Ratios and Proportional Relationships: Grade 7
Hook 1 7.RP.A.1
Prompt: What is the same about each model? What is different?
Asking kids to decipher double number lines may be a lot to chew for some but that's okay. If that's the case, then ask kids to be descriptive. Have them describe exactly what they see and then build from there. Both models have two lines. One is comparing soda to money the other is comparing time to distance. Maybe someone notices the top model has five vertical lines and the bottom has four. Ask them why they think this is. In other words, use inquiry to lead into your direct instruction on how to use a double number line to calculate unit rates. On the other hand, if kids can decipher the models on their own, the discussion can steer in a more abstract direction and lead to creating proportions from the models. In either case, when students begin applying proportions to solve problems, labels are immensely important to help them create the proper ratios. Double number line models provide visuals for students to see those labels and begin the habit of using them to create correct ratios and proportions.
Hook 1 7.RP.A.2
Prompt: What makes a relationship "proportional"?
Students have the fish visual and the tables to contemplate their ideas. This is a great opportunity to introduce into the class discussion the vocabulary of "constant of proportionality" and "constant rate". Many students assume they are the same, which appears to be true in proportional relationships. However, the constant of proportionality is calculated by dividing an output (y-value) by its given input (x value). In other words, we can write in "x5" between each input and output in the proportional table. The constant rate is the number being added to each y-value as the x-value increases by 1. This can be shown, on both tables, unlike the constant of proportionality, by drawing loops outside the table connecting each y-value with a +5. This is the essence of proportional vs. linear but non-proportional and a great opportunity to apply the common core vocabulary to teach an important concept of linear relationships.
Video: Earth Compared to Universe (3:33)
Hook 2 7.RP.A.2
Prompt: Which graph is proportional? Why?
Students hate the question "Why?". It forces them to think deeply about what they are doing. Making students write in math class on a regular basis is a great way to overcome this obstacle. When students realize that a rigorous understanding of the concepts at hand is the expectation they stop taking short cuts and begin thinking deeply. One method to help students understand how to answer the question "Why?" is to discuss the use of proofs in mathematics. Presenting students the opportunity to be mathematicians who are capable of defending their own thinking is an engaging way to challenge students to discuss mathematics. Students may prove their answer using a table, the constant of proportionality, the y-intercept or perhaps in another way. Share these ideas with students to deepen their understanding of proportionality and to promote the idea of finding multiple paths to a solution.
Video: Video Game Code (4:06)
Hook 3 7.RP.A.2
Prompt: What is the same about the relationships? What is different?
The focus here, according to the standard, is to identify the constant of proportionality in tables, graphs and equations. Obviously, the table's constant of proportionality differs from that of the equation and the graph. However, this hook allows students to review all their knowledge of linear relationships and the vocabulary associated with these concepts, which is always a worthy use of time. Variables, positive and linear relationships, slopes, proportionality, y-intercepts and of course the constant of proportionality can all be reviewed, reflected upon and discussed.
Video: Misleading Graphs (4:09)
Hook 3 7.RP.A.2
Prompt: Explain what r represents.
The idea that r can represent any unit rate may be difficult for students but it's okay to let them struggle for a bit looking for connections. To help them, deconstruct the graph. Ask them to think up a relationship between variables that might be represented by a linear graph. (Perhaps, even add labels to the graph for clarity.) Then ask them, what would the 1 represent in the coordinates (1,r)? What would the 2 represent in (2,r)? Try to get them thinking in real world terms in order for them to grasp the meaning of r, 2r, 3r and so on. Once they've made a concrete connection, discuss other possibilities for r. In the end, whether it's through small group work or whole class discussion, students should be able to articulate the relationship between the coordinate (1,r) and a unit rate.
Hook 1 7.RP.A.3
Prompt: Write 4 part-to-whole ratios.
The key to applying proportions to solve problems is understanding ratios. Using labels can be an incredibly effective tool towards this goal. "Part" and "Whole" should be recognizable ratio labels for students to begin with but if it helps, offer more specific labels to struggling students. Sale Price to Whole Price. Discount Percent to Whole Price Percent. Students who need the specificity of the labels should make this specificity a part of their practice when creating ratios and proportions. There's no doubt some students will struggle and compare percents to prices or vice versa, or just be plain stuck, but the discussion that ensues from this confusion should offer them solid tools to moving forward in their work with proportions.
Video: Parts of the World (3:41)
Hook 2 7.RP.A.3
Prompt: Create tape diagrams by attaching or overlapping the pieces, labels and values.
Model 1: A 20% tip.
Model 2: A 20% discount.
Students may struggle to begin due to the ambiguous and, perhaps, novel nature of the hook's task. However, eliciting their prior knowledge can get them started. Ask them first to match labels to percents. Once they've done this, attaching the label and percent to the proper values inside their strips should begin to click for them. Building a model to represent a tip or discount provides students with a visual to make sense of the additive and subtractive nature of each transaction. A completed model also allows students to discuss alternative ways to calculate these operations in a single step (finding 80% or 120%) rather than as two-step problems. Finally, the model labels may be used by students to create ratios and proportions and further solidify or deepen their conceptual understanding of the mathematics involved in solving such problems.