Equivalent representations of 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 36 fraction, decimal and percentage trio cards, number line strips, 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, and cut out the trio cards and the number line strips before Day 1. Both come back on several days.
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

One amount can wear several names. Three quarters of the way from 0 to 1 is 3/4, 6/8, 0.75 and 75%: different names, one point on the number line. Equivalent fractions come from cutting the same amount into more, smaller pieces, so the top and the bottom are multiplied by the same number. A fraction becomes a decimal by dividing the top by the bottom, or by rewriting it over 10, 100 or 1000, and a decimal becomes a percentage by multiplying by 100. Numbers carry on past 1: 11/4 = 2 3/4 = 2.75. To compare a mixed set, rename everything into one form.

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 fractions

Fractions are written with a slash, 3/4, and mixed numbers with a space, 2 3/4, read “two and three quarters”. A fraction is in simplest form when no whole number bigger than 1 divides both its top and its bottom. Calculators are not needed this week; the divisions are short.

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 gives one point three names; Day 2 explains why different fractions can name the same point; Day 3 moves fluently between the three forms; Day 4 carries the forms past 1; and Day 5 uses them to compare and order. It builds on the fractions, decimals and percentages of Year 6 and prepares the rounding, the operations with fractions and the percentage problems later in Year 7.

Materials for the week (one trip)

• From the classroom: pencils, coloured pencils, scissors, small envelopes or paper clips for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 36 trio cards, twelve values each as a fraction, a decimal and a percentage (Days 1, 3 and 5).
• Cut-out sheet 2, one per pair: three number line strips, 0 to 1 in tenths, 0 to 1 in eighths and 0 to 3 in quarters (Days 1, 4 and 5).
• No calculators and no other equipment to buy.

Words for the word wall

rational number, fraction, numerator, denominator, equivalent, simplest form, decimal, tenths, hundredths, thousandths, percentage, per cent, improper fraction, mixed number, number line, convert, compare, order

Dear families

This week in maths, Year 7 is learning that one amount can be written several ways: 3/4, 0.75 and 75% are the same point on a number line. We make equivalent fractions, convert between fractions, decimals and percentages, and use one form to compare and order mixed sets.

Try this at home

• Look for a percentage in a shop, on a battery icon or in a sports score. What fraction and decimal is it?
• Ask: which is more, 3/5 of a pizza or 0.55 of it? (3/5 = 0.6, so 3/5.)
• Ask: why is 2/4 the same as 1/2? Fold a sheet of paper to show it.
• Ask: where is 2 1/2 on a ruler, and what is it as a decimal?

Three names this week

Fill one row a day with an amount you met and its three names.

Printed from the free seegongsik Equivalent representations of rational numbers teaching pack · seegongsik.com/au/y7/number/AC9M7N04/pack

Three names for one point

One point on the number line can be named as a fraction, a decimal and a percentage. Students find the missing names, mark points given in any form, and spot the name that does not belong.

We are learning to

• name a point between 0 and 1 as a fraction, a decimal and a percentage,
• mark a point on a number line from any of its three names,
• explain why 0.5 and 50% are equal but 5% is not.

Success criteria

• I can write 3/5 as 0.6 and as 60%, and mark it on a line from 0 to 1.
• I can say which of 0.4, 2/5 and 4% is the odd one out, and why.

You need

The trio cards from cut-out sheet 1 and the 0 to 1 strip from cut-out sheet 2 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after trio snap; start the second with Part A.

Watch for these ideas

• Reading a decimal digit by digit as a percentage: 0.05 is 5%, not 50%. Multiply by 100.
• Writing 1/5 as 0.15 or 0.5, using the digits of the fraction. Divide 1 by 5, or write 2/10.
• Thinking a fraction with bigger numbers is bigger: 6/10 and 3/5 are the same point.

Answers

• Part A: 1/5 = 0.2 = 20%; 4/5 = 0.8 = 80%; 7/20 = 0.35 = 35%; 7/10 = 0.7 = 70%; 1/20 = 0.05 = 5%; 9/10 = 0.9 = 90%; 3/20 = 0.15 = 15%; 6/25 = 0.24 = 24%.
• Part B: A (0.3) at 0.3, B (3/4) at 0.75, C (50%) at 0.5, D (1/5) at 0.2, E (0.95) at 0.95, F (65%) at 0.65.
• Part C: 5% (the others are 0.5); 7% (the others are 0.7); 0.25 (the others are 0.4); 0.08 (the others are 0.8).
• Part D: halfway between 1/4 and 1/2 is 3/8 = 0.375 = 37.5%, since 0.25 and 0.5 average to 0.375.

Three names for one point

A. Fill in the missing names

B. Mark the points

Mark each point with its letter: A = 0.3,   B = 3/4,   C = 50%,   D = 1/5,   E = 0.95,   F = 65%.

C. Odd one out

Circle the one that names a different point. Then write the point the other two name.

5%     1/2     0.5    The others are ________

0.7     7/10     7%    The others are ________

2/5     0.25     40%    The others are ________

0.08     80%     4/5    The others are ________

D. Stretch

What number is exactly halfway between 1/4 and 1/2? Write it as a fraction, a decimal and a percentage. ____________________

Equivalent fractions

Cutting the same amount into more, smaller pieces gives an equivalent fraction: the top and the bottom are multiplied by the same number. Students find missing numbers, write fractions in simplest form, and decide whether two fractions name the same point.

We are learning to

• make an equivalent fraction by multiplying or dividing the top and the bottom by the same number,
• write a fraction in simplest form,
• decide whether two fractions name the same point, and say how we know.

Success criteria

• I can find the missing number in 2/7 = 10/□ and say what I multiplied by.
• I can write 20/32 in simplest form and check it.

You need

A sheet of scrap paper per student to fold, the worksheet. No calculators.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Adding the same number to the top and bottom: 2/3 is not 4/5. Only multiplying or dividing keeps the point.
• Stopping before simplest form: 12/16 halves to 6/8, which halves again.
• Thinking bigger numbers mean a bigger fraction: 16/20 and 4/5 are the same point.

Answers

• Part A: 8, 24, 3, 10, 4, 3.
• Part B: 12/16 = 3/4 (divide by 4); 30/45 = 2/3 (divide by 15); 14/35 = 2/5 (divide by 7); 25/40 = 5/8 (divide by 5); 18/30 = 3/5 (divide by 6); 56/64 = 7/8 (divide by 8).
• Part C: 2/3 and 6/9 yes; 3/4 and 4/5 no (4/5); 3/8 and 6/16 yes; 2/5 and 4/10 yes; 5/6 and 10/15 no (2/3); 4/7 and 12/21 yes.
• Part D: 3/4 = 15/20, since 3 + 4 = 7 and 35 ÷ 7 = 5. Multiplying top and bottom by 5 cuts every piece into 5 smaller pieces and takes 5 times as many.

Equivalent fractions

A. Find the missing number

B. Simplest form

Divide the top and the bottom by the same number until only 1 divides both.

C. The same point?

Tick yes if the two fractions name the same point. Show how you checked.

2/3 and 6/9    Yes □    No □   ______________________

3/4 and 4/5    Yes □    No □   ______________________

3/8 and 6/16    Yes □    No □   ______________________

2/5 and 4/10    Yes □    No □   ______________________

5/6 and 10/15    Yes □    No □   ______________________

4/7 and 12/21    Yes □    No □   ______________________

D. Stretch

Find a fraction equal to 3/4 whose top and bottom add up to 35. ____________ Why does multiplying the top and the bottom by the same number not change the point? ______________________________

Converting between the forms

Students move between the three names: a fraction becomes a decimal by rewriting it over 10, 100 or 1000 or by dividing; a decimal becomes a percentage by multiplying by 100; and a percentage is a fraction over 100, to be simplified.

We are learning to

• convert a fraction to a decimal and a percentage,
• convert a decimal or a percentage to a fraction in simplest form,
• work with amounts over 100%.

Success criteria

• I can write 7/8 as 0.875 and as 87.5%, and show the step.
• I can write 0.85 as 17/20 and as 85%.

You need

The trio 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 relay; start the second with Part A.

Watch for these ideas

• Multiplying by 100 by adding zeros: 0.375 × 100 is 37.5, not 0.37500.
• Writing 0.08 as 8/10. It is 8 hundredths: 8/100 = 2/25.
• Thinking a percentage cannot pass 100%. 1.2 of a whole is 120%.

Answers

• Part A: 7/20 = 35/100 = 0.35 = 35%; 3/25 = 12/100 = 0.12 = 12%; 9/50 = 18/100 = 0.18 = 18%; 1/8 = 125/1000 = 0.125 = 12.5%; 5/8 = 625/1000 = 0.625 = 62.5%; 7/4 = 175/100 = 1.75 = 175%.
• Part B: 0.45 = 9/20 = 45%; 0.08 = 2/25 = 8%; 0.375 = 3/8 = 37.5%; 1.2 = 6/5 = 1 1/5 = 120%.
• Part C: 64% = 64/100 = 16/25 = 0.64; 5% = 5/100 = 1/20 = 0.05; 150% = 150/100 = 3/2 = 1 1/2 = 1.5; 2.5% = 2.5/100 = 1/40 = 0.025.
• Part D: 1/3 = 0.333... = 33⅓%; 2/3 = 0.666... = 66⅔%. The division never ends: the same digit comes back every time.

Converting between the forms

A. Fraction to decimal and percentage

B. Decimal to fraction and percentage

C. Percentage to fraction and decimal

Write each fraction in simplest form.

D. Stretch: thirds

Divide 1 by 3, and 2 by 3. What happens? Write 1/3 and 2/3 as decimals and as percentages as best you can. ______________________

Beyond one

Rational numbers carry on past 1. Students write them as improper fractions, mixed numbers and decimals, place them on a line from 0 to 3, and order a mixed set; the stretch carries the line below zero.

We are learning to

• change an improper fraction into a mixed number and a decimal, and back,
• place numbers past 1 on a number line from any of their names,
• order a mixed set of numbers past 1.

Success criteria

• I can write 11/5 as 2 1/5 and as 2.2, and mark it on a line.
• I can say why 2 3/4 means 2 + 3/4.

You need

The 0 to 3 strip from cut-out sheet 2 (one per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after strip placing; start the second with Part A.

Watch for these ideas

• Reading 2 3/4 as 2 × 3/4. It means 2 + 3/4.
• Turning 13/5 into 13.5. Divide: 13 ÷ 5 = 2 remainder 3, so 2 3/5 = 2.6.
• Muddling 1 1/5 with 1.5. One fifth is 0.2, so 1 1/5 = 1.2.

Answers

• Part A: 9/4 = 2 1/4 = 2.25; 13/5 = 2 3/5 = 2.6; 7/2 = 3 1/2 = 3.5; 23/10 = 2 3/10 = 2.3; 1 3/8 = 11/8, 2 2/3 = 8/3.
• Part B: A (5/4) at 1.25, B (2.5) at 2.5, C (1 3/4) at 1.75, D (0.75) at 0.75, E (11/4) at 2.75, F (150%) at 1.5.
• Part C: 1 1/2 (1.5), 1.6 (1.6), 170% (1.7), 7/4 (1.75).
• Part D: from the left, −1.25, −3/4, −1/2, 1/4; −3/4 is smaller than −1/2, being further left.

Beyond one

A. Three names past one

Write as improper fractions: 1 3/8 = ______     2 2/3 = ______

B. Mark the points

Mark each point with its letter: A = 5/4,   B = 2.5,   C = 1 3/4,   D = 0.75,   E = 11/4,   F = 150%.

C. Smallest first

7/4,   1.6,   1 1/2,   170%:   ____________________________________

D. Stretch: below zero

Mark −1/2, −1.25, −3/4, 1/4 on the line. Which is smaller, −1/2 or −3/4?

____________________________________________

Compare and order

To compare a fraction, a decimal and a percentage, rename them all into one form. Students order mixed sets, compare amounts in context, and find a number between two others.

We are learning to

• rename a mixed set into one form so it can be compared,
• order fractions, decimals and percentages, including equal ones,
• use the three forms to answer questions in context.

Success criteria

• I can put 0.7, 3/4, 72% and 7/10 in order, smallest first.
• I can say which form I changed everything into, and why.

You need

The worksheet, the trio cards and strips, and the unit’s self-check quiz on the board. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Comparing digits across forms: 5/8 looks smaller than 0.6 because 5 < 6. Rename first: 0.625.
• Missing an equal pair. 0.6 and 3/5 are the same point and sit together in the order.
• Rounding too soon: 2/3 is 0.666..., so it is more than 0.65 even though 0.66 looks close.

Answers with working

1. 0.6 (0.6), 3/5 (0.6), 62% (0.62), 5/8 (0.625); 0.6 and 3/5 are equal.
2. 7/20 = 0.35, more than 0.3.
3. 9/25 = 36/100 = 36%.
4. 2/3 = 0.666..., more than 0.65, so the 2/3 tank.
5. 17/20 = 85%, 42/50 = 84%, 84% = 84%; 17/20 is best, and the other two are equal.
6. 5/4 (1.25), 128% (1.28), 1.3 (1.3), 1 1/3 (1.333...).
7. for example 5/8 = 0.625, between 0.6 and 0.666....
8. 1 + 40% of 1 = 1.4 = 7/5 = 1 2/5.

Compare and order

Change each set into one form first. Show your working.

Problems (7 and 8 are the stretch)

1. Put in order, smallest first: 0.6,   5/8,   62%,   3/5. Are any equal?
2. Which is more of a pizza: 7/20 of it or 0.3 of it? Show how you know.
3. In a class survey, 9 of 25 students walk to school. What percentage is that?
4. One water tank is 0.65 full and another is 2/3 full. Which is fuller?
5. Three test results are 17/20, 42/50, 84%. Which is the best result? Are any the same?
6. Put in order, smallest first: 1.3,   5/4,   128%,   1 1/3.
7. Stretch. Find a fraction between 3/5 and 2/3. Show that it is between them.
8. Stretch. A point is 40% of the way from 1 to 2 on a number line. Write it as a decimal, an improper fraction and a mixed number.

Trio cards

Cut out the 36 cards. Each value appears three times: as a fraction, a decimal and a percentage. Mix them before playing.

Fractions

Decimals

Percentages

Teacher note: the twelve trios are 1/10 = 0.1 = 10%, 1/8 = 0.125 = 12.5%, 1/5 = 0.2 = 20%, 1/4 = 0.25 = 25%, 3/10 = 0.3 = 30%, 3/8 = 0.375 = 37.5%, 2/5 = 0.4 = 40%, 1/2 = 0.5 = 50%, 3/5 = 0.6 = 60%, 7/10 = 0.7 = 70%, 3/4 = 0.75 = 75%, 4/5 = 0.8 = 80%.

Number line strips

Cut out each strip along the dashed border. Lay trio cards above the strip, or point to places on it.

What we know: Equivalent representations of rational numbers

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

1. Write 3/5 as a decimal and as a percentage. ______________________
2. Write two fractions equal to 4/6. Make one of them the simplest form. ______________________
3. Write 0.35 as a fraction in simplest form and as a percentage. ______________________
4. Write 140% as a decimal and as a mixed number. ______________________
5. Mark 0.7, 3/4, 65% on the line. Which is the biggest? ________
6. Write 11/4 as a mixed number and as a decimal. ______________________
7. Put in order, smallest first: 0.45,   3/8,   42%,   1/2. ______________________
8. Is 0.3 equal to 1/3? Explain. ______________________

Answers and marking guide

Answers with working

1. 3/5 = 6/10 = 0.6 = 60%.
2. 2/3 (simplest, dividing by 2) and 8/12 (multiplying by 2); any fraction with top and bottom in the ratio 2 to 3 is right.
3. 0.35 = 35/100 = 7/20 = 35%.
4. 140% = 1.4 = 7/5 = 1 2/5.
5. 0.7 at 0.7, 3/4 at 0.75, 65% at 0.65; the biggest is 3/4.
6. 11/4 = 2 3/4 = 2.75.
7. 3/8 (0.375), 42% (0.42), 0.45 (0.45), 1/2 (0.5).
8. No: 1 ÷ 3 = 0.333..., which never ends, so 1/3 is a little more than 0.3 (and 0.3 = 3/10).

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and renames a mixed set into one form before comparing.

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: three names, equivalent fractions, converting, past one, and comparing.