Spots for M.A.T.H. Profession​al Development Events

1. 1

2. Spots for M. A. T. H.™
Professional Development
School Year ‘14-‘15
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3. Agenda
• Understanding the program philosophy
• Getting acquainted with your material
• Books
• Teaching Materials
• Practice Cards
• Posters
• Daily Routine Materials
• Modeling a sample lesson
• The Goal of the Common Core
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4. Our Goal
To help all students develop real math wisdom
This includes:
• An understanding of numbers and math concepts
• The ability to manipulate numbers
• The ability to make generalizations with
mathematics
Or to solve a problem like 36 + 23
mentally, by breaking up the 23 into
tens and ones so it would be calculated
• Fluency in basic math facts, which is so important
To be able to solve problems with
teen for future sums, math such success
as 8 + 5, using the
“make a ten” strategy (at a Grade 1 level).
That is to say, If I can solve 6 + ___ = 10,
• Proficiency in solving word problems
as 36 + 20 = 56 + 3 = 59 (at a Grade 2 level).
then I can solve 46 + ___ = 50.
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5. Research on Early Math Education
Research done by the National Institute for Child
Health Development shows that early math
success is critical to math success both in upper
grades and in life. Students in the first grade who
have failed to: acquire an understanding of the
number system, the relationship between a
numeral and the quantity it represents, and then
to manipulate these numbers and make
calculations, will most likely never catch up. When
tested in the seventh grade, these students
scored far behind their peers. (NICHD, Feb. 2013).
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6. The Challenge
How can we help our students become
mathematical thinkers while teaching
them to solve a problem like 9 - 6?
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7. The Challenge
• Math is a challenging abstract subject, built on
concepts and strategies. It has its own language and
a host of symbols: digits, >, <, operation symbols, etc.
• How can we teach six-year-old children to
manipulate numbers?
• How can we teach so that children learn to make
connections?
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8. The Spots for M.A.T.H. Solution
• Through the use of innovative tools:
 Spots for M.A.T.H. Dot Cards
 The Open Number Line
 Puzzle-Piece Models for Solving Word Problems
• A predictable and unique program
progression
• A progressive practice system
We can help all students develop real math
wisdom.
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9. The Dot Cards
Predictable images of numbers and operations,
which are easy to visualize confidently, are
used to overcome the abstract challenge.
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10. Dot Cards 1-10
• These show the quantity of numbers 1-10,
using black dots in a specific format.
14372561890
1 2 3 4 5 6 7 8 9 10
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11. Spots for Math Dot Cards
vs. Other Types of Ten Frames
DecaDots® is a trademark of ETA hand2mind and is not affiliated with Spots for M.A.T.H.
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12. Teen Numbers
Math educator Kathy Richardson has observed
just how hard it is for children to understand the
numbers 11 through 20 in terms of place value.
She summarizes her many years of working with
and observing children attempting this hurdle as
follows: “Children who have not yet learned that
numbers are composed of tens and ones think
of the numerals that are used to write particular
numbers as the way you 'spell' them.
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13. Teen Numbers
From the child's point of view, it just happens
that we need a 1 and a 5 to write fifteen and a 1
and a 2 to write twelve. It is not obvious to
young children that the numerals describe the
underlying structure of the number” (p. 26).
Richardson, K. (2003). Assessing Math Concepts: Ten Frames. Rowley, MA:
Didax.
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14. Teen Dot Cards 11-19
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15. When and how are the Dot Cards used?
• Teacher models the concept or strategy using
Magnetic Dry-Erase Dot-Boards with magnetic
counters.
• Students use Dot Boards and counters, and
they practice in their book.
• Then the Concept Representation Dot-Cards
are used in lesson warm-ups for practice and
reinforcement.
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16. Stages of Learning
Effective mathematics instruction tells us to move
from the concrete to abstract. All too often, these
processes are seen as single entities. i.e. On day one
use actual object, day 2 representational objects
(chips), day 3 drawing, day 4 abstract and from then
on all abstract…
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17. Transitioning the Stages of Learning
Our program is unique in that we see these
experiences as a coordinated holistic approach.
With this as our belief, Spots for M.A.T.H. offers the
teacher with specific tools, namely our set of
patented Dot Boards which is used to introduce
concepts and strategies. Our corresponding
Concept Representation Dot Cards which is used to
reinforce the concepts and strategies in order to
facilitate the transition from concrete to abstract
experiences. Using these tools, we foster greater
understanding, fluency and internalization of given
concepts.
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18. Magnetic Dry-Erase Dot Boards
What’s inside?
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19. Magnetic Dry-Erase Dot Boards with
black-and-white magnetic counters
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20. Modeling a Concept with Magnetic Dry-
Erase Dot Boards and Magnetic Counters
7 + 2 = 9 7 - 1 = 6
20
7 - 6 = 1

21. Modeling a Concept with Magnetic Dry-
Erase Dot Boards and Magnetic Counters
• The make-a-ten strategy
9 + 8 = 17
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22. Modeling the Concept with
Magnetic Dry-Erase Dot Boards
13 - 5 = 8 22 13 - 9 = 4

23. Students’ Blank Dot Boards and
Black-and-White Foam counters
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24. Concept-Representation Dot Cards
What’s Inside?
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25. Addition Dot Cards 1-10
The greater addend is shown first, with black
dots; the lesser addend is shown second, with
white dots.
3 + 1 = 4 4 + 2 = 6 5 + 3 = 8 6 + 4 = 10
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26. Subtraction Dot Cards 1-10
The subtrahend (the number subtracted) is shown by
circling and crossing off the appropriate number of dots.
When it is a small number, the dots are crossed off the
top.
10 – 1 = 9 9 – 2 = 7 26

27. Subtraction Dot Cards 1-10
When the subtrahend is a large number, the
dots are crossed off the bottom.
7 – 6 = 1
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28. Teen Addition Dot Cards
• Used for addition with teen sums to 19. The
greater addend is shown first, with black dots;
the lesser addend is shown second, with white
dots.
9 + 5 = 14 8 + 7 = 15 28

29. Teen Subtraction Dot Cards
• When subtracting a small number, dots are
crossed off starting from the “ones side.”
14 - 6 = 8 29

30. Teen Subtraction Dot Cards
When subtracting a large number (10, 9, 8, and some-times
7), they are crossed off from the “ten side.”
14 - 8 = 6 30

31. FAQ
• Must children cross off dots the way we tell
them to?
• What if a student of mine will want to cross
off dots differently?
• What does “sometimes 7” mean? Why not all
the time?
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32. How would you subtract 7?
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33. FAQ
• Will my students still use their fingers to help
them add or subtract?
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34. Lesson Index
Chapter 1
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35. 35

36. Lesson Index
Chapter 2
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37. 37

38. Lesson Index
Chapter 3
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39. 39

40. Lesson Index
Chapter 4
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41. 41

42. Lesson Index
Chapter 5
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43. 43

44. Lesson Index
Chapter 6
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45. 45

46. Lesson Index
Chapter 7
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47. 47

48. Lesson Index
Chapter 8
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49. 49

50. Lesson Index
Chapter 9
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51. 51

52. Using the Number Line to Extend Thinking
Strategies to Two Digit Numbers and Beyond
When it comes to calculating with larger
numbers mentally, it becomes hard to visualize
the amounts, as we must think of quantity
images of all the tens and ones we had, and
then how many we are adding on. At this point
it’s much more helpful to think of a number line
beginning at a specific point, and then jumping
by tens and by ones.
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53. Using the Number Line cont.
There is much research showing that the brain actually
thinks of the larger units first; that is, if you would ask a
student to solve two-digit addition before he or she was
taught a formal process for such equations, the child
would think of the tens first! The algorithm actually asks
us to work against our understanding of numbers! So its
crucial to first develop number sense and the ability to
calculate mentally, and then to transfer it to the
algorithm – the formal paper and pencil process.
Number line Classroom Banner 1-100
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54. Student Book, Pages 49 and 83
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55. Student Book, Pages 121 and 125
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56. Student Book, Pages 141 and 153
56

57. The Program Progression
Students see clearly how one skill builds on
another.
6 + 1 = 7 6 + 3 = 9 6 - 1 = 5 6 - 2 = 4 57

58. The Program Progression
Predictability and patterns help students generalize strategies
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59. The Program Progression cont.
Concepts are built and layered over time.
Chapter 4: Teen Numbers Chapter 5: Decade Numbers Chapter 6: Two-Digit Nu5m9bers

60. The Program Progression cont.
• In each chapter we compare the numbers to each other
(e.g., 15 to 50 or 51).
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61. The Program Progression cont.
Money skills are inserted throughout the chapters as a
problem solving application of the concepts presented.
This helps teach students to generalize skills.
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62. Problem Solving
Problem solving is an integral component
needed to be successful with mathematics.
We don’t leave this skill to chance, as part of
our goal to develop a solid foundation, we
offer a formal methodical meta-cognitive
approach to problem solving. We have a
scaffolding mapping tool - the puzzle piece
model which gives students the framework of
how to solve story problems.
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63. Puzzle-Piece Models
Math Puzzles are a great tool to
help organize the parts of a
number sentence. The top
piece shows the whole number,
and the two bottom pieces
show the parts of that number.
The puzzles are used to show
the relationship between
numbers, as in number
families. They are also used to
help solve various types of
story problems. 63

64. Puzzle-Piece Models cont.
Using puzzles, the numbers can be easily
organized, making it simple to identify the missing
component – whether the whole or a part – and
then solve accordingly: For a missing whole, add
the two parts; for a missing part, subtract the part
we already know from the whole. The puzzle
pieces are always the same size. They are not
scaled according to quantity. This helps students
stay focused on organizing the numbers into their
respective parts, as opposed to trying to figure
out what size the pieces should be.
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65. Puzzle-Piece Models for
Problem Solving
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66. From Understanding to Internalization
Common Core State Standards for Mathematics
2.OA.2: Fluently add and subtract within 20 using
mental strategies. By end of Grade 2, know from
memory all sums of two one-digit numbers. Research
indicates that students internalize facts and develop
fluency by repeatedly using the strategies that make
sense to them. Research indicates that teachers can
best support students’ memory of the sums of two
one-digit numbers through varied experiences
including making 10, breaking numbers apart, and
working on mental strategies.
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67. From Understanding to Internalization cont.
These strategies replace the use of repetitive timed tests
in which students try to memorize operations as if there
were not any relationships among the various facts.
When teachers teach facts for automaticity, rather than
memorization, they encourage students to think about
the relationships among the facts. (Fosnot & Dolk, 2001)
It is no coincidence that the standard uses the term
“know from memory” rather than “memorize.” The
former describes an outcome, whereas the latter might
be seen as describing a method of achieving that
outcome. So, no, the standards are not dictating timed
tests. (McCallum, October 2011)
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68. The Practice System
Lesson Warm-Up with Drop-Its Form #1
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69. The Practice System, cont’d.
Lesson Warm-Up with Drop-Its Form #2
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70. The Practice System
4
+ 2
5
+ 3
Lesson Warm-Up with Drop-Its Form #3
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71. The Practice System, cont’d.
Double-Sided Number Sentence Wipe-Off
Boards (optional product)
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72. SMART Board® - Interactive Whiteboard
• All student lesson pages are available for the
Interactive Whiteboard (IWB). They are
compatible with SMART Board® Interactive
Whiteboards and any other interactive surface
or projector.
• Additional Components: Dot Boards, Counters,
Concept Representation Dot-Cards and Drop-It
forms.
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73. Focus Standards and Facts Fluency
Practice Book
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74. Maximizing the Learning Experience
• The daily routine
• Ongoing visual reinforcement
– Banners
– Math window
• Teacher’s Resource Book
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75. Daily Routine Material
Spots for M. A. T. H.
Magnetic Money House
Hundred Number Pocket Chart with
100 Clear Pockets, & Pattern Marke75rs

76. Ongoing Visual Reinforcement
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77. Included in your “Classroom
Kit” is the Teacher’s Resource
Book, it is a 148-page binder
that provides copy masters for
teachers to use throughout
the year. It includes:
• Family letters (to keep the
families informed of and
involved in all that the class is
learning)
• Drop-Its forms (used in the
lesson warm-up section to
develop fluency and for
ongoing assessment)
• Cutouts (drawings that are
meant to be cut, for the
teacher to use, such as a frog
cutout to model jumping on
the number line)
• Lesson Handouts (which are
used by students to enhance
the lessons)
• Assessment Forms
• Reproducible Game Cards and
Boards
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78. The Spots for M.A.T.H. Lesson Format
78

79. Model Lesson
Chapter 2 Lesson 5: Adding Three
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80. Lesson Goal:
• CCSS 1.OA.6 Add and subtract within 20.
• Goal: Students will use Addition Dot Cards to
demonstrate adding three.
• Materials Needed: Dot Board; black and white
magnetic counters; blank Dot Boards (cut from
the last page of the student book); student
counters.
Common Core State Standard
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81. Lesson Warm-Up:
Flash all +1 and +2 Addition Dot Cards. Have the class identify
the number sentence of each card in unison.
(Remember to show each card for only a few seconds! )
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82. Introductory Statement:
• Yesterday, we learned to add one and two using
our Addition Dot Cards. Today we will use
Addition Dot Cards to add three.
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83. Thinking Trigger:
• How did we add one and two using our Dot
Cards? [Place a sample of each on the board.
Have class identify the equation each one
shows.] How do you think we will add three
with the Dot Cards? [Allow time for
suggestions. Remove the cards.]
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84. Concept Development:
I. Adding three
Place Dot Card 4 on the board and use magnetic counters to model adding three. As
you place the white counters, count on: We begin with 4 and we add on 5, 6, and 7.
Ask: How many black dots are on the card? [4] How many white dots did I add? [3] How
many do we have in all? What number does this look like? [7] What addition sentence
can we write for what we did? [4 + 3 = 7] [Show Dot Card 7 and point out that the
formation is the same as the 4 + 3 Dot Card on the board.]
4 + 3 = 7
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85. Concept Development:
• Present Dot Card 6 and model adding three
magnetic counters. Ask: How many do we
have in all? What number does this look like?
[9] What addition sentence do we have now?
[6 + 3 = 9] [Show Dot Card 9 and compare.]
• Continue in the same way for 5 + 3.
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86. Concept Development:
• Show the class the +3 Addition Dot Cards and
read the equations together.
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87. Concept Development:
II. Adding by counting on
• Now let’s do something different. [Write 6 + 3 on the
board]. Let’s solve this without using Dot Cards and
counters. We can use the banner and pretend. With what
number do we start? [6] Let’s look at Dot Card-6 on the
banner. [Point to Dot Card-6.] How many more do we need
to put on? [3] Let’s pretend to put on three more counters.
We begin with 6 and we count on 7, 8, and 9. There are
nine in all. [Write in the sum.]
• In the same way, model solving 3 + 3 and 7 + 3.
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88. Student Teacher:
• Divide the class into pairs. Have each partner write
an addition sentence with +3 on their number
sentence wipe off boards. Then have the partners
work together to show the number sentences on
their Dot Cards. Have each set of partners show their
work to another set of partners and explain what
they did.
• Be sure counters are placed correctly, from left to
right, so that the correct format for each number is
shown.
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89. Conclusion:
• We see that we can solve “plus 3” addition
sentences by adding three white dots to our
Dot Cards and seeing what new Dot Cards we
get.
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90. Using The Book: Pages 41-42
Page 41:
Place Dot Card 3 on the board. Model adding three
counters. Ask: What addition sentence do we have?
First we had ___ [3], then we added ___ [3]. Which Dot
Card do we have now? [6] 3 + 3 = 6. [Write the addition
sentence under the card.]
3 + 3 = 6 90

91. Using The Book: Page 41
• Read the directions. Have the class find the first
example in their books. Show that example 1 is the
same as you modeled on the board. Say: In the book
they also have Dot Card 3 with three white counters.
Fill in the addition sentence : 3 + 3 = 6.
• Examples 2-5: In the same way, continue with
example 2 through 5. Have the class complete the
section independently while you circulate to offer
help as needed. Review the answers together.
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92. Using The Book: Page 41
• Examples 6-9: Say: This is a new kind of
practice for us. [Read the directions.] Look at
example 6. It is done for us. What is the
number sentence? [6 + 3 = 9] The book has a
line drawn to the matching Addition Dot Card.
It is the one next to the yellow square. Trace
the connecting line and write in the sum.
• In this way, complete the section together.
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93. Using The Book: Pages 41
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94. Using The Book: Page 42
• Examples 1-6: Read the directions. Read the first number
sentence together. Ask: Which Dot Card matches this
sentence? Why? [Wait for answers.] In the book, the correct
Dot Card is already circled for us.
• In a similar way, read examples 2 and 3. Place the correct
Addition Dot Card on the board, and remind the students
to circle the correct one in their books. Have the class
complete the page independently while you offer
assistance as necessary. Review the answers together.
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95. Using The Book: Page 42
• Examples 7-12: Have students complete this
section independently. Students may choose
to draw dots or just pretend adding dots to
help them add. Review the section together.
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96. Using The Book: Pages 42
96

97. Closing Statement:
• Ask: What did we learn to do today in math
class? [Accept relevant answers.] Today we
learned how to add three using our Dot Cards.
When we add three white dots to the Dot
Card, we can see how many we have
altogether. Tomorrow we will use Dot Cards to
tell math stories.
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98. Changes in Instruction
"The Common Core demands significant shifts in
the way we teach. Each teacher must adopt
these shifts so that students remain on track
towards success in college and careers. These
shifts in instruction will require that many
teachers learn new skills and reflect upon and
evolve in their classroom practices"
(engageny.org).
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99. The Goals of the Common Core
• To develop students who are proficient in
mathematics
• To teach with deep conceptual understanding
and practice to acquire fluency of facts and
procedures
• We can’t be satisfied with students just being
quick “counters”
• Your efforts will affect the results of grade 3
state testing
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100. Managing Your Time
• How long does a lesson take?
• Can I skip lessons? Or parts of a lesson?
100

101. Pacing Guide for School Year ‘14- ‘15
Based on a four-days-per-week schedule (excluding off days).
Each lesson should take one day. An extra day is allotted for assessment and chapter introduction.
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102. Plan for Grade 2
• Transition with a Stepping Up Review Unit as
the first chapter for grade 2 is included, along
with a Teacher’s Edition.
• Then students will continue with their current
grade-2 program, until Spots for M.A.T.H. will
officially release their grade-2 book!
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103. The Results בס"ד
Students:
• Develop true number sense
• Master their math facts
• Acquire thinking strategies
• Generalize their learning
• And most important, students develop a
confident, “can do!” attitude toward math.
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