##
* *Reflection: Developing a Conceptual Understanding
Patterns in the graph - Section 3: Exploration

I am sometimes tempted to move on as soon as a single student has come to a correct conclusion. I get so excited when someone comes up with an insight, I forget that hearing or seeing it once is not enough for the rest of the students in the class. Even that student needs to be able to investigate, verify, confirm, and see it in another way before he/she has a firm understanding.

I don't want them to just * be able to* plot points from a table to a graph.

**I want them to understand the relationships between the numbers in the table, the points on the graph, and what they represent.** I think working in a group and hearing each others responses to purposeful questions help them build meaning.

**"what does this [number] represent?"****"where do we find it on the graph?" (along a single axis - one dimension)****"what does the [tile floor] look like at this point on the graph?"**

**I want my students to understand** **how each of the points on the graph are related to each of the others.**

**"how are the points related to each other?"****"what is the same about each point on the graph?"****"what patterns do you notice on the graph?"****"is that pattern true for all the points on the graph?"**

**I want them to understand how the graph represents the constant change represented by the ratio.**

**"how do we get from one point to the next?"****"could you use the pattern to predict where other points might be located on the graph?"****"why does that make sense?"****"does it make sense to connect the dots? why/why not?"**

Discussing these in their collaborative 'math family groups' helps them incorporate all the different viewpoints for a more complete understanding of proportional relationships.I have sentence starters taped to each of their desks to help them conduct productive group discussions.

I am always listening for their ideas and questions to help me plan future lessons. In this lesson I would specifically listen for any discussion of the origin (0,0).

*Developing a Conceptual Understanding: Don't rush it!*

# Patterns in the graph

Lesson 1 of 10

## 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.

*46 minutes*

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.

*expand content*

#### Warm up

*15 min*

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.

*expand content*

#### Exploration

*31 min*

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.

*expand content*

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- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Patterns in the graph
- LESSON 2: What is it trying to tell us?
- LESSON 3: Keep it in proportion
- LESSON 4: Recognizing proportional relationships in a graph
- LESSON 5: Are they proportional?
- LESSON 6: Writing arguments
- LESSON 7: Clarify & Correct arguments
- LESSON 8: Which is blackest the sequel
- LESSON 9: Scaling up, scaling down, scaling all around
- LESSON 10: Ratio assessment day