1. Math CAMPPP 2011 PLENARY 3 Thinking About Multiple Representations
Productive Tasks
Fostering Students’ Conjectures
Ruth Beatty & Cathy Bruce

2. Multiple Representations
Grounding an understanding of algebraic relationships in concrete, abstract and graphical representations allows students to construct symbolic expressions
For instance, an understanding of both linear growing patterns and graphs can support students in solving equations of the form ax+b=cx+d
At about this point (or perhaps earlier?) I would like to give participants a graphing task – given a graph of 5x+3 how many rules can you think of that will give you a trend line that intersects at (3,18). Why do we get a particular pattern of rules:
0x+18
X+15
2x+12
3x+9
4x+6
5x+3 6x
What if we had 4x+6 and 3x+7? Where would the point of intersection be?
What if we had 4x+6 and 3x+8,
What if we had 4x+6 and 3x+9
What if we had 4x+6 and 3x+10?
What is the same? What is varying?
What if we had 4x+6 and 2x+10?
What do you notice about all of the graphs created?
How can considering the rules and the graphs help you think about predicting the point of intersection?

3. Multiple Representations
When students are exposed to multiple representations, they can abstract mathematical principals within the learning context
The more representations they have, the more intricate their networks of understanding become
Students can then move between concrete and less concrete representations

4. What do we mean by “multiple representations”?
We don’t mean: Numbers, pictures and words
We are talking about how the representations illustrate, deepen, and connect student understanding
What does the square tile representation illustrate about “steepness” for example? (relationship between tile building and graphing)
Insert Movie of Ali talking about steepness related to growth (squares, and line through gesture)

5. Let’s Start With a Task
Given a graphical representation of 5x+3, how many rules can you think of that would have a trend line that intersects at (3,18)?
A lot of the time instruction tends to focus on asking students to interpret graphs. What is more beneficial is to have students create graphs from rule, as well as interpret rules from graphs. In this way, students learn to be flexible in terms of taking a point-wise approach (considering specific points on a graph) as well as a global approach (considering the graph holistically – that is – as the representation of a function). Also, until students develop a deep understanding of graphs it is not recommended that students use a graphing calculator, which can become a black box (equations go in, graphs come out but with little understanding of the connection between the two).

6. If…
If we look at rules that have whole numbers, what pattern do you see (let’s include 0)?
0x+18
1x+15
2x+12
3x+9
4x+6
5x+3
6x+0

7. What if…
What if we had 4x+6 and 3x+7? Where would the point of intersection be?
Can you predict it?
Make a graph…

8. Now what if… What if we had the following rules: 4x+6 and 3x+8
Predict the point of intersection for each
Create a graph for each pair
What do you notice?
What is changing? What is staying the same?

9. Now what if.. The rules were 4x+6 and 2x+10?
How can considering these rules, and the graphical representations, help you to predict the intersection point of the trend line for any two pattern rules?

10. Conjectures
What needs to be true of two rules if their trend lines are to intersect in the first quadrant?
What part of the rule determines where the line starts (or is at on the y-axis?)
What part of the rule determines the slope (or “how steep” the line is?
Conjecturing is a powerful way of establishing mathematical generalizations!

11. Sequence of Tasks
The following tasks were designed to allow students to explore relationships among pattern rules through working with graphs
Students made conjectures as they completed each task, and refined these as they gained experience

12. Task 1 Tiles = position number x2+6 Tiles = position number x3+5
What would a graph of these two rules look like?
Tiles = position number x2+6
Tiles = position number x3+5
The two rules were written on a white board.

13. Task 1
Predict what the graphical representation for these two pattern rules would look like:
number of tiles = position number x5+3
number of tiles = position number x6+2

14. Constructing a Situated Abstraction
Ilse: At the first position they’re going to be the same because…um…that rule adds up to 8 and that rule adds up to 8. They’re going to be the same value.
Amy: It’s like if you were building two patterns, based on the two rules, they would both have the same amount of tiles at that position. So that’s what it looks like when you graph it, both the lines would have a dot at 8 tiles for position 1. 1 2 3
number of tiles = position number x5+3 1 2 3
Ilse and Amy did not have to build patterns to know this.
number of tiles = position number x6+2

15. Constructing a Situated Abstraction
Anne: The x5+3 is going to start higher because the plus 3 is a higher number so it would start higher but not grow as fast. The x6+2 would be steeper because x6 is a higher value, and so would mean a steeper line on the graph because it’s growing faster.
Jack: So they will intersect like pow, smack, boom! They’re going to intersect at 1 and then keep going!
Anne and Jack, on the other hand, use their understanding of linear graphs to predict the behaviour of the trend lines given the values of the parameters

16. Multiple Representations
This line [3x+5] and this line [2x+6] are going to intersect for sure because “times 3 plus 5” for [x-value] one is going to be 8 and for “times 2 plus 6” it’s also going to be 8.
So, the zeroth position is going to be 5 for this one [drawing a point at 5 on the y-axis] and 6 for this one [drawing a point at 6 on the y-axis] and then it’s going to go something like this [sketches the two trend lines].
But I know it’s going to intersect here [pointing at (1, 8)] for sure because for position 1, both answers are 8!
In this video the student uses both his knowledge of linear growing patterns, and his knowledge of the relationship between pattern rules and trend lines on a graph, to predict the point of intersection. It is interesting to note that in the video, when the student points to the pattern rule, he refers to it as “this line”, even though no line has yet been drawn. This is an example of seamlessly moving between representations.
This also illustrates the difference between active vs. passive graphing. Passive graphing is creating a graph for the sake of creating a graph. Active graphing involves using the graph to explore the relationships among pattern rules. Developing conjectures and testing in order to refine conjectures. As students refine their thinking they generate more sophisticated approximations of formal mathematics.

17. First Set of Conjectures
If two rules have values that add to the same amount, they will intersect at an x-value of 1
To intersect, one rule has to have a higher multiplier and a lower constant, and the other rule has to have a lower multiplier and a higher constant

18. Testing the Second Conjecture
What if one rule has a higher multiplier AND a higher constant than the other rule?
Will the trend lines intersect?
For example, 1x+2 and 3x+4.

19. Graph of 1x+2 and 3x+4
They don’t intersect…but…they’re not parallel…

20. Revised Conjecture 2
If one rule has a higher multiplier AND a higher constant, they will intersect “somewhere behind zero…like an x-value of negative 2!”
If two rules have different values for the multiplier and the constant, they will intersect somewhere
If two rules have different values for the multiplier and the same constant, they will intersect on the y-axis (have the same y-intercept)

21. Task 2
For the given graph, how many rules can you think of that will have trend lines that intersect at position [x value] 3?

22. Guess and check strategy
These students thought of different rules and then graphed them to check which ones “worked”
Rules that didn’t work were crossed out

23. Simultaneously guessing and checking rules by drawing the trend lines
This student started to include rules that had a negative constant

24. Coming up with equations
This student, after plotting the trend lines for one rule [x0+18] realized that all the rules would have to have multipliers of 3 (because the point is at x-value 3) and all the rules would have to equal 18 (the y-value)
She created three equations with a box indicating an unknown constant

25. Ordered equations
This student noticed the numeric pattern that as the value of the multiplier increased by 1, the value of the constant decreased by 3
She extended this pattern to list rules with a negative multiplier (one of which she has graphed, since it was a new concept for her)

26. Ordered equations
These students also figured out the numeric pattern and then explored rules that had a negative multiplier (one of which they graphed) or a negative constant (on of which they graphed)

27. Task 2
By the end of task 2, students had a deeper understanding of the relationship between pattern rules and trend lines
Most students created equations with two unknown variables: 3 x ___ + ____ = 18
Some students began playing with negative values for the multiplier and the constant, and then explored what this could look like on a graph
By the end of task 2 students primarily worked with pattern rules as equations, and used the graph as a means of checking their answers

28. New Conjecture
If the multipliers of two rules differ by one, then the difference between the value of the constants is same as the x-value of the point of intersection

29. number of tiles = position number x3+5
intersection
Pete: For each one (trend line) you have to think about how far apart they’re starting on the graph and how long it will take them come together. So if the rules are x3+5 and x4+2, they start three spaces apart and get together by one space each time, so it would take them to the third position to intersect. {
1 space apart {
2 spaces apart {
3 spaces apart
Students develop a sense of the difference between where the trend lines “start” on the y-axis, as indicated by the values of the constant, and the rate at which they “come together” as indicated by the multipliers. For the two rules above, the trend lines start 3 spaces apart, but come together by 1 space each time, therefore they intersect at (3,14). Students figure this out both numerically (by comparing the values of the parameters) and by considering the trajectories of the trend lines.

30. Multiple Representations
number of tiles = position number x3+6
number of tiles = position number x5+2
intersection
Alan: So then say you have multipliers that are two apart. The trend lines would come together by 2 spaces each time. So if they start 4 spaces apart, and then come together by 2 each time, they’ll intersect at position 2. {
2 spaces apart {
4 spaces apart
For the two rules above, the trend lines start 4 spaces apart, but come together by 2 spaces each time, therefore they intersect at (2,12).

31. Final Conjecture
“For any two rules that will have trend lines that intersect in the first quadrant, if you know how far apart they ‘start’ by comparing the value of the constants, and the rate at which they come together by comparing the value of the multipliers, you can predict where the trend lines will intersect – or the point at which the two rules will have the same position value (x) and tiles value (y).”

32. Solving Equations
If you think about it, solving an equation like 3x+5=x+16 is like thinking about the position number (x) where two pattern would have the same number of tiles
It’s also like thinking about the x-value where two trend lines would intersect on a graph, if you knew how far apart they “started” and how quickly they were coming together

33. Balancing Equations
An explanation of how to solve equations of the form ax+b=cx+d
difference of 3
2x+16 = 5x+1
15 ÷ 3 = 5
difference of 15

34. Solving Equations
In this video, John is able to work only with equations, with numbers greater than those he worked with previously
Given his understanding, the values of the equations can be larger than might be possible when working with pattern tiles
He can check his calculations by using the graph – but in this example he does not need to create the entire graph – he uses only one point on the graph to indicate the point of intersection

35. Solving Equations
In this example, he can think about rational numbers with respect to the point of intersection by predicting the point of intersection for x4+5 and x6+4
“The difference between 4 and 6 is 2, and the difference between 5 and 4 is 1. So 1 divided by 2 is 0.5, so I think they’re going to intersect at x-value of a half – halfway between 1 and 2.”

36. Story Problem
An extended task during which students consider linear relationships in a specific context
This kind of problem has chapters that pose interrelated problems

37. Grandma Splendido’s Garden
Students have to predict and explain the changes in two representations (pictorial/figural and graphical) that occur as a result of changing the rule parameters

39. Video Clip

40. Extended Tasks
This kind of task offers an opportunity for students to explore the interrelationships among pattern rules, patterns and graphs

41. Supporting Student Thinking
Allowing students to explore the connections amongst different representations – pattern rules, graphs, and emergent equations
Including both active and passive graphing activities

42. Supporting Student Thinking
Encouraging/promoting the offering and evaluating conjectures
Is it always the case that this is true?
Can you think of a counter-example?
If we introduce a new idea, how does that affect the conjectures we already have? [For instance, students had a conjecture that the constant was responsible for “where the trend line starts.” When they started using a 4-quadrant graph, this was amended to “the constant is responsible for the y-intercept”]

43. Supporting Student Thinking
Ask good questions, and help them to ask their own good questions
For instance what part of the rule is responsible for the slope?
What part of the rule is responsible for the y-intercept?
What is the point of intersection?
How can you determine whether two rules will have trend lines that intersect?
How can you determine the point of intersection for the trend lines of any two rules?