Download Understanding fractions with a dozen counters

Understanding fractions with a dozen counters
Students find it simple to think of a dozen (12) objects as a unit. So that will be used below.
Another unit that is handy is the score (20). It can be used for fifths, tens, quarters and halves.
Comparing fractions
There are two basic types that should be clearly understood.
• For fractions with the same denominator, the larger the numerator the larger the fraction.
2
3
Using a dozen as the unit, there are very many examples of this, such as 4 (6 counters) and 4
(9 counters).
• For fractions with the same numerator, the larger the denominator the smaller the fraction.
For many students this is counter-intuitive, and requires care and many examples. Once a child
can explain to you why a fraction with a larger denominator must be smaller than another (with
the same numerator) you can be sure the child understands.
1
1
1
Using the dozen as the unit, compare 2 with 3 . Clearly 2 is bigger.
1
1 1
1 1
1 3
3
Try these: Choose the larger. 2 and 4 , 6 and 8 , 3 and 4 , 8 and 12 .
Mixed numbers and improper fractions
2
Use the unit (dozen) to show 13 . How many thirds are there in total?
If you have six quarters, how many units (1s) are there, and what fraction is left?
Equivalent fractions
1
Find other fractions that have the same number of counters as 2 (that is, 6 out of a dozen)?
3
Find three other fractions that are the same as 4 .
2
Find three other fractions that are the same as 3 .
4
Find six other fractions that are the same as 4 .
Multiply by whole numbers (‘lots of’)
Multiplying by a whole number always means finding that number of the quantity.
2
1
What fraction is the same as 2 lots of 6 ? (It should be clear that it is 3 .)
1
What fraction is the same as 3 lots of 12 ?
1
What fraction is the same as 4 lots of 6 ?
Multiply by fractions (‘fraction of’)
Multiplying by a fraction always means finding that fraction of the quantity.
1 1 2 4 10 1 1
Find one half of these fractions: 2 , 3 , 3 , 6 , 12 , 6 , 8 .
1
1
1
1
Here is half of 3 , or 2 x 3 . You can see it by getting 3 first, and then getting half of it.
1 1 2 4 10 1
Find one third of these fractions: 2 , 3 , 3 , 6 , 12 , 6
1 1
8
Find one quarter of these fractions: 2 , 3 , 12
2
3
3
2
3
Find 3 of 4 and 4 of 3 .
1
1
3
Find 4 of 3 and 3 of 4 .
Add fractions
1
2
1
1
1
+ 3 : Get 2 of one dozen, and 3 of another dozen.
5
Put the fraction parts side by side so you can see the total. What fraction of a dozen is it? (6 !)
Use the same method add these fractions of a dozen.
1
4
1
6
1
+2
2
+3
1
4
1
2
1
+3
2
+3
1
6
3
4
2
+3
2
+3
1
4
5
6
1
+6
2
+3
1
4
2
+3
Subtract fractions
1
2
1
1
1
1
– 3 . Get 2 of a dozen. Get 3 of another dozen. Subtract the smaller from the larger. It is 6 .
Use the same method (fractions of a dozen) for these:
2
3
2
3
1
1
2
1
12 – 3
–2
–6
1
–4
1
2
2
3
3
4
1
–6
2
–3
1
4
5
6
1
–6
2
–3
2
3
1
–4