MO 10 min
A 30 min
C/D 20 min
60 min / Math Learning Goals
· Identify common student struggles (errors, misconceptions, and partial conceptions 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 / Materials
· BLM QR.2.1
· BLM QR.2.1 teacher notes
· BLM QR.2.2, 3
· Sticky notes (new colour)
· BLM QR.2.4
· PPT QR DAY 2
Whole Group à Gallery Walk
Participants identify student struggles and post alongside corresponding curriculum cluster &/or lesson goal. Use sticky notes. / Curriculum expectation clusters from About Quadratic Functions 1
Create cards of BLM QR.2.1
Minds On…
Small Group à Activity
Each participant receives one card from BLM QR.2.1. Take participations through activity using instructions given on BLM QR.2.1 (teacher notes) steps 1–7.
Whole Group à Discussion
“How did/Why did/Did this activity address one or more big ideas?”
Small Group à Task
Participants discuss struggles that they had during the activity.
Record anticipated struggles that their students might have when completing BLM QR.2.1.
Follow up by recording questions they can ask to reveal those struggles and to help scaffold for some struggling students.
Whole Group à Discussion
Open the discussion by asking participants to share the connections they see between the identified struggles and the big ideas. (Could link the struggles back to the clusters and the sticky notes from the first part – I think we were going to do that…)
Whole Group à Lecture
Using the ppt., connect to Marian’s work by sharing what the research says about student struggles with quadratics.
Action!
Small Group à Task
In small groups, participants discuss the following questions:
· 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?
Individual à Exit Card
Collect Exit cards as teachers leave – read these in preparation for tomorrow’s activities.
Consolidate Debrief
Home Activity or Further Classroom Consolidation
Using manipulative of choice (e.g. linking cubes or colour tiles) complete BLM QR.2.2. Cut out algebra tiles from BLM QR.2.4. / One copy per participant of
BLM QR.2.2
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = 2x2 + 8x – 24
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = 2(x + 6)(x – 2)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = 2(x+2)2 – 32
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = ½ x2 + 4x + 6
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = ½(x + 6)(x + 2)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = ½(x+4)2 – 2
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = –3x2 + 12x – 9
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –3(x – 1)(x – 3)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –3(x–2)2 + 3
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = -2x2 – 4x + 6
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –2(x – 1)(x + 3)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –2(x+1)2 + 8
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = - ½ x2 – 1x + 1.5
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –½ (x – 1)(x + 3)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –½ (x+1)2 + 2
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = – 2 x2 +16x – 24
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = – 2 (x + 6)(x + 2)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –2 (x+4)2 + 8
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = –2x2 + 8x – 6
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –2(x – 1)(x – 3)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –2(x–2)2 + 2
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = – ½x2 – 2x + 6
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –½(x + 6)(x – 2)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –½ (x+2)2 + 8
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = – 2x2 – 8x
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = -2x(x + 4)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = –2(x+2)2 + 8
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = – ½x2 – 4x
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = – ½x(x + 8)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = – ½(x+4)2 + 8
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation
y = 8x2 + 32x
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = 8x(x + 4)
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value /
y = 8(x+2)2 – 32
State:
a) The direction of opening
b) The vertex
c) The step pattern
d) The zeros
e) The y-intercept
f) The axis of symmetry
g) Does it have a Max/min?
h) The optimal value
BLM QR.2.1 – Investigating Different Forms of a Quadratic Relation (Teacher Notes)
Lesson Goal: Different representations of relationships highlight different characteristics or behaviours and can serve different purposes.
Explain to the class that “Quadratic equations have many forms… we’re going to look at three of them today to see if any of these forms can help us determine particular characteristics about the graph without having to do to much work”
Example
y = 3x2 + 6x – 9 y = 3 (x + 3)(x – 1) y = 3(x + 1)2 – 12
Ask the question: “How could you justify that all of the above are quadratic equations?”
Introduce activity…
For the graph and equation you have been given:
1. State:
a. The direction of opening
b. The vertex
c. The step pattern
d. The zeros
e. Y-intercept
f. The axis of symmetry
g. Does it have a Max/min?
h. The optimal value
2. Find the people in the class with the same zeros as you… make observations (i.e. compare graphs, equations, etc) <1 min>
3. Find the people in the class with the same vertex as you… make observations <1 min>
4. Separate into just the people who have exactly the same parabola as you (if not already there)
5. In their new groups, on chart paper, write the 3 equations that represent your graph, draw the graph, and list the vertex, step pattern, zeros, and y-intercept . You may wish to give them the following example to work with: Ask them why this information gives all the other details of the graph (i.e. Q: why not state the optimum value? A: b/c it’s part of the vertex…)
y = 3x2+6x – 9 y = 3 (x – 1)(x + 3) y = 3(x + 1)2 – 12/ Vertex (-1, -12)
Step: 3, 9, …
Zeros (0, -3), (0, 1)
y – int = –9
6. Have students post charts using some organizational strategy (i.e. put the ones with the same vertex together, or the ones with the same zeros together) it’s helpful to use magnets so that they can move the charts around.