Patterns in the graph

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Objective

SWBAT recognize patterns of regularity in the graphs of proportional relationships.

Big Idea

The ideas of scale factor and equivalent ratios show up in the graph as the constant of proportionality.

Intro & Rationale

The key to this lesson is not rushing through it. It is really important to follow student thinking and explore the ideas they come up with. Allowing students to discover the patterns on their own helps them internalize the information. Taking the time to let them describe and show the patterns on their own also helps them make sense of the information in the graph. I hope that the more they focus on the regularity in the graph the more likely they will be to relate it to the constant ratio. I ask a lot of questions like

  • "who can explain that in another way?"
  • "who can see what she is talking about?"
  • "does this remind you of anything?", "can you show us what you mean on the graph?"

These questions help them make connections between the table and the graph which is important for them in future lessons to see how both relate to the constant of proportionality.

Warm up

15 minutes

The warm up asks students to check with their math family groups and go over their homework  together. This gives them access to peer instruction. As they check in with each other I circulate to look at the their tables to see how many students need more help figuring out the scale factor. If the majority of students need support then this lesson will be more direct instruction. If there are enough students who did the work correctly I will ask a series of questions to help lead their instruction.

I will display the table from the homework and work through the questioning in the top half of the intervention sheet. If there are enough students who understand I will ask them to circle the numbers in the table that gave them a clue about the scale factor and have them explain. Then I give students a couple of minutes to double check or finish the table in their homework.

Then we go through the questions in the bottom half of the intervention sheet and give them time to finish the rest of their homework.

Exploration

31 minutes

This exploration starts with the ratio table they helped to create in a previous lesson (Let's get organized). It shows how many black, white, and total tiles are needed for different areas using a certain tile pattern. As a reminder I ask them what the ratio of black to white tiles is in the table (1:3) and remind them that this is what it looks like in a table and now they will see what it looks like in a graph.

Each student gets a graph with axes already drawn and the horizontal axis labeled "black tiles". I ask what the numbers on the horizontal axis are counting (black tiles), so the vertical axis must be....(white tiles), which we then label. I draw their attention to the labels, but, because they enter with such diverse levels of graphing skills, I don't want to get bogged down teaching those right now. I highlight in the table the top two rows to show that we will only be using the numbers that represent black and white tiles for this graph. I have them graph the first point with me. I circle one number at a time and ask what it represents. Since the 1 represents the number of black tiles I ask which direction I should go on the graph. I don't expect students to know and they will probably respond with both directions. I then just ask what the horizontal axes is counting (black tiles) and that should be enough for them to know that we count across 1 on the graph. Since the 3 represents the number of white tiles we count up 3 then place our first point. I think it is important to for students to make the connection between the context and the representation (table & graph), because it helps them make sense of what the representations are telling them.

I ask students to then try to graph the rest of the information in the table. I circulate and ask students to go up and plot the point on my graph under the document camera for the class.

Then I ask students to look for and try to describe any patterns they see in the graph.

The first thing they may notice is that the points form a straight line and I expect some students to have connected the dots. I expect them to notice that their appears to be a gap in the dots. At this point I stop and let them know that we will explore that gap a bit more tomorrow and let them start their homework graph. I stop here because some students will have noticed the ratio in the graph and I don't want to rush through this at the very end. A few of my students may be describing the patterns in the graph, but I want to save this discussion for when we have more time to explore it.