Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 1 of this unit) Write 3/5 as a fraction with a denominator of 35.
Answer
21/35. 35 ÷ 5 = 7, and 3 × 7 = 21. - (Lesson 2 of this unit) Write 17/3 as a mixed number.
Answer
5 2/3. 3 × 5 = 15, and 2 is left over. - (Unit 1, lesson 5) What is the lowest common multiple of 6 and 8?
Answer
24. It is the smallest number that both 6 and 8 divide exactly.
Words used in this lesson
- common denominator
- a bottom number that two fractions can both be written over. For example: 12 is a common denominator for 3/4 and 5/6.
- lowest common multiple
- the smallest number that is a multiple of two numbers. For example: the lowest common multiple of 4 and 6 is 12.
- benchmark
- a friendly number to compare against. For example: 1/2 is a benchmark.
- order
- to arrange from least to greatest. For example: order 1/3, 1/4, 1/2.
- greater than
- further along the number line, more. For example: 3/4 is greater than 2/3.
The same size of part
Asha lights 3 of the 5 diyas in her row. Ravi lights 5 of the 8 diyas in his row. Both rows are the same length, so each diya lights an equal stretch of its row. Whose row is more lit? Predict first: 3 of 5 or 5 of 8? Then test it: set the first fraction to 3/5 and the second to 5/8. The figure recuts both rows into stretches of the same length so you can count them again.
3/4 and 2/3 both recut into 12 parts: 9/12 and 8/12. 9 is greater than 8, so 3/4 is greater than 2/3.
Show me the steps
- 3/5 and 5/8 over 40: 24/40 and 25/40, so 3/5 < 5/8.
- 2/3 and 3/4 over 12: 8/12 and 9/12, so 2/3 < 3/4.
- 1/2 and 7/12 over 12: 6/12 and 7/12, so 1/2 < 7/12.
- 7/10 and 7/12 over 60: 42/60 and 35/60, so 7/10 > 7/12. Same top, smaller parts in the second.
- 3/4 and 3/4 are the same fraction, so they are equal.
Counts of parts only compare when the parts are the same size. To compare two fractions, rewrite both over a common denominator, a bottom number that both bottoms divide into. The lowest common multiple of the two bottoms gives the smallest numbers. Then compare the numerators (tops): the fraction with more parts is greater.
Two shortcuts. If the tops are equal, the fraction with the bigger bottom is smaller, because its parts are smaller. And n/d is more than 1/2 exactly when 2 × n is more than d.
On a number line, the greater fraction is the one further to the right.
Back to the lamps: 3/5 = 24/40 and 5/8 = 25/40, so Ravi's row is just a little more lit.
A common mistake: “1/9 is greater than 1/6, because 9 is greater than 6.” The bottom number says how many parts the whole is cut into. More parts means smaller parts: 1/9 of a bar is less than 1/6 of the same bar.
Ordering several fractions
Place three fractions on one line. Each marker sits at its own value, whatever the bottom number is.
On this zoomed line 7/12 is furthest left, then 3/5, then 5/8, so 7/12 < 3/5 < 5/8.
Show me the steps
- 7/12, 3/5 and 5/8 sit close together: 7/12 < 3/5 < 5/8.
- 2/3, 3/4 and 5/6: 2/3 < 3/4 < 5/6.
- 4/9, 1/2 and 5/12: 5/12 < 4/9 < 1/2.
- 7/10, 5/7 and 3/4: 7/10 < 5/7 < 3/4. The gaps are small.
- 2/3 and 6/9 are the same point: 5/8 < 2/3 = 6/9.
- 7/8, 8/9 and 9/10 are all close to 1: 7/8 < 8/9 < 9/10.
Worked examples
Which is greater, 5/6 or 7/9?
- The bottoms are 6 and 9. The lowest common multiple is 18.
- 5/6 = 15/18 (multiply by 3) and 7/9 = 14/18 (multiply by 2).
- 15 is more than 14, so 5/6 is greater.
Is 11/24 more or less than 1/2?
- Double the top: 2 × 11 = 22.
- 22 is less than the bottom, 24, so 11/24 is less than 1/2. (1/2 would be 12/24.)
Your go. The first step is given. Which is greater, 7/8 or 11/14?
- The lowest common multiple of 8 and 14 is 56.
- Now write both over 56 yourself, compare, then check.
Check the answer
7/8 = 49/56 and 11/14 = 44/56, and 49 is more than 44, so 7/8 is greater.
Check before you go on
1. Which is greater, 4/5 or 7/9?
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. What is the lowest common denominator for 3/4 and 5/6?
3. Four children eat parts of identical bars: Lin ate 3/4, Omar 5/8, Pia 2/3 and Rey 7/12. Written over 24, how many 24ths did the child who ate the most eat?
Level 2 · Building
4. Put 2/3, 3/5 and 7/10 in order, smallest first. Written over 30, what is the top number of the middle fraction?
5. How many of these fractions are more than 1/2: 3/7, 5/9, 4/8, 6/11, 7/15? Do not convert them.
6. Red won 7 of its 9 games. Blue won 5 of its 7 games. Written over 63 games, how many 63rds greater is Red's share than Blue's?
Level 3 · Stretch
7. A fraction with a bottom number of 20 lies between 3/5 and 7/10. What is its top number?
8. 3/n is greater than 5/8, and n is a whole number. What is the greatest whole number n can be?
Something to think about
Compare 1/2 with 2/3, then 3/4 with 4/5, then 7/8 with 8/9. In each pair the top and bottom of the second fraction are one more than the first. What do you notice? Does it always happen when the fraction is less than 1?
One way to think about it
In each pair the second fraction is greater: 1/2 < 2/3, 3/4 < 4/5, 7/8 < 8/9. Look at what is missing from a whole. 1/2 is missing 1/2, 2/3 is missing 1/3. Adding one to both keeps the same number of missing parts but cuts the whole into more, smaller parts, so less is missing and the fraction is nearer 1. Yes, this happens for any fraction less than 1. For a fraction more than 1 it goes the other way: 3/2 is more than 4/3.
Exit ticket
9. Marlee says 3/8 is greater than 1/3, because 8 is greater than 3. Which reply is right?