In this lesson, students will learn how to identify the constant of proportionality from a graph. Students will answer questions related to a proportional graph to help them understand the constant of proportionality. Through student discussions, they will work to recognize lattice points to help them find the constant of proportionality. You can expect this lesson with additional practice to take one `45`-minute class period.
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In this lesson, the warm-up is intended to be an opportunity for students to discover how to find the constant of proportionality from a graph. Give students a copy of the guided notes so they have all of the graph readily available. Allow students time to work on their own to answer the questions.
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Once students have had an opportunity to answer the questions, ask students to compare their answers with a peer. You will want to circulate to listen to student conversations about the points chosen for the third question. Keep in mind that you do not want to make corrections to student errors yet; however, you can ask guiding questions for groups that seem to be struggling.
Allow students to volunteer their answers and explain how they got there. Once students have explained how they found the cost per shirt, it is important to bring up the constant of proportionality from the graph. Students will use different words to describe the answer to the last question, such as “rate” or “cost per shirt”. You’ll want to mention that all of these words are describing what we call the constant or proportionality. We call it a “constant” because it represents a constant rate. Each point on the graph tells us the same rate.
To help solidify the discussion from the warm up, you can show students the next slide and give them time to write down the important information.
As students are writing, it may be helpful to give them additional examples of independent and dependent variables, such as miles per hour or cost per gallon. You can also relate it back to the warm up. Before moving on, make sure students understand that finding the constant of proportionality from a graph will require them to write a ratio of `y` to `x` and simplify as needed.
With this next example, allow students to try to identify the constant of proportionality from the graph. Students can work independently or with a partner.
If students are struggling, you can remind them that they should identify the independent and dependent variables, then determine a lattice point they can use.
When reviewing the problem with the class, make sure you ask students to identify the point they used to find the constant of proportionality. This can help students recognize that it doesn’t matter what point they choose as long as they remember to write the ratio as `y` over `x`.
Next, you’ll want to help students with the concept of lattice points. Display the slide below. Ask students to pick a point to use to find the constant of proportionality from the graph.
Some students might try to estimate or guess a point. You’ll want to highlight each lattice point and show students that those are the points where we can find the exact coordinates, so we need to use one of those to find the constant of proportionality.
With this last example for identifying the constant of proportionality from a graph, allow students to work independently. Then allow them to check their work with a partner. If students disagree with their partner, they should explain their reasoning. This can help catch students’ misconception of writing the ratio as `x` over `y` instead of `y` over `x`. For some students, it may be helpful to include the context when possible to help them set up the ratio.
After you’ve completed the examples with the whole class, it’s time for some independent practice! ByteLearn gives you access to tons of practice problems for identifying the constant of proportionality from a graph. Check out the online practice and assign to your students for classwork and/or homework!
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