Number Sense and Estimation: Making the Connection

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Number Sense and EstimationMaking the
Connection

• KATM Annual Conference
• October 14, 2005
• Liberal, KS.
• Dr. David S. Allen
• Melisa J. Hancock
• Kansas State University

Time 130
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Race to 20

• Two players
• Each player can count 1 or 2 numbers at a time
• First player says .1
• Second player can say 2 or 2..3
• Player to say 20 wins!!!!

Time 145
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Dealing In Horses!
A woman bought a horse for 50 and sold it for
60. She then bought the horse back for 70 and
sold it again for 80. What do you think was the
financial outcome of these transactions?
Time 200
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Recognizing Deficiencies in Student Thinking with
Respect to Number

• Personal Experience
• Student/Teacher Frustrations
• Over Reliance on Algorithms
• Sense Making vs. Following Prescribed Methods
• 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 205
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Group Discussion

• Brainstorming session!!!!!
• What characteristics would you expect to see in
  your students that would represent number sense?
• When considering your students.What do you see
  that is missing with respect to number sense
  development?

Time 220
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A Student With Number Sense Will

• look at a problem holistically before
  confronting details.
• Example
• 1 2/3 3/4 1/3

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A Student With Number Sense Will
look for relationships among numbers and
operations and will consider the context in which
a question is posed. 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.
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A Student With Number Sense Will

• choose or invent a method that takes advantage
  of his or her own understanding
• Example
• 75 of the class of 30 students need to
• 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

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A Student With Number Sense Will

• use benchmarks to judge number magnitude
• Example
• 2/5 of 49 is less than half of 49

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A Student With Number Sense Will

• recognize unreasonable results for
  calculations in the normal process of reflecting
  on answers.
• Example
• 3.2 X 4.8 cant possibly be 153.6 since the
  answer must be about 3 X 5 or 15

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Number Sense A Definition

• Number sense can be described as a good
  intuition about numbers and their relationships.
  It develops gradually as a result of exploring
  numbers, visualizing them in a variety of
  contexts, and relating them in ways that are not
  limited by traditional algorithms.
• (Van de Walle, 2005)

Time 235
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!!!!!!!!!TEST!!!!!!!!!

• 426
• 258
• 741

983 651 348
289 438 561
387 523 476 508 491
294 862 137 493 506
489 291 620 379 708






825 481 246 518 753
536 176 294 705 823
763 864 710 135 289



Time 250
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Recommendations for Addressing Number Sense In
the Classroom

• 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.
• Think-aloud is a good strategy to model a
  difficult task for students.
• This is a great opportunity to incorporate
  writing into the math classroom.
• When estimating avoid dictating procedures.

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Beginning Number Sense in K-2

• Experiencing Numbers in the Early Grades
• Spatial Relationships
• One and Two More/Less Than
• Benchmark Numbers
• Part-Part-Whole Relationships

Time 255
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Four Relationships for Early Numbers 0-9

• Spatial Relationships

Back
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Four Relationships for Early Numbers 0-9

• One More/Two More/One Less/Two Less
• 6 7 8

Back
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Four Relationships for Early Numbers 0-9

• Benchmarks to 5 and 10

Back
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Four Relationships for Early Numbers 0-9

• Part-Part-Whole

Back
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Number Sense in K-2

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

Time 255
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Estimation

• Estimates are useful when exact answers are
  impossible, unrealistic, or unnecessary.
• First step in developing estimation skill is to
  learn to recognize whether a particular situation
  requires an exact answer or an estimate, and the
  degree of accuracy 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.

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About How Many Cans?
CAT FOOD
CAT FOOD
1.
2.
3.
4.
5.
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About How Many Cans?
CAT FOOD
CAT FOOD
6.
7.
8.
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More or Less Than Half of a Group
Where more than half of the dots white? If you
think the number of white dots was more than 5,
give me a thumbs up!
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More or Less Than Half of a Group
Where more than half of the dots white? If you
think the number of white dots was more than 5,
give me a thumbs up!
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More or Less Than Half of a Group
Each picture contains 20 dots. Give me a thumbs
up if more than half of the dots are white.
Each picture contains 20 dots. Give me a thumbs
up if less than half of the dots are white.
Each picture contains 20 dots. Give me a thumbs
up if the number of white dots and gray dots are
equal.
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More or Less Than Half of a Group
Each picture contains 20 dots. Give me a thumbs
up if more than half of the dots are white.
Each picture contains 20 dots. Give me a thumbs
up if less than half of the dots are white.
Each picture contains 20 dots. Give me a thumbs
up if the number of white dots and gray dots are
equal.
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Addressing Number Sense in 3-4

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

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About How Many Objects Will Fit?
How did you decide?
Base 10 longs Base 10 units Small cubes Small
lids Square tiles Counters Pattern blocks Thumb
tacks Coins Paper clips Dried beans Macaroni
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About How Many Do You See?
Estimate the number of books in this bookcase.
There are 10 books shown below. Now how many do
you think are in the bookcase?
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About How Many Equals?
Put this in the calculator 0 Start Number. How
many times will you press before you reach or
pass the Goal?
Start
Start
Number Goal
Number Goal

• 5 30
• 5 51
• 5 77
• 6 58
• 5. 6 97

6. 6 115 7. 7 48 8. 7 91 9. 8 65 10. 8 119
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Over or Under?
Look at each sum. Is the sum Over 100? Under
100? About 100

• 54 26
• 28.76 81.79
• 31.45 66.85
• 54.73 45.27
• 5. 14.44 84.66

6. 33 24 51 7. 17.84 26.99 45.25
8. 33.33 33.34 33.35 9. 22.46 53.18
19.58 10. 13.26 49.27 37.48
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Identifying Compatible Numbers Summing to 100
25 82 65 8 14 37 21 88 51 30 86 92 35 18 75 70
49 12 79 63
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Addressing Number Sense in 4-6

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

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How Full Is It?
1
0
1
0
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What Percent Are.?
Estimate what percent of each set of cubes is
white. Each set contains 20 white cubes.
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What Percent Are..?
Estimate what percent of each set of cubes is
white. Each set contains 20 white cubes.
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Calculators for Every Student, Every Day

• Students and teachers must learn to make wise
  choices about when to use calculators for
  tedious computationsand when to use mental
  mathfor simple computations and estimations.
  They learn this only by making such choices
  independently and on a regular basis.

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Will You Do It in Your Head?
7 x 400
48 51

• 15 x 40

19 x 27
15 x 7 x 2
600 x 600
24 x 5 x 2
34 x 100
25 x 480
48 x 51
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Will You Do It in Your Head?
1 0.045
0.5 0.25

• 0.1 x 87

0.25 x 123
0.632 0.5
1.001 0.45
0.01 x 450
400 x 0.5
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Will You Do It in Your Head?
10 of 18
50 of 48

• 5 of 90

1 of 600
25 of 480
0.1 of 600
99 of 75
10 of 450
75 of 60
15 of 40
66.66 of 45
33.33 of 17
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Broken Calculator Activities

• Only Working Keys are 3, 8, X, -,
• Display all the numbers 1 - 10

• 7, 8, and 9 keys are broken
• Work the following problems
• 625 192
• 138 80
• 89 19
• 875 125

• Is there only one answer?
• Which way uses fewest key strokes?

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Close, Far, In-between
219
364
457
Closest two numbers?
About how far apart are ______ and _______?
Closest to 250?
To 300?
To 400?
Name a multiple of 25 between two of these.
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References
All activities in this presentation were taken
from the following sources.

• Burns, M. (2000). About Teaching Mathematics A
  K-8 Resource (2nd Edition). Sausalito, CA Math
  Solutions Publications.
• McIntosh, Reys, Reys (1997). Number Sense
  Simple Effective Number Sense Experiences (Grades
  1-2). Parsippany, NJ Dale Seymour Publications.
• McIntosh, Reys, Reys (1997). Number Sense
  Simple Effective Number Sense Experiences (Grades
  3-4). Parsippany, NJ Dale Seymour Publications.
• 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.

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Number Sense and EstimationMaking the
Connection

• KATM Annual Conference
• October 14, 2005
• Liberal, KS.
• Dr. David S. Allen
• Melisa J. Hancock
• Kansas State University

Time 130