Transcript Secondary Math - 20 Level Focus

September 9 – TMSS
Hanover Room
9:00
Welcome & Introductions
9:15
Mathematics 20 Courses – What’s New
What are we finding thus far?
Resources?
StudentsAchieve?
9:40
The Painted Cube
A Mathematical Inquiry
From Spatial Reasoning to Algebraic Generalization
10:30
Refreshment Break and Networking
10:40
Unpacking and Rubric Development
Needs to be shared across our teachers!!
Noon
Lunch Break
12:45
Convention 2011 – 2012 (Sharolyn Simoneau)
1:00
Unpacking and Rubric Development con’t
2:15
Being Confident about Confidence Intervals
Developing an Understanding of New Concepts
Exploring, Discussing and Summarizing
2:50
Closure
Lunch provided
Enjoy the activities.
Engage in the mathematics.
Be an active participant:
listen,talk,question, explore,
persist, wonder, predict
summarize, synthesize.
The more the student becomes the teacher
and the more the teacher becomes the learner,
then the more successful the outcomes.
(John Hattie, 2009, “Visible Learning”)
Please introduce yourself:
name, school, etc.
share an implementation story, anecdote
or experience
What are we finding thus far?
Resources, etc.?
StudentsAchieve?
20 Level Courses … FM20, WA20, PC20
20 Level Textbooks …FM20, WA20, PC20
30 Level Courses … FM30, WA30, PC30,
Modified Courses … Math 11, Math 21
Calculus 30
Ministry Exams for FM30, WA30, PC30
Prototype Exams for FM30, WA30, PC30
Foundations of Mathematics 20
Workplace and Apprenticeship 20
Pre-Calculus 20
Foundations of Mathematics 20
Course Information
The Foundations of Mathematics pathway is designed to provide students with the
mathematical knowledge, skills and understandings required for post secondary
studies. Content in this pathway will meet the needs of students intending to
pursue careers in areas that typically require a university degree, but are not math
intensive, such as humanities, fine arts, and social sciences. Students who
successfully complete this course will be granted a grade 11 credit. Students must
successfully complete the common course, Foundation and Pre-calculus 10, prior
to taking this course. This course is a prerequisite to Foundations of Mathematics
30.
Topics Include:
 inductive and deductive reasoning
 proportional reasoning
 properties of angles and triangles
 sine and cosine laws
 normal distributions
 interpretation of statistical data
 systems of linear inequalities
 characteristics of quadratic functions
Workplace and Apprenticeship 20
Course Information
The Workplace and Apprenticeship pathway is designed to provide students with
the mathematical knowledge, skills and understandings needed for entry into some
trades-related courses and for direct entry into the work force. Students who
successfully complete this course will be granted a grade 11 credit. Students must
successfully complete Workplace and Apprenticeship 10 prior to taking this course.
This course is a prerequisite to Workplace and Apprenticeship 30
Topics Include:
 preservation of equality
 surface area, volume and capacity
 right triangles
 3 dimensional objects
 personal budgets
 compound interest, credit and related topics
 slope
 proportional reasoning
 representing data using graphs
Pre-Calculus 20
Course Information
The Pre-calculus pathway is designed to provide students with the mathematical
knowledge, skills and understandings required for post secondary studies. Content
in this pathway will meet the needs of students intending to pursue careers that will
require a university degree with a math intensive focus. Students who successfully
complete this course will be granted a grade 11 credit. Students must successfully
complete the common course, Foundations and Pre-calculus 10, prior to taking this
course. This course is a prerequisite to Pre-calculus 30.
Topics Include:
 absolute value
 radicals
 rational expressions and equations
 trigonometric ratios
 sine and cosine laws
 factoring polynomial expressions
 quadratic functions and equations
 inequalities
 arithmetic sequences and series
 geometric sequences and series
The Painted Cube
A Mathematical Inquiry
From Spatial Reasoning
to Algebraic Generalization
Demonstrate understanding of inductive
and deductive reasoning including: analyzing
conjectures, analyzing spatial puzzles and
games, providing conjectures, solving
problems.
Demonstrate the ability to analyze puzzles
and games that involve numerical reasoning
and problem solving strategies
What are we curious about?
What do we want to explore?
How can we begin?
Painted Cube Problem
A large cube, made up of small unit cubes, is dipped
into a bucket of orange paint and removed.
a) How many small cubes will have 1 face painted orange?
b) How many small cubes will have 2 faces painted orange?
c) How many small cubes will have 3 faces painted orange?
d) How many small cubes will have 0 faces painted orange?
e) Generalize your results for an n x n x n cube.
3 x 3 x 3 cubes
4 x 4 x 4 cubes
5 x 5 x 5 cubes
Size
3x3x3
4x4x4
5x5x5
1 face
painted
2 faces
painted
3 faces
painted
0 faces
painted
Size
3x3x3
4x4x4
5x5x5
1 face
painted
2 faces
painted
6
12
3 faces
painted
8
0 faces
painted
1
Size
1 face
painted
2 faces
painted
3x3x3
6
12
8
1
4x4x4
24
24
8
8
5x5x5
3 faces
painted
0 faces
painted
Size
1 face
painted
2 faces
painted
3 faces
painted
0 faces
painted
3x3x3
6
12
8
1
4x4x4
24
24
8
8
5x5x5
54
36
8
27
Size
3x3x3
1 face
painted
2 faces
painted
1x6 =6
1 x 12 = 12
3 faces
painted
1x8=8
0 faces
painted
1x1x1=1
Size
1 face
painted
2 faces
painted
3 faces
painted
0 faces
painted
3x3x3
1x6 =6
1 x 12 = 12
1x8=8
1x1x1=1
4x4x4
4 x 6 = 24
2 x 12 = 24
1x8=8
2x2x2=8
Size
1 face
painted
2 faces
painted
3 faces
painted
0 faces
painted
3x3x3
1x6 =6
1 x 12 = 12
1x8=8
1x1x1=1
4x4x4
4 x 6 = 24
2 x 12 = 24
1x8=8
2x2x2=8
5x5x5
9 x 6 = 54
3 x 12 = 36
1x8=8
3 x 3 x 3 = 27
Painted Cube Problem
A 10 x 10 x 10 cube made up of small unit cubes is dipped
into a bucket of orange paint and removed.
a. How many small cubes will have 1 face painted orange?
_______________________________________________
b. How many small cubes will have 2 faces painted orange?
_______________________________________________
c. How many small cubes will have 3 faces painted orange?
_______________________________________________
d. How many small cubes will have 0 faces painted orange?
_______________________________________________
Painted Cube Problem
A 10 x 10 x 10 cube made up of small unit cubes is dipped
into a bucket of orange paint and removed.
a. How many small cubes will have 1 face painted orange?
6 faces …. an 8 x 8 square on each face …..6 x 64 = 384
b. How many small cubes will have 2 faces painted orange?
12 edges …. 8 on each edge …. 12 x 8 = 96
c. How many small cubes will have 3 faces painted orange?
8 vertices …… always one per vertex ….. 8 x 1 = 8
d. How many small cubes will have 0 faces painted orange?
an 8 x 8 x 8 cube is hidden inside …. 8 x 8 x 8 = 512
Painted Cube Problem
An n x n x n cube made up of small unit cubes is dipped
into a bucket of orange paint and removed.
a. How many small cubes will have 1 face painted orange?
_______________________________________________
b. How many small cubes will have 2 faces painted orange?
_______________________________________________
c. How many small cubes will have 3 faces painted orange?
_______________________________________________
d. How many small cubes will have 0 faces painted orange?
_______________________________________________
Painted Cube Problem
An n x n x n cube made up of small unit cubes is dipped
into a bucket of orange paint and removed.
a. How many small cubes will have 1 face painted orange?
6 ( n – 2 )²
b. How many small cubes will have 2 faces painted orange?
12 ( n – 2 )
c. How many small cubes will have 3 faces painted orange?
8
d. How many small cubes will have 0 faces painted orange?
(n - 2)³
Faces Painted
Exponent in Generalization
3
0
2
1
1
2
0
3
Geometrically, using cubes and patterns...
13
2face
faces
painted
painted
(n – 2)X(n – 2)
(n – 2)
“square”
8 Corners
N3 = 8
6 Faces
12
Edges
N2 = 12(n – 2)
N1 = 6(n – 2)2
2 Faces
Painted
N2 = 12(n – 2)
1 Face
Painted
N1 = 6(n – 2)2
0 Faces
Painted
N0 = (n – 2)3
Graphically, using Excel...
600
Faces Painted
500
No. Faces Painted
400
Cube #
3 faces painted
300
2 faces painted
1 face painted
0 faces painted
200
100
0
1
2
3
4
5
Cube No.
6
7
8
A large cube is constructed from individual unit
cubes and then dipped into paint. When the paint
has dried, it is disassembled into the original unit
cubes. You are told that 486 of these unit cubes
have exactly one face painted.
How many unit cubes were used to construct the
large cube?
How many of the unit cubes have …. two faces
painted, three faces painted, no faces painted?
As teachers of mathematics, we want our
students not only to understand what they
think but also to be able to articulate how
they arrived at those understandings.
(Schuster & Canavan Anderson, 2005



We need to unpack outcomes and develop
rubrics for all 3 courses.
We will try to share the work-load across the
teachers.
Supports available
◦ Templates (Curriculum Corner or handouts)
◦ Curricular Documents (online or in print)
◦ Textbook Resources

Please forward completed documents to
myself for posting on Curriculum Corner.
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 realworld phenomena, students begin to view
mathematics as useful, relevant, and
integrated.
(FM 20 – Page 15)

Porcupine Plain
◦ October 24 & 25

Sharolyn Simoneau:

We have until 2:15 pm.
Outcome FM 20.7
Demonstrate understanding of the
interpretation of statistical data, including:
• confidence intervals
• confidence levels
• margin of error.
Note: It is intended that the focus of this outcome be on
interpretation of data rather than on statistical calculations.
Opinion polls from a sub group (sample)
of a larger population
Quality control checks in large scale
manufacturing / production lines
A poll determined that 81% of people who live
in Canada know that climate change is
affecting Inuit people more than the rest of
Canadians. The results of the survey are
considered accurate within ±3 % points, 19
times out of 20.
A cereal company takes a random sample
from their production line to check the
masses of the boxes of cereal. For a sample
of 200 boxes, the mean mass is 542 grams,
with a margin of error of ±1.9 grams. The
result is considered accurate 95% of the time.
TORONTO (Reuters) - The Conservatives have a lead of about
9 points over the Liberals in an opinion poll released on
Saturday, April 11 hovering around levels that could give
them a majority in the May 2 federal election. The Nanos
Research tracking poll of results over three days of surveys
put support for the Conservatives at 40.5 percent, barely
changed from 40.6 in Friday's poll. Support for the main
opposition Liberals was at 31.7 percent, up slightly from 31.1
percent, while the New Democratic Party fell to 13.2 percent
from 14.9 percent.
The daily tracking figures are based on a three-day rolling
telephone sample of 1,001 decided voters and is considered
accurate to within 3.1 percentage points, 19 times out of 20.
CON
LIB
NDP
BLQ
GRN
E-2008
37.6%
26.2%
18.2%
10.0%
6.8%
Mar 15
38.6%
27.6%
19.9%
10.1%
3.8%
May 01
37.1%
20.5%
31.6%
5.7%
3.8%
E-2011
39.6%
18.9%
30.6%
6.0%
3.9%
random sampling of a large population,
reflection of a normal distribution,
a sample mean is calculated to represent the
population mean,
sample mean extrapolated to the population,
95%, 99%, 90% confidence levels, confidence
interval, margin of error
Individually read the material presented.
Share and discuss the ideas with your table group.
Choose two ideas from your table group to report
to the large group.
Remember to choose a recorder and reporter for
your group.
Group sharing will begin at 2:40 pm.
Comments
Questions
Thank you and best wishes as you
conclude the school year.