September 9 – TMSS Hanover Room. 9:00 Welcome & Introductions 9:15Mathematics 20 Courses –...

Secondary Mathematics

Learning Project Day

September 9 – TMSSHanover Room

9:00 Welcome & Introductions

9:15 Mathematics 20 Courses – What’s NewWhat are we finding thus far?Resources?StudentsAchieve?

9:40 The Painted CubeA Mathematical InquiryFrom Spatial Reasoning to Algebraic Generalization

10:30 Refreshment Break and Networking

10:40 Unpacking and Rubric DevelopmentNeeds to be shared across our teachers!!

Noon Lunch Break Lunch provided

12:45 Convention 2011 – 2012 (Sharolyn Simoneau)

1:00 Unpacking and Rubric Development con’t

2:15 Being Confident about Confidence IntervalsDeveloping an Understanding of New ConceptsExploring, Discussing and Summarizing

2:50 Closure

Agenda

Enjoy the activities.

Engage in the mathematics.

Be an active participant:

listen,talk,question, explore, persist, wonder, predictsummarize, synthesize.

Invitation to the Spirit of the Day

Shared Responsibility

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?

Introductions

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

Secondary Mathematics Update

Foundations of Mathematics 20

Workplace and Apprenticeship 20

Pre-Calculus 20

Mathematics 20Course Summaries

Foundations of Mathematics 20

Course InformationThe 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 InformationThe 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 InformationThe 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

Activity 1

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.

Outcome FM 20.2

Demonstrate the ability to analyze puzzles and games that involve numerical reasoning and problem solving strategies

Outcome WA 20.2

What are we curious about?

What do we want to explore?

How can we begin?

Inquiry

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

Explore Possibilities

Begin with Simpler Models

Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3

4 x 4 x 4

5 x 5 x 5


Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3 6 12 8 1

4 x 4 x 4

5 x 5 x 5


Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3 6 12 8 1

4 x 4 x 4 24 24 8 8

5 x 5 x 5


Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3 6 12 8 1

4 x 4 x 4 24 24 8 8

5 x 5 x 5 54 36 8 27


Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3 1 x 6 = 6 1 x 12 = 12 1 x 8 = 8 1 x 1 x 1 = 1


Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3 1 x 6 = 6 1 x 12 = 12 1 x 8 = 8 1 x 1 x 1 = 1

4 x 4 x 4 4 x 6 = 24 2 x 12 = 24 1 x 8 = 8 2 x 2 x 2 = 8


Size 1 face painted

2 faces painted

3 faces painted

0 faces painted

3 x 3 x 3 1 x 6 = 6 1 x 12 = 12 1 x 8 = 8 1 x 1 x 1 = 1

4 x 4 x 4 4 x 6 = 24 2 x 12 = 24 1 x 8 = 8 2 x 2 x 2 = 8

5 x 5 x 5 9 x 6 = 54 3 x 12 = 36 1 x 8 = 8 3 x 3 x 3 = 27

Painted Cube ProblemA 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 ProblemA 10 x 10 x 10 cube made up of small unit cubes is dipped into a bucket of orange paint and removed.


6 faces …. an 8 x 8 square on each face …..6 x 64 = 384


12 edges …. 8 on each edge …. 12 x 8 = 96


8 vertices …… always one per vertex ….. 8 x 1 = 8


an 8 x 8 x 8 cube is hidden inside …. 8 x 8 x 8 = 512

Painted Cube ProblemAn n x n x n cube made up of small unit cubes is dipped into a bucket of orange paint and removed.


_______________________________________________


_______________________________________________


_______________________________________________


_______________________________________________

Painted Cube ProblemAn n x n x n cube made up of small unit cubes is dipped into a bucket of orange paint and removed.


6 ( n – 2 )²


12 ( n – 2 )


8


(n - 2)³

An Interesting Comparison

Faces Painted Exponent in Generalization

3 0

2 1

1 2

0 3

Animated power point of the painted cube problem …..

Geometrically, using cubes and patterns...

2 faces painted1 face painted

Painted Cube Problem…

3 faces painted

N2 = 12(n – 2) N1 = 6(n – 2)2N3 = 8

8 Corners 12 Edges6 Faces

(n – 2)(n – 2)X(n – 2)

“square”

2 Faces Painted

N2 = 12(n – 2)

Painted Cube Problem…Using Finite Differences

1 Face Painted

N1 = 6(n – 2)2


0 Faces Painted

N0 = (n – 2)3


Graphically, using Excel...

Faces Painted

0

100

200

300

400

500

600

1 2 3 4 5 6 7 8

Cube No.

No. Faces P

ain

ted

Cube #

3 faces painted

2 faces painted

1 face painted

0 faces painted

Painted Cube Problem…Graphically

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?

Another Way to Pose the Problem

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

Refreshment Break

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.

Unpacking and Rubrics

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.

(FM 20 – Page 15)

Lunch Break

Porcupine Plain◦ October 24 & 25

Sharolyn Simoneau:

Convention 2011 – 2012

We have until 2:15 pm.

Unpacking con’t

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.

Being Confident about Confidence Intervals

Opinion polls from a sub group (sample) of a larger population

Quality control checks in large scale manufacturing / production lines

Typical Uses of Confidence Intervals

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.

Examples of Confidence Statements

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.


Popular Vote Shift2011 Federal Election

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

Key Concepts to Explore

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.

Process for Activity

Comments

Questions

Thank you and best wishes as you conclude the school year.

Closing and Adjournment

September 9 – TMSS Hanover Room. 9:00 Welcome & Introductions 9:15Mathematics 20 Courses –...

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