1. Beyond Slopes and Points
David Harris
WestEd/K - 12 Alliance Escondido USD
NEA Teacher Ambassador Training
February 19, 2014

2. Objective
We will “speak graph.” That is, we will be able to describe something common in life in “graph” and not even notice that we have switched languages!
This objective may be a bit cute” but it is part of the A-ha near the end of the initial activity and will be used for the second part (Write and Trade) as a shorthand for interpreting a graph beyond just describing the slope or points.

3. Observationn
Looking at this object, assume you were describing it to someone who had common knowledge of the world but did not know what this was and could not see it as you describe it.
Looking at this object, assume you were describing it to someone who had common knowledge of the world but did not know what this was and could not see it as you describe it. How would you describe it?
After some time for participants to observe, ask them to share some of their observations and chart responses into two charts: qual and quant.
Discussion is about describing in any form or level of precision in language.
Note the corny visual math joke in the title at your own peril….. Powers of observation.

4. Sort of Describe It
•Take what you have in the envelope and sort by putting together those you see as belonging together.
(No need to over think this.)
Allow a minute or two for them to sort. They should see that there is not a secret here. The most obvious categorization is ok (wide and narrow) DO NOT give away the subject of ratio. There needs to be the opportunity for various descriptions before the mathematical one.
People may sort a few ways, we will focus on the most common one later but it is ok to have them consider a bit at their tables to help discuss descriptions of their categories later.

5. Consider that a new rectangle is dropped on your table.
I Want to Belong
Consider that a new rectangle is dropped on your table.
Describe what would make it fit into each, but not every, category.
This is where we massage the conversation looking for ratio. “Skinny” rectangles versus “not skinny” have a greater ratio between the dimensions. The “less skinny” category has a smaller ratio of length to width. A square category has a ratio of 1:1 – there are no squares in the set.
Finding a new rectangle, you would determine the category based on the ratio of sides and not the length of any one side, regardless of how small or big the area or perimeter is.
Often, to clarify wide versus narrow, I point to different rectangles in the room (ceiling tiles, doors, patterns on the wall) and ask the whole group to say if is wide or narrow before we go deeply into describing what defines the category.
Often people talk about the difference in dimensions. This is almost true but 2X0.5 has a difference of 1.5 and would be “narrow” while 120 X 115 has a difference of 5 and would be wide.

6. Common Two Categories
This is just a picture to verify what are the categories and to show that in a class of students, this is where they almost all go.

7. What about a wide rectangle?
Wide vs. Narrow
Without concern for units, what could be the dimensions of a narrow rectangle?
What about a wide rectangle?
Describing the category with example and number.
This is a chance for people to “make up” data. If they don’t know what the question means you may refer to previous examples in the room or of the rectangles. Actual measurement is not the point. One can give these answers with no units, just assuming the same unit for length and width. You may even say that to explain what this prompt is suggesting for their thinking.
Take at least 4 for each category.

8. Classroom Example: Wide vs. Narrow
Describing the category with example and number. This is a picture of the white board from a 7th grade science class.

9. Categorization by Shape
• Required using more than one measure to describe a category of rectangles
• Used a ratio of two measures as a category’s defining characteristic
• One length alone would not define a category – it is a relationship between variables
Note that in math and surely in science there are concepts that are commonly talked about where we may overlook the fact that they are in fact ratios because we report their value as a single number such as density or speed.
It is also true that when we use the coordinate grid it is nearly always about a relationship between two variables. One alone, does not a Cartesian relationship make. That is where we are heading next.
0.8 ≤ ratio ≥ 1.25 is “fat”, for example.
If Square is different, then fat is 0.8 ≤ ratio ≥ 1.25 is “fat” where ratio ≠1.

10. Graphic Stack of a Category
On the piece of paper you will find an x - y coordinate grid. For the sake of time we will define the length as the longer side of a rectangle.
Take one “narrow” rectangle and place it on the grid such that one vertex is tucked into the origin of the graph.
In discussion, come to the conclusion that we want the long side of the paper to be length and that length needs to be on one axis while width is on the other. This part of the activity is pretty directed because the data points are found physically, not mathematically. The questions about this process is the content piece.

11. And Your Point Is . . .?
What data points could you derive from this display?
Which of these points describes the shape best?
There are 4 vertices on the rectangle. One is the origin and that only says we are measuring from zero. The two points on axes give us a single dimensions at a time (0, length) and (width, 0). The one point not on an axis will actually be (length, width) and represent the shape. In fact all points not on an axis represent a length and width for some rectangle if we define the axes and length and width. It will be important to discuss this as it is not a rectangle sitting in a cartesian plane – We are using a rectangle to help us define points in the plan that are in fact a description of a ratio, a relationship between length and width just as can later be done with time and distance or mass and volume.

12. Plot
Plot the data points from each rectangle of the category for which (x, y) represents (length, width). There do not need to be any numbers to do this.
If there is one, sketch a line of best fit.
Let people draw whatever they wish. Questions about appropriateness of the line come next.
They may not see that this is a direct variation – the line must come through the origin. Other wise it implies there are rectangles with one dimension.

13. So … Should your line of best fit pass through the origin?
Justify your answer.
Discussion should be left to participant justification first. Do all lines of best fit suggest a trend line that would include (or have as a limit) the origin? When would it not? How does context tie into this decision?
Once one suggests it should indeed go through (0, 0) because there cannot be a 1-dimensional rectangle, does that not also mean there are no zero-dimensional rectangles? This leads to the discussion of domain and range. Theoretically, the equations for these lines of best fit do suggest linear functions that pass through the origin.
-The actuality is a matter of domain and range and can’t be ignored for fear of incorrectly suggesting it does include (0, 0) or that it does not so the line can be “aimed” anywhere we wish to capture data points in the scatter plot.
Domain>0 Range >0

14. Repeat
Follow the same steps for the other set of rectangles. You may make a new set of axes or use the same one as before.
-This time there should be some expectation about the pattern of data. This is also a chance to make sure anyone who got lost during the graphing gets to see it done again.
••Assessment – Do they draw the line of best fit trending towards the origin but not touching? In their discussions, do they note how the difference between the categories is reflected on the graph.

15. What if… What if there had been a few squares?
Show me what that would look like. (You may do it in the air if you wish).
••Fun assessment (“speaking graph”) Ask, What if there had been a few squares – How would that category look? Show me? People will show or describe a 45˚ angle, thus describing a square by the relationship in (graphic form) of the two dimensions.
Response: Wow! If we had started the day with “what would a square look like?” how many of you would have made that arm motion (model the 45˚ arm). You SPOKE GRAPH when describing a square!

16. Let’s Talk Math
Other Connections

17. Equations
What could be an equation for the line of best fit for each category we have described? Justify your answer.
This may be analyzed with tables, graphs. Many will note it implies inequalities since they are not on the line, but near. It is a range of data, for example l/w ≥1/2 for narrow and ½≤ l/w <1 for wide but not square. Square would be l/w = 1. This is not necessary. One can simply say it is close to the ratio that is the unit ratio for the line of best fit. That is a description that is a focus from 6th and 7th grade CCSSM.

18. Mathematical Similarity
Two shapes are called similar if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.
The ratio of corresponding sides of two similar shapes is equal to k, a constant of proportionality.
Note how pure the definition is. It still rings true but the line of best fit turns it into this versus a more casual definition. Their categories are fine. However, to quantify if a rectangle fits, the ratio that is the lines slope would be the marker and you would have to decide how much difference in ratio you would accept as the category – just like in some relations. The line of best fit (trend line) approximates the idea behind finding the mean, although it is not. If we were to describe its slope and position (in the first quadrant and not at the origin) we would be describing what the rectangles in this category are close to being. You are defining “narrow” and “wide” mathematically – quantifying and representing as a single measure. This is what measures of center do.

19. Imagine that each set of rectangles had been similar by mathematical definition.
How would this activity have been the same for participants?
How would this activity have been different for participants?

20. What if . . .
There was a category of rectangles all with the same length? Same width?
What if there were no categories to discern? Is there anything to graph?
Vertical and horizontal lines.
The equations would have only one variable because…..it is not a relationship of the two.
Second question – there would still be points. Each data point is a specific rectangle. A line represents rectangles with the same relationship. Lack of a line suggests no relationship just as in any rough data analysis.

21. What if …
Slide, if needed.

22. Our Storyboard
Often we ask participants to take a science concept such as density or acceleration or speed (whatever) and make a story that would parallel what is shown above. Note that the first two are about categorization of a shape so those need to change. The task is to be able to have the same pictures accurately depict a graphic representation of a lesson in the same pattern.
The storyboard is to pictorially remember the pattern of the lesson. Careful to say that the rectangles had a stage that was unusual and would not be replicated, the part where we put the actual rectangles on the graph.

23. 8th Grade
How does the activity address this standard? What part of the standard does it not address?

24. 8th Grade Expressions and Equations
Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
How does the activity address this standard? What part of the standard does it not address?
How does this connect to 6th and 7th grade (may need to look this up)

25. 8th Grade
How does the activity address this standard? What part of the standard does it not address?
How could soeone say it is more of a part of the function domain than the EE, vice versa? What is the difference in the standards?

26. Algebra 1 Critical Area #4
Building upon their prior experiences with data, students explore a more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.
Heading to Algebra or if this were part of an algebra class.

27. Mathematical Practice Standards

28. 1. Make sense of problems and persevere in solving them.
MP Standards
Students:
Teachers:
1. Make sense of problems and persevere in solving them.
Analyze information and explain the meaning of the problem
Actively engage in problem solving (develop, carry out, and refine a plan)
Show patience and positive attitudes
Ask if their answers make sense
Check their answers with a different method
Pose rich problems and/or ask open-ended questions
Provide wait time for processing/finding solutions
Circulate to pose probing questions and monitor student progress
Provide opportunities and time for cooperative problem solving and reciprocal teaching
3. Construct viable arguments and critique the reasoning of others.
Make conjectures to explore their ideas
Justify solutions and approaches
Listen to the reasoning of others, compare arguments, and decide if the arguments of others makes sense
Ask clarifying and probing questions
Provide opportunities for students to listen to or read the conclusions and arguments of others
Establish a safe environment for discussion
Avoid giving too much assistance (e.g., providing answers or procedures)
4. Model with mathematics.
Apply prior knowledge to new problems and reflect
Use representations to solve real life problems
Apply formulas and equations where appropriate
Ask questions about the world around them and attempt to attach meaningful mathematics to the world
Pose problems connected to previous concepts
Provide a variety of real-world contexts
Use intentional representations
Provide students the space to ask questions and pose problems about the world around them

29. Science and Engineering Practices (NGSS)
Practice 4 Analyzing and Interpreting Data: Construct, analyze, and interpret graphical displays of data, use graphical displays to identify temporal and spatial relationships, distinguish between causal and correlational relationships in data, analyze and interpret data to provide evidence for phenomena, consider limitations of data analysis (e.g., measurement error). Practice 5 Using Mathematics and Computational Thinking: Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions.

30. Communicating by Graph
Write and Trade
Communicating by Graph

31. Write
Each team will use a similar storyline to create an example similar to the rectangle story but in a science-related context.
Write and annotate your storyboard.
Do not share your story with other groups … yet.
Hand out the work page that has the storyboard and the steps below. This is just to scaffold the work. It may actually not be necessary.

32. Trade (and Write)
On the chart paper, create the graphic representation of your “story.” Label the axes.
Swap papers.
From the graph only, create your own story for the graph given to you. You may use equations if you wish.
People can have fun with this. Encourage them to do so. In the end, Because they are going to match back up, it is fun to see how the same graph can be two different stories but the mathematics remains the same. The equation sentence is just to differentiate for those that can. No worries for those uncertain of this step.

33. Tell Me a Story, My Friends
Match back up with the ones you have swapped papers.
Choose one graph at a time. Each tell your story for the graph. The ones who wrote the second story can explain how the graph suggested their story.
Have fun comparing stories!
For example, You and I swapped our graphs. That means now we each have a story for both graphs. Choose a graph, tell the two stories (including how you got your idea fro the graph).
Facilitation of group share would be about relationships. In either case (either story for the same graph), is the relationship between x and y the same? How is that the story? Why would we care to know this relationship?
Note how this process assesses through production of and analysis of the full story. By the time they compare they will have done the work many times and heard it from other students as well as hopefully given their own voice to this process.