M ATHEMATICS T eachable M oment as a T eachable M oment M eaning C hoice D iversity T rust T ime 1

MMATHEMATICSATHEMATICSas a TTeachable eachable MMomentoment

MMeaningeaning

CChoicehoice

DDiversityiversity

TTrustrust

TTimeime

1

Student Projects

Student Written Problems and Solutions

Sports / Pets / Cooking

Date / Special Days / Season / Weather

Place (Home / Community / School)

Games

Discussion in pairs, small groups and as a class

Create

Meaning

2

Choices provide meaning through a sense of:

ControlControl

CommitmenCommitmentt ChallengeChallenge

Projects, Student Written Problems, “How Many Ways” sheets,

and Discussion are all embedded with choice.

Give Choices

3

Encourage alternative strategies.

Ask “How?” not “Why?”

Give students choice in the order, methods, strategies and topics.

Make sure all students are involved in creating rules and sharing strategies.

Value Diversity

Diversity should be treated as a positive factor in the classroom.

We need to:We need to:

4

Create a

Climate of

TrustAsk open-ended questions and

value diverse strategies for solving problems.

Ask students to explain, discuss and show – especially when their answers are correct.

Value errors as opportunities to investigate conceptual understanding and create

new understandings.

5

6

Ensure There

is Adequate

TimeTeach the curriculum in an integrated manner

so that there are opportunities to review every major theme or

skill set.

Integrate the intended learning outcomes (ILOs)

into major themes and evaluate over the

whole year.

Value accuracy over speed.

Avoid:Races

Contestsand

Strictly Timed Basic Fact Tests

Review of

Silent

Mouthing

Use the “silent mouthing technique:

When students make errors give them hints, suggest that they are close, acknowledge that they are a step ahead or say, “That is the answer to a different question.”

Student FeedbackStudent Feedbackto give: to give:

Slower processors and complex thinkers the time they need to do the question.

7

Review of

Place Value

Place value should be taught at least once a week but preferably a place value connection should be made almost every day.

The connections to algebraic thinking should be made (collecting like terms) as this will pay off when doing operations with fractions and algebraic expressions.

8

Organization of

the CURRICULUM

All four strands (ŸNumber Sense,

ŸSpatial Sense, ŸProbability and Data Sense

and ŸPattern and Relationship

Sense) should be covered every month

(every week in Primary).

Problem solving often embeds three of the strands depending on whether the problem has a focus on spatial relationships or data relationships. It is usually preferable to introduce a new topic through a problem. The Japanese teachers use this technique effectively.

9

Making Meaning with the WEEKLY GRAPH

Graphing is a tool for

making meaning

if the data is collected

from the students.

The “Weekly Graph” is

intended to be student

driven

by the fourth week

at the latest.

Eventually the

“Weekly Graph”

becomes a day for

teaching proportional

thinking, decimals,

fractions, percents,

graphing, patterns and

relations, and

probability.

10

Watch for the “big ideas”Watch for the “big ideas”in the video.in the video.

What teaching techniques are effective?

What Mathematical concepts are covered?

11

How manyILOS were covered in

the previous video clip?

How do we find the time to teach

this way?

12

NEWNEW Strategies for Strategies for OLDOLD Ideas Ideas

Intermediate

Students

Where do we find the time to teach this way?

If students are taught this way, how will they do on the FSA tests?

13

Multi-step Divisionand

Decimal Fractions

The first few times multi-step division is taught it should be done as a whole class.

The errors made should be used as opportunities to investigate conceptual understanding.

Placement of the decimal in the quotient should be done by asking,

“Where does it make sense to put the decimal so that the answer makes sense?”

14

PProcess for rocess for

TTeachingeaching

11 ÷ ÷ 99 If possible, do multi-step division on grid paper

(cm graph paper works well).

If grid paper is not available, use lined paper turned sideways so that the lines become grids for keeping the numerals

in the correct position.

15

)9 10

-01

.0

0

1

- 91

.0

0

1

- 91

)9 0. 0 0 0 0 0 0 00.1 1 1 1 1 1 1

19 0.1

Ÿ=

19 0.1=or

11 ÷ ÷ 99

16

Many algorithms are culture specific time savers that create accuracy.

The multi-digit regrouping system we use for subtracting is based on the principle of equivalence and is done differently in parts of Europe.

Some Europeans use a system that depends on the principles of balance and equivalence.

17

Many algorithms are culture specific time savers that create accuracy.

The algorithm we use for multi-digit multiplication has changed considerably over the years.

In the middle ages we used a box or window method.

In the fifties we moved the second product over one space which paralleled the way we multiplied using adding machines. Now we add a zero

for the second product. 18

Algorithms in the 21st Century

Algorithms should be developed through discussion with learners because the purpose of teaching algorithms is to develop understanding.

The focus should be on accuracy, then on efficiency.

The most efficient algorithm today is always based on today’s technology.

The most efficient algorithm today is the calculator or the computer but we do need to understand the underlying concept or we don’t know if the answer makes sense. 19

FRACTIONS are RICH in PATTERNS

Working at your table or in your group, assign different members of the group to find the decimal fraction for:

17

, 27

, 37

, 47

, 57

, 67

, 77

, 87

,

Do you notice

any patterns?

How many remainders did it take before you

achieved a repeating decimal pattern?

20

Common Fractions

SimplestForm

DecimalEquivalent

PercentageEquivalent

14 = 2

8= 3

12 = 416 = 5

20 = 40100

On your “Memorable Fractions” sheet please write in all the fractions studied in the video.

Include some of the equivalent fractions for these.

For example:(the first fraction

illustrated in the video)

There were three other fractions in the problem. There was one fraction from the graph.

WWEEKEEK OONENEof theof the

WWeekly eekly GGraphraph

21



Have the students draw a bar graph of the results.

Do not give students criteria for creating a good graph. Discuss the results and focus on the fact that graphs are supposed to give you a lot of information at a glance. This means that the graph should be neat, have a title and a legend (if necessary).

In the end, the class will have developed assessment criteria from a meaningful context by having students notice what makes a graph a good communication tool.

Self-evaluation is often the most effective.

22

Have students discuss (write) what they know about the class by analyzing the data (graph).

Use the think/pair/share method to create discussion, then share as a group (valuing diversity, creating trust and developing meaning through choice).

Can they think of any questions or extensions?

Use these for further research.



23

WWEEKEEK TTWOWOof theof the


Collect data.

Decide which fractions (decimals and percents) you wish to study. If you are worried about coloring in the hundreds squares for a tricky fraction, leave this part until the next day and try it yourself. Enter the fractions on the Memorable Fraction sheet.

Draw a circle graph of the data.

Review the criteria.

24

Can be rich in CURRICULUM Connections

If the number of voters in the class is: 12

151820243036or

Do the following:

Have the students find the prime factorization of 360 and the prime factorization of the number who voted (e.g. 30).

Write the equation in fractional form:36030

2 x 2 x 2 x 3 x 3 x 52 x 3 x 5

=

25

Find the ones.

36030

2 x 2 x 2 x 3 x 3 x2 x 3 x 5

= 5

This principle was used in the video to make equivalent fractions – in particular:

25

2020

40100

x =

PPRINCIPLE of RINCIPLE of ONEONE

14

520

52 x 2 x 5

==

26

PPRINCIPLERINCIPLE of of

EQUIVALENCEEQUIVALENCE

Throughout the video and on the “Memorable

Fractions” sheet, the students have been

making equivalent fractions and have learned

that every fraction can be expressed as an

infinite number of common fractions, exactly

one decimal fraction and one percentage

fraction. It can also be expressed as a ratio.

27

PPRINCIPLERINCIPLE of of BALANCEBALANCE

In the video one student noticed that when

equivalent fractions are generated, both the

numerator and denominator have to be multiplied

by the same number.

This is also an example of the Principle of One as:

14

520

55

x ==x

28

Please solve the Please solve the following:following:Don’t forget to show your steps.

2x + 5 = 31

2x + 5 – 5 = 31 – 5

2x = 26

2x = 262 = 2

x x == 13 13

29

PPRINCIPLERINCIPLE of of ZEROZERO

This step is necessary for equation solving and is the only principle that is not generated in doing the “Weekly Graph”.

It should have been generated much earlier in the primary grades when doing the “How Many Ways Can You Make a Number” activity during Calendar Time.

How many different ways can you make __________ ?

+ - x ÷

30

How How Many Many WaysWays

??

31

How How Many Many WaysWays

??

32

Mark Criteria

Mark Criteria

Where any sentence contains theAddition operation

Where any sentence contains theSubtraction operation

Where any sentence contains theMultiplication operation

Where any sentence contains theDivision operation

Where any sentence containsmore than two terms(e.g. 2 x 3 + 5 = 10)

Where any sentence containsmore than two operations(e.g. 2 x 3 + 4 = 10)

Where any sentence contains a numbermore than the goal number(in this case 10)

Where any sentence contains a numbersubstantially greater than the goal number(in this case 50 or 100)

Where any group of sentences showsevidence of a pattern(e.g. 1 + 9, 2 + 8, 3 + 7)

Where any sentence shows knowledge of the power of zero(e.g. 6 – 6 + 10 = 10 or 10 + 0 = 10)

Where any sentence usesdoubling and halving to generate new questions(e.g. 4 x 6 = 24, 2 x 12 = 24, 1 x 24 = 24)

Where any sentence shows knowledge ofthe power of one(e.g. 6 ÷ 6 + 9 = 10 or 10 x 1 = 10)

Where any sentence shows knowledge ofthe commutative principle(e.g. 6 + 4 = 10 and 4 + 6 = 10)

Where any sentence shows knowledge ofthe number Note: this applies only for numbers greater than 10, such as 24. In upper intermediate grades, award marks for exponential notation also.(e.g. 20 + 4 = 24 and 2 x 10 + 4 = 24)

Where any sentence contains brackets, such as: (3 + 2) + (3 + 2) + (3 + 2) + (3 + 2) + 4 = 24

Where any sentence contains exponents, square roots, factorials, or fractions.Note: there should be no expectation of the demonstration of exponents, square roots or factorials before grade six, but their use should be acknowledged and rewarded where a student chooses to employ such operations in earlier grades.

How Many Different Ways Can You Make a Number?How Many Different Ways Can You Make a Number?

1

11

1

1

1

1

1

1

1

1

1

1

1

1

1

33

PPRINCIPLESRINCIPLES of of

EQUATION SOLVINGEQUATION SOLVING

Principle of ZeroZero

Principle of OneOne

Principle of EquivalenceEquivalence

Principle of BalanceBalanceThese four principles should be generated by and attributed to students. They are all you need to solve most equations and work with rational expressions throughout high school.

34

ConnectionConnectionss

ALGEBRAICALGEBRAIC

THINKINGTHINKINGtoto

The principle of one and the principle of balance are used in rationalizing radical expressions.

The principles of one and balance can be used to generate an easy to remember algorithm for dividing fractions.

35

PPRINCIPLE of RINCIPLE of ONEONE

2357

1

23

57

x= 2

357

x =

7575

x

x= 14

15

We refer to this as Invert and Multiply which has no other

application in mathematics.

The Principle of OnePrinciple of One has many applications.

36


ALGEBRAICALGEBRAIC

THINKINGTHINKINGtoto

Equivalence is used in all facets of mathematics.Balance is used in equation solving as well as multiplication and division of rational expressions.

The Principle of Zero is extensively in simplifying rational expressions.

All four principles are used in

equation solving.

37

Probability can be introduced during the “Weekly Graph” process.

Probability was introduced in the first session when playing “hangman” which is an activity students love to play.

Probability sense is an important skill we use in everyday life.

38

In your group, have one person shuffle the red deck (cards numbered 1 to 10) and a different person shuffle the blue deck.

TTen-en-FFramerame

PProbabilityrobability

Place the decks face

down side-by-side.

Predict the sum if you were to turn over the top two cards.

Collect the predictions from the whole group.

39

Was the most common prediction a 7?

TTen-en-FFramerame

PProbabilityrobability

Turn over the two decks and find all the combinations that equal 77.

It often is.

What is the probability of turning over two cards whose sum is 77?

How many did you get?

Check with other groups to see how many they got.

Discuss the reason for your answers until you come to a consensus.

What would the probability be for getting 66? 55?

Now watch the video. 40

NEWNEW Strategies for Strategies for OLDOLD Ideas Ideas

Intermediate

Students

Which ILOs were covered in the activity?

What are some connected or follow-up activities that you could use?

41


ADDINGADDING

FACTSFACTStoto

Introducing the ten-frame cards this way allows grade four to eight students to look at numbers in a new way and learn to add visually without counting.

The games shown in the video are called “Solitaire 10” and “Concentration 10”. Some students in intermediate grades have difficulty adding, and this is a new way to learn an old concept of making tens.

42


““ALL THEALL THE

FACTS” FACTS” SheetSheettoto

All the Addition Facts You Ever Need to Know (B)

8 + 2 = 6 + 0 = 2 + 9 = 8 + 0 =

8 + 3 = 6 + 9 = 7 + 3 = 1 + 1 =

6 + 8 = 9 + 3 = 8 + 5 = 5 + 5 =

3 + 1 = 3 + 4 = 5 + 4 = 7 + 5 =

1 + 2 = 4 + 9 = 9 + 1 = 6 + 4 =

7 + 4 = 2 + 2 = 3 + 3 = 2 + 3 =

9 + 9 = 3 + 5 = 0 + 1 = 1 + 7 =

0 + 9 = 2 + 5 = 6 + 7 = 2 + 0 =

3 + 6 = 4 + 1 = 4 + 0 = 8 + 8 =

0 + 3 = 1 + 5 = 2 + 4 = 7 + 0 =

1 + 8 = 4 + 4 = 6 + 5 = 2 + 6 =

5 + 0 = 6 + 1 = 2 + 7 = 1 + 9 =

4 + 8 = 5 + 9 = 6 + 6 = 8 + 9 =

7 + 8 = 3 + 7 =

Column 1 Column 2 Column 3 Column 4

Name left to learn

All the Addition Facts You Ever Need to Know (B)

8 + 2 = 6 + 0 = 2 + 9 = 8 + 0 =

8 + 3 = 6 + 9 = 7 + 3 = 1 + 1 =

6 + 8 = 9 + 3 = 8 + 5 = 5 + 5 =

3 + 1 = 3 + 4 = 5 + 4 = 7 + 5 =

1 + 2 = 4 + 9 = 9 + 1 = 6 + 4 =

7 + 4 = 2 + 2 = 3 + 3 = 2 + 3 =

9 + 9 = 3 + 5 = 0 + 1 = 1 + 7 =

0 + 9 = 2 + 5 = 6 + 7 = 2 + 0 =

3 + 6 = 4 + 1 = 4 + 0 = 8 + 8 =

0 + 3 = 1 + 5 = 2 + 4 = 7 + 0 =

1 + 8 = 4 + 4 = 6 + 5 = 2 + 6 =

5 + 0 = 6 + 1 = 2 + 7 = 1 + 9 =

4 + 8 = 5 + 9 = 6 + 6 = 8 + 9 =

7 + 8 = 3 + 7 =

Column 1 Column 2 Column 3 Column 4

43


SUBTRACTIONSUBTRACTION FACTSFACTS

toto

Visual tools are powerful.

After just this one lesson, which may take two or three days to complete, most students when asked to visualize how to make ’15’ with the cards will say, “Get a ten and a five”.

When they say, “Six”, ask them

how they see the six.

They should say, “One and Five”.

This tool works for subtracting 9, 8 and 5, which is almost half of the subtracting facts.

Now ask them to cover up or

take away nine.

44


PROBABILITYPROBABILITYwithwith

All of the fractions generated in the video

were for ‘what you would expect to get’.

This is called the “Expected Probability”.

What we are really interested in is

the “Experimental Probability”.

The next step is to have each pair or students

do 100 trials each and compare the Expected

Probability to the Experimental Probability.

The difference explains why people gamble.

45


TECHNOLOGYTECHNOLOGYwithwith

If each student in the class does 100 trials and then the data is put on a spreadsheet, it is clear that while some students will win if they pick their favourite number, others will lose.

However, the experimental results for the whole class will usually mirror the expected probability.

Government figures the odds, pays less than the expected probability, and makes lots of money.

Gambling then is a tax on the under-educated, often the poor.

46



A B C D EErica Kaeli Sarah Alyssa J essie

1 2 2 1 0 1

2 1 1 3 2 1

3 2 2 3 4 2

4 4 4 5 5 4

5 2 5 5 2 4

6 7 5 7 6 3

7 6 10 8 11 4

8 7 5 5 8 10

9 6 10 6 8 14

10 18 13 8 7 15

11 11 9 11 9 9

12 13 12 8 9 11

13 4 5 8 6 6

14 8 5 2 5 4

15 3 2 9 8 6

16 0 3 4 4 4

17 2 6 4 3 2

18 3 1 1 2 0

19 1 0 2 1 0

20

21

22

100 100 100 100 100

47



Do the same activity with six-sided, ten-sided, or twelve-sided dice.

Probability of getting a specific number or color of SmartiesTM or other candies on Halloween or Valentines Day.

Probability of a new student in class wanting a specific kind of pizza, liking a certain pop star, or wearing a certain kind of clothing.

48


DECIMAL DECIMAL FRACTIONS FRACTIONS PROJECTPROJECT

to theto the

Take out the Decimal Fractions Project sheet.Enter all the fractions and decimals collected so far.

Find the prime factorization of the denominator for each fraction (use fractions in their lowest terms only.)

Do you see any patterns?

49

SSTUDENTTUDENT FFRACTIONRACTION DDECIMALECIMAL IINVESTIGATIONNVESTIGATION

SSHEETHEETFraction Decimal

Check if aRepeatingDecimal

Check if aTerminating

Decimal Prime Factorization of the Denominator

25

0.4 5

200.25

310

0.3 19

0.1 9 = 3 x 3

5 = 5

10 = 2 x 5

so 4 = 2 x 25 20

1 4

=

50


NUMBER LINESNUMBER LINEStoto

0 1

Draw a line from 0 to 2.

58

Place the fraction

on the line.58

110

2

14

12

34

121

The important issue when connecting number lines to

rational numbers is to create reference points

(tenths, quarters and halves).

19

Place the fraction

on the line.19

27

Place the fraction

on the line.27

85

Place the fraction

on the line.85

83%

Place 83%on the line.

150%

Place 150%on the line.

0.7

Place 0.7 on the line.

51


NUMBER LINESNUMBER LINEStoto

Sometimes it is important to have the number lines drawn vertically so that the student makes the connection to a thermometer.

Then it is easy to introduce the idea of integers and negative integers in a natural context.

0

52

MAKING MEMORIESMAKING MEMORIES

In the last session the “Norman” story was introduced as a way to create a metaphor (based on scientific theory

about the way we create memories) about how Norman learned to add 8 + 7 and other numbers by breaking the number up and using doubles.

Other students were asked if they did the question in different ways and five responded.

How can this story be used in a classroom when there is a student who yells out answers or interrupts with what he or she considers interesting comments?

53

MAKING MEMORIESMAKING MEMORIES

What have you mylenized over the

course of the two videos?

Please take 2 minutes of silence to write out a list.

When the two minutes are up, the facilitator will ask you to share

a strategy or concept you learned that you feel

will be useful.

This writing and then sharing helps

“re-mylenize” your learning.

54

Create a Class List with some or all of the following headings:

EEVALUATINGVALUATINGWWeeklyeekly GGraphraph AActivitiesctivities

Name

Collects &Organizes

Data

Creates aCIRCLE Graphfrom Raw Data

Creates aBAR Graph

from Raw Data

Creates aPICTOGRAPHfrom Raw Data

InterpretsGraphs

CreatesQuestions or

Word-Problemsfrom Data

Tim

Bethany

J eevan

Morgan

Chris

55

Create Criteria for Create Criteria for each Headingeach HeadingCollects &

OrganizesData

Creates aCIRCLE Graphfrom Raw Data

Creates aBAR Graph

from Raw Data

Creates aPICTOGRAPHfrom Raw Data

InterpretsGraphs

CreatesQuestions or

Word-Problemsfrom Data

Example:Creates a CIRCLE Graph from Raw

Data.

Can create a circle graph using percentages and

includes headings and legend. The graph is

easy to interpret (neat and complete).

Can create a circle graph using

a circle graph frame and includes headings

and legend. The graph is easy to

interpret (neat and complete).

Can create a circle graph using

a circle graph frame but is missing

headings or legend.The graph is difficult

to interpret(may not be neat or

complete).

Serious errors and hardly gets

started.

56

CCircle ircle GGraphraph

Number of Siblings

Zero siblings

One sibling

Two siblings

Three siblings

27 students in the class told how many siblings they have.

DataData

HeadinHeadingg

LegendLegend

NeatnesNeatnesss

3

125

7

57

EVALUATINGDecimals / Fractions / Percentage

Example for Multi-age Grade 6/7(Grade 6 gets a 4 in the 3 category) Is fluent among the three forms of a rational

number, both repeating and terminating common fractions, including most of halves, thirds, quarters, fifths, sixths, eighths, tenths, twentieths, fortieths, fiftieths, hundredths, and thousandths.

Makes occasional errors with tenths and hundredths.

Can give the three forms for tenths and hundredths.

Is fluent for the three forms for halves, quarters, fifths, tenths and hundredths.

58

Example for Multi-age Grade 6/7(Grade 6 gets a 4 in the 3 category)Given a set of ordinary or special

dice or a spinner, can create a data set and interpret both the expected and experimental probability.

Creates a data set and interprets but makes some errors (not fundamental).

Gets a good start and creates a data set but not both of expected and experimental. Barely gets started if at

all, needs a lot of help.

4

3

2

1

59

TeachingTeaching ClassesClasses

There is some research that shows that students in multi-age classes demonstrate superior learning.

This may result from the fact that the teacher knows she has to individualize more because of the spread of ability.

In fact, this is true for all classes even when they are streamed.

I have found it most effective when teaching a multi-age class to teach to the top grade and evaluate

the lower grade at their own level.

60

EVALUATIONEVALUATIONWork with someone at your table to create criteria for

at least one of the Intended Learning Outcomes that

you will be evaluating.

Keep in mind that the creation of criteria is always a

process of negotiation between you, the curriculum and

your context (class and school).

If you involve the students in creation of the criteria,

they often create criteria that has a high standard of

expectation for excellence.

61

and

Good Problem Solvers:

Get started Get unstuck Persevere Can solve problems in more

than one way Self-correct

62

FACILITATINGProblem Solving

Use the think / pair / share method.

Give problems that are multi-step and take note of student strategies.

Record the strategies, slowly building up a list.

Discuss the efficacy and efficiency of the various strategies that students use.

63

FACILITATINGProblem Solving

Use model problems and have students write problems using the frame as a model.

Encourage the use of mathematical vocabulary by giving bonus marks.

Encourage the use of mathematical vocabulary by creating a word wall or a glossary in student workbooks.

64

STRATEGIES forGetting Unstuck

Look for a pattern

Make a model

Draw a diagram

Create a table, chart or list

Use logic

Create a simpler related problem

Work backwards

Seek help from a peer, the internet, a book65

EVALUATING Problem SolvingEVALUATING Problem SolvingExample for Multi-age Grade 4/5(grade 4 gets a 4 by achieving at the 3 level)

Occasionally gets started and

perseveres, uses at least one

strategy for getting unstuck

over the term, occasionally

self-corrects, solves one-step

problems.

Usually gets started, usually

perseveres, uses at least two

different strategies for getting

unstuck over the term, sometimes

self-corrects, occasionally solves

multi-step problems.

Almost never gets started, gives

up easily (demonstrates learned

helplessness), never self-

corrects, occasionally solves

simple one-step problems.

Always gets started, perseveres,

uses at least three different

strategies for getting unstuck

over the term, self-corrects,

solves multi-step problems.

66

IIMPLEMENTATIONMPLEMENTATION

Take the time to make a plan for

implementation.

What obstacles do you

perceive?

What help do you need?What help do you need?67

Fuson, Karen C., Kalchman, Mindy and Bransford, John D., Chapter 5, “Mathematical Understanding: An Introduction in How Students Learn Mathematics in the Classroom”, Ed. Donovan, Susanne and Bransford, John D., National Academies Press, Washington, D.C. 2005

Buschman, Larry E.E... Mythmatics” Teaching Children Mathematics, Vol.12, No.3, Oct. 2005, p136 –143

Calkins, Trevor “Mathematics as a Teachable Moment” Grades K-3, Power of Ten Educational Consulting Ltd, Victoria, B.C. 2004

Calkins, Trevor “Mathematics as a Teachable Moment” Grades 4 - 6, Power of Ten Educational Consulting Ltd, Victoria, B.C. 2004

Silver, Edward A and Cai, Jinfa. “Assessing Students’ Mathematical Problem Posing” Teaching Children Mathematics, Vol.12, No.3, Oct. 2005, p129 -135

Bibliography

68

Slide presentation created by:

Trevor CalkinsPower of Ten Educational Consulting

809 Kimberley PlaceVictoria, B. C.

V8X 4R2www.poweroften.ca

Power Point presentation constructed by:

Karen HendersonP. O. Box 18

Shawnigan Lake, B. C.V0R 2W0

69

Documents

M ATHEMATICS T eachable M oment as a T eachable M oment M eaning C hoice D iversity T rust T ime 1