1. Using Dot Plates and Ten Frames in the Primary Classroom
Know Your Numbers
Using Dot Plates and Ten Frames in the Primary Classroom

2. Subitizing
Subitizing is the ability to recognize dot arrangements in different patterns.
Since children begin to learn these patterns by repetitive counting they are closely connected to their understanding of the particular number concept. Quantities up to 10 can be known and named without the routine of counting. This can help children in counting on (from a known patterned set) or learning combinations of numbers (seeing a pattern of two known smaller patterns).
Young children should begin by learning the patterns of dots up to 6. Students should also associate the dot patterns to numbers, numerals, finger patterns, bead strings, etc. You can then extend this to patterns up to 10 when they are ready.
Subitizing is a fundamental skill in the development of number sense, supporting the development of conservation, compensation, unitizing,counting on, composing and decomposing of numbers.

4. Two Types of Subitizing
Perceptual Subitizing:
Recognition of number patterns
Conceptual Subitizing
Recognition of number pattern as composite of parts and whole
Spatial (dot arrangements), temporal (attaching number to sounds), Kinesthetic (finger patterns), rhythmic (hand clapping)
Spatial arrangements of sets influences how difficult they are to subitize (rectangular easiest → linear → circular →scrambled most difficult
Different arrangements lead to different decompositions of that number
Oral Numbers: Generally, students are ready to use oral numbers if they know that a set
does not alter the number of objects in the set.
Numerals: Generally, the use of numerals with dot patterns should occur after children have
made solid connections between the oral names for numbers and sets.

5. Dot Card Activities
Activities:
1. Counter Match (Materials: dot plates or dot cards, paper plates, counters)
Students place one counter on top of each dot (dot plate or card). They
compare the number of counters to the number of dots. Students dump
counters onto an empty plate and compare the number of counters to the
number of dots on the dot plate.
2. Double Counter Match (Materials: dot plates, paper plates, variety of counters)
Place two empty plates, one on either side of a dot plate or card. Students
make equivalent sets in each plate using a different type of counter. Students
describe how all three plates compare.
3. Make the Pattern (Materials: dot cards, numeral cards, paper plates, counters)
Hold up a dot card and have your students make the same pattern they see on
their own plate using counters. Ask them how many dots they see and how they
see them. To extend, place two empty plates down. Place a dot card in the
center. Students build a set that is one less on one plate and one more on the
other. Do the same activity by holding up numeral cards.

6. 4. Dot Card Flash (Materials: dot cards, hole-punched cards, bingo chips, overhead)
Flash a dot card then hide, or briefly display on an overhead a hole-punched
card, an overhead dot card or bingo chips. Students state the number, hold up
a dot card or numeral, or construct the arrangement.
5. Dot Card Match (Materials: dot cards)
Students sort different arrangements of the same number. Discuss the
number of dots in each group; which group has the most, least, etc.
6. Number/Numeral Match (Materials: dot cards)
Teacher states a number or holds up a numeral card and students find the
corresponding dot card.
7. Which One is Out? (Materials: dot cards)
Students determine which card does not belong in a set where all but one
represent the same number.
8. Dot Card Trains (Materials: dot cards)
Students arrange a random set of dot cards in order (from 1-6 and back down).
Extend to trains from 1-10.
9. Concentration (Materials: 2 sets of dot cards or plates)
Place dot cards face down in a 5x4 array. Students take turns turning over two
cards trying to find a match.

7. 10. Dot Card Challenge (Materials: 2 sets of dot cards in 2 colours)
Each student gets 1 set of cards. Each student turns over the top card of
their pile and identifies the amount. The student with the larger number takes
both cards.
11. Addarama (Materials: 2 sets of dot cards in 2 colors)
Each student gets 1 set of cards. Each student turns over their top card. Both
students add the two dot cards together. The first student to say the total
amount out loud gets both cards. To extend, have each student turn over two
cards and find the total of their cards. The student with the greatest amount
takes all the cards.
12. Finger Dot Match (Materials: dot cards, numeral cards, finger cards)
Teacher holds up fingers (i.e. 2) and asks students how many fingers.
Students imitate and state number. Students then find a dot card with that
many dots. Teacher then holds up 2 fingers and one more. Students imitate
and state number. Students find a plate with 3 dots. Continue with other
finger patterns to 10.
13. Clothespin Match (Materials: dot plates or cards, clothespins)
Students choose a dot plate and attach the corresponding number of
clothespins on the edge of the card.
14. Popsicle Stick Match (Materials: popsicle sticks, dot cards)
Students match popsicle sticks with different dot arrangements on dot cards.
15. Dice/Card Match (Materials: dice, dot cards)
Roll the die or dice and have students find a dot card with the same amount.

8. Five Frame and Ten Frame Activities
Introduction
Five frames and ten frames are one of the most important models to help students
anchor to 5 and 10.
Five frames are a 1x5 array and ten frames are a 2x5 array in which counters or dots
can be placed to illustrate numbers.
The five frame helps students learn the combinations that make 5. The ten-frame
helps students learn the combinations that make 10. Ten-frames immediately model
all of the facts from 5+1 to 5+5 and the respective turnarounds. Even 5+6, 5+7 and
5+8 are quickly seen as two fives and some more when depicted with these powerful
models.
For students in kindergarten or early first grade who have not yet explored a ten
frame, a good idea is to anchor to five by beginning with a five frame.

9. Starting with Five Frames
Activities:
1. Building Sets (Materials: blank five frame mat, counters)
Call out a number to the students, such as 4, and have them show that amount on their mat. They may place the counters in any manner. Ask if they can place the 4 counters down in a different way. Try other numbers from 0-5. Have your students make observations about their placement of counters.
- It has a space in the middle.
-It’s two and two.
Numbers greater than 5 are shown with a full five-frame and additional
counters on the mat but not on the frame.

10. 2. Roll and Build (Materials: five frame cards, dice)
Students roll one die or two dice and build that amount on their five frame
mat.
3. Memory (Materials: two sets of five frame cards)
Place the five frame cards face down in an array. Students take turns turning
over two cards. They identify the amount on each card. If they are the same
they take both cards. Play goes to the next players.
4. Challenge (Materials: two sets of five frame cards in 2 colours)
Each student gets 1 set of cards. Each student turns over the top card of
their pile and identifies the amount. The student with the greater amount
takes both cards.
5. Five Frame Flash (Materials: large five frame cards)
Flash a five frame card to your students and ask them to identify how many
dots they saw. To challenge students ask them to identify one more or one less
than the amount of dots. To extend, have them tell you how many empty spaces
there are or how many more are needed to make 5.

11. 6. Five Frame Trains (Materials: at least two sets of five frame cards)
Students sequence a random set of five frame cards in order from 1 to 5 and
then back to 1, etc. Students practice counting forwards and backwards out
loud. Extend by turning over one card in the train and have students identify
which number was turned over.
7. Make 5 (Materials: two sets of five frame cards)
Place the cards face up in an array. Students try to find two cards that
together total 5. To challenge students turn the cards face down.
8. Dice Match (Materials: die, five frame cards)
Roll the die and have students find the five frame card that has the same
amount. If they roll a 6, they roll again.

12. Starting with Ten Frames
1. Building Sets (Materials: blank ten frame mats, double ten frame mats, counters)
Call out a number from 1-10 and have students build that amount on their ten
frame. Students fill the first row first. Call out a different number and have
students build the new number. Observe to see which students can simply add
or remove counters and those that must begin from 1. Continue with different
amounts. Extend to a double ten frame building numbers to 20.
2. Roll and Build (Materials: ten frame cards, dice)
Students roll two dice and build that amount on their ten frame mat.
3. Memory (Materials: two sets of ten frame cards)
Place the ten frame cards face down in an array. Students take turns turning
over two cards. They identify the amount on each card. If they are the same
they take both cards.
4. Challenge (Materials: two sets of ten frame cards in 2 colours)
Each student gets 1 set of cards. Each student turns over the top card of
their pile and identifies the amount. The student with the greater amount
takes both cards.

13. 5. Ten Frame Flash (Materials: large ten frame cards)
Flash a ten frame card to your students and ask them to identify how many
dots they saw. To challenge students ask them to identify one more or one
less than the amount of dots. To extend, have them tell you how many empty
spaces there are or how many more are needed to make 10.
6. Ten Frame Trains (Materials: at least two sets of ten frame cards)
Students sequence a random set of ten frame cards in order from 1 to 10 and
then back to 1, etc. Students practice counting forwards and backwards out
loud. Extend by turning over one card in the train and having students identify
which number was turned over.
7. Make 10 (Materials: two sets of ten frame cards)
Place the cards face up in an array. Students try to find two cards that
together total 10. To challenge students turn the cards face down.
8. Dice Match (Materials: dice and ten frame cards)
Roll the dice and have students find the ten frame card that has the same
amount. If they roll 11 or 12, they roll again.

14. Making a difference with Subtraction

15. INTRODUCTION
What is subtraction?
According to Van de Walle & Lovin, subtraction is not “take away”. Referring to
subtraction as “take away” is not entirely correct and is way overused. When children
learn this definition of subtraction, they have difficulty later when the structure of
the problem is other than take away (Separate, Part-Part-Whole, Compare).
Consider this definition of subtraction: In a part-part whole model, when the
whole and one of the parts are known, subtraction names the other part.
If you start with a set of 9 and remove a set of 4, the two sets that you know are the
sets of 9 and 4. The expression 9 – 4 names the five left. So, nine minus four is five.
Subtraction (and division) cause children greater difficulty than addition (or
multiplication).
Here is the BIG QUESTION: How do you get a child to fact mastery?
Keep in mind that mastery of a basic fact means that a child can give a quick
response within about three seconds without having to count (non-efficient).
Mastery has very little to do with how much drill or with drill techniques.
But, is there a place for drill and practice? According to Van de Walle & Lovin, drill is
appropriate when:
(a) the subtraction concepts have been meaningfully developed
(b) students already have developed (not mastered) flexible and useful procedures
(c) speed and accuracy are needed
Reminder: Drill is not a way of developing ideas.

16. Now for the ANSWER to the BIG QUESTION of how to get a child to fact mastery.
According to Van de Walle, there are three steps:
1) Develop a strong understanding of number relationships and operations.
Conceptual understanding is the comprehension of mathematical concepts,
operations, and relations. Students with conceptual understanding know more than
isolated facts and methods.
2) Develop efficient strategies for students to recall facts through practice.
Children need to construct efficient tools to master basic facts.
3) Provide drill in the use and selection of those strategies once they have been
developed. 3
- Are we guilty of getting the “cart ahead of the horse”, so to speak when it comes to
pushing for mastery by drilling, drilling, drilling without the conceptual understanding
and efficient strategies in place?
- Fact mastery can happen by the end of grade 3 if appropriate development takes
place in the primary grades.
- This session will focus on the three steps necessary in helping children develop fact
mastery:
1) an understanding of number relationships and operations
2) efficient strategies
3) drill

17. II. NUMBER SENSE: NUMBER RELATIONSHIPS/OPERATIONS/PROPERTIES
A. Early Counting and Number
B. Early Number Sense
C. Relationships Among Numbers 1 – 10
D. Relationships for Numbers 10 – 20
E. Numbers to 100: Early Introduction
F. Numbers Sense and the Real World
G. Operation Concepts
H. Properties
Number relationships play a significant role in fact mastery.
For example, knowing how numbers are related to 5 & 10 helps students master
facts such as (think of a ten-frame) and ( since 9 is 1 away from 10,
take 1 from 6 to make = 15).
Children must understand numbers in terms of part-whole relationships
(important conceptual achievement for children) to decompose (break-apart) wholes
and recombine parts.
It’s important to develop early number concepts and number sense.

18. 5 A. EARLY COUNTING AND NUMBER
1) The Relationships of More, Less, and Same
- Children should construct sets of counters as well as make comparisons choices between two given sets.
- ACTIVITY: Make Sets of More/Less/Same Van de Walle & Lovin (2006)
Make about eight cards with sets of 4 to 12 objects. Give students a set of small counters and some word cards
labeled More, Less, and Same. Next to each card have students make three collections of counters: a set that is
more, one that is less, and one that is the same. Have labels prepared for the sets.
2) Early Counting
- The meaning attached to counting is the key conceptual idea on which all other number concepts are developed.
Van de Walle & Lovin (2006)
- Children must understand the cardinality principle - understanding that the last count word indicates the amount
of the set. Before developing cardinality, children may count again. Fosnot and Dolk (2001)
- To develop their understanding of counting, involve children in game activities that involve counts and comparisons.
3) Counting On and Counting Back
- Many children are better at counting on than counting backward. One reason is that children have had more practice in
counting on. However, it is very important to have children practice counting backward.
- Do frequent short practice drills. For example, when children are lining up for lunch, recess, etc. You can count 1-5 & 5-1; 1-10
& 10-1; 1-20 & 20-1 depending on the level of student. Clapping can be done as the count goes up and back.
- ACTIVITY: Real Counting On Van de Walle & Lovin (2006)
This “game” for two children requires a deck of cards with numbers 1-7,a die, a paper cup, and some counters. The first
player turns over the top number card and places that number of counters in the cup. The card is placed next to the cup as a
reminder of how many are there. The second child rolls the die and places counters NEXT to the cup. Working together they
decide how many counters in all. A record sheet with columns for “In the Cup”, “On the Side”, and “In All” is an option. The
largest number in the card deck can be changed if needed. 5

19. B. EARLY NUMBER SENSE
- Early number sense development should be given much attention.
- Remember, first comes the concept of cardinality and counting skills.
Then more relationships can be created for children to develop number sense.
C. RELATIONSHIPS AMONG NUMBERS 1 THROUGH 10
- There are four different types of relationships that children can and should
develop with numbers:
1) Spatial relationships
2) One and two more, one and two less
3) Anchors or “benchmarks” of 5 and 10
4) Part-part-whole relationships
- At first you will notice a lot of counting, but counting will become less and
less.
- A closer look at the four relationships:
1) Spatial Relationships: Patterned Set Recognition
- Recognize sets of objects in patterned arrangements; tell how many without
counting (subitizing); significant milestone for a child.
- Quantities up to 10 can be known without having to count. This can help
in counting on (from a patterned set you know) or learning combinations
of numbers (seeing two smaller patterns you know) or being able to
decompose quantities into small groups (e.g., 5 can be thought of as a
group of 2 and a group of 3).
- ACTIVITY: Dot Plate Flash Van de Walle & Lovin (2006)
(Dot plates are a good set of materials to use in pattern recognition
activities.) Hold up a dot plate for only 1 to 3 seconds. “How many?
How did you see it?” A set of dot-pattern dominoes will work also.
ATTENTION: DOT PATTERNS FOR DOT PLATES - Go to:

20. 2) One and Two More, One and Two Less
- It is important for children to learn one and two more and one and two less.
- ACTIVITY: One-Less-Than Dominoes
Act. 2.10, p44 Van de Walle & Lovin (2006)
Use the dot-pattern dominoes or a standard set to play “one-less-than”
dominoes. Play in the usual way, but instead of matching ends, a new domino
can be added if it has an end that is one less than the end on the board.
A similar game can be played for two less, one more, or two more.
3) Anchoring Numbers to 5 and 10
- Children need to be able to relate a given number to 5 & 10.
- This is useful in thinking about different combinations of numbers.
- The most common and maybe the most important model for this relationship
is the ten-frame. This helps students organize things in terms of 5 and 10.
- Ten-frame: Always fill the top row first, starting on the left.
Then the bottom row can be filled.
- For young children (K & early 1), start with a five-frame.
Example: Show children a five-frame with 3 dots.
Ask, “How many dots do you see?”
Ask, “How many more to make five?”
- ACTIVITY: Ten-Frame Flash Cards Van de Walle & Lovin (2006)
Note: This is a variation of ten-frames that will provide drill to reinforce the
5 & 10 anchors. Flash ten-frame cards to the class or group, and see how fast
children can tell how many dots are shown. This is a fast-paced activity, takes
only a few minutes, can be done any time, and is a lot of fun if you encourage
speed.
IMPORTANT VARIATIONS:
1) Saying the number of spaces on the card instead of the no. of dots.
2) Saying one more than the number of dots (or two more, and also less
than).
3) Saying the “ten fact” – for example, “Six and four make ten”.
TEN-FRAME CARDS:
Each card contains a ten-frame and dots drawn in the frames.
A set of 20 cards consists of a 0 card, a 10 card, and two each of the
numbers 1 to 9.

21. ATTENTION:
To download full-sized copies of Blackline Masters that are referenced throughout the Van de Walle & Lovin (2006) book: Teaching Student-Centered Mathematics, K-3, go to:
For a large set of Ten-Frame cards, view the Winnipeg School Division Numeracy Project:
4) Part-Part-Whole Relationships (focus is on combinations)
- The ability to think about a number in terms of parts is a major milestone
in the development of number.
- Of the number relationships that were just mentioned – spatial
relationships, one and two more/one and two less, anchors or “benchmarks”
of 5 and 10, part-whole ideas are the most important.
- There is a clear connection between part-part-whole relationships and
addition/subtraction ideas.
- ACTIVITY: Build It in Parts Van de Walle & Lovin (2006)
NOTE: Children are focusing on the combinations.
Provide children with connecting cubes. The task is to see how many
different combinations for a particular number they can make using two
parts. Make each bar with two colors. Keep the colors together.
- MISSING-PART ACTIVITIES - a special and important variation of
part-part-whole activities. They provide the greatest way to focus on the
combinations for a number. They are very important in preparing children
for subtraction facts. With a whole of 9 but with only 4 showing, the child can
later learn to write 9 – 4 = 5.
- ACTIVITY: Covered Parts Van de Walle & Lovin (2006)
(Similar to a Math Recovery activity)
A set of counters is counted out, and the rest are put aside. One child places
the counters under a margarine tub (bowl) or a foam cover. The child then
pulls some out into view. For example, if 6 is the whole and 4 are showing, the
other child says, “Four and two is six”. If the hidden part is unknown, that
part is immediately shown.

22. - ACTIVITY: Missing-Part Cards Van de Walle & Lovin (2006)
For each number 4 to 10, make missing-part cards on strips of 3-by-9-inch
tagboard (sentence strips will work too). Each card has a numeral for the
whole and two dot sets with one set covered by a flap. For the number 8, you
need nine cards with the visible part ranging from 0-8 dots. Students use the
cards as in “Covered Parts” saying, “Four and two is six” for a card showing
four dots and hiding two.
- ACTIVITY: I Wish I Had Van de Walle & Lovin (2006)
Hold out a bar of connecting cubes. Say, “I wish I had six”. The children
respond with the part that is needed to make 6. Counting-on can be used to
check. The game can focus on a single whole, or the “I wish I had” number can
change each time.
- Missing Subtrahend Task (Math Recovery)
Example: Briefly display and then screen (cover) 8 red counters.
Say, “Here are 8 red counters.” Then ask the child to look away while 3
counters are removed. Say, “There were 8 red counters and I removed some.
Now there are only 5. How many did I remove?” Note: If a child has difficulty
with totally screened (covered) collections, use partial screening.
- Research says that when kindergarten and first-grade children regularly
solve problems, they learn addition and subtraction facts.
- Note: More dot card activities can be found on p53 in Teaching Student-
Centered Mathematics, K-3, Van de Walle & Lovin (2006).

23. D. RELATIONSHIPS FOR NUMBERS 10 TO 20
1) A Pre-Place-Value Relationship with 10
- A set of ten should play a major role in children’s initial understanding of
numbers between 10 and 20. When children see a set of four with a set of
ten, they should know without counting that the total is 14.
2) Extending More and Less Relationships
3) Double and Near-Double Relationships
- Doubles and near-doubles are simply special cases of the general part-part-
whole construct. Double 6 is 12; Near doubles – 13 is double 6 and 1 more.
- Relate the doubles to special images. Thornton (1982) helped first graders
connect doubles to these visual ideas:
Double 3 is the bug double: three legs on each side.
Double 4 is the spider double: four legs on each side.
Double 5 is the hand double: two hands.
Double 6 is the egg carton double: two rows of six eggs.
Double 7 is the two-week double: two weeks on the calendar.
Double 8 is the crayon double: two rows of eight crayons in a box.
Double 9 is the 18-wheeler double: two sides, nine wheels on each side.
- This is usually considered a strategy for memorizing basic addition facts,
but it is good to develop these relationships before they are concerned with
memorization of facts.
E. NUMBERS TO 100: EARLY INTRODUCTION
- Most important at K-1 is for students to become familiar with the counting
patterns to 100. A hundreds chart is a must for every primary classroom.
- TEACHERS IN GRADE 3 WITH STUDENTS WHO HAVE NOT MASTERED
ADDITION AND SUBTRACTION FACTS NEED TO LOOK CAREFULLY AT
STUDENT UNDERSTANDING OF NUMBER RELATIONSHIPS. WITHOUT
THESE RELATIONSHIPS, THE STRATEGIES WILL BE DIFFICULT.
- At this point more drill will not work; students need to work on number
relationships.

24. - Technology – KCM Website www.kymath.org
Click on INTERVENTION; click on RESOURCES, scroll down to
“COUNT ME IN TOO”; click on hundreds board and/or other activities.
- ACTIVITY: Missing Numbers Van de Walle & Lovin (2006)
Give students a hundreds chart on which some of the number cards have been
removed or covered. Their task is to replace the missing numbers in the chart.
Start with only a random selection of individual numbers removed. Later, remove
sequences of several numbers from three or four different rows. Finally, remove
all but one or two rows or columns. Eventually, have children replace all of the
numbers in blank chart.
F. NUMBER SENSE AND THE REAL WORLD
- Relationships of numbers to real-world quantities and measures and the use of
numbers in simple estimations can help children develop the flexible, intuitive
ideas about numbers that are most desired. Van de Walle & Lovin (2006)
- Estimation & Measurement – A good way for children to think about real
quantities is to connect numbers with measures of things.
- Young children have a hard time understanding the word “about”.
G. OPERATION CONCEPTS
- Relationship between Addition and Subtraction – Inverse Operations
- Addition names the whole in terms of the parts and subtraction names a
missing part.
- Understanding inverse operation will make subtraction facts much easier to
learn.
- Before working on mastery of subtraction facts, check on students’ mastery
of addition facts and see if they are starting to make connections between the
two operations.

25. - Avoid key words! Van de Walle & Lovin (2006)
Three arguments against the key word approach:
1) Key words are misleading. They may suggest an operation that is incorrect.
EXAMPLE: Maxine took the 28 stickers she no longer wanted and
gave them to Brittany. Now Maxine has 73 stickers left.
How many stickers did Maxine have to begin with?
2) Many problems have no key words. If children have been taught to look
for key words, they have no strategy for solving the problem. Looking for
key words encourages students to ignore the meaning and structure of the
problem. (The most important approach to solving any word/story problem
is to analyze its structure – to make sense of it.) Mathematics is about
reasoning and making sense.
3) Evidence suggests that the general meaning of a problem rather than
specific words or phrases determines both the difficulty of the problem
and the processes students use to solve it. Normally, difficulty with word
problems do not happen because students cannot read the words but
because they cannot make sense of the mathematical relationships
expressed by those words.
- Using Model-Based Problems (counters, drawings, number lines)
1) Addition
- In part-part-whole activities both of the parts should be distinct even
after the parts are joined. FOR CHILDREN TO SEE THE
RELATIONSHIP BETWEEN THE TWO PARTS AND THE WHOLE, THE
IMAGE OF THE PARTS (5 and 3) MUST BE KEPT AS TWO SEPARATE
SETS. EXAMPLE: Join or separate 2 bars of connecting cubes.
A number line represents some real conceptual difficulties for first and
second graders. Its use as a model at that level is generally NOT
recommended. A number line measures distances from zero the same way
a ruler does. In the early grades, children focus on the dots or numerals
on a number line instead of the spaces. However, if arrows (hops) are
drawn for each number in an exercise, the length concept is more clearly
illustrated. To model the part-part-whole concept of 5 + 3, start by
drawing an arrow from 0 to 5, indicating, “This much is five”. Do NOT point
to the dot for 5, saying “This is five.” Van de Walle & Lovin (2006)

26. 2) Subtraction
- In a part-part-whole model, when the whole and one of the parts are
known, subtraction names the other part.
ACTIVITY: Missing-Part Subtraction Van de Walle & Lovin (2006)
A number of counters is placed on a mat. One child separates the counters
into two parts while the other child hides his or her eyes. The first child
covers one of the two parts showing only the other part. The second child
says the subtraction sentence. For example, “Nine minus four (the visible
part) is five (the covered part).” The covered part can be shown if
necessary for the child to say how many are there. Both the subtraction
equation and the addition equation can be written.
H. PROPERTIES
1) Addition & Subtraction
- Commutative Property – order property
EXAMPLE: Use story problems that have the same addends but different
orders. The context should be the same.
- The commutative property for addition reduces the number of facts from
100 to 55.

27. III. EFFICIENT STRATEGIES
- What is an efficient strategy? It is one that can be done mentally and quickly.
- Counting is not efficient. If drill is done when counting is the only strategy used, all
you get is faster counting.
A. STRATEGIES FOR ADDITION FACTS:
When working on addition facts, be sure to include number relationship
activities.
1) One-More-Than and Two-More-Than Facts
- 36 facts with 1 and 2
ACTIVITY: One-/Two-More-Than Dice Van de Walle & Lovin (2006)
Label a die +1, +2, +1, +2, “one more”, and “two more”. Label another die
4, 5, 6, 7, 8, and 9. After each roll of the dice, children say the complete
fact: “Four and two is six.”
2) Facts with Zero
- 19 facts with zero
- Use word problems that involve zero.
- In facts with zero, put zero first in some facts and second in some facts.
0 + 3 = 3 and = 3
Can use counters and a part-part-whole mat to model the facts.
3) Doubles
- 10 doubles facts
- pretty easy to learn
- a very good way to learn the near-doubles (addends one apart)
- used as anchors for other facts

28. 4) Near Doubles
- also called the “doubles-plus-one” facts; include all combinations where
one addend is one more than the other
- strategy - double the smaller numbers and add 1
- STUDENTS MUST KNOW DOUBLES BEFORE YOU FOCUS ON THIS ONE.
- An idea for introducing the strategy:
Write some near-doubles facts on the board vertically and horizontally
varying which addend is smallest. After students work the problems,
discuss their ideas for efficient methods. If no one used a near-double
strategy, model the “near doubles” thinking. 14
ACTIVITY: Double Dice Plus One Van de Walle & Lovin (2006)
Roll a single die with numerals or dot sets and say the complete double-plus-
one fact. That is, for 7, students should say, “Seven plus eight is fifteen.”
- Figure Near-doubles facts Van de Walle & Lovin (2006)
Prepare a doubles sheet/board and have near-doubles mini-cards cut
apart. Put the near-doubles mini-cards on the double fact that helps.
5) Make-Ten Facts
- 30 facts have at least one addend of 8 or 9
- One strategy - build onto the 8 or 9 up to 10 and then add on the rest.
For 6 + 8, start with 8, then add 2 more to make 10; that leaves four which
makes 14.
- Before using this strategy, children need to be able to think of the numbers
11 to 18 as 10 and some more.
- ACTIVITY: Make 10 on the Ten-Frame Van de Walle & Lovin (2006)
Use a mat with two ten-frames. Either use flash cards next to the ten-
frames or give a fact orally. The students should model each number in the
two ten-frames first and then determine the easiest way to show (without
counting) the total. One way to model the number is to place counters on the
frame showing either 8 or 9. Have students explain what they did. Focus on
the idea that 1 (or 2) can be taken from the other number and put with the 9
(or 8) to make 10. Then you have 10 and whatever is left.
Materials needed: little ten-frame cards

29. Other Strategies and the Last Six Facts
It would be very helpful to spend some time with word problems where these
facts are the addends before trying to develop any strategies. Pay attention to
the ideas that students use in figuring out the answers.
6) Doubles Plus Two, or Two-Apart Facts
- 3 facts have addends that differ by 2:
- Two possible relationships that might be helpful:
a. Move from the idea of the near-doubles to double plus 2. For example,
4 + 6 is double 4 and 2 more.
b. Take 1 from the larger addend and give it to the smaller one. The 5 + 3
fact is changed into the double 4 fact – double the number in between. 15
7) Make-Ten Extended
- Three facts have 7 as one of the addends. The make-ten strategy can be
used for these facts, too.
- For 7 + 5, the idea is 7 and 3 more makes 10 and 2 left makes 12.
It may be a good idea to suggest this idea at the same time that you
introduce the make-ten strategy.
- Other countries teach an addition strategy of building through 10 in the
first grade. Many second graders in the U.S. do not know what 10 plus any
number is.
8) Counting On
-Efficient up to 10
- Most widely promoted strategy
- Reasons not to encourage this strategy:
1) Many times it is used with facts where it is not efficient, such as
2) It is a lot more procedural that conceptual.
3) It isn’t necessary if students use other strategies.

30. 9) Ten-Frame Facts
- The ten-frame model is valuable in seeing certain number relationships.
- The ten-frame helps children learn the combinations that make 10.
- Ten-frames immediately model all of the facts from (1 + 5) to
- Even 5 + 6, 5 + 7, and can be seen as two fives and some more
when these models are used.
C. STRATEGIES FOR SUBTRACTION FACTS:
- Subtraction facts prove to be more difficult that addition.
- The “count-count-count” approach has not been found to be helpful in subtraction
fact mastery.
- Repeat use of a mental strategy will make it automatic.
1) Subtraction as Think-Addition
- This is extremely significant for mastering subtraction facts, but addition
facts must be mastered first.
- If the important relationship between parts and wholes - between addition
and subtraction - can be made, subtraction facts will be much easier.
- Word problems that promote think-addition are those that sound like addition
but have a missing part. 16
- CONNECTING SUBTRACTION TO ADDITION KNOWLEDGE:
Van de Walle & Lovin (2006)
Count out 13 and cover. Count and remove 5. Keep these in view.
Think: “Five and what makes thirteen?” left minus 5 is 8.
Uncover. 8 and 5 is 13.
Part-Whole Models Van de Walle & Lovin (2006)
Join or separate using:
1) two colors, 2) a part-part-whole mat, 3) two bars of connecting cubes,
4) a design with squares, or 5) two hops on a number line (note the whole hop).

31. 2) Subtraction Facts with Sums to 10
- Part-Whole Relationships – For example, 10 is the whole and the numbers
making up the parts are: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5.
- Learn addition facts first
- Understand important basic number relationships
3) The 36 “Hard” Subtraction Facts: Sums Greater Than 10
- Build Up Through – 9
Start with 9 and work up through 10: 9 and 1 more is 10, and 4 more
makes 5. This group includes all facts where the part or subtracted number is
either 8 or 9. Examples are 13 – 9 and 15 – 8.
ACTIVITY: Build Up Through the Ten-Frame Van de Walle & Lovin (2006)
Draw a ten-frame with 9 dots. Talk about how you can build numbers between
11 and 18, starting with 9 in the ten-frame. Stress the idea of one more to get
to 10 and then the rest of the number. Repeat for a ten-frame showing 8.
Next, with either the 8 or 9 ten-frame in sight, call out numbers from 11 to 18,
and have students explain how they can figure out the difference between that
number and the one on the ten-frame. Later, use the same approach but show
fact cards to connect this idea with the symbolic subtraction fact.
- Back Down Through – 6
10 is used by working backward from 15 – a take away process: Take away 5 to
get 10, and 1 more leaves 9. This strategy is really take-away and not think-
addition. It is useful for facts where the ones digit of the whole is close to the
number being subtracted. For 14 – 6, just take off 4 and then take off 2 more
to get 8. This is working backward with 10 as a “bridge”.

32. - Think – Addition – 6
It is quite common to hear “double 6”, a think-addition approach.
Think-addition is one of the most powerful ways to think about
subtraction facts.
- Extend Think-Addition
It is important to listen to children’s thinking as they attempt to answer
subtraction facts that they have not mastered. If they are not using
Build Up Through 10, Back Down Through 10, or Think-Addition, they are
probably using the inefficient method of counting.
Activities for the Think-Addition strategy can be used for ALL of the
subtraction facts, not only the “hard facts”:
1. Missing-Number Cards Van de Walle & Lovin (2006)
Three numbers shown with one circled or with one number missing.
Why do these numbers belong together? Why is one circled?
Which number is missing? How can you tell what it is?
2. Missing-Number Worksheets Van de Walle & Lovin (2006)
Facts with one number missing; Use for drill exercises.
4) Doubling or Doubles Facts (see Math Matters book – p66)
- Doubles and near doubles are simply special cases of the general part-part-
whole construct.
Example: 13 – 7 as = 14, so 6 because 13 is 1 less than 14.

33. To achieve fact mastery:
IV. EFFECTIVE DRILL/STRATEGY SELECTION
A. Difference between drill and practice
1) Drill – repetitive non-problem-based activity.
2) Practice – problem-based activities where students develop/use strategies
that are meaningful.
B. Avoid Premature Drill
1) Do not introduce drill too soon. If a child is not ready for drill, they will use
inefficient methods, such as counting.
C. Practice Strategy Selection (deciding on an appropriate strategy)
1) Prepare a list of facts selected from two or more strategies and then, one fact
at a time, ask children to name a strategy that would work for that fact.
Have them explain why they chose the strategy and show its use.
D. Approach to instruction from development to drill
1) Make strategies explicit in the classroom – list new strategies in the classroom
as students develop them (as they solve word problems) and give the strategies
names.
2) Drill Established Strategies – Use flash cards: make several sets using all of
the facts that fit that strategy; do other activities/games
3) Individualize – listen to your students
4) Practice Strategy Selection – do strategy selection drills.
V. WRAPPING IT UP
To achieve fact mastery:
1) improve students’ conceptual understanding of subtraction by focusing on number
relationships and operations
2) help students develop efficient strategies
3) provide effective drill

34. WEBSITES
Dot Cards for “Dot Plates” Activity in Teaching Student-Centered Mathematics, K-3
Full-sized Blackline Masters for Teaching Student-Centered Mathematics, K-3
Winnipeg School Division, Numeracy Project
Dot Card Activities; Five-Frame and Ten-Frame Activities
pdf

35. Five-Frame/Ten-Frame Activities
Introduction
Five frames and ten frames are one of the most important models to help students anchor to 5 and 10.
Five frames are a 1x5 array and ten frames are a 2x5 array in which counter or dots can be placed to illustrate numbers.
The five frame helps students learn the combinations that make 5. The ten-frame helps students learn the combinations that make 10. Ten-frames immediately model all of the facts from 5+1 to 5+5 and the respective turnarounds. Even 5+6, 5+7, and 5+8 are quickly seen as two fives and some more when depicted with these powerful models.
For students in kindergarten or early first grade who have not yet explored a ten frame,
a good idea is to anchor to five by beginning with a five frame.
Starting with Five-Frames
Activities:
1. Building Sets (Materials: blank five-frame mat, counters)
Call out a number to the students, such as 4, and have them show that amount on
their mat. They may place the counters in any manner. Ask if they can place the 4
counters down in a different way. Try other numbers from 0-5. Have your students
make observations about their placement of counters.
- It has a space in the middle.
- It’s two and two.
Numbers greater that 5 are shown with a full five-frame and additional counters on
the mat but not on the frame.
2. Roll and Build (Materials: five-frame cards, dice)
Students roll one die or two dice and build that amount on their five frame mat .
3. Memory (Materials: two sets of five-frame cards)
Place the five frame cards face down in an array. Students take turns turning over two
cards. They identify the amount on each card. If they are the same they take both
cards. Play goes to the next players.
4. Challenge (Materials: two sets of five-frame cards in 2 colors)
Each student gets 1 set of cards. Each student turns over the top card of their pile
and identifies the amount. The student with the greater amount takes both cards.

36. 5. Five Frame Flash (Materials: large five-frame cards)
Flash a five frame card to your students and ask them to identify how many dots they
saw. To challenge students ask them to identify one more or one less that the amount
of dots. To extend, have them tell you how many empty spaces there are or how many
more are needed to make 5.
6. Five Frame Trains (Materials: at least two sets of five-frame cards)
Students sequence a random set of five frame cards in order from 1 to 5 and then
back to 1, etc. Students practice counting forwards and backwards out loud. Extend
by turning over one card in the train and have students identify which number was
turned over.
7. Make 5 (Materials: two sets of five-frame cards)
Place the cards face up in an array. Students try to find two cards that together total
5. To challenge students, turn the cards face down.
8. Dice Match (Materials: die, five-frame cards)
Roll the die and have students find the five-frame card that has the same amount.
If they roll a 6, they roll again.

37. Starting with Ten-Frames
Activities:
1. Building Sets (Materials: blank ten-frame mates, double ten-frame mats, counters)
Call out a number from 1-10 and have students build that amount on their ten frame.
Students fill the first row first. Call out a different number and have students build
the new number. Observe to see which students can simply add or remove counters
and those that must begin from 1. Continue with different amounts. Extend to a
double ten-frame building numbers to 20.
2. Roll and Build (Materials: ten-frame cards, dice)
Students roll two dice and build that amount on their ten-frame mat.
3. Memory (Materials: two sets of ten frame cards)
Place the ten-frame cards face down in an array. Students take turns turning over
two cards. They identify the amount on each card. If they are the same, they take
both cards.
4. Challenge (Materials: two sets of ten-frame cards in 2 colors)
Each student gets 1 set of cards. Each student turns over the top card of their pile
and identifies the amount. The student with the greater amount takes both cards.
5. Ten-Frame Flash (Materials: large ten-frame cards)
Flash a ten-frame card to your students and ask them to identify how many dots they
saw. To challenge students, ask them to identify one more or one less that the amount
of dots. To extend, have them tell you how many empty spaces there are or how many
more are needed to make 10.
Winnipeg School Division Numeracy Project

38. 6. Ten-Frame Trains (Materials: at least two sets of ten-frame cards)
Students sequence a random set of ten frame cards in order from 1 to 10 and then
back to 1, etc. Students practice counting forwards and backwards. Extend by
turning over one card in the train and having students identify which number was
turned over.
7. Make 10 (Materials: two sets of ten-frame cards)
Place the cards face up in an array. Students try to find two cards that together
total 10. To challenge students, turn the cards face down.
8. Dice Match (Materials: dice and ten-frame cards)
Roll the dice and have students find the ten-frame card that has the same amount. If
they roll 11 or 12, they roll again.
9. What’s the Difference (Materials: at least three sets of ten-frames, 50 counters)
5 cards are spread out face down and the rest are placed in a pile face down. The
students take turns turning over the top card in the pile as well as one of the spread
cards. They then determine the difference between the two cards and take that
amount of counters. The card that was turned over from the spread pile is flipped
over again. Play continues until all of the cards from the pile have been used. The
player with the most counters wins. Observe to see what strategies students are
using to find the difference and to get the most counters.

39. 11. Fish (Materials: at least two sets of ten-frames)
10. Ten-Frame Difference Challenge (Materials: two sets of ten-frame cards in 2
colors, 50 counters)
Students play the game like the traditional “War” game. Each student turns over the
top card from their pile. Each identifies their amount. The student with the largest
takes as many counters from the pile as the difference between the two cards. Play
continues until all the counters are gone. The winner is the player with the most
counters.
11. Fish (Materials: at least two sets of ten-frames)
Students play in groups of 2 to 4. Deal each player 4 cards. Spread the rest in the
center like a fish pond. Students take turns asking another if they have a card with
an amount that is the same as one of their cards. If they have the card, they give it
to the player. If they do not, they draw a card from the pile. Play continues until one
player gets rid of all their cards, or all the cards are matched.
Fish (Materials: ten-frame cards)
Play the game like “Fish”. The object of the game is to ask the other students for a
card that will add to yours to make a sum of ten.
13. Ten Plus/Nine Plus (Materials: 2 sets of ten-frames in 2 colors)
Place one ten-frame card face up in the center. Place the other cards in a pile face
down. Students take turns turning over the top card and adding it to “10”. Play the
game “Nine Plus” like “Ten Plus” only use the nine card as the card to add the other
numbers to.
Numeral/Ten-Frame Match (Materials: two sets of ten-frame cards, 0-20
numeral cards)
Spread the 0-20 numeral cards face up in order. Students take turns turning over
two ten-frame cards and finding the total. If the numeral card match is there, they
take the card. The winner is the player with the most numeral cards.