The main point of this lesson is to show my students what proportional relationships look like in a graph. I could just tell them to look for a graph through the origin, but I don't feel this is a powerful or lasting method. I think my students understand concepts and retain information better when they discover patterns and make connections for themselves. I also think it is important for them to spend time explaining it in their own way.
Multiplicative thinking is crucial to understanding proportion, but my students struggle with multiplicative thinking. Since additive thinking is prerequisite, the warm up in this lesson is designed to help students make connections between additive and multiplicative reasoning. It helps the students to improve their capacity to think multiplicatively.
In the Rose Ratios Warm Up students consider a vase that holds red and white roses. The vase contains three red roses for every two white roses. Students need to decide how many flowers might be in the vase.
I expect some students may stop at the answer 5 while others will fill a page with possibilities. Because it is productive for students to consider the nature of a complete answer to this problem, I ask my class to discuss their solutions with their math family groups. If there is an entire group who came up with 5 as the only solution, I will direct them to read the problem carefully. I will ask, "Does it say that the vase holds "exactly" three red and "exactly" two white roses?"
For students who are considering other possibilities, I will wait for them to begin to ask, "When can we stop?" I will respond, "How big do you think the vase could be?" Though it is not the primary purpose of the lesson, I want them to reason about constraints, an important habit when making sense of and determining the reasonableness of solutions (MP1).
As noted above, I expect students to be working at different levels in my classroom. I am ready to help them build their capacity for multiplicative thinking with some scaffolding questions (see Warm Up Rose Ratios Notes). During a lesson like this one, I accept all student explanations, but I try to model explanations in a way that allows students to notice the patterns and connections between additive and multiplicative thinking (MP8).
This exploration picks up where yesterday's Keep it in proportion lesson left off. Students are in the process of determining which of five graphs (A, B, C, D, & E) which show a proportional relationship. (Yesterday, they decided that E does, but B does not.) Today we are revisiting the question with respect to Graphs A, C, and D.
Most of the time, some of my students think all three graphs show proportional relationships because they are all straight lines. There is less certainly about Graphs A and D because "they don't start at zero." The appearance of a "gap" or "break" in the pattern from point to point leaves some students unsure about Graph C.
Today, we will begin by having students discuss each graph separately in their Math Family Groups. After a short group discussion, each group will share their thinking with the whole class.
My students keep getting better and better at sharing their ideas in group discussions, but I am prepared to jump start conversations with some guiding questions:
At this point in the year, I expect most if not all of the ideas and arguments to come from students. I model the form for discussion with the visible thinking resource. If the conversation doesn't go as I expect, I have prepared some targeted questions for each graph:
As we close out the lesson, the students are given scenarios matching tile ratios in which each person built a tile floor using a given number of black and white tiles. They are asked to figure out who used the same proportions of black and white tiles shown in each of the proportional graphs and tell whether each person used the same ratio as Graph C, Graph D, Graph E or none of the above.
The matching proportions to graphs homework builds on the work that we have done over the course of the last two days identifying proportional relationships using graphs. I expect students to use one of two different strategies as they work on these problems:
One error I expect to see is students plotting points on the graph, but not extending the lines, which makes it difficult to see which points fall on the lines (See helping her figure it out for herself). In this case I would ask them what they could do to see if Garrett's design, for instance, uses the same proportion shown in graph D (see would that help you).
I anticipate that many students may use the "stairs" pattern (e.g., what could you try). If they try to extend the line by hand I ask how they could make sure it is straight. They will either say the "stairs" pattern or a ruler.
Another error I expect to see is students simplifying the ratios, but forgetting to answer the question. I would just remind them to take another look at what they are supposed to be figuring out.