Task Teacher Guide
Be sure to read the teacher guide prior to running the task. When you’re ready to run the task, use the tabs at the top of the page to navigate through the lesson.
In This Purposeful Practice…
Students will explore equivalent ratios and reveal rates through partitive division.
Intentionality…
The purpose of the Day 2 activities is to reinforce key concepts from Day 1. Students will engage in a string of related problems through a math talk and will have an opportunity to complete independent purposeful practice. The math talk and purposeful practice serve to develop a deeper understanding of the following big ideas.
- 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.
Math Talk
Present the following composed unit ratios one at a time. Create a context to define the units for each ratio. Focus on ratios with two distinct units at this time. Ask students to determine and justify one or more equivalent ratios for each composed unit. Equivalent ratios can be revealed by scaling the composed unit up or down. For each ratio, reveal the rate by leveraging partitive division.
8 : 4
14 : 2
27 : 3
32 : 8
6 : 4
As you’ll notice with this set of animated Math Visual Prompts, we begin by introducing a composed unit ratio of 8 apples to 4 baskets which provides an extremely low floor for students to access and enter the problem.
It will almost seem obvious to many students that the 8:4 ratio they are given of apples to baskets is equivalent to 2:1.
What students (and many educators) may not realize, is that while we have scaled this composed unit ratio from 8:4 down to 2:1 by “fourthing” the ratio, we can also quickly reveal a rate of 2 apples per basket through partitive division.
The goal here is to give students an opportunity to work with ratios using their intuition. Also we aim to help them realize that while we can scale any ratio up or down to find an infinite number of equivalent ratios, scaling to a unit ratio provides access to reveal a rate through partitive division (without really having to divide at all).
We also emerge the double number line as a tool for thinking as well as a tool to represent thinking.
Although there are five (5) ratios to work through in this animated Math Visual Prompt sets where we use partitive division via fair sharing (distributing apples equally to each basket) as well as through scaling in tandem on the double number line, be sure to dive into the final ratio where we reveal a fractional rate of apples per basket.
Through partitive division and the fair sharing strategy, you can quickly see that each basket will receive 1 and 1 half apples, revealing a rate of 1 ½ apples per basket:
Scaling in tandem by halving twice (or “fourthing”) reveals an equivalent ratio of 1.5:1 apples to baskets:
Consider watching this silent solution animation to help you prepare to facilitate this math talk utilizing a double bar model.
Facilitator Note: From each of these ratios, two rates can be derived. For example 8 apples : 4 baskets, the two rates can be revealed through partitive division:
8 apples ÷ 4 baskets = 2 apples/basket
and
4 baskets ÷ 8 apples = ½ basket/apple.
You can learn more about rates, ratios and the roadmap to proportional relationships through the course: The Concept Holding Your Students Back.
Purposeful Practice
While Students Are Practicing…
Students will have an opportunity to independently complete practice questions related to ratios and rates.
As students are working, you might consider using this opportunity to pose purposeful questions and document student thinking for the purpose of formative assessment.
Pay close attention to the strategies that students are using.
Are students:
- Using concrete materials to fair share?
- Using a concrete or visual array?
- Partitioning or scaling a bar model?
- Using a double number line?
- Using a ratio table?
- Counting on?
- Repeated addition?
- Multiplicative thinking?
While students are working independently, you might also consider strategically pulling students individually or in small groups to offer guided instruction in order to move them along their developmental continuum.
Questions: Ratios and Rates
Login/Join to access the Purposeful Practice as well as the Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
Download Editable/Printable Handout
Become a member to access purposeful practice to display via your projector/TV, download the PDF to upload to your LMS and/or print for students to have a physical copy
Resources and Downloads
Oh No! You Must Be Logged In!
Download a printable lesson plan, video/images, and presentation slides in Keynote and PowerPoint format to make the experience as smooth as possible by becoming a member of the Make Math Moments Academy.
Lesson Tip Sheet
Download the lesson plan in PDF format so you can keep it handy and share with colleagues.
Videos & Images
Download the videos, images, and related media files to your computer to avoid streaming.
Keynote Slides
Download in Apple Keynote format to avoid streaming video and run the lesson smoothly.
PowerPoint Slides
Download in Microsoft PowerPoint format to avoid streaming video and run the lesson smoothly.
Printable Handout
Download/edit the handout so you can keep it handy and share with colleagues.
Explore The Entire Unit of Study
This Make Math Moments Task was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, and extensions to elicit and emerge mathematical models and strategies.
Click the links at the top of this task to head to the other related lessons created for this unit of study.
How many apples per basket?
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
1. Consider the…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
2. Jane is picking…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
3. Complete the…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
4. Complete the…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
5. Grace says…
Login/Join to access the full teacher guides, downloadable slide decks and printable handouts for this lesson and all problem based units.
Download Printable PDF Handout
Become a member to access purposeful practice to display via your projector/TV, download the PDF to upload to your LMS and/or print for students to have a physical copy.