Students will explore the ratio of flowers to flower boxes. They will scale the composed unit in tandem and reveal a rate.

In this task, students will determine different equivalent ratios of flowers to flower beds. Students will scale the composed unit in tandem and reveal a rate through partitive division. Some of the big ideas that may emerge in today’s task include:

• There are two types of ratios; composed unit and multiplicative comparison;
• A composed unit is often (not always) a ratio with two distinct units; 
• A composed unit can be scaled in tandem; 
• Equivalent ratios are derived from the same rate; 
• When you divide a composed unit through partitive division, you reveal a rate.

Show students the following video:

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Replaying the video can be helpful here.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up includes:

• I notice 4 rectangular boxes.
• I wonder if they are flower boxes.
• I notice flowers.
• I notice a tray of flowers that flashed on the screen.
• I notice six flowers in that tray. 
• I notice more trays of flowers appear on the screen. 
• I wonder how many flowers there were.

After we have heard students and demonstrated that we value their voice, we can land on the first question we will challenge them with:

How many flowers did you see? 

Make an estimate

You might consider replaying the Spark video to give students another opportunity to visualize the structure of the nested array of flowers.

We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to determine a number of flowers that would be reasonable before determining a more precise answer. Consider asking students to think about a number of peas that would be "too low" and a number that would be "too high" before asking for their best estimate in order to help them come up with a more reasonable estimate. Encourage students to share their estimates, however avoid sharing their justification just yet. We do not want to rob other students of their thinking.

Show the following image and prompt students to:

Update your estimate without counting each individual flower.

While students are updating their estimate, listen and observe as students work to make updates. What strategies and mathematical models are students leveraging? Are they:

• Counting by ones?
• Skip counting?
• Repeatedly adding?
• Using spatial reasoning?
• Using an algorithm?

Reveal the number of flowers.

Celebrate student estimates that were very close to the actual number of peas using a routine of your choice as we head into the sense making portion of this lesson.

Now that we have confirmed that there are 72 flowers to be planted, begin playing the following animated video while verbally sharing the context using a script similar to this one:

The gardener has planted all 72 flowers in the four flower boxes in the front of the house, ensuring that every flower box has the same number of flowers. 

In the backyard, there are six more identical flower boxes. How many flowers will he need to fill the six flower boxes in the backyard? Keep in mind that the gardener wants to ensure that all flower boxes are exactly the same.

Consider pausing the final frame of the video or leave the following image up on the screen for students to begin their productive struggle:

Be sure to remind students that they are not to use a calculator to determine the number of flowers for six flower boxes as using that tool will rob them of this mathematical experience. Students should use a mathematical model in order to communicate their thinking.

Consolidate your intentionally selected student solutions. Look for similarities in students' approaches along the developmental continuum of counting, to addition and eventually multiplicative thinking.

Consider how students went about representing this ratio, a composed unit of flowers to boxes. 

Explore all of the equivalent ratios revealed through student solutions. 

72 flowers : 4 boxes

36 flowers : 2 boxes

18 flowers : 1 box

108 flowers : 6 boxes

Ask the students the following question:

What do all of these ratios have in common?

Students might notice that they all include two distinct units (flowers and boxes). They might also notice that all of the ratios are equivalent, they are derived from the same rate (18 flowers/box). Meaning, that nested within each of those equivalent ratios are a different number of iterations of the same rate of flowers per box. 

Here is a short silent solution animation video that will be helpful as you prepare to facilitate this consolidation:

Many students likely solved this problem by exposing the rate. When you divide 72 flowers by 4 boxes through partitive division (fair-sharing), you reveal a rate of flowers per box. 

Key takeaways from today’s lesson:

• 72 Flowers : 4 Boxes is a ratio.
• This ratio is considered a composed unit, we have two distinct units, and they need to be scaled in tandem (meaning if I double the number of boxes, I have to double the number of flowers). 
• All other ratios that are derived from the same rate (in this case 18 : 1) are equivalent ratios. 
• A rate can be revealed through partitive division.

Show students the following reveal video:

Here is a screenshot of the final frame:

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

Consolidation Prompt #2:

We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.