In this lesson students are working with both part to part and part to whole ratios. We pay special attention to the "hidden" information that is implied within ratios. For example we may be given information both "parts", but not the whole or about one part and the whole, but not the other part. This is where it becomes so important to clarify or define the terms of the ratio and figure out what each number represents in order to avoid the most common mistake of just writing a ratio with the two given numbers even if they do not represent the quantities being compared.
We also explore what equivalent ratios look like on a graph as a preview to scaling up with ratios.
This warm up Warm up homework check.docx asks students to take a look at their homework Another kind of ratio.docx together. In their homework they were supposed to write part:whole ratios and I suspect that most if not all of them will write only the one for which information is given, for example: if Suzie got 7 out of 10 answers correct, they probably would write the ratio of correct answers (7/10), but not notice that there is another part:whole ratio that is kind of hidden (the ratio of answers she got wrong).
As groups work together I expect them to also help each other correct mistakes they made in the homework. While I circulate I am looking for groups who are unsure what to do. I may need to tell them that they are only given information about one part. I may point out that in problem 1 they are given a black and white tile pattern and asked to write a ratio about the black part, but there is also a white part, so they can write ratios for each part.
This activity has students matching given ratios (part:part and part:whole) to tile floor designs. They must find all the ratios that match each design and label them to show what is being compared or "counted". Clearly defining what the numbers in the ratio represent helps students to make sense of what a ratio is. It also helps them develop the vocabulary that allows them to use ratios in argumentation.
As I circulate I am listening for students who are working together and explaining their reasoning or who are asking each other questions. I like to share some of their ideas with the class. I may say "Mason just noticed a ratio that matches more than one of the designs", or "Jaedon is talking about simplifying or equivalence", or "I like how Hallie is using our sentence frame to help her explain".
When students finish matching the ratios to the tile floor designs I ask them to circle the floor designs that have the same or equivalent ratio of black tiles. (Floors 1, 4 & 5)
Next I want to show them what those equivalent ratios would look like on a graph. I hand out a graph with the x-axis already labeled "Black tiles". I keep the y-axis unlabeled because we are going to look at the black:total tile ratio and the black:white tile ratio.
We graph the equivalent part:whole ratios together as a class, one at a time. My students come to me with very spotty graphing skills, so starting off by doing these together gives them a feel for the mechanics, but allows us to focus on the interpretation part. Once we have plotted all for points, I ask them what we can do with the dots. Many of them will already have connected them. I ask them to take a moment with their group and figure out how they would describe the line. (straight).
Next we take one of the ratios that was not equivalent so they can see that it does not fall on the same line.
I do the same for the black:white ratios.