per pair, for dealing out shares.
• No calculators and no other equipment to buy.

Words for the word wall

ratio, part, whole, part to part, part to whole, order, equivalent ratio, simplest form, common factor, highest common factor, scale, share, bar model, ratio table, double number line, missing amount, mix, stronger, weaker, recipe

Dear families

This week in maths, Year 7 is learning about ratios: comparing amounts by parts, as in 1 part cordial to 4 parts water. We write ratios, simplify them, share amounts in a ratio, and scale recipes and mixes up and down.

Try this at home

• Mix a drink at 1 part cordial to 4 parts water. How much water goes with 3 cups of cordial? (12 cups.)
• Find a recipe and ask: what is the ratio of two of its ingredients? What would it be for twice as many people?
• Share 20 grapes between two people in the ratio 3 : 1. How many does each get? (15 and 5.)
• Ask: is 2 : 3 the same as 3 : 2? When would the difference matter?

Ratios this week

Fill one row a day with a ratio you noticed: in the kitchen, on a label, in a game or in the street.

Printed from the free seegongsik Ratios teaching pack · seegongsik.com/au/y7/number/AC9M7N08/pack

What a ratio says

A ratio compares amounts by parts. Students write ratios from bead strings in the order the words ask, tell a part-to-part comparison from a part-to-whole one, and turn a part-to-whole ratio into a fraction.

We are learning to

• write a ratio for a collection, in the order the words ask,
• tell a part-to-part ratio from a part-to-whole ratio,
• write a part-to-whole ratio as a fraction of the whole.

Success criteria

• I can write round to square beads for a string and say why the order matters.
• I can say what fraction of a class is girls when girls to boys is 4 : 3.

You need

The worksheet and coloured pencils. Two-colour counters help students who want to build a string.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the bead string; start the second with Part A.

Watch for these ideas

• Writing the parts in the wrong order. Read the words: round to square puts round first.
• Reading 3 : 4 as the fraction 3/4 of the beads. A fraction needs the whole: 3/7.
• Thinking a ratio gives the actual amounts. 1 : 4 could be 2 and 8, or 5 and 20.

Answers

• Part A: 5 : 2, 2 : 5, 5 : 7; 2 : 6, 6 : 2, 2 : 8; 4 : 3, 3 : 4, 4 : 7; 5 : 5, 5 : 5, 5 : 10.
• Part B: 1 part to whole, 2 part to part, 3 part to whole, 4 part to part, 5 part to part, 6 part to whole.
• Part C: boys 2/7; sand 3/4; yellow 5/9; dogs 3/10.
• Part D: red is 2/7, not 2/5: 5 counts only the yellow, so the whole is 2 + 5 = 7 parts. A string of 15: 9 round and 6 square.

What a ratio says

A. Write the ratio

Count the round and the square beads on each string.

B. Part to part or part to whole?

Write “part to part” or “part to whole” in the box.

C. From a ratio to a fraction

Boys to girls in a choir is 2 : 5. What fraction of the choir is boys? ________

Sand to cement in a mortar mix is 3 : 1. What fraction of the mix is sand? ________

Red to yellow lollies in a jar is 4 : 5. What fraction of the lollies are yellow? ________

Cats to dogs at an animal shelter is 7 : 3. What fraction of these animals are dogs? ________

D. Stretch

A jar has red and yellow lollies in the ratio 2 : 5. Jo says 2/5 of the lollies are red. Explain the mistake and give the right fraction. ________________________________

Draw a string of 15 beads in which round to square is 3 : 2.

Equivalent ratios

Multiplying or dividing both parts of a ratio by the same number gives an equivalent ratio: the same split in bigger or smaller amounts. Dividing both parts by their highest common factor gives the simplest form.

We are learning to

• make equivalent ratios by scaling both parts by the same number,
• write a ratio in simplest form,
• decide whether two ratios are equivalent.

Success criteria

• I can write three ratios equal to 4 : 7.
• I can simplify 16 : 20 to 4 : 5 and say what I divided by.

You need

The ratio family cards from cut-out sheet 1 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the ratio families; start the second with Part A.

Watch for these ideas

• Adding the same number to both parts: 2 : 5 and 4 : 7 are not equal. Equivalent ratios multiply or divide.
• Stopping too soon: 18 : 12 becomes 9 : 6, which still divides by 3.
• Simplifying amounts in different units. Write 2 hours as 120 minutes first.

Answers

• Part A: blue paint (L) to white paint (L): 3 : 8, 6 : 16, 9 : 24, 15 : 40, 30 : 80; boys to girls: 48 : 36, 24 : 18, 12 : 9, 8 : 6, 4 : 3.
• Part B: 2 : 3, 3 : 2, 5 : 8, 1 : 4, 5 : 3, 3 : 4, 2 : 7, 11 : 5.
• Part C: 1 yes, both 3 : 4; 2 no, 2 : 5 and 4 : 7; 3 yes, both 2 : 3; 4 no, 5 : 2 and 15 : 8; 5 no, 2 : 3 and 3 : 5; 6 yes, both 3 : 2.
• Part D: 45 : 120 = 3 : 8; 15 : 25 = 3 : 5; 2 : 3 : 5. Cards: the families simplify to 2 : 3, 5 : 2, 3 : 4, 4 : 1, 1 : 3, 3 : 5.

Equivalent ratios

A. Complete the ratio tables

B. Write in simplest form

C. Equivalent or not? Write yes or no.

D. Stretch

Write each ratio in simplest form.

45 minutes to 2 hours ________     1.5 : 2.5 ________     12 : 18 : 30 ________

Sharing in a ratio

To share a total in a ratio, count the parts, find one part by dividing the total by the number of parts, then give each share its parts. The shares always add back to the total, which is the quickest check there is.

We are learning to

• share a total in a given ratio with a bar model,
• find the value of one part,
• check that the shares add back to the total.

Success criteria

• I can share $35 in the ratio 2 : 5 and get $10 and $25.
• I can say why 2 : 5 means 7 equal parts.

You need

30 counters, paper clips or blocks per pair, the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the bar model; start the second with Part A.

Watch for these ideas

• Dividing the total by one number of the ratio, such as $35 ÷ 2. Add the parts first: 2 + 5 = 7.
• Giving the shares in the wrong order. The first number belongs to the first person named.
• Not checking. If the shares do not add back to the total, something has gone wrong.

Answers

• Part A: 4 parts of 6, so 6 and 18; 9 parts of $5, so $20 and $25; 7 parts of 6, so 30 and 12.
• Part B: 1. $42 and $48; 2. 600 mL and 400 mL; 3. 16 and 12.
• Part C: no. 24 + 27 = 51, not 54. One part is 54 ÷ 9 = 6, so 24 and 30.
• Part D: the 20 is 4 parts, so one part is 5: 15 red and 35 blue. $120 in 1 : 2 : 3: $20, $40, $60.

Sharing in a ratio

A. Share with a bar

Write the total over the bar, the value of one part in each box, then the two shares.

B. Word problems

1. Ana and Bea share $90 in the ratio 7 : 8. How much does each get? ________________
2. A 1 L jug of punch is orange juice and lemonade in the ratio 3 : 2. How many millilitres of each? ________________
3. A class of 28 votes 4 : 3 for a beach trip over a museum trip. How many votes does each trip get? ________________

C. Check and fix

Sam shared 54 lollies in the ratio 4 : 5 and wrote 24 and 27. Is Sam right? If not, fix the shares. ________________________________

D. Stretch

Red and blue counters are in the ratio 3 : 7. There are 20 more blue than red. How many of each are there? ________________

Share $120 in the ratio 1 : 2 : 3. ________________

Finding a missing amount

When one amount is known, scale the whole ratio by the same factor to find its partner. A double number line or a ratio table keeps the two amounts in step, and finding one part first works when the factor is not a whole number.

We are learning to

• find a missing amount by scaling a ratio,
• use a double number line and a ratio table,
• scale a recipe up and down.

Success criteria

• I can find the orange juice for 15 cups of mango juice when orange to mango is 2 : 3.
• I can scale a recipe for 4 people to one for 6 people.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after one part; start the second with Part A.

Watch for these ideas

• Adding instead of multiplying: with 2 : 3, 8 cups of sugar needs 12 cups of flour, not 9.
• Mixing up the two amounts. Label both lines before writing any numbers.
• Scaling only some ingredients of a recipe. Every amount uses the same factor.

Answers

• Part A: cups of cordial 3, 5; cups of water 10, 20. red paint (L) 4, 8; white paint (L) 9.
• Part B: 1. 300 g; 2. 8 leaders; 3. 7 buckets.
• Part C: for 18, multiply by 1.5: self-raising flour 450 g, eggs 3, milk 375 mL, berries 225 g. For 6, halve: self-raising flour 150 g, eggs 1, milk 125 mL, berries 75 g.
• Part D: 7.5 ÷ 3 = 2.5, so 12.5 cups of soda. 450 ÷ 6 = 75 g a serve, so 750 g.

Finding a missing amount

A. Fill the empty boxes on the double number lines

B. Find the missing amount

1. Peanuts to raisins in a trail mix are 3 : 2 by weight. How many grams of raisins go with 450 g of peanuts? ________
2. A camp needs 2 leaders for every 15 students. How many leaders are needed for 60 students? ________
3. Sand to cement in a mortar mix is 4 : 1. How many buckets of cement go with 28 buckets of sand? ________

C. Scale the recipe

D. Stretch

A punch uses 3 cups of juice to 5 cups of soda. How much soda goes with 7.5 cups of juice? ________

A recipe for 6 serves needs 450 g of rice. How much rice is needed for 10 serves? ________

Ratio problems

Students compare mixes part against whole, then solve problems from the whole week.

We are learning to

• compare two ratios as fractions of their wholes,
• solve ratio problems with several steps,
• check an answer against the story.

Success criteria

• I can say which of 1 : 6 and 2 : 11 is the stronger mix, and why.
• I can choose to share or to scale, and check my answer.

You need

Mix cards (cut-out sheet 2, one set per pair), the worksheet, the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the mix cards; start the second with problem 1.

Watch for these ideas

• Comparing only the cordial. 2 : 11 has more cordial than 1 : 6, but more water too.
• Reading a scale as an amount. In 1 : 20, the 20 is a multiplier, not 20 cm.
• Sharing when the story asks for scaling. Ask: is the total known, or one part?

Answers with working

1. A is 3/13, about 0.23, of the drink; B is 1/4 = 0.25; B is stronger.
2. one part is 16 ÷ 4 = 4, so 20 seniors and 36 members.
3. 7 parts of $12: Ari $36, Theo $48.
4. 21 × 20 = 420 cm = 4.2 m.
5. scale by 4: 8 of sand and 12 of gravel, 24 buckets altogether.
6. 1500 ÷ 10 = 150 mL a part: 300 mL, 450 mL, 750 mL.
7. 400 mL cordial and 1600 mL water; 1 : 5 needs 2000 mL water, so add 400 mL.
8. the 6 red fill the gap of 1 part, so one part is 6: 18 blue (12 red at first).

Mix cards, weakest first: G 1 : 4, B 2 : 7, I 1 : 3, D 2 : 5, F 3 : 7, K 4 : 9, A 1 : 2, H 3 : 5, J 5 : 8, C 2 : 3, L 3 : 4, E 4 : 5.

Ratio problems

For each problem, decide first: is the total known, or one part? Then share or scale, and check.

Problems (7 and 8 are the stretch)

1. Cordial and water are mixed 3 : 10 in jug A and 1 : 3 in jug B. Which jug is stronger? Show why.
2. In a club the ratio of juniors to seniors is 4 : 5. There are 16 juniors. How many members does the club have?
3. Ari and Theo share $84 in the ratio 3 : 4. How much does each get?
4. A model car is built at a scale of 1 : 20. The model is 21 cm long. How long is the real car, in metres?
5. Concrete for a path uses cement, sand and gravel in the ratio 1 : 2 : 3. With 4 buckets of cement, how many buckets of sand and of gravel are needed, and how many buckets altogether?
6. A smoothie is banana, yoghurt and milk in the ratio 2 : 3 : 5 by volume. How many millilitres of each make 1.5 L of smoothie?
7. Stretch. A 2 L jug of cordial is mixed 1 : 4, cordial to water. How much water must be added to make the mix 1 : 5?
8. Stretch. A bag has red and blue counters in the ratio 2 : 3. When 6 red counters are added, the ratio becomes 1 : 1. How many blue counters are in the bag?

Ratio family cards

Cut out the 24 cards. Day 2: sort them into six families of four equal ratios, then write the simplest form of each family.

Mix cards

Cut out the 12 cards. Each card is a jug of cordial mixed with water. Day 5: lay them out from the weakest mix to the strongest. Compare the cordial with the whole jug, not with the water alone.

What we know: Ratios

Work on your own, without a calculator. Show your working.

1. A packet has 5 red and 7 yellow jelly beans. Write red to yellow as a ratio, and the fraction of the jelly beans that are red. ________________
2. Write 18 : 24 in simplest form. ________
3. Circle every ratio equal to 4 : 5:  8 : 10, 6 : 7, 12 : 15, 16 : 25
4. Share $72 in the ratio 5 : 3. ________________
5. Cordial and water are mixed 1 : 6. How many cups of water go with 3 cups of cordial? ________
6. A recipe for 6 people uses 240 g of flour. How much flour is needed for 9 people? ________
7. Which mix is stronger, 3 : 5 or 4 : 7 (cordial to water)? Show why. ________________
8. Two numbers are in the ratio 2 : 7, and one is 25 more than the other. What are they? ________________

Answers and marking guide

Answers with working

1. 5 : 7; red is 5/12 of the packet.
2. 3 : 4, dividing both by 6.
3. 8 : 10 and 12 : 15; the others simplify to 6 : 7 and 16 : 25.
4. 8 parts of $9: $45 and $27.
5. Scale by 3: 18 cups of water.
6. 240 ÷ 6 = 40 g a person, so 360 g.
7. 3 : 5 is 3/8 = 0.375; 4 : 7 is 4/11, about 0.364; 3 : 5 is stronger.
8. The difference is 5 parts, so one part is 5: 10 and 35.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and can check an answer by simplifying it back to the ratio.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: writing ratios, equivalent ratios, sharing, scaling, and problems.