How to Why TMC14

5/22/2018 How to Why TMC14

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Shifting Our Mindsets and

Our Actions fromRemembering HOW to

Understanding WHY

Steve Leinwand

TMC14Jenks, OK

[email protected] www.steveleinwand.com

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mailto:[email protected]://www.steveleinwand.com/http://www.steveleinwand.com/mailto:[email protected]


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And what message do far toomany of our students get?

(even those in Namibia!)

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Ready??

What is 8 + 9?

17 Bing Bang Done!

Vs.

Convince me that 9 + 8 = 17.Hmmmm.

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8 + 9 =17know it cold

10 + 7add 1 to 9, subtract 1 from 87 + 1 + 9decompose the 8 into 7 and 1

181add 10 and adjust

16 + 1double plus 1

203round up and adjust

Whos right? Does it matter?5


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4 + 29 =How did you do it?

How did you do it?Who did it differently?

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Sothe problem is:

If we continue to do what weve

always done.

Well continue to get what weve

always gotten.

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Where is the opportunity to

learn?Where is the sense-making?

Does anyone benefit from asheet like this?

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95

- 48 How did you do it?or

Convince me that 95-48=47.

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In other words,

our questions make all thedifference.

(no pun intended)

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Mathematics

A set of rules to be learned and memorized

to find answers to exercises that have limited

real world value

OR

A set of competencies and understanding

driven by sense-making and used to get

solutions to problems that have real world

value

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And Alt apps and mult reps emerge

from this why/convince me

Effective teachers of mathematics elicit, value, andcelebrate alternative approaches to solvingmathematics problems so that students are taughtthat mathematics is a sense-making process for

understanding why and not memorizing the rightprocedure to get the one right answer.

Effective teachers of mathematics provide multiplerepresentationsfor example, models, diagrams,number lines, tables and graphs, as well as symbols

of all mathematical work to support thevisualization of skills and concepts.

Also know as rational, doable DIFFERENTIATION!14


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Adding and SubtractingIntegers

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Remember How

5 + (-9)

To find the difference of two integers, subtract

the absolute value of the two integers and then

assign the sign of the integer with the greatest

absolute value

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Understand Why

5 + (-9)

- Have $5, lost $9

- Gained 5 yards, lost 9

- 5 degrees above zero, gets 9 degrees colder

- Decompose 5 + (-5 + -4)

- Zero pairs: x x x x x O O O O O O O O O

- On number line, start at 5 and move 9 to the left

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Lets laugh at the absurdity of the

standard algorithm and the one right

way to multiply

58

x 47

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3 5

58x 47

406

232_2726

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How nice if we wish tocontinue using math to sort

our students!

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So whats the alternative?

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Multiplication

What is 3 x 4? How do you know?


What is 3 x 47? How do you know? What is 13 x 40? How do you know?



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3 x 4

Convince me that 3 x 4 is 12.

4 + 4 + 4

3 + 3 + 3 + 3 Three threes are nine and three more for the

fourth

3

4

12

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3 x 40

3 x 4 x 10 (properties)

40 + 40 + 40

12 with a 0 appended

3

40

120

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3 x 47

3 (40 + 7) = 3 40s + 3 7s

47 + 47 + 47 or 120 + 21

3

40 7

120 21

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58 x 47

40 7

50

8

58

x 47

56

350

320

2000

2726

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Why bother?

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Just do it:Siti packs her clothes into a suitcase and it

weighs 29 kg.

Rahim packs his clothes into an identical

suitcase and it weighs 11 kg.Sitis clothes are three times as heavy as

Rahims.

What is the mass of Rahims clothes?What is the mass of the suitcase?


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The old (only) way or RemHow:

Let S = the weight of Sitis clothes

Let R = the weight of Rahims clothes

Let X = the weight of the suitcase

S = 3R S + X = 29 R + X = 11

so by substitution: 3R + X = 29

and by subtraction: 2R = 18so R = 9 and X = 2


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Or using a model to support UndWhy:

www.thesingaporemaths.com

11 kg

Rahim

Siti

29 kg
http://www.thesingaporemaths.com/http://www.thesingaporemaths.com/


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Multiplying Decimals

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Remember How

4.39x 4.2

We dont line them up here.

We count decimals.

Remember, I told you that youre not allowed

to that thatlike girls cant go into boysbathrooms.

Let me say it again: The rule is count the

decimal places.32


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But why?How can this make sense?

How about a context?

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Understand Why

So? What do you see?34


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Understand Why

gallons

Total

Where are we?35


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Understand Why

4.2 gallons

Total

How many gallons? About how many?

$

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Understand Why

4.2 gallons

$ 4.39

Total

About how much? Maximum?? Minimum??37


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Understand Why

4.2 gallons

$ 4.39

Total184.38

Context makes ridiculous obvious, and breeds sense-making.

Actual cost? So how do we multiply decimals sensibly? 38


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Solving Simple LinearEquations

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3x + 7 = 22

How do we solve equations:

Subtract 7 3 x + 7 = 22

- 7 - 7

3 x = 15

Divide by 3 3 3

Voila: x = 5

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3x + 7

1. Tell me what you see: 3 x + 7

2. Suppose x = 0, 1, 2, 3..

3. Lets record that:

x 3x + 7

0 7

1 10

2 13

4. How do we get 22?

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3x + 7 = 22

Where did we start? What did we do?

x 5

x 3 3x 15 3

+ 7 3x + 7 22 - 7

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3x + 7 = 22

X X X IIIIIII IIII IIII IIII IIII II

X X X IIIII IIIII IIIII

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Lets look at a silly problem

Sandra is interested in buying party favors for

the friends she is inviting to her birthday

party.

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party. The price of the fancy straws she

wants is 12 cents for 20 straws.

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wants is 12 cents for 20 straws. Thestorekeeper is willing to split a bundle of

straws for her.

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straws for her. She wants 35 straws.

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straws for her. She wants 35 straws. How

much will they cost?

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So?

Your turn. How much?

How did you get your answer?

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Putting it all together one way

Good morning class.

Todays objective: Find the surface area of rightcircular cylinders.

Open to page 384-5.

3

Example 1:

4 S.A.= 2rh + 2 r2

Find the surface area.

Homework: Page 385 1-19 odd

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Putting it all together another way

Overheard in the ER as the sirens blare:

Oh my, look at this next one. Hes completely burned

from head to toe.

Not a problem, just order up 1000 square inches ofskin from the graft bank.

You have two possible responses:

- Oh goodthat will be enough.OR

- Oh godwere in trouble.

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Which response, oh good or oh my ismore appropriate?

Explain your thinking.

Assuming you are the patient, how much skinwould you hope they ordered up?

Show how you arrived at your answer and be

prepared to defend it to the class.

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Exit slip: Sketch an object and its

dimensions that has a surface area of about

100 square inches?

Homework: How many square cm of skin do

you have and be prepared to show how you

arrived at your answer.

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The CCSSM Trojan Horse:

SMP 3: Construct viable

arguments and critique

the reasoning of others

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Questions?

Comments?

Thank You!

Documents

How to Why TMC14