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MCTM Future Primary Math
- 1. The Future of Primary Math:
More Understanding/Less Counting
by Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com
MCTM
Saturday, May 5, 2012
Duluth, Minnesota
1000 100 10 1
30
7
30
7
PowerPoint Presentation
RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
- 3. Verbal Counting Model
From a child's perspective
Because we’re so familiar with 1, 2, 3, we’ll use letters.
A=1
B=2
C=3
D=4
E = 5, and so forth
3 © Joan A. Cotter, Ph.D., 2012
- 10. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B
10 © Joan A. Cotter, Ph.D., 2012
- 11. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B C D E
11 © Joan A. Cotter, Ph.D., 2012
- 12. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B C D E
What is the sum?
(It must be a letter.)
12 © Joan A. Cotter, Ph.D., 2012
- 13. Verbal Counting Model
From a child's perspective
F
+E
K
A B C D E F G H I J K
13 © Joan A. Cotter, Ph.D., 2012
- 14. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
14 © Joan A. Cotter, Ph.D., 2012
- 15. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G
F
+D
15 © Joan A. Cotter, Ph.D., 2012
- 16. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G
F
+D
D
+C
16 © Joan A. Cotter, Ph.D., 2012
- 17. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G
F
+D
D C
+C +G
17 © Joan A. Cotter, Ph.D., 2012
- 18. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
E
+
G
I
F
+
+D
D C
+C +G
18 © Joan A. Cotter, Ph.D., 2012
- 19. Verbal Counting Model
From a child's perspective
H
–E
Subtract with your fingers by counting backward.
19 © Joan A. Cotter, Ph.D., 2012
- 20. Verbal Counting Model
From a child's perspective
J
–F
Subtract without using your fingers.
20 © Joan A. Cotter, Ph.D., 2012
- 21. Verbal Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
21 © Joan A. Cotter, Ph.D., 2012
- 22. Verbal Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
What is D × E?
22 © Joan A. Cotter, Ph.D., 2012
- 23. Verbal Counting Model
From a child's perspective
L
is written AB
because it is A J
and B A’s
23 © Joan A. Cotter, Ph.D., 2012
- 24. Verbal Counting Model
From a child's perspective
L
is written AB
because it is A J
and B A’s
huh?
24 © Joan A. Cotter, Ph.D., 2012
- 25. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB
because it is A J
and B A’s
25 © Joan A. Cotter, Ph.D., 2012
- 26. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J
and B A’s
26 © Joan A. Cotter, Ph.D., 2012
- 27. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s
27 © Joan A. Cotter, Ph.D., 2012
- 28. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s (two 1s).
28 © Joan A. Cotter, Ph.D., 2012
- 29. Calendar Math
August
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
29 © Joan A. Cotter, Ph.D., 2012
- 30. Calendar Math
Calendar Counting
August
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
30 © Joan A. Cotter, Ph.D., 2012
- 31. Calendar Math
Calendar Counting
August
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
31 © Joan A. Cotter, Ph.D., 2012
- 32. Calendar Math
Calendar Counting
August
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
32 © Joan A. Cotter, Ph.D., 2012
- 33. Calendar Math
Septemb
Calendar Counting
1234567
August
89101214
1 2
113
11921
15112628
8
122820
67527
9
3 4 5 6
10 11 12 13 14
7
2234
20
15 16 17 18 19 20 21
29
3
22 23 24 25 26 27 28
29 30 31
33 © Joan A. Cotter, Ph.D., 2012
- 34. Calendar Math
Septemb
Calendar Counting
1234567
August
89101214
1
113
11921
2
15112628
122820
8
67527
9
3 4 5 6
10 11 12 13 14
7
2234
20
15 16 17 18 19 20 21
29
3
22 23 24 25 26 27 28
29 30 31
This is ordinal counting, not cardinal counting.
34 © Joan A. Cotter, Ph.D., 2012
- 35. Calendar Math
Partial Calendar
August
1 2 3 4 5 6 7
8 9 10
35 © Joan A. Cotter, Ph.D., 2012
- 36. Calendar Math
Partial Calendar
August
1 2 3 4 5 6 7
8 9 10
Children need the whole month to plan ahead.
36 © Joan A. Cotter, Ph.D., 2012
- 37. Calendar Math
Septemb
Calendar Patterning
1234567
August
89101214
1 2
113
11921
15112628
8
122820
67527
9
3 4 5 6
10 11 12 13 14
7
2234
20
15 16 17 18 19 20 21
29
3
22 23 24 25 26 27 28
29 30 31
Patterns are rarely based on 7s or proceed row by row.
Patterns go on forever; they don’t stop at 31.
37 © Joan A. Cotter, Ph.D., 2012
- 38. Minnesota Standards
Number Sense
K: Represent quantities using whole numbers and
understand relationships among whole numbers.
1–2: Understand place value and relationships
among whole numbers.
With the counting model, how difficult are the
associated benchmarks for children to master?
38 © Joan A. Cotter, Ph.D., 2012
- 39. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
39 © Joan A. Cotter, Ph.D., 2012
- 40. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
40 © Joan A. Cotter, Ph.D., 2012
- 41. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
41 © Joan A. Cotter, Ph.D., 2012
- 42. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
42 © Joan A. Cotter, Ph.D., 2012
- 43. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
43 © Joan A. Cotter, Ph.D., 2012
- 44. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
44 © Joan A. Cotter, Ph.D., 2012
- 45. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
• Add and subtract whole numbers up to 6, using objects.
45 © Joan A. Cotter, Ph.D., 2012
- 46. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
46 © Joan A. Cotter, Ph.D., 2012
- 47. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
47 © Joan A. Cotter, Ph.D., 2012
- 48. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
48 © Joan A. Cotter, Ph.D., 2012
- 49. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
49 © Joan A. Cotter, Ph.D., 2012
- 50. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
50 © Joan A. Cotter, Ph.D., 2012
- 51. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
• Represent whole numbers up to 20 in various ways.
51 © Joan A. Cotter, Ph.D., 2012
- 52. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
52 © Joan A. Cotter, Ph.D., 2012
- 53. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
53 © Joan A. Cotter, Ph.D., 2012
- 54. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
54 © Joan A. Cotter, Ph.D., 2012
- 55. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
55 © Joan A. Cotter, Ph.D., 2012
- 56. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
56 © Joan A. Cotter, Ph.D., 2012
- 57. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
• Represent whole numbers up to 20 in various ways.
57 © Joan A. Cotter, Ph.D., 2012
- 67. Research on Counting
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
67 © Joan A. Cotter, Ph.D., 2012
- 68. Research on Counting
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
68 © Joan A. Cotter, Ph.D., 2012
- 69. Research on Counting
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
• Adults, ages 18-50, from Boston.
Edward Gibson and Michael Frank, MIT, 2008.
69 © Joan A. Cotter, Ph.D., 2012
- 70. Research on Counting
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
• Adults, ages 18-50, from Boston.
Edward Gibson and Michael Frank, MIT, 2008.
• Baby chicks from Italy.
Lucia Regolin, University of Padova, 2009.
70 © Joan A. Cotter, Ph.D., 2012
- 71. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
71 © Joan A. Cotter, Ph.D., 2012
- 72. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
• They consistently group in 5s.
72 © Joan A. Cotter, Ph.D., 2012
- 73. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
73 © Joan A. Cotter, Ph.D., 2012
- 74. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
74 © Joan A. Cotter, Ph.D., 2012
- 75. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
• Children who can subitize perform better in
mathematics long term.—Butterworth
75 © Joan A. Cotter, Ph.D., 2012
- 76. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
• Children who can subitize perform better in
mathematics long term.—Butterworth
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
76 © Joan A. Cotter, Ph.D., 2012
- 77. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
• Children who can subitize perform better in
mathematics long term.—Butterworth
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
• Subitizing seems to be a necessary skill for
understanding what the counting process means.—
Glasersfeld
77 © Joan A. Cotter, Ph.D., 2012
- 79. Visualizing Quantities
“Think in pictures, because the
brain remembers images better
than it does anything else.”
Ben Pridmore, World Memory Champion, 2009
79 © Joan A. Cotter, Ph.D., 2012
- 80. Visualizing Quantities
“The role of physical manipulatives
was to help the child form those
visual images and thus to eliminate
the need for the physical
manipulatives.”
Ginsberg and others
80 © Joan A. Cotter, Ph.D., 2012
- 81. Visualizing Quantities
Japanese criteria for manipulatives
• Representative of structure of numbers.
• Easily manipulated by children.
• Imaginable mentally.
Japanese Council of
Mathematics Education
© Joan A. Cotter, Ph.D., 2012
- 82. Visualizing Quantities
Visualizing also needed in:
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
© Joan A. Cotter, Ph.D., 2012
- 83. Visualizing Quantities
Visualizing also needed in:
• Reading • Architecture
• Sports • Astronomy
• Creativity • Archeology
• Geography • Chemistry
• Engineering • Physics
• Construction • Surgery
© Joan A. Cotter, Ph.D., 2012
- 92. Visualizing Quantities
Early Roman numerals
1 I
2 II
3 III
4 IIII
5 V
8 VIII
© Joan A. Cotter, Ph.D., 2012
- 94. AN ALTERNATIVE
to learning place value:
Subitizing (groups of five)
Math Way (of number naming)
Place Value Cards
Trading (with 4-digit numbers)
94 © Joan A. Cotter, Ph.D., 2012
- 101. Grouping in Fives
Yellow is the Sun
Yellow is the sun.
Six is five and one.
Why is the sky so blue?
Seven is five and two.
Salty is the sea.
Eight is five and three.
Hear the thunder roar.
Nine is five and four.
Ducks will swim and dive.
Ten is five and five.
–Joan A. Cotter
© Joan A. Cotter, Ph.D., 2012
- 104. Grouping in Fives
Recognizing 5
5 has a middle; 4 does not.
© Joan A. Cotter, Ph.D., 2012
- 124. Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
124 © Joan A. Cotter, Ph.D., 2012
- 125. Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
Object of the game:
To collect the most pairs that equal ten.
125 © Joan A. Cotter, Ph.D., 2012
- 126. Go to the Dump Game
6+ = 10
126 © Joan A. Cotter, Ph.D., 2012
- 127. “Math” Way of Naming Numbers
127 © Joan A. Cotter, Ph.D., 2012
- 128. “Math” Way of Naming Numbers
11 = ten 1
128 © Joan A. Cotter, Ph.D., 2012
- 129. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
129 © Joan A. Cotter, Ph.D., 2012
- 130. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
130 © Joan A. Cotter, Ph.D., 2012
- 131. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
131 © Joan A. Cotter, Ph.D., 2012
- 132. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
132 © Joan A. Cotter, Ph.D., 2012
- 133. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
133 © Joan A. Cotter, Ph.D., 2012
- 134. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2 21 = 2-ten 1
13 = ten 3
14 = ten 4
....
19 = ten 9
134 © Joan A. Cotter, Ph.D., 2012
- 135. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2 21 = 2-ten 1
13 = ten 3 22 = 2-ten 2
14 = ten 4
....
19 = ten 9
135 © Joan A. Cotter, Ph.D., 2012
- 136. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2 21 = 2-ten 1
13 = ten 3 22 = 2-ten 2
14 = ten 4 23 = 2-ten 3
....
19 = ten 9
136 © Joan A. Cotter, Ph.D., 2012
- 137. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2 21 = 2-ten 1
13 = ten 3 22 = 2-ten 2
14 = ten 4 23 = 2-ten 3
.... ....
19 = ten 9 ....
99 = 9-ten 9
137 © Joan A. Cotter, Ph.D., 2012
- 138. “Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
138 © Joan A. Cotter, Ph.D., 2012
- 139. “Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7
139 © Joan A. Cotter, Ph.D., 2012
- 140. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
140 © Joan A. Cotter, Ph.D., 2012
- 141. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
141 © Joan A. Cotter, Ph.D., 2012
- 142. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
142 © Joan A. Cotter, Ph.D., 2012
- 143. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
143 © Joan A. Cotter, Ph.D., 2012
- 144. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
144 © Joan A. Cotter, Ph.D., 2012
- 145. Math Way of Naming Numbers
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
145 © Joan A. Cotter, Ph.D., 2012
- 146. Math Way of Naming Numbers
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
• Asian children learn mathematics using the
math way of counting.
146 © Joan A. Cotter, Ph.D., 2012
- 147. Math Way of Naming Numbers
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
• Asian children learn mathematics using the
math way of counting.
• They understand place value in first grade;
only half of U.S. children understand place
value at the end of fourth grade.
147 © Joan A. Cotter, Ph.D., 2012
- 148. Math Way of Naming Numbers
• Only 11 words are needed to count to 100 the
math way, 28 in English. (All Indo-European
languages are non-standard in number naming.)
• Asian children learn mathematics using the
math way of counting.
• They understand place value in first grade;
only half of U.S. children understand place
value at the end of fourth grade.
• Mathematics is the science of patterns. The
patterned math way of counting greatly helps
children learn number sense.
148 © Joan A. Cotter, Ph.D., 2012
- 149. Math Way of Naming Numbers
Compared to reading:
149 © Joan A. Cotter, Ph.D., 2012
- 150. Math Way of Naming Numbers
Compared to reading:
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
150 © Joan A. Cotter, Ph.D., 2012
- 151. Math Way of Naming Numbers
Compared to reading:
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
• Just as we first teach the sound of the letters, we
must first teach the name of the quantity (math way).
151 © Joan A. Cotter, Ph.D., 2012
- 152. Math Way of Naming Numbers
“Rather, the increased gap between Chinese and
U.S. students and that of Chinese Americans and
Caucasian Americans may be due primarily to the
nature of their initial gap prior to formal schooling,
such as counting efficiency and base-ten number
sense.”
Jian Wang and Emily Lin, 2005
Researchers
152 © Joan A. Cotter, Ph.D., 2012
- 153. Math Way of Naming Numbers
Traditional names
4-ten =
forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
- 154. Math Way of Naming Numbers
Traditional names
4-ten =
forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
- 155. Math Way of Naming Numbers
Traditional names
6-ten = sixty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
- 156. Math Way of Naming Numbers
Traditional names
3-ten = thirty
“Thir” also
used in 1/3,
13 and 30.
© Joan A. Cotter, Ph.D., 2012
- 157. Math Way of Naming Numbers
Traditional names
5-ten = fifty
“Fif” also
used in 1/5,
15 and 50.
© Joan A. Cotter, Ph.D., 2012
- 158. Math Way of Naming Numbers
Traditional names
2-ten = twenty
Two used to be
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
- 159. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
© Joan A. Cotter, Ph.D., 2012
- 160. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
newspaper paper-news
© Joan A. Cotter, Ph.D., 2012
- 161. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
newspaper paper-news
box-mail mailbox
© Joan A. Cotter, Ph.D., 2012
- 162. Math Way of Naming Numbers
Traditional names
ten 4
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 163. Math Way of Naming Numbers
Traditional names
ten 4 teen 4
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 164. Math Way of Naming Numbers
Traditional names
ten 4 teen 4 fourtee
n
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 165. Math Way of Naming Numbers
Traditional names
a one left
© Joan A. Cotter, Ph.D., 2012
- 166. Math Way of Naming Numbers
Traditional names
a one left a left-one
© Joan A. Cotter, Ph.D., 2012
- 167. Math Way of Naming Numbers
Traditional names
a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2012
- 168. Math Way of Naming Numbers
Traditional names
two left
Two
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
- 169. Math Way of Naming Numbers
Traditional names
two left twelve
Two
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
- 180. Composing Numbers
3-ten 7
30
7
Notice the way we say the number, represent the
number, and write the number all correspond.
© Joan A. Cotter, Ph.D., 2012
- 199. Counting by 2s and 5s
Counting by 2s
© Joan A. Cotter, Ph.D., 2012
- 200. Counting by 2s and 5s
Counting by 2s
2
© Joan A. Cotter, Ph.D., 2012
- 201. Counting by 2s and 5s
Counting by 2s
2 4
© Joan A. Cotter, Ph.D., 2012
- 202. Counting by 2s and 5s
Counting by 2s
2 4 6
© Joan A. Cotter, Ph.D., 2012
- 203. Counting by 2s and 5s
Counting by 2s
2 4 6 8
© Joan A. Cotter, Ph.D., 2012
- 204. Counting by 2s and 5s
Counting by 2s
2 4 6 8 10
© Joan A. Cotter, Ph.D., 2012
- 205. Counting by 2s and 5s
Counting by 2s
2 4 6 8 10
12
© Joan A. Cotter, Ph.D., 2012
- 206. Counting by 2s and 5s
Counting by 2s
2 4 6 8 10
12 14
© Joan A. Cotter, Ph.D., 2012
- 207. Counting by 2s and 5s
Counting by 2s
2 4 6 8 10
12 14 16
© Joan A. Cotter, Ph.D., 2012
- 208. Counting by 2s and 5s
Counting by 2s
2 4 6 8 10
12 14 16 18
© Joan A. Cotter, Ph.D., 2012
- 209. Counting by 2s and 5s
Counting by 2s
2 4 6 8 10
12 14 16 18 20
© Joan A. Cotter, Ph.D., 2012
- 210. Counting by 2s and 5s
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
- 211. Counting by 2s and 5s
Counting by 5s
5
© Joan A. Cotter, Ph.D., 2012
- 212. Counting by 2s and 5s
Counting by 5s
5 10
© Joan A. Cotter, Ph.D., 2012
- 213. Counting by 2s and 5s
Counting by 5s
5 10
15
© Joan A. Cotter, Ph.D., 2012
- 214. Counting by 2s and 5s
Counting by 5s
5 10
15 20
© Joan A. Cotter, Ph.D., 2012
- 215. Counting by 2s and 5s
Counting by 5s
5 10
15 20
25
© Joan A. Cotter, Ph.D., 2012
- 216. Counting by 2s and 5s
Counting by 5s
5 10
15 20
25 30
© Joan A. Cotter, Ph.D., 2012
- 218. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 219. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 220. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 221. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
EVEN!
© Joan A. Cotter, Ph.D., 2012
- 222. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 223. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 224. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 225. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 226. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
ODD!
© Joan A. Cotter, Ph.D., 2012
- 231. Fact Strategies
Complete the Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 232. Fact Strategies
Complete the Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 233. Fact Strategies
Complete the Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 234. Fact Strategies
Complete the Ten
9 + 5 = 14
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 239. Fact Strategies
Two Fives
8+6=
10 + 4 = 14
© Joan A. Cotter, Ph.D., 2012
- 242. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 243. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 244. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 245. Fact Strategies
Going Down
15 – 9 = 6
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 246. Fact Strategies
Subtract from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
- 247. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 248. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 249. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 250. Fact Strategies
Subtract from 10
15 – 9 = 6
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 252. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 253. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 254. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 255. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 256. Fact Strategies
Going Up
15 – 9 =
1+5=6
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 257. Fact Strategies
Multiplication
6× 4=
(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
- 258. Fact Strategies
Multiplication
6× 4=
(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012
- 260. Place Value
Two aspects
Static
© Joan A. Cotter, Ph.D., 2012
- 261. Place Value
Two aspects
Static
• Value of a digit is determined by position
© Joan A. Cotter, Ph.D., 2012
- 262. Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.
© Joan A. Cotter, Ph.D., 2012
- 263. Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value at each position
is ten times greater than previous position.
© Joan A. Cotter, Ph.D., 2012
- 264. Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value at each position
is ten times greater than previous position.
• Place value cards show this aspect.
© Joan A. Cotter, Ph.D., 2012
- 265. Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value at each position
is ten times greater than previous position.
• Place value cards show this aspect.
Dynamic
© Joan A. Cotter, Ph.D., 2012
- 266. Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value at each position
is ten times greater than previous position.
• Place value cards show this aspect.
Dynamic
• Ten ones = 1 ten; ten tens = 1 hundred; ten
hundreds = 1 thousand, ….
© Joan A. Cotter, Ph.D., 2012
- 268. Trading
Thousands
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 269. Trading
Hundreds
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 270. Trading
Tens
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 271. Trading
Ones
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 272. Trading
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 273. Trading
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 274. Trading
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 275. Trading
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 276. Trading
Adding
1000 100 10 1
8
+6
14
© Joan A. Cotter, Ph.D., 2012
- 277. Trading
Adding
1000 100 10 1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2012
- 278. Trading
Adding
1000 100 10 1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2012
- 279. Trading
Adding
1000 100 10 1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2012
- 280. Trading
Adding
1000 100 10 1
8
+6
14
Same answer
before and after
trading.
© Joan A. Cotter, Ph.D., 2012
- 281. Trading
Bead Trading Activity
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 282. Trading
Bead Trading Activity
1000 100 10 1
Object: To get a
high score by
adding numbers on
the green cards.
© Joan A. Cotter, Ph.D., 2012
- 283. Trading
Bead Trading Activity
1000 100 10 1
7
Object: To get a
high score by
adding numbers on
the green cards.
© Joan A. Cotter, Ph.D., 2012
- 284. Trading
Bead Trading Activity
1000 100 10 1
7
Object: To get a
high score by
adding numbers on
the green cards.
© Joan A. Cotter, Ph.D., 2012
- 285. Trading
Bead Trading Activity
1000 100 10 1
6
© Joan A. Cotter, Ph.D., 2012
- 286. Trading
Bead Trading Activity
1000 100 10 1
6
© Joan A. Cotter, Ph.D., 2012
- 287. Trading
Bead Trading Activity
1000 100 10 1
6
© Joan A. Cotter, Ph.D., 2012
- 288. Trading
Bead Trading Activity
1000 100 10 1
6
Trade 10 ones
for 1 ten.
© Joan A. Cotter, Ph.D., 2012
- 289. Trading
Bead Trading Activity
1000 100 10 1
6
© Joan A. Cotter, Ph.D., 2012
- 290. Trading
Bead Trading Activity
1000 100 10 1
6
© Joan A. Cotter, Ph.D., 2012
- 291. Trading
Bead Trading Activity
1000 100 10 1
9
© Joan A. Cotter, Ph.D., 2012
- 292. Trading
Bead Trading Activity
1000 100 10 1
9
© Joan A. Cotter, Ph.D., 2012
- 293. Trading
Bead Trading Activity
1000 100 10 1
9
Another trade.
© Joan A. Cotter, Ph.D., 2012
- 294. Trading
Bead Trading Activity
1000 100 10 1
9
Another trade.
© Joan A. Cotter, Ph.D., 2012
- 295. Trading
Bead Trading Activity
1000 100 10 1
3
© Joan A. Cotter, Ph.D., 2012
- 296. Trading
Bead Trading Activity
1000 100 10 1
3
© Joan A. Cotter, Ph.D., 2012
- 297. Trading
Bead Trading Activity
• In the Bead Trading activity trading
10 ones for 1 ten occurs frequently;
© Joan A. Cotter, Ph.D., 2012
- 298. Trading
Bead Trading Activity
• In the Bead Trading activity trading
10 ones for 1 ten occurs frequently;
10 tens for 1 hundred, less often;
© Joan A. Cotter, Ph.D., 2012
- 299. Trading
Bead Trading Activity
• In the Bead Trading activity trading
10 ones for 1 ten occurs frequently;
10 tens for 1 hundred, less often;
10 hundreds for 1 thousand, rarely.
© Joan A. Cotter, Ph.D., 2012
- 300. Trading
Bead Trading Activity
• In the Bead Trading activity trading
10 ones for 1 ten occurs frequently;
10 tens for 1 hundred, less often;
10 hundreds for 1 thousand, rarely.
• Bead trading helps the child experience the
greater value of each column from left to right.
© Joan A. Cotter, Ph.D., 2012
- 301. Trading
Bead Trading Activity
• In the Bead Trading activity trading
10 ones for 1 ten occurs frequently;
10 tens for 1 hundred, less often;
10 hundreds for 1 thousand, rarely.
• Bead trading helps the child experience the
greater value of each column from left to right.
• To detect a pattern, there must be at least three
examples in the sequence. Place value is a pattern.
© Joan A. Cotter, Ph.D., 2012
- 302. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
© Joan A. Cotter, Ph.D., 2012
- 303. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 304. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 305. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 306. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 307. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 308. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 309. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 310. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 311. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 312. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 313. Trading
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 314. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 315. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 316. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 317. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 318. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 319. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 320. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 321. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 322. Trading
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 323. Trading
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 324. Trading
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 325. Trading
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 326. Trading
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
6396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 327. Trading
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
6396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 328. Minnesota Standards
Number Sense
K: Represent quantities using whole numbers and
understand relationships among whole numbers.
1–2: Understand place value and relationships
among whole numbers.
With this alternate model, how difficult are the
associated benchmarks for children to master?
328 © Joan A. Cotter, Ph.D., 2012
- 329. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
329 © Joan A. Cotter, Ph.D., 2012
- 330. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
330 © Joan A. Cotter, Ph.D., 2012
- 331. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
331 © Joan A. Cotter, Ph.D., 2012
- 332. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
332 © Joan A. Cotter, Ph.D., 2012
- 333. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
333 © Joan A. Cotter, Ph.D., 2012
- 334. Minnesota Standards
Kindergarten
Represent quantities using whole numbers and
understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
• Add and subtract whole numbers up to 6, using objects.
334 © Joan A. Cotter, Ph.D., 2012
- 335. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
335 © Joan A. Cotter, Ph.D., 2012
- 336. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
336 © Joan A. Cotter, Ph.D., 2012
- 337. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
337 © Joan A. Cotter, Ph.D., 2012
- 338. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
338 © Joan A. Cotter, Ph.D., 2012
- 339. Minnesota Standards
Grade 1
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
• Represent whole numbers up to 20 in various ways.
339 © Joan A. Cotter, Ph.D., 2012
- 340. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
340 © Joan A. Cotter, Ph.D., 2012
- 341. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
341 © Joan A. Cotter, Ph.D., 2012
- 342. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
342 © Joan A. Cotter, Ph.D., 2012
- 343. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
343 © Joan A. Cotter, Ph.D., 2012
- 344. Minnesota Standards
Grade 2
Understand place value and relationships among
whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
• Represent whole numbers up to 20 in various ways.
344 © Joan A. Cotter, Ph.D., 2012
- 346. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
346 © Joan A. Cotter, Ph.D., 2012
- 347. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
347 © Joan A. Cotter, Ph.D., 2012
- 348. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
348 © Joan A. Cotter, Ph.D., 2012
- 349. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
349 © Joan A. Cotter, Ph.D., 2012
- 350. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
350 © Joan A. Cotter, Ph.D., 2012
- 351. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
351 © Joan A. Cotter, Ph.D., 2012
- 352. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
352 © Joan A. Cotter, Ph.D., 2012
- 353. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
353 © Joan A. Cotter, Ph.D., 2012
- 354. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
354 © Joan A. Cotter, Ph.D., 2012
- 355. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
355 © Joan A. Cotter, Ph.D., 2012
- 356. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
356 © Joan A. Cotter, Ph.D., 2012
- 357. Research Highlights
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
357 © Joan A. Cotter, Ph.D., 2012
Notes de l'éditeur
- Show the baby 2 bears.
- Show the baby 2 bears.
- Show the baby 2 bears.
- Show the baby 2 bears.
- Stairs
