# The Florist Shop

## Task

The florist can order roses in bunches of one dozen and lilies in bunches of 8. Last month she ordered the same number of roses as lilies. If she ordered no more than 100 roses, how many bunches of each could she have ordered? What is the smallest number of bunches of each that she could have ordered? Explain your reasoning.

## IM Commentary

This task provides a context for some of the questions asked in "6.NS Multiples and Common Multiples." A scaffolded version of this task could be adapted into a teaching task that could help motivate the need for the concept of a common multiple.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task helps illustrate Mathematical Practice 1, “Make sense of problems.” Students are asked to analyze the problem and look for efficient ways to represent and solve it. Students translate the verbal description into a representation, solve, and then explore with their class the efficiency of their solution compared to other possible solutions. This task could be used in a classroom situation where students begin with the basic understanding of two whole numbers less than or equal to 12 and find their least common multiples by listing their multiples and/or using prime factorization. The teacher would then pose multiple context situations like this task to help students analyze a problem situation, decide on an entry point and look for a way to represent and solve it.

## Solution

The florist could have ordered any multiple of 12 roses that is less than 100:

12, 24, 36, 48, 60, 72, 84, or 96.

The florist could have ordered any multiple of 8 lilies that is less than 100:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96

If she ordered the same number of each kind of flower, she must have ordered a common multiple of 8 and 12, shown in the table below:

Number of each kind of flower | 24 | 48 | 72 | 96 |
---|---|---|---|---|

Number of bunches of roses | 2 | 4 | 6 | 8 |

Number of bunches of lilies | 3 | 6 | 9 | 12 |

The number of bunches of each are shown in the second and third rows. We can find the number of bunches of roses by dividing the number of flowers by 12, and we can find the number of bunches of lilies by dividing the number of flowers by 8.

The smallest number of each she could have ordered was 2 bunches of roses and 3 bunches of lilies.

## The Florist Shop

The florist can order roses in bunches of one dozen and lilies in bunches of 8. Last month she ordered the same number of roses as lilies. If she ordered no more than 100 roses, how many bunches of each could she have ordered? What is the smallest number of bunches of each that she could have ordered? Explain your reasoning.