Don’t Be So Symbol Minded Problem Solving, Reasoning, and Sense Making in

the Core Standards Environment

Jim RubilloJim RubilloJRubillo@verizon.netJRubillo@verizon.net

An Important Notice

The opinions expressed in this presentation are solely those of the presenter.

They are offered for your consideration and reflection.

They are aligned with the speaker’s lifelong motto:

“Seldom right, but never in doubt.”

What Do You See?

A List of Specific Content Skills

or

A Plan with a Purpose?

Primary Purpose of the Endeavor!

Ensuring

College and Workforce Readiness

for ALL Students

While You Are Teaching Them Math, Teach Them … (Judy Spitz, CIO, Verizon)

How to see the big picture. How to see the forest for the trees.

How to be good at story telling. Understand the difference between leadership

and power. “Leadership is the ability to get people to move in a

consistent direction when you have no power over them.”

Think non-linearly, but execute in a linear fashion.

These Standards endeavor to follow such a design, not only by stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value or the laws of arithmetic to structure those ideas.

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning.

Standards for Mathematical Practice

SHOW ME!

What is PROBLEM SOLVING?

Engaging in an activity for which the method of solution

is not known in advance.

A Problem for You!

Prove that NO number in the following “Fibonacci-Like” sequence is divisible

by 5:

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207,

3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, . . .

A Fundamental Example

I have quarters, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have in my hand?

15 cents75 cents20 cents25 cents30 cents35 cents40 cents45 cents50 cents55 cents60 cents65 cents70 cents

Solve the following using only the numerals used in our base ten number system: 0,1,2,3,4,5,6,7,8,9

In the equation:

The same letter stands for the same digit and different letters stand for different digits. There are no repeated digits. There are ten letters and ten digits.

What is the value of the product ?

E I G H TT W O

F O U R

T H R E E

Solution: F, O, U, and R can not equal zero, why? The value of T must equal zero, why? The values of E, I, G, H, F, O, U, R, and W can scramble the values 1,2,3,4,5,6,7,8,and 9.

Basic Skills are important, but so is the understanding when it is useful!YES, we want a 4th grader to know

how to correctly complete:

4 26

But, can you think of four unique situations where knowing how to

compute

would be useful to a 4th grader?

6 R2 6.54 26 or 4 26

Situation A: There are 4 students and 26 cookies.

If the cookies are equally divided among the students, how many

cookies should each student receive?

26 4 = ?

Situation B: You have 26 quarters. You go to the bank and ask the teller to swap your quarters for $1 bills. How many one

dollar bills will you receive?

26 4 = ?

Situation C: There are 26 students going on a

class field trip. We are driving to the site in cars. We can place a

maximum of four students in each car. What is the minimum number of

cars required for the field trip?

26 4 = ?

Situation D: There are 26 students in a

classroom. The teacher wants to arrange the students’ into four rows? How many students are in each row?

26 4 = ?

Is the “Standard” Algorithm Just a Set of Rote Procedures?

124 x13 72 24__ 312

Why Does It Work?

20 + 420 + 4

x 10 + 310 + 3

1212

6060

4040

+ 200+ 200

312312

An Alternate Algorithm?

The Link to Area

24

x 13

1212

6060

4040

+ + 200200

312

3 10

20

4

The Far Too Typical Experience!

1. Here is an equation: y = 3x + 4

2. Make a table of x and y values using whole number values of x and then find the y values,

3. Plot the points on a Cartesian coordinate system.

4. Connect the points with a line.

Opinion: In a student’s first experience, the equation should come last, not first.

Situation 3:The Mirror Problem

Parts

Corner

Edge

Center

A company makes “bordered” square mirrors. Each mirror is constructed of 1 foot by 1 foot square mirror “tiles.” The mirror is constructed from the “stock” parts. How many “tiles” of each of the following stock tiles are needed to construct a “bordered” mirror of the given dimensions?

The Mirror Problem

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft

3 ft x 3 ft

4 ft x 4 ft

5 ft x 5 ft

6 ft x 6 ft

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft

4 ft x 4 ft

5 ft x 5 ft

6 ft x 6 ft

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft

5 ft x 5 ft

6 ft x 6 ft

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft 4 8 4

5 ft x 5 ft

6 ft x 6 ft

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft 4 8 4

5 ft x 5 ft 4 12 9

6 ft x 6 ft

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft 4 8 4

5 ft x 5 ft 4 12 9

6 ft x 6 ft 4 16 16

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft 4 8 4

5 ft x 5 ft 4 12 9

6 ft x 6 ft 4 16 16

7 ft by 7 ft 4 20 25

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft 4 8 4

5 ft x 5 ft 4 12 9

6 ft x 6 ft 4 16 16

7 ft by 7 ft 4 20 25

8 ft by 8 ft 4 24 36

9 ft by 9 ft

10 ft x 10 ft

The Mirror ProblemMirror Size

Number of2 borders tiles

Number of1 border tiles

Number ofNo border tiles

2 ft x 2 ft 4 0 0

3 ft x 3 ft 4 4 1

4 ft x 4 ft 4 8 4

5 ft x 5 ft 4 12 9

6 ft x 6 ft 4 16 16

7 ft by 7 ft 4 20 25

8 ft by 8 ft 4 24 36

9 ft by 9 ft 4 28 49

10 ft x 10 ft

The Mirror Problem

Mirror Size

Number of“Tiles”

(2 borders)

Number of“Tiles”

(1 border)

Number of“Tiles”

(No borders)

TotalNumber

of “Tiles”

2 ft x 2 ft 4 0 0 4

3 ft x 3 ft 4 4 1 9

4 ft x 4 ft 4 8 4 16

5 ft x 5 ft 4 12 9 25

6 ft x 6 ft 4 16 16 36

7 ft by 7 ft 4 20 25 49

8 ft by 8 ft 4 24 36 64

9 ft by 9 ft 4 28 49 81

10 ft x 10 ft 4 32 64 100

The Mirror Problem

side tiles

2 0

3 1

4 4

5 9

6 16

7 25

8 36

9 49

10 64

11 81

1 2 3 4 5 6 7 8 9 10 11

The Mirror Problem

1 2 3 4 5 6 7 8 9 10 11

The Mirror Problem Mirror Size

Number of“Tiles”

(2 borders)

Number of“Tiles”

(1 border)

Number of“Tiles”

(No borders)

TotalNumber

of “Tiles”

2 ft x 2 ft 4 0 0 4

3 ft x 3 ft 4 4 1 9

4 ft x 4 ft 4 8 4 16

5 ft x 5 ft 4 12 9 25

6 ft x 6 ft 4 16 16 36

7 ft by 7 ft 4 20 25 49

8 ft by 8 ft 4 24 36 64

9 ft by 9 ft 4 28 49 81

8 ft by 8 ft 4 32 64 100

: : : : :

n ft by n ft 4 4(n-2) (n-2)2 n2

7

6

5

4

3

2

1 7

0 3

0 1 2 3 4 5 6 7 8

PENCILS

ERASERS

Total Cost Table: Example 1

7

6

5

4 28

3

2 24

1

0

0 1 2 3 4 5 6 7 8

PENCILS

ERASERS

Total Cost Table Example 2

The Graph Tells a Story!Interpretation is More Important than Drawing

You are at the movie and you buy a cup of popcorn. The graph shows the level of the popcorn in your cup. What happened?

You are at the movie and you buy a cup of popcorn. The graph shows the level of the popcorn in your cup. What happened?

The Graph Tells a Story!Interpretation is More Important than Drawing

What are Today’s Big Ideas?

• Problem Solving and Reasoning should be a vital part of every mathematics lesson. “Why?” and “How do you know that?” should be the most frequently asked teacher questions.

• There are at least five ways of representing a mathematical concept. Each way can enhance and extend our understanding.

Key to Implementing Any Set of Standards!

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the

reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated

reasoning.

ENCORE! The Standards for

Mathematical Practice

Keys to Implementing the Standards!

Integrate the Standards of Mathematical Practice

Collaboration/Teamwork among Teachers

Articulation through the Grades

Assessment to Improve Teaching and Learning

8Q + 2Q = ?email: JRubillo@verizon.net

The Mirror Problem

Mirror Size

Number of“Tiles”

(2 borders)

Number of“Tiles”

(1 border)

Number of“Tiles”

(No borders)

TotalNumber

of “Tiles”

2 ft x 2 ft

3 ft x 3 ft

4 ft x 4 ft

5 ft x 5 ft

6 ft x 6 ft

7 ft by 7 ft

8 ft by 8 ft

9 ft by 9 ft

10 ft x 10 ft