Tripping Over Stumbling Blocks

Learning Goals• Finish the work from yesterday• Identify common student struggles (errors,

misconceptions, and partial understandings, etc) that are grounded in experience and/or research.

• Realize that different representations of relationships highlight different characteristics or behaviours, and can serve different purposes.

• Highlight the importance of being conscious and explicit about the instructional decisions; specifically, the need for simultaneous introduction & exploration of different models

Notes From the Exit Cards

• One way that the big ideas informed my thinking today is…– they are difficult to integrate into the current

thinking– they allow for more meaningful connections to

be drawn between concepts– they overlap so that expectations could go

under more than one– struggling with how to express them in

student friendly language

Notes From the Exit Cards

• A question that I might ask of my students to address that big idea is…– Write an equation of a quadratic that has a

vertex in the third quadrant– What does x=5 mean?– Given a graph/table/expression/algebra tile

representation, which are quadratic? How do you know?

Notes From the Exit Cards

• When I teach factoring, my students seem to struggle most with…– algebraic manipulation (selecting the numbers

to generate the correct sum and product)– connecting the algebraic to the concrete– selecting the appropriate method– connection to solving equations – zeros/roots– completing the square– multiplication facts

Big Ideas for Algebraic Reasoning• Algebraic reasoning is a process of describing and analyzing

(e.g., predicting) generalized mathematical relationships and change using words and symbols

• Comparing mathematical relationships helps us see that there are classes of relationships and provides insight into each member of the class.

• Different representations of relationships (e.g., numeric, graphic, geometric, algebraic, verbal, concrete/pictorial) highlight different characteristics or behaviours, and can serve different purposes.

• Limited information about a mathematical relationship can sometimes, but not always, allow us to predict other information about that relationship. 

Big Ideas of Quadratics• The graphical, algebraic, numerical,

geometric or verbal representation of a quadratic function reveals different information about the concrete/pictorial representation of the quadratic.

• Consideration of the graphical representation allows understanding of the predictable affect of parameters on the function. (something about transformations and combining functions)

Action – Connecting to the BI• How well do the big ideas encompass the

expectations

• Did you think any of the expectations fit under more than 1 big idea?

• How do concepts develop over the grades

• What other characteristics did you notice about your clusters?

• Is there anything that we can now add to /remove from our group Frayer model?

Action – Lesson GoalsMany people fail in life, not for lack of ability

or brains or even courage but simply because they have never organized their

energies around a goal. Elbert Hubbard

• Link your lesson goal from last night and link it to a cluster of expectations

• Meet with other like-minded colleagues

• Write your lesson goal(s) on a large piece of paper and post near the cluster

Consolidate – Gallery Walk

• Using the sticky notes, provide feedback to your colleagues on their lesson goals

• Some guiding questions:– How would you assess student achievement

of the lesson goal?– Within the stated goal, can you think of a

variety of ways for students to reach and/or demonstrate their understanding?

Minds On – Stumbling Blocks

• Think about common student – misconceptions– stumbling blocks– errors– partial understandings

• Write one per sticky note

• Post the sticky note beside the appropriate curriculum cluster or big idea or lesson goal

Minds On – Revisit Lesson Goal

• Reconvene with your group for lesson goals

• Using the sticky notes to supplement your own consideration of the struggles students normally encounter, revise and/or add to your lesson goal

• Repost your lesson goal

Common Struggles with QuadraticsStudents may• view a graph as a sketch through three or four points and

never consider trends in families of functions• rarely consider ways in which the graph provides visual

insight into the behaviour of the function• fail to connect the graphical representation to other

representations• not comprehend the beauty/power and significance of the

graph (because there is an overdependence on the procedure of graphing)

Common Struggles with Quadratics (continued)Students may• use symbols before they understand the meanings

behind the symbols• have a tendency to use standard form over vertex form• misinterpreting the point (b,c) as the vertex of the

standard form y=ax2+bx+c• lack understanding about when and how to use the

vertex (h,k) with the vertex form, y=x(a-h)2+k, which is demonstrated through students using the same process (algebraic to graphical) regardless of the task

• not recognize the role or significance of the x-intercepts

Common Struggles with Quadratics (continued)

Students may• have two different, potentially conflicting algebraic and

graphical schemes in the cognitive structure (the phenomenon of compartmentalization)

• have two simultaneous evoked images for one concept, which results in cognitive conflict (e.g., quadratic function evokes quadratic formula and parabola)

• not understand the relevance of a solution to an equation• lack understanding of why a parabola horizontally or vertically

shifts

Suggested Instructional StrategiesTeachers should• monitor student thinking to expose misconstructs and to

allow deeper connections to be established• be open and prepared to communicate with students in

terms of confronting, discussing, and dealing with conflicting schemes

• engage students in judging reasonableness of the answer and appropriate types of solutions, which requires students to understand the number system

Suggested Instructional StrategiesTeachers should• engage students in connecting various representations

of a function• avoid sequential presentation of the forms and instead

simultaneously introduce vertex and standard form, emphasize the differences between them and reveal the underlying thoughts that generated vertex form

• connecting the problem to the graph and the vertex to avoid isolated and unconnected knowledge in the student’s cognitive structure

Suggested Instructional Strategies (continued)Teachers should• have students solve problems graphically before

algebraically in order to build better connections between the graph and the equation (algebraic to graphic doesn’t result in the same depth of understanding)

• use a well-designed set of tasks or examples that emphasize the distinction between the quadratic formula and quadratic function

• expose students to quadratic equations that cannot be factored in order to expand their understanding that factoring is not always the appropriate strategy

Suggested Instructional Strategies (continued)Teachers should• focus on quantitative relationships that can be

represented by both linear and quadratic functions• examine the ways in which quadratic growth differs from

linear growth

Action - Activity

Activity Goal:

Recognize that different representations of relationships

highlight different characteristics or behaviours and can serve different

purposes

Action – Activity continuedEveryone is going to receive a graph of a quadratic relation with its equation. The equations are given in 3 different forms (examples shown below) and we will be looking to see what information each form gives, if any, about a parabola.

y = 3(x + 2)2 – 7 y = 5 (x + 4)(x – 2)

y = – 2x2 + 4x + 2 y = 5 x (x – 2)

Do you believe that all of the above relations would give parabolic graphs? Why or why not?

Action – Activity continuedInstructions:

1.Complete the information requested below the graph given to you.

2.Form a group with all the people who have the same zeros - compare your graphs (similarities? differences?)

3.Form a new group with all the people who have the same vertex - compare your graphs (similarities? differences?)

4.Form a group with only the people who have the exact same graph.

Action – Activity continuedInstructions:

5.On chart paper record the important info about your 3 graphs.

Eg.

Action – Activity continuedActivity Consolidation?

As a class decide how the chart papers should be arranged to help make patterns clear.

What information about the graph of a parabola does each form of the equation give?

Follow-up exit card: which form do you think is most useful and why?

this activity address

How did Why did Did

the big ideas?

the lesson goal?

Discussion

• Share the connections you see between the identified struggles and – the big ideas– the curriculum expectations– the suggested instructional strategies– the lesson goals

Action - Activity

Using the colour tiles or cube-a-links, construct a unique square at your table.

Order these at your table by side length. Construct additional square(s) if necessary to generate at least three squares with sequential side lengths.

Stack the tiles vertically on the chart paper.

Relationships1. A rectangle with a width equal to twice the length2. A square that always has an additional 1 unit

square attached 3. A rectangle with a width equal to three times the

length4. A rectangle with width that is 2 units longer than

the length5. A rectangle with a width that is 3 units shorter

than the length6. A square that always has 1 square unit removed7. A rectangle with a width equal to half the length

Discussion Questions• What relationship do you see between the width

and the area?• How might students connect this learning to their

understanding of linear relations?• What connections can you make between the

geometric model and the graphical model?• How might this help students understand

transformations?• What other insights did you have?

this activity address

How did Why did Did

the big ideas?

the lesson goal?

Consolidate – Think Back

• Which specific struggles would this activity potentially address for students?

• What could be done to address the others?

• How did the use of manipulatives add to your problem solving strategy? understanding? ability to connect representations?

Learning Goals• Finish the work from yesterday• Identify common student struggles (errors,

misconceptions, and partial understandings, etc) that are grounded in experience and/or research.

• Realize that different representations of relationships highlight different characteristics or behaviours, and can serve different purposes.

• Highlight the importance of being conscious and explicit about the instructional decisions; specifically, the need for simultaneous introduction & exploration of different models

Consolidate – Exit Card

• Complete the exit card

• Cut out algebra tiles from BLM QR.2.4