The activities in the lesson are intended to support initial, informal conversations about the key ideas in proportional relationships before the next lesson introduces the terms for those ideas. At the same time, there are opportunities to review work from grade 6 in representing ratios with tables and diagrams.
The tasks are intentionally not well-posed, that is, they do not have exact solutions. They are designed to give students an opportunity to think about how we can bring a mathematical lens to better understand common perceptual experiences (MP4), such as things that taste or look the same or different. Other possibilities include experiments with mixtures of paint, looking at videos of vehicles moving at different constant speeds, looking at faucets or other water sources that flow at different rates, and so on. The focus is on examination of a feature that can be represented as a unit rate (flavor, color intensity, speed, etc.) and beginning to analyze differences in that feature in terms of the two quantities involved (drink mix and water, two paint colors, time and distance, and so on). In the next lesson, this will be identified as the key idea motivating the concept of a proportional relationship. Students may recognize, from their work in grade 6, associated quantities as equivalent ratios and reason in terms of scale factors and unit rates.
The second activity provides a bridge from students’ work with scale drawings in the previous unit. In the first activity, students are given the relevant measurements; in the second, they are asked to think about how to quantify what they see, in particular, what measurements might help describe the picture.
The amount of time students spend on these activities can be adjusted based on the results of the diagnostic assessment. This lesson can be used to support just-in-time review of any ratio concepts from grade 6 that students struggled with in the diagnostic assessment.
- Choose and create representations to compare ratios in the context of recipes or scaled copies.
- Coordinate (orally) different representations of a situation involving equivalent ratios, e.g., discrete diagrams, tables, or double number line diagrams.
- Determine which recipes or geometric figures involve equivalent ratios, and justify (orally, in writing, and through other representations) that they are equivalent.
Let’s remember what equivalent ratios are.
Make three mixtures:
- 1 cup of water with \(1\frac12\) teaspoons of powdered drink mix
- 2 cups of water with \(\frac12\) teaspoon of powdered drink mix
- 1 cup of water with \(\frac14\) teaspoon of powdered drink mix
Students will need three small cups each; they just need a few sips of the mixture in each cup.
- I can use equivalent ratios to describe scaled copies of shapes.
- I know that two recipes will taste the same if the ingredients are in equivalent ratios.
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\boldcdot\frac12 = 4\) and \(6\boldcdot\frac12 = 3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
cups of water number of lemons 8 6 4 3