Year 7 · Unit 5 Fractions · Lesson 1 of 7

Same amount, different names

A fraction is a point on the number line. Cut every part into smaller equal parts and the point does not move, so one amount has many names. Then go the other way to the simplest name.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Unit 1, lesson 5) What is the greatest common factor of 12 and 30?
    Answer6. The factors of 12 are 1, 2, 3, 4, 6, 12, and 6 is the largest that also divides 30.
  2. (Unit 1, lesson 3) Write all the factor pairs of 28.
    Answer1 × 28, 2 × 14 and 4 × 7.
  3. (Unit 2, lesson 1) A bar has 20 equal parts and 15 are shaded. What percentage is shaded?
    Answer75%. Each part is 100 ÷ 20 = 5%, and 15 × 5 = 75.
Words used in this lesson
fraction
a number of equal parts out of a whole cut into equal parts. For example: 3 parts out of 4 is 3/4.
numerator
the top number: how many parts are counted. For example: in 3/5 the numerator is 3.
denominator
the bottom number: how many equal parts the whole is cut into. For example: in 3/5 the denominator is 5.
equivalent fractions
different fractions that name the same amount. For example: 1/2 and 3/6.
simplest form
a fraction where only 1 divides both top and bottom. For example: 3/4 is the simplest form of 42/56.
common factor
a number that divides two numbers exactly. For example: 7 is a common factor of 21 and 28.

One amount, two names

Zainab drinks 2/3 of a bottle of water at lunch. Kwame drinks 4/6 of an identical bottle. Zainab says, “We drank the same.” Kwame says, “No, mine has bigger numbers.” Predict first: who is right? Then test it. The bottle is a bar: once you have a prediction, set the fraction to 2/3 and slide the cut to 2.

1/2
1
1/2the fraction
1/2after cutting
2parts in the whole now

1/2 has 1 shaded part out of 2. Nothing is cut yet: slide to cut every part smaller.

Show me the steps
  1. 1/2 with every part cut into 3: 3/6. The shaded end sits at the same point.
  2. 2/3 with every part cut into 2: 4/6.
  3. 2/5 with every part cut into 4: 8/20.
  4. 3/8 with every part cut into 5: 15/40, which is 40 thin parts with 15 shaded.
  5. 7/10 with every part cut into 1: nothing is cut, so it is still 7/10.

A fraction n/d means n of the d equal parts of a whole. The top number is the numerator (it counts parts) and the bottom number is the denominator (it names the size of the part). A fraction also names a point on the number line between 0 and 1 (or beyond). Cutting every part into k smaller equal parts gives k times as many parts and k times as many shaded, so n/d = (n × k)/(d × k). The parts are smaller, but there are more of them, and the shaded end does not move.

Going the other way, dividing the top and the bottom by the same number that divides both gives an equivalent fraction with smaller numbers. Simplest form is reached when only 1 divides both numbers.

Multiplying or dividing the top and bottom by the same number does not change the amount. Adding the same number to the top and bottom changes the amount, unless the fraction is 1 already.

Back to the bottles: 4/6 is 2/3 with every third cut into two, so Zainab and Kwame drank the same amount. Kwame's numbers are bigger only because his parts are smaller.

A common mistake: “Add 2 to the top and the bottom: 1/2 = 3/4.” 1/2 has parts that are halves; 3/4 has parts that are quarters and takes three of them. Adding 2 to both changed the amount from half a bar to three quarters of it. Only multiplying (or dividing) the top and bottom by the same number keeps the amount.

Going the other way: simplest form

To simplify, divide the top and the bottom by a number that divides both. Try some in the figure: it only lets you use a number that divides both. Then find the single number that gets to simplest form in one step.

18/30
3
6/10result
not in simplest form yetsimplest form?

18/30 divided by 3 on top and bottom is 6/10. Both 6 and 10 still divide by 2, so it is not in simplest form yet.

Show me the steps
  1. 18/30 divided by 3 gives 6/10. Both still divide by 2.
  2. 18/30 divided by 6 gives 3/5 in one step. Nothing but 1 divides both 3 and 5: simplest form.
  3. 24/32 divided by 8 gives 3/4.
  4. 15/40 divided by 4 is not allowed: 4 does not divide 15.
  5. 36/60 divided by 12 gives 3/5, the greatest common factor of 36 and 60.

The number that gets there in one step is the greatest common factor of the top and bottom from Unit 1. Dividing by a smaller common factor is fine too, but then you may need to divide again.

Worked example: finding a missing number

Write 4/7 as a fraction with a denominator of 35.

  1. What was 7 multiplied by to make 35? 35 ÷ 7 = 5.
  2. Multiply the top by 5 as well: 4 × 5 = 20.
  3. Check by cutting: 7 parts each cut into 5 gives 35 parts, and 4 parts become 20. 4/7 = 20/35.

Simplify 42/56.

  1. Both are even: divide by 2 to get 21/28.
  2. 21 and 28 are both multiples of 7: divide by 7 to get 3/4.
  3. Only 1 divides both 3 and 4, so 3/4 is the simplest form. (Dividing by 14, the greatest common factor, does it in one step.)

Your go. The first step is given. Write 5/6 as a fraction with a denominator of 54.

  1. 54 ÷ 6 = 9, so the bottom was multiplied by 9.
  2. Now do the top yourself, then check.
Check the answer

5 × 9 = 45, so 5/6 = 45/54. Check: 45 ÷ 9 = 5 and 54 ÷ 9 = 6.

Check before you go on

1. Which fraction is equivalent to 3/5?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. 7/8 = ☐/40. What number goes in the box?

3. 20/36 = 5/☐. What number goes in the box?

Level 2 · Building

4. A tray holds 24 mango pieces. Nalini takes 7/12 of the tray. How many pieces is that?

5. Bea says 4/9 = 6/11, because she added 2 to the top and 2 to the bottom. Which reply is right?

6. Write 40/64 in simplest form. What is the top number of your answer?

Level 3 · Stretch

7. 6/9 = ☐/15. What number goes in the box?

8. A fraction is equivalent to 3/8. Its top and bottom add up to 66. What is its top number?

Something to think about

Find a fraction that is strictly between 1/3 and 1/2. Then find another between your fraction and 1/2. Could you keep going for ever?

One way to think about it

1/3 is 2/6 and 1/2 is 3/6, and there is no whole number between 2 and 3, so cut finer: 4/12 and 6/12 have 5/12 between them. Cut finer again and there is room again. However close two fractions are, another fits between them, so there is no “next” fraction after 1/3.

Exit ticket

9. Dina divides the top and bottom of 20/28 by 2 and gets 10/14. She says she has finished. What should you say?