# Fractions: Counting by three-fifths

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*Challenge*:

Count by 3/5's and explore patterns in the sequence of groups of three-fifths.

*Hint*:

- Count by 3/5s.
- Make a list of groups of three-fifths.
- What do you notice about the numbers?
- Organize them into rows with each row starting with a whole number equivalent.
- What patterns do you notice?

*Discussion*:

*Count orally*: three-fifths, six-fifths, nine-fifths, twelve-fifths, fifteen-fifths, eighteen-fifths, ...

*Record* the group and number of fifths in each group ...

Number of groups | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Total |

Number of groups | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Total |

Patterns for the denominator ...

Patterns for the numerator ...

What about group 100?

Group ...

What patterns can you find when you arrange them in a grid starting a new row for each whole number equivalent. Like below.

## Sample with some notes

Some patterns:

- Denominators are all fifths.
- Numerators ar 0, 3, 6, 9, 12, ...
- Numerators add three each tme.
- Numerators are multiple of three.
- Numerators are odd, even, odd, even, ...
- You add 15 going down each column.
- You add 18/5 diagonally.
- Five groups of three (15/5) are the same as three.
- Ten groups of 3/5 are the same as six.
- Number in ones place in each column alternate. (0,5; 3,8; 6,1; 9,4; 2,7).

What will a graph look like for the relationship for groups and number of fifths?

What will be the difference if the number of groups are graphed as x-values and totals as y-values vs. the number of groups as y-values and totals as x-values.

## Graph