In this lesson students learn that two quantities have a proportional relationship if they maintain the same/equal ratio. Another way of saying this is, "If the comparison using multiplication (or division) stays the same, then there is a proportional relationship between two variables." Since this is new terminology, I write this information on the board and refer to it throughout the lesson.
Students spend the majority of today's lesson finding patterns using graphs and making connections using to their existing knowledge of ratios. I believe that it is important for students to discover patterns for themselves, and, to articulate patterns in their own words when a graph shows a proportional relationship (MP2).
In my classroom, students always get excited when they discover a pattern. The emotional response makes the learning last! It is also how they learn to persevere and make sense (MP1).
The Finding Ratios on a Graph Warm-up asks students to look at three lines (A, B, & C) on the same graph. Each graph represents the number of black and white tiles in a pattern for a floor plan. Students are to identify the ratio of black-to-white tiles for each pattern.
I expect that my students may ask if they are supposed to write a ratio for each point or each line. I want them to figure out the answer to this question for themselves. So, I tell them to do what makes sense, and, to ask themselves that question again after they have looked at the numbers.
Even though many of my students "know" that the ratios for each of the points on a given line are the same, many may still label points on a line with different ratios. I am prepared to identifying this issue by pointing out that some students have written two or three different ratios for each line, while others have written just one. If I sense puzzlement, I will I ask the students to discuss this discrepancy in their groups. Hopefully this conversation will lead us down a productive path towards the idea of equivalent ratios.
After the groups have discussed the labeling of ratios, I will ask:
These questions usually get my students on the road to a meaningful explanation of how they know whether two ratios are equivalent. It also helps me to identify and plan support for those learners who are still having difficulty simplifying ratios. It is good motivation for these students, when they recognize that it is often only necessary to use one (simplest) ratio for each line.
I expect that Graphs A & C will be more difficult than Graph B. The placement of points or the fact that some points are missing can confuse students. As a result, I prepare some specific notes for scaffolding students work on these tasks (see finding ratios on a graph scaffolding questions).
As we bring this activity to a close, I tell students that this is what the graph of an equivalent ratio looks like. Then I ask, "Do you see any patterns in the graph?" As students are looking for patterns, I circulate, listen, and make note of who is coming up with observations like:
I am also on the lookout for exemplary work on the graph. I would like to complete this section of the class by having one or more students show their work under the document camera (see finding ratios on a graph patterns students notice).
For our next activity, students are given 5 different graphs (A, B, C, D, & E) to study and interpret with respect to the pattern relating the two variables. I will ask, "Which graphs show a proportional relationship or a constant ratio of black-to-white tiles?" The task provides a good opportunity for students to engage in productive argumentation, so I ask them to work in groups on the task. As they work I encourage students to explain their reasoning to each other and supply evidence (MP3). I ask students to make connections to the definitions on the board as well as the patterns that were discussed during the Warm-up.
As students discuss in their math family groups, I circulate and ask what they are thinking about the graphs. I expect a quick consensus on graphs E and B. Graph E is a straight line that passes through zero and the pattern from point to point is "up 1, over 3." At this early point in their learning, I like to clarify with students so they don't forget that going "up" or "over" means that white and black tiles are being added in the constant ratio. I also ask how else they could show me that the ratio was staying the same. (all the points simplify to the same ratio). Graph B passes through zero, but is not straight. If students are not making a full explanation of their reasoning, I ask for more evidence that Graph B does not show a constant ratio. Considering (or using) non-examples is an important mathematical practice (MP2, MP3).
To bring the class to a close, we finish with a whole group discussion of why we think the remaining graphs (A, C, & D) may or may not show a proportional relationship.
Typically, some students will think all three graphs show proportional relationships because they are all straight lines. Many are unsure about Graph A and Graph D because they don't start at zero. Some are unsure about Graph C because it appears to have a "gap" or "break" in the pattern from point to point.
I tell them they can take it home and we will pick up the conversation tomorrow. I think it is important not to rush this or the entire argumentation experience is lost.