Post on 11-Jan-2016
Secondary MathematicsWorkshops
Nov. 22 – Saskatoon
Nov. 26 – Regina
Nov. 30 - Prince Albert
ed.bourassa@gov.sk.ca
Agenda
9:00 Welcome, Introductions, Orientation to the Day
9:30 Activity 1: A Farmer’s DilemmaMathematics in the Workplace Connection: Goals, Processes and Outcomes
10:30 Refreshment Break and Networking
10:50 Activity 2: A Visit to the Welding ShopEffective Communication in Mathematics Connection: Instruction and Assessment
Noon Lunch Break
1:00 Activity 3: The Ambiguous RectangleInquiry and Conjecture in MathematicsConnection: Motivation and Persistence
2:00 Activity 4: The Geometry of the Star BlanketGeneralization in MathematicsConnection: Patterns and Logical Thinking
3:00 Round Table
3:20 Closure
Orientation to the Day
The day is structured around four mathematical activities that connect to outcomes in WA 10 and FPC 10.
The activities, although intriguing in their own right, are meant to be a vehicle to encourage discussion about other topics.
As we reflect on the activities, we will have an opportunity to share our collective wisdom.
Introductions
Please introduce yourself:
name, SD, FN
position, role, responsibility,
a brief comment about WA 10, FPC 10 implementation ….
or … your analysis of the Grey Cup!!
Invitation to the Spirit of the Day
Enjoy the activities and engage in the mathematics they represent.
Be an active participant:
listen, talk, question, explore, persist, wonder, summarize, synthesize.
Activity 1A Farmer’s Dilemma
Mathematics in the Workplace
Joe is a 56 year-old farmer. In 2004 his wheat harvest was larger than expected and, as a result, he ran out of bin space. In a panic to complete the harvest he augered several truck loads of grain directly onto the ground forming a cone shaped pile.
His dilemma …. Leave the grain out over winter or purchase more bins.
He decided to purchase more grain bins to get his wheat under cover before winter. Unsure of how many truck loads he had dumped on the ground, he needed to calculate the number of bushels of wheat in the pile in order to purchase the correct number and size of grain bins.
A Farmer’s Dilemma – Task 1
Discuss what information Joe will need, and how he might get that information.
Use the chart provided to record your information.
A Farmer’s Dilemma – Task 1
Information Needed Ways to Get This Information
A Farmer’s Dilemma – Task 1
Information Needed Ways to Get This Information
might need a formula or two !!! textbook, teacher, parent, web, student
might need some measurements !!! ???????
??????
A Farmer’s Dilemma – Task 1
Useful Information Ways to Get This Information
various formulae textbook, teacher, parent, web, student
circumference of the pile of wheat pace it, string, tape measure, trundle wheel
diameter and radius estimate, referent, tape measure, formula
angle of incidence with the ground estimate, home made device, from grain bins
height of pile estimate, pole probe, referent, trigonometry
slant height (slant distance) tape measure, trigonometry, Pythagorean Th.
linear conversions SI ↔ Imperial textbook, teacher, parent, web, student
volume conversions .. ex .. cubic feet to bushels textbook, teacher, parent, web, student
grain bin sizes community, teacher, parent, web, student
A Farmer’s Dilemma – Task 2
If the circumference of the pile was 60 meters, and the cone was shaped with an angle of 35 degrees, how many bushels would be in the pile? How many bins might he purchase?
Or ……
If the circumference of the pile was 195 feet and the slant height was 38 feet, how many bushels would be in the pile? How many bins might he purchase?
A Farmer’s Dilemma – Task 2
Calculation of Volume in Bushels Labeled Sketch
Calculation of Number of Bins Other Information or Reflections
Useful Information: Formulae
C = πd
d = 2r
V (cyl) = (π)(r2)(h)
V (cone) = (π)(r2)(h) 3
a2 + b2 = c2
Sin(θ) = opposite side
hypotenuse
Cos(θ) = adjacent side
hypotenuse
Tan(θ) = opposite side
adjacent side
Useful Information: Measurements
SI Measure
c = 60 m
d = 19.10 m
r = 9.55 m
h = 6.7 m
s = 11.67 m
Imperial Measure
c = 195 feet
d = 62.10 feet
r = 31.05 feet
h = 21.78 feet
s = 37.92 feet
Useful Information: Measurements
SI Measure
c = 60 m
d = 19.10 m
r = 9.55 m
h = 6.7 m
s = 11.67 m
v = 640 cubic meters
bushels ≈ 18 000
number of bins …….
Imperial Measure
c = 195 feet
d = 62.10 feet
r = 31.05 feet
h = 21.78 feet
s = 37.92 feet
v = 21 907 cubic feet
bushels ≈ 18 000
number of bins …….
A Farmer’s Dilemma – Task 3
Connection to Outcomes and Processes:
Use the curriculum documents provided.
Identify the outcomes and processes that this activity support. The indicators may be helpful in this task.
Discuss your findings in table groups and be prepared to share with the large group.
A Farmer’s Dilemma – Task 3
Outcomes:
Processes:
A Farmer’s Dilemma – Task 3
Connection to Outcomes and Processes:
FP10.3 c, i, j, k, l
FP10.4 c, d
Processes: CN, PS, C
Refreshment Break
Contextualization and making connections to the experiences of learners are powerful processes in developing mathematical understanding. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant, and integrated.
(WA 10 – Page 14)
Activity 2A Visit to the Welding Shop
Effective Communication in Mathematics
Andrew, a grade 10 student, visits the local welding shop to observe how the workers in this shop fabricate cone-shaped hopper bottoms for grain bins.
Following his visit and a discussion with the workers, Andrew is required to write a report describing the mathematics used in the process of building the hopper bottoms.
WA 10.5, 10.6, 10.8
A Visit to the Welding Shop – Task 1
Read Andrew’s project report and discuss its strengths and weaknesses with a
partner.
Record the key points of your discussion below and be prepared to share these with the large group.
A Visit to the Welding Shop – Task 2
Read the article, “Pushing the Vocabulary“, about effective communication in mathematics.
With a partner, discuss Andrew’s work in reference to the information in the article. In the space below, record at least two observations or connections between Andrew’s work and the article.
A Visit to the Welding Shop – Task 3
Use the rubrics provided and the summary of the 8 big ideas about
assessment.
Assess Andrew’s work. Record your assessment and any additional comments in the space below
Lunch Break
Students need opportunities to read about, represent, view, write about, listen to and discuss mathematical ideas using both personal and mathematical language and symbols. These opportunities allow students to create links among their own language, ideas, prior knowledge, the formal language and symbols of mathematics, and new learning.
(FPC 10 – Page 14)
Observations and Reflections
Try just one thing differently tomorrow
Open up the activity and the assessment
Perplex them
Implementation is a a counter-cultural endeavor
The traditional to reform continuum
Stress and emotional well-being
Activity 3The Ambiguous Rectangle
Inquiry and Conjecture
FP10.7(c) WA10.2(b)
Sometimes a task or a problem may be so complex or confusing that it creates within us “cognitive dissonance”.
We need to develop the “habit of mind” needed to persist through the dissonance.
Activity 3The Ambiguous Rectangle
“Cognitive dissonance” is a psychological phenomenon that refers to the discomfort we feel when there is a discrepancy between what we already know or believe, and new information or perceptions.
A “habit of mind” is a disposition toward behaving intelligently when confronted with a problem … and the answer or the process needed to get to the answer is not immediately known.
Activity 3The Ambiguous Rectangle
Cut the square into four pieces as indicated in the diagram.
Arrange these four pieces to form a rectangle with a height of 5 units.
Compare the area of the square and the area of the rectangle, both made from the same pieces.
The Ambiguous Rectangle – Task 1
Discuss why the areas of the square and the rectangle are different? Is there a logical reason why they differ by one square unit? Share your thoughts and ideas.
Thoughts …
Is the rectangle, really a rectangle?
What guarantees that it is a rectangle?
Considering the area of the rectangle is different, might there be “suspicious” aspects in the interior of the rectangle ….. ?
Slope of a Line is the Ratio of Height to Length or Rise to Run
Slope of the rectangle’s diagonal:
m = 5/13 = 0.384
Slope of parts of the diagonal:
m = 2/5 = 0.4
m = 3/8 = 0.375
The Ambiguous Rectangle – Task 2Conjecture is Important to Mathematical Inquiry
Are you left wondering anything about this activity?
Is there anything else you would like to explore?
What if? What about? I’m wondering?
There is more …….!
Consider the three measurements used in this activity: 5, 8, 13
What do you notice?
There is more …….!
Consider the three measurements used in this activity: 5, 8, 13
They belong to an important mathematical sequence called the Fibonacci Sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144 …..
There is more …….!
The Fibonacci Sequence:
1, 1, 2, 3, 5, 8,13, 21, 34, 55, 89,144, 233 …
Take any triplet in the sequence: a, b, c.
Calculate (a)(c) – (b)(b) for any triplet.
The Ambiguous Rectangle – Task 3
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144 …
Calculate (a)(c) – (b)(b) for several triplets a,b,c
Record your observations and conjectures, if any ……
The Ambiguous Rectangle – Task 3
Calculate (a)(c) – (b)(b) for several triplets a,b,c
Record your observations and conjectures, if any ……
The Ambiguous Rectangle – Task 4
What would be an assessable assignment or product for this inquiry?
Activity 4Geometry of the Star Blanket
Generalization in Mathematics
Discovering and understanding patterns is an important mathematical skill. One of the goals of algebraic thinking is to generalize our understanding of patterns.
The First Nations Star Blanket is made of an interesting geometric pattern using the parallelogram.
FP 10.8 (d,s)
The Star Blanket – Task 1
Use the information and diagram found on the handout provided for this activity. Complete the first task outlined in the paragraph located below the diagram.
The Star Blanket – Task 1
Level Transition Total in Level
1 8
2 8 + 24
3
4
5
6
7
8
The Star Blanket – Task 1
Level Transition Total in Level
1 8 8
2 8 + 24 32
3 32 + 40 72
4 72 + 56 128
5 128 +
6
7
8
The Star Blanket – Task 1
Level Transition Total in Level
1 8 8 8x1
2 8 + 24 ( 8+16) 32 8x4
3 32 + 40 (24+16) 72 8x9
4 72 + 56 128
5 128 + 72 200
6 200 + 88 288
7 288 + 104 392
8 392 + 120 512 8x64
The Star Blanket – Task 1
Level Transition Total in Level
1 8 8 8x1x1
2 8 + 24 (8+16) 32 8x2x2
3 32 + 40 (24+16) 72 8x3x3
4 72 + 56 (40+16) 128
5 128 + 72 200
6 200 + 88 288
7 288 + 104 392
8 392 + 120 512 8x8x8
The Star Blanket – Task 2
So, now ….. Can we generalize how many parallelograms there are at level “L”?
In point form or in a paragraph, describe the pattern and the generalization.
The Star Blanket – Task 2
So, now ….. Can we generalize how many parallelograms are at level “n”?
Level “L” has: L x L x 8 parallelograms
The Star Blanket – Task 3
Create a table of values for this relationship and graph the ordered pairs.
How might this activity connect to and support outcome FP 10.8
L 1 2 3 4 5 6 7 8
T
The Star Blanket – Task 3
Create a table of values for this relationship and graph the ordered pairs.
How might this activity connect to and support outcome FP 10.8
L 1 2 3 4 5 6 7 8
T 8 32 72 128 200 288 392 512
Star Blanket - Parallelograms Per Level
0
100
200
300
400
500
600
0 1 2 3 4 5 6 7 8 9
Level
To
tal
Par
alle
log
ram
s
Round Table Discussion
Comments Questions Follow Up
If you would like a copy of this power point to use with other teachers or with your
students, please email me at:
ed.bourassa@gov.sk.ca