# Conceptualizing Number Sense: What do we know about it? What is it

Document Sample

```					   Number Sense and Estimation:
Making the Connection

1
2
KATM Annual Conference
0011 0010 1010 1101 0001 0100 1011
October 14, 2005

4
Liberal, KS.
Dr. David S. Allen
Melisa J. Hancock
Kansas State University   Time: 1:30
Race to 20
0011 0010 1010 1101 0001 0100 1011

• Two players
• Each player can count 1 or 2 numbers at a

1
2
time
• First player says ….1

4
• Second player can say 2 or 2..3…
• Player to say 20 wins!!!!

Time: 1:45
Dealing In Horses!
0011 0010 1010 1101 0001 0100 1011

A woman bought a horse
for \$50 and sold it for

1
2
\$60. She then bought the
horse back for \$70 and

4
sold it again for \$80.
What do you think was
the financial outcome of
these transactions?
Time: 2:00
Recognizing Deficiencies in Student
Thinking with Respect to Number
0011 0010 1010 1101 0001 0100 1011

• Personal Experience
–   Student/Teacher Frustrations

1
2
–   Over Reliance on Algorithms
–   Sense Making vs. Following Prescribed Methods

4
–   Place Value Concepts (0 as Placeholder)
• Base 10 Operations
• Remediation typically involves re-teaching the procedure
– Lack of Understanding of Large Numbers
– Operations and Numbers Between 0 and 1
Time: 2:05

Number/Operations                                                    Algebra                                                            Geometry                                 Data and Probability

B1               B2           B3           B4               B1               B2                 B3           B4                B1               B2                  B3           B4            B1            B2

K                K            K            K                K                K                  K             k                K                 K                  K            K             K                K

1                1            1            1                1                1                  1             1                1                 1                  1            1              1               1

A            2            2            2                    A            2                  2                 A            2                 2                  2            2              2                   A

B            3            3            3                    B                  A            3                 B            3                       A            3            3              3                   B
C            4            4                A            2                      B            4                 C            4                       B        A                4         A                        C

2                5        A                    B                A                  C        A                     D            5                       C            1        A                  1                   C

0011 0010 1010 1101 C0001 0100 1011
A     A  1            B   3                                                                         1                 E            6                       D            2            1              2                   D

B                B        2                D                C            4                  2                 F        A                           E            3            2              3                   E

C                C        3                E                D        A                                        G            1                 3                                                                  F

A                        D        4                F                E            1                                    H                A                   A                                                            G

1                    E                         G                F            2                                    I                B                   B                                                            H

A                F                     4                    G                  A                          2                2                       C                                                        2

B        A                             5                3                      B                      A                                            D                                                        3

C            1                         6                4                3                                1                                  4                                                                  A

1
2
2                    A                         A        A                          A                              A                                    A                                                            B

A                B                         B            1                      B                              B                                    B                                                            C

B                C                         C                A                                                 C                              5                                                                  D
C                D                         D                B                                                 D                                    A                                                            E

2                         7                    C                                                 E                                    B                                                    A

A                 A                        D                                                 F                          A                                                                  1

4
B                     1                    E                                                 G                              1                                                                  A

C                         A                F                                                 H                                    A                                                            B

3                             B            2                                                     I                                    B                                                            C
C                                                              2                                        C                                                            D

D                                                                                                       D                                                            E

E                                                                                                       E                                                            F

2                                                                                                           F                                                            G

G                                                            H

H                                                        2

2                                                                  A

3                                                                  B

C
D
E

3

Number/Operations                                                    Algebra                                                            Geometry                                 Data and Probability

B1               B2           B3           B4               B1               B2                 B3           B4                B1               B2                  B3           B4            B1            B2

K                K            K            K                K                K                  K             k                K                 K                  K            K             K                K

1                1            1            1                1                1                  1             1                1                 1                  1            1              1               1

A            2            2            2                    A            2                  2                 A            2                 2                  2            2              2                   A

B            3            3            3                    B                  A            3                 B            3                       A            3            3              3                   B
C            4            4                A            2                      B            4                 C            4                       B        A                4         A                        C

2                5        A                    B                A                  C        A                     D            5                       C            1        A                  1                   C

0011 0010 1010 1101 C0001 0100 1011
A     A  1            B   3                                                                         1                 E            6                       D            2            1              2                   D

B                B        2                D                C            4                  2                 F        A                           E            3            2              3                   E

C                C        3                E                D        A                                        G            1                 3                                                                  F

A                        D        4                F                E            1                                    H                A                   A                                                            G

1                    E                         G                F            2                                    I                B                   B                                                            H

A                F                     4                    G                  A                          2                2                       C                                                        2

B        A                             5                3                      B                      A                                            D                                                        3

C            1                         6                4                3                                1                                  4                                                                  A

1
2
2                    A                         A        A                          A                              A                                    A                                                            B

A                B                         B            1                      B                              B                                    B                                                            C

B                C                         C                A                                                 C                              5                                                                  D
C                D                         D                B                                                 D                                    A                                                            E

2                         7                    C                                                 E                                    B                                                    A

A                 A                        D                                                 F                          A                                                                  1

4
B                     1                    E                                                 G                              1                                                                  A

C                         A                F                                                 H                                    A                                                            B

3                             B            2                                                     I                                    B                                                            C
C                                                              2                                        C                                                            D

D                                                                                                       D                                                            E

E                                                                                                       E                                                            F

2                                                                                                           F                                                            G

G                                                            H

H                                                        2

2                                                                  A

3                                                                  B

C
D
E

3
Group Discussion
0011 0010 1010 1101 0001 0100 1011
Brainstorming session!!!!!

• What characteristics would you expect to

1
2
see in your students that would represent
number sense?

4
do you see that is missing with respect to
number sense development?
Time: 2:20
A Student With Number Sense Will:
0011 0010 1010 1101 0001 0100 1011

look at a problem holistically before
confronting details.

Example:                               1
2
4
1 2/3 + 3/4 + 1/3 =
A Student With Number Sense Will:
0011 0010 1010 1101 0001 0100 1011
look for relationships among numbers and
operations and will consider the context in which
a question is posed.

1
2
4
Example:
When buying 4 notebooks priced at 39 cents each, the
student with \$2 might reason that she has enough money,
since each notebook costs less than half a dollar.
A Student With Number Sense Will:
0011 0010 1010 1101 0001 0100 1011

choose or invent a method that takes
advantage of his or her own understanding

1
2
Example:
• 75% of the class of 30 students need to

4
agree on a plan for a trip.
•    A student may reason that 75% is 50%
plus 25% or half plus half of that
•    15 + 8 = 23 students must agree
A Student With Number Sense Will:
0011 0010 1010 1101 0001 0100 1011

use benchmarks to judge number magnitude

Example:
2/5 of 49 is less than half of 49   1
2
4
A Student With Number Sense Will:
0011 0010 1010 1101 0001 0100 1011

recognize unreasonable results for
calculations in the normal process of

1
2

4
Example:

3.2 X 4.8 can’t possibly be 153.6 since the
Number Sense: A Definition
0011 0010 1010 1101 0001 0100 1011

“Number sense can be described as a good
intuition about numbers and their relationships.

1
2
It develops gradually as a result of exploring
numbers, visualizing them in a variety of

4
contexts, and relating them in ways that are not
(Van de Walle, 2005)

Time: 2:35
!!!!!!!!!TEST!!!!!!!!!
426   983           289
0011 0010 1010 1101 0001 0100 1011

258   651           438          387     294     489
+ 741 + 348         + 561          523     862     291

2
476     137     620
508
+ 491 +    1
493
506
379
+ 708

4
825        536       763
481        176       864
246        294       710
518        705       135
+ 753 +      823 +     289
Time: 2:50
Number Sense In the Classroom
0011 0010 1010 1101 0001 0100 1011

• Conduct a number sense activity every day.
• Address number sense in all strands.
• When talking about number sense the teacher
should avoid doing all the talking.
1
2
– Think-aloud is a good strategy to model a difficult task

4
for students.
• This is a great opportunity to incorporate writing
into the math classroom.
• When estimating avoid dictating procedures.
Beginning Number Sense in K-2
0011 0010 1010 1101 0001 0100 1011

• Experiencing Numbers in the Early Grades
–   Spatial Relationships

1
2
–   One and Two More/Less Than
–   Benchmark Numbers

4
–   Part-Part-Whole Relationships

Time: 2:55
Four Relationships for Early
Numbers 0-9
0011 0010 1010 1101 0001 0100 1011

Spatial Relationships

1
2
Back                              4
Four Relationships for Early
Numbers 0-9
0011 0010 1010 1101 0001 0100 1011

One More/Two More/One Less/Two Less

1
2
Back
6                      7
4
8
Four Relationships for Early
Numbers 0-9
0011 0010 1010 1101 0001 0100 1011

Benchmarks to 5 and 10

1
2
Back                                 4
Four Relationships for Early
Numbers 0-9
0011 0010 1010 1101 0001 0100 1011
Part-Part-Whole

1
2
Back                                    4
Number Sense in K-2
0011 0010 1010 1101 0001 0100 1011

•   Estimation: Comparing to Benchmarks
•   Measurement
•
•
Mental Computation
Multiple Representation      1
2
•
•
Number Relationships
Relative Size

4    Time: 2:55
Estimation
0011 0010 1010 1101 0001 0100 1011
• Estimates are useful when exact answers are
impossible, unrealistic, or unnecessary.
• First step in developing estimation skill is to learn to

1
2
recognize whether a particular situation requires an
exact answer or an estimate, and the degree of accuracy

4
needed.
• One valuable estimation technique is relating the
estimate to a referent or benchmark. People with good
number sense use a variety of personal benchmarks.
CAT FOOD
About How Many Cans?    CAT FOOD

0011 0010 1010 1101 0001 0100 1011

1.

2.

3.
1
2
4.

5.

4
CAT FOOD
About How Many Cans?    CAT FOOD

0011 0010 1010 1101 0001 0100 1011

6.

7.
1
2
8.

4
More or Less Than Half of a Group
0011 0010 1010 1101 0001 0100 1011

Where more than half of the dots
1
2
white? If you think the number of

4
white dots was more than 5, give
me a thumbs up!
More or Less Than Half of a Group
0011 0010 1010 1101 0001 0100 1011

Where more than half of the dots
1
white? If you think the number of

2
4
white dots was more than 5, give
me a thumbs up!
More or Less Than Half of a Group
0011 0010 1010 1101 0001 0100 1011

Each picture contains 20 dots.
Give me a thumbs up if more
number of white dots and
the
1
2
less than
half half of the dots are white.
than of the dots are white. gray
dots are equal.

4
More or Less Than Half of a Group
0011 0010 1010 1101 0001 0100 1011

Each picture contains 20 dots.
the
less than
Give me a thumbs up if more
number of white dots and
1
half half of the dots are white.
than of the dots are white. gray

2
4
dots are equal.
0011 0010 1010 1101 0001 0100 1011

•   Estimation
•   Measurement
•
•
Mental Computation
Multiple Representation     1
2
•
•
Number Relationships
Relative Size

4
About How Many Objects Will Fit?
Base 10 longs
0011 0010 1010 1101 0001 0100 1011
How did
Base 10 units                         you
Small cubes                         decide?
Small lids

1
2
Square tiles
Counters

4
Pattern blocks
Thumb tacks
Coins
Paper clips
Dried beans
Macaroni
About How Many Do You See?
0011 0010 1010 1101 0001 0100 1011

1
Estimate the number of books in this bookcase.

2
4
There are 10 books shown below. Now how
many do you think are in the bookcase?
0011 0010 1010 1101 0001 0100 1011
Put this in the calculator: 0 + Start Number.
How many times will you press = before you reach or pass the Goal?

1
2
Start                           Start
Number       Goal               Number      Goal

1. 5            30             6. 6            115

4
2. 5            51             7. 7            48
3. 5            77             8. 7            91
4. 6            58             9. 8            65
5. 6            97             10. 8           119
Over or Under?
0011 0010 1010 1101 0001 0100 1011
Look at each sum. Is the sum:
Over \$100? Under \$100? About \$100

1. \$54 + \$26
1
6. \$33 + \$24 + \$51

2
4
2. \$28.76 + \$81.79           7. \$17.84 + \$26.99 +\$45.25
3. \$31.45 + \$66.85           8. \$33.33 + \$33.34 + \$33.35
4. \$54.73 + \$45.27           9. \$22.46 + \$53.18 + \$19.58
5. \$14.44 + \$84.66           10. \$13.26 + \$49.27 + \$37.48
Identifying Compatible Numbers:
Summing to 100
0011 0010 1010 1101 0001 0100 1011
25 82 65       8   14

37 21 88 51 30
1
2
86 92 35 18 75

70 49 12 79 63     4
0011 0010 1010 1101 0001 0100 1011

•   Estimation
•   Measurement
•
•
Mental Computation
Multiple Representation     1
2
•
•
Number Relationships
Relative Size

4
How Full Is It?
1
0011 0010 1010 1101 0001 0100 1011

1
2
0

1

0
4
What Percent Are….?
Estimate what percent of each set of cubes is white.
0011 0010 1010 1101 0001 0100 1011
Each set contains 20 white cubes.

1
2
4
What Percent Are…..?
Estimate what percent of each set of cubes is white.
0011 0010 1010 1101 0001 0100 1011
Each set contains 20 white cubes.

1
2
4
Calculators for Every Student,
Every Day
0011 0010 1010 1101 0001 0100 1011

Students and teachers must learn to make wise
choices about when to use calculators – for

1
2
tedious computations—and when to use mental
math—for simple computations and estimations.

4
They learn this only by making such choices
independently and on a regular basis.
7 x 400
0011 0010 1010 1101 0001 0100 1011
48 + 51
15 x 40                       19 x 27
450 : 45                      15 x 7 x 2
1
2
4
600 x 600                     399 : 3
24 x 5 x 2                    34 x 100
48 x 51                       25 x 480
1 – 0.045
0011 0010 1010 1101 0001 0100 1011
0.5 – 0.25
0.1 x 87                      0.25 x 123
0.632 – 0.5
1
1.001 – 0.45

2
4
52 – 0.1
0.01 x 450                        :
8 – 0.5
:                           376 – 0.001
:
400 x 0.5                      3 – 0.25
:
10% of \$18
0011 0010 1010 1101 0001 0100 1011
50% of 48
5% of 90                       1% of \$600
25% of \$480
1
0.1% of \$600

2
99% of 75
15% of \$40
33.33% of \$17                     4
10% of 450
75% of \$60
66.66% of 45
Broken Calculator Activities
• Only Working Keys are 3, 8, X, -, =
0011 0010 1010 1101 0001 0100 1011
– Display all the numbers 1 - 10

• 7, 8, and 9 keys are broken

1
2
– Work the following problems
•   625 + 192=

4
•   138 + 80=
•   89 + 19 =
•   875 – 125 =
• Is there only one answer?
• Which way uses fewest key strokes?
Close, Far, In-between
0011 0010 1010 1101 0001 0100 1011
219

457                                   364

1
2
4
Closest two numbers?              About how far apart are
Closest to              ______ and _______?
250?       To 300?
To 400?              Name a multiple of 25
between two of these.
References
All activities in this presentation were taken from the following sources.

0011 0010M. (2000). About Teaching Mathematics: A K-8 Resource (2nd Edition).
Burns, 1010 1101 0001 0100 1011
Sausalito, CA: Math Solutions Publications.

McIntosh, Reys, & Reys (1997). Number Sense: Simple Effective Number Sense
Experiences (Grades 1-2). Parsippany, NJ: Dale Seymour Publications.

1
McIntosh, Reys, & Reys (1997). Number Sense: Simple Effective Number Sense
Experiences (Grades 3-4). Parsippany, NJ: Dale Seymour Publications.

2
4
McIntosh, Reys, & Reys (1997). Number Sense: Simple Effective Number Sense
Experiences (Grades 5-6). Parsippany, NJ: Dale Seymour Publications.

Van de Walle, J. A. (2003). Elementary and Middle School Mathematics:
Teaching Developmentally (5th Edition). Boston, MA: Allyn & Bacon.
Number Sense and Estimation:
Making the Connection

1
2
KATM Annual Conference
0011 0010 1010 1101 0001 0100 1011
October 14, 2005

4
Liberal, KS.
Dr. David S. Allen
Melisa J. Hancock
Kansas State University   Time: 1:30

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 0 posted: 7/19/2012 language: pages: 46
How are you planning on using Docstoc?