Student Book, Volume I: Chapters 1-4 (158 pages); Volume II: Chapters 5-8 (156 pages) Teacher’s Edition, Volume I and II Teachers Resource Book (Blackline Masters) Focus Standards and Facts Fluency Practice Book
What’s the problem if a student puts out nine fingers and then hides away six? It may become embarrassing as students get older. The strategy is not extendable to problems with larger numbers, such as 13 – 6.
Jean Piaget’s four stages of children’s cognitive development: Sensory-motor learning Preoperational learning Concrete operational learning Operational learning
Generally, when we say “all,” it indicates that we want to include lower achieving or less capable students. This program is not necessarily geared for children with special needs, although its strong visual base and fine progression makes it very adaptable for them. The thinking level involved here is actually very high, and the fact is that many average and above-average students have been underserviced until now, as they were not helped to develop real number sense. We were satisfied if they found the sum of a number by counting; we did not develop their thinking abilities. When we say we can help all students develop real math wisdom, we refer to gaining a greater quality of achievement for all.
As the NCTM points out, "The ten-frame uses the concept of benchmark numbers (5 and 10) and helps students develop visual images for each number." Using a ten-frame, students can easily see, for example, that 6 is 1 more than 5 and 4 less than 10, or that 8 can be seen as "5 and 3 more" and as "2 away from 10." Once students are able to visualize the numbers 1 through 10, they begin to develop mental strategies for manipulating those numbers, all within the context of the numbers' relationship to ten.
With the Dot Cards as our base, we can call our approach a “quantity-based” approach as opposed to a digit-based approach. These cards are the most basic element of the program. The students learn to recognize each Dot Card as an image without counting the dots. Once students learn what each Dot Card represents with instantaneous recognition, they can use these cards to build their understanding of progressively more abstract math concepts.Math becomes so simple that even when working with the complicated process of the “make a ten” strategy, one wonders, “What's so hard about math?”
As you may know, ten frames are an ancient Japanese mathematical model. Ten frames in and of themselves are helpful but are not visually dynamic or powerful enough to help students internalize concepts.Christine Losque devised putting the dots in sets of two. (Her product is called Deca-Dots.) The division of the card into 5 red and 5 green is our innovation. Having the 5 as a bench mark helps differentiate between similar-looking numbers, such as the even numbers 4, 6, and or the odd numbers 5, 7, and 9.
If children don’t understand the underlying structure of teen numbers, how can we expect them to decompose 14 when solving 14 - 6 as 14 - 4 - 2?
These are a basic application of the 1-10 Dot Cards. These Dot Cards develop a strong foundation for place value: we have 1 ten and a specific number of ones to represent the teen number. Using this system, the percentage of young students who don’t understand the concept of place value, or who reverse numbers, is really very low. We had one student who her digits in mirror image but wrote them in the correct places!
The material used to teach concepts with, is referred to as Dot Boards; they are magnetic dry-erase boards. The material used for practice and reinforcement is referred to as Dot cards, like flash cards.
A Dot Board is used to show the first number in the number sentence. The teacher models the operation by adding white counters or crossing off dots.
We turn over the white counter to its black side to show we need it to make a ten .Then the students are taught to imagine that the dot needed to make a ten “fly over” to the tens side while the rest stay white in their place on the ones side. Focusing on the white dots, it is clear to see the sum is 17.
This is taught in chapter 7. Working with Dot Boards has a number of advantages. It saves time because the teacher does not have to count out the first quantity in the equation- its just printed on the board. Also, the images allow students to solve the problem without too much teacher explanation. When students solve problems in this manner they can create a mental image of the process - which would be harder to do if they were working with counters.
You will find this page at the end of your Student Book. [SHOW STUDENT DOT CARDS]You will receive a packet with 600 foam counters. Every student needs 20 of them. [SHOW COUNTERS]Students use these when concepts are introduced. We recommend that teachers collect the materials after use so students don’t loose them.
Reviewing with our Dot Cards provides reinforcement in the most time-efficient manner; students develop fluency of the facts as well as internalizing the thinking framework for how to solve similar problems. Thus, when a 9 - 6 Dot Card is presented, the student is practicing the specific example while the concept of “subtracting a lot” is being reinforced. Also, practicing with Dot Cards takes less time than practicing with other manipulatives.Practicing on digit level, with flashcards or the like, is “rote drill.” This gives students the message, “Get the answer fast,” so they just count, and we end up creating great “counters” instead of great “thinkers.”
At this point the red and green are embedded in the subconscious. The focus is on the black and the white.
There is much debate in the research regarding how first-graders should learn to subtract equations such as 9 – 7. Should we have them count back six numbers? Or should we teach them to count forward (six…7, 8, 9)? Research suggests that they learn to count back, because that is consistent with the concept of subtracting as taking away. However, the Common Core wants children to understand that we can count on to subtract. With our Dot Cards, this is clear and elegantly simple.The Subtraction Dot Cards show the “shape” of the number we are subtracting. This model makes it clear that we are taking away, yet we can also clearly see the amount that is left, encouraging counting up.FAQWhy is the seventh dot not crossed off?Why are the dots in the equation 9 - 2 circled and crossed off with 1 big X?ANSWERS: When we subtract a “lot,” as in 9 – 7, we want to think of the 7 as a group of seven that we are taking away. The circle indicates that whatever is in this circle is being subtracted, even though the X isn’t actually covering the seventh dot.Because we have two methods of subtracting – from the top for a little and from the bottom for a lot – we want students to think, “Do I need to subtract from the top or from the bottom?” We don’t want them to have to think about, “Do I need one big X or a few X’s?”
This is taught in Chapter 7.
This question reflects your experience with rules on a digit level. When we present math as a “quantity-based” subject, our “rules” are based on number sense, which simply makes sense. Of course, we can’t force children to cross off nine dots on the “tens” side, but why would she/he want to do it differently?When it comes to subtracting 7, we help students arrive at their own conclusions regarding from which numbers is pays to subtract from the “tens” side and from which numbers it pays to subtract from the “ones” side. Let’s look at our teen Dot Cards and see: What do you think? How would you subtract 7?
Research suggests that students should have reference to a numbered number line prior to working with an open, or empty, number line. The banner exposes the students to this more advancedrepresentation of number.In fact, the Common Core is requiring modeling with open number lines in grades 2 and up. Teaching this to our students in this format will give them a head start for future success in math.
After students have learned about the concept of addition with the Dot Cards, they learn that they can think of a number line. For subtraction, they first learn the concept of subtracting a little (i.e., a small amount-such as 9 - 2), then they learn to subtract on the number line; then they learn to subtract a lot, such as 9 - 6. (We don’t model 9 - 6 on the number line, because we feel it might confuse the students.) In these chapters they work with a numbered number line. in chapter 4 we introduce the open number line.
In Chapter 4 we introduce the number line with no numbers. The students learn that we can make one big jump instead of individual jumps! And we can make two jumps when working with 3 addends! Students learn about this way of thinking by filling in modeled number lines. The “On Your Own” section gives students an opportunity to draw the jumps themselves.
We then apply the strategy of adding in two steps to making two jumps. This will help the students the following year with solving problems such as 39 + 7. I’ve included p. 153 to point out that we make students aware there are several approaches and each one is correct, and some students actually choose the number line!After learning to add with Dot Cards,students have also been shown how to represent the make-a ten strategy with a “break-apart number sentence. All three methods; Dot cards, number lines and the break- apart number sentence. are referenced on this page.
The puzzle piece modelprovides an iconic graphic organizer for the parts of a story. Each number sentence consists of three variables; two of them are known and one is unknown. With the puzzle-piece model, students can learn to organize their variables as the “whole” or “parts,” and they can then write a number sentence which they can solve.This concludes the presentation of our unique tools. The next factor that enables students to develop real math wisdom is the program progression.
The program follows a unique progression. Lessons and chapters are carefully scaffolded, to help students to absorb ideas gradually over a span of time, and to guide them to make connections and generalizations.
Every concept is presented over at least 3 lessons to help students absorb the idea. First they learn to add to 10, then to 9, then to 8. Thus, in this case, when they get to adding to 7 and 6, they know just what to do. This helps them to develop confidence in their abilities.
Three chapters address place value to aid in concept assimilation.In Chapter 4 we learn about numbers with a ten a some ones. In Chapter 5 we learn about numbers with many tens. In Chapter 6 we learn about numbers with many tens and many ones. In each chapter we compare the numbers to each other (e.g., 15 to 50 or 51). We also apply what we know about numbers to ten: to add or subtract with teen numbers (e.g., 12 + 6 or 17 - 5), with decade numbers (e.g., 20 + 60 or 70 - 50), or two-digit numbers (e.g., 82 + 6 or 47 – 5. For any of these examples, we think of 2 + 6 or 7 - 5 to help us).
Dot Cards for Five and Ten are related to the nickel and dime.
In Chapter 1 we flash Dot Cards 1-10. We flash the card for only one second! We can flash it again, but we must make sure that students don’t count the dots. Instead, they should use the structure of the Dot Card to help them identify the card (e.g., 1 or 2 more than 5, or 1 or 2 less than 10).
With form #3, we begin our weaning process; from reliance on the Dot Card to calculating at the digit level.
These boards are used in concept development and can also be used instead of the Drop-Its forms to provide for a change. The puzzle-piece model is used for problem solving.
This student workbook provides additional practice for the focus standards, which are: conceptual understanding of place value, fluency with facts through 10, modeling with math, and proficiency with word problems. At first, in Chapters 2 through 4, the practice pages have Dot Cards either next to the examples or at the bottom of the page. Later on, there are no Do Cards on the page. There are Dot Cards on the inside of the back cover for students to refer to if they wish. This pushes children to be less dependent on actually needing to look at the cards. Instead, they can visualize the cards or think about the process of solving the examples.
In addition to offering a strong suite of teaching tools and a Teacher’s Edition with step-by-step instruction for an interactive teaching style in order to achieve our goal of developing real math wisdom, we have several other components to maximize the learning experience.
The goal of the daily routine is to expose students to place-value concepts prior to learning about place value and then to reinforce the concepts after they have been taught. In the Teacher’s Edition you will find instructions on how to implement the daily routine. These routines do not necessarily need to be carried during math class and should not take more than one or two minutes. Take care not to this routine into a 10-15-minute mini-lesson. Also, if you have been doing any other daily math routine, you need to compare the benefits of the two routines and decide which one to use . These routines are more effective than for example , a routine of rote counting to 100. Don’t worry! By the time your students will leave first grade they will know how to count!
These banners ensure ongoing learning even when math class is not going on! The Teacher’s Edition indicates when and how to use these aids so students become accustomed to them and refer to them constantly. The 1-10 Dot Card banner should be hanging in your classroom beginning with first day of class. It will help with developing automatic recognition of the Dot Cards. The Math Windowposter provides additional reinforcement of one or two specific math facts at a time. The Teacher’s Edition will help guide you as to indicates when to hang the banners and which facts to place in the math window when.
The Teacher’s Edition guides you by indicating when to use what.
Each lesson follows a predictable sequence. This helps the students organize their learning and saves precious learning time as the students know what to expect. We will go through the different components in the model lesson.
These are the se pages for the lesson I will model. The program is so engaging, students don’t want to miss a lesson! We had a case of a student with behavior challenges who started the math lesson class without her books. When it came to using the math book, she kept on looking at her neighbor’s open book. Her teacher told her that her book was waiting for her and that she could come to take it if she wanted to. She actually went to get the book! She didn’t want to lose out!
This is a quick activity that reinforces fluency of facts and strategies previously learned.
This is the introductory Statement: it’s A statement used to link previous learning to current lesson.
Now we have the Thinking Trigger: A question or idea that challenges the students and encourages them to tell their own ideas about how to solve problems or apply their prior knowledge.
Concept Development: A step by step guide to developing the lesson concept through use of focused questions, teacher modeling, student practice and discussion prompts.
Student Teacher: An opportunity for students to practice and concretize their learning by practicing new skills and/or discussing mathematical problem solving processes with peers.
Conclusion: A summary statement to recap the lesson.
In the te we indicate how to do the se pages, together as a class or if we expect the students to do the work independently
Closing statement: A lesson ending segment that asks students to reflect on the lesson taught and connects this lesson to the lesson that follows.
*1 Students have a good sense of the value of a digit. *3 Students acquire thinking strategies, such as the “make a ten” strategy, or how to subtract 9 - 2 vs. 9 - 6, at the same time as they practice for fluency. Reviewing with the Dot Cards is the most efficient way to maximize classroom time spent on review. *5 The lessons are carefully scaffolded, and enough practice is provided for concepts and strategies to get internalized, so it’s no wonder that Spots for M.A.T.H. students develop a positive attitude toward their ability with mathematics.
Spots for M.A.T.H. Professional Development Events
Spots for M. A. T. H.™
School Year ‘14-‘15
• Understanding the program philosophy
• Getting acquainted with your material
• Teaching Materials
• Practice Cards
• Daily Routine Materials
• Modeling a sample lesson
• The Goal of the Common Core
To help all students develop real math wisdom
• An understanding of numbers and math concepts
• The ability to manipulate numbers
• The ability to make generalizations with
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.
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).
How can we help our students become
mathematical thinkers while teaching
them to solve a problem like 9 - 6?
• 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
• How can we teach so that children learn to make
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
• A progressive practice system
We can help all students develop real math
The Dot Cards
Predictable images of numbers and operations,
which are easy to visualize confidently, are
used to overcome the abstract challenge.
Dot Cards 1-10
• These show the quantity of numbers 1-10,
using black dots in a specific format.
1 2 3 4 5 6 7 8 9 10
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.
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.
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:
When and how are the Dot Cards used?
• Teacher models the concept or strategy using
Magnetic Dry-Erase Dot-Boards with magnetic
• 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
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…
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
Magnetic Dry-Erase Dot Boards
Magnetic Dry-Erase Dot Boards with
black-and-white magnetic counters
Modeling a Concept with Magnetic Dry-
Erase Dot Boards and Magnetic Counters
7 + 2 = 9 7 - 1 = 6
7 - 6 = 1
Modeling a Concept with Magnetic Dry-
Erase Dot Boards and Magnetic Counters
• The make-a-ten strategy
9 + 8 = 17
Modeling the Concept with
Magnetic Dry-Erase Dot Boards
13 - 5 = 8 22 13 - 9 = 4
Students’ Blank Dot Boards and
Black-and-White Foam counters
Addition Dot Cards 1-10
The greater addend is shown first, with black
dots; the lesser addend is shown second, with
3 + 1 = 4 4 + 2 = 6 5 + 3 = 8 6 + 4 = 10
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
10 – 1 = 9 9 – 2 = 7 26
Subtraction Dot Cards 1-10
When the subtrahend is a large number, the
dots are crossed off the bottom.
7 – 6 = 1
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
9 + 5 = 14 8 + 7 = 15 28
Teen Subtraction Dot Cards
• When subtracting a small number, dots are
crossed off starting from the “ones side.”
14 - 6 = 8 29
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
• Must children cross off dots the way we tell
• What if a student of mine will want to cross
off dots differently?
• What does “sometimes 7” mean? Why not all
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.
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
The Program Progression
Students see clearly how one skill builds on
6 + 1 = 7 6 + 3 = 9 6 - 1 = 5 6 - 2 = 4 57
The Program Progression
Predictability and patterns help students generalize strategies
The Program Progression cont.
Concepts are built and layered over time.
Chapter 4: Teen Numbers Chapter 5: Decade Numbers Chapter 6: Two-Digit Nu5m9bers
The Program Progression cont.
• In each chapter we compare the numbers to each other
(e.g., 15 to 50 or 51).
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.
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.
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
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.
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.
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)
The Practice System
Lesson Warm-Up with Drop-Its Form #1
The Practice System, cont’d.
Lesson Warm-Up with Drop-Its Form #2
The Practice System
Lesson Warm-Up with Drop-Its Form #3
The Practice System, cont’d.
Double-Sided Number Sentence Wipe-Off
Boards (optional product)
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
• Additional Components: Dot Boards, Counters,
Concept Representation Dot-Cards and Drop-It
Focus Standards and Facts Fluency
Maximizing the Learning Experience
• The daily routine
• Ongoing visual reinforcement
– Math window
• Teacher’s Resource Book
Daily Routine Material
Spots for M. A. T. H.
Magnetic Money House
Hundred Number Pocket Chart with
100 Clear Pockets, & Pattern Marke75rs
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
• Drop-Its forms (used in the
lesson warm-up section to
develop fluency and for
• 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
• Assessment Forms
• Reproducible Game Cards and
Chapter 2 Lesson 5: Adding Three
• 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
Common Core State Standard
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! )
• Yesterday, we learned to add one and two using
our Addition Dot Cards. Today we will use
Addition Dot Cards to add three.
• 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.]
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?  How many white dots did I add?  How
many do we have in all? What number does this look like?  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
• Present Dot Card 6 and model adding three
magnetic counters. Ask: How many do we
have in all? What number does this look like?
 What addition sentence do we have now?
[6 + 3 = 9] [Show Dot Card 9 and compare.]
• Continue in the same way for 5 + 3.
• Show the class the +3 Addition Dot Cards and
read the equations together.
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?  Let’s look at Dot Card-6 on the
banner. [Point to Dot Card-6.] How many more do we need
to put on?  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.
• 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
• Be sure counters are placed correctly, from left to
right, so that the correct format for each number is
• 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
Using The Book: Pages 41-42
Place Dot Card 3 on the board. Model adding three
counters. Ask: What addition sentence do we have?
First we had ___ , then we added ___ . Which Dot
Card do we have now?  3 + 3 = 6. [Write the addition
sentence under the card.]
3 + 3 = 6 90
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.
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.
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.
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.
• 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.
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"
The Goals of the Common Core
• To develop students who are proficient in
• To teach with deep conceptual understanding
and practice to acquire fluency of facts and
• We can’t be satisfied with students just being
• Your efforts will affect the results of grade 3
Managing Your Time
• How long does a lesson take?
• Can I skip lessons? Or parts of a lesson?
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.
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!
The Results בס"ד
• 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.