Beyond Counting by Ones
Mathematical Activities for
Developing Number Sense and Reasoning
in Young Children
Dr. DeAnn Huinker
University of Wisconsin-Milwaukee
February 2000
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Thinking in Groups
Seeing twoness and threeness
Hide three pennies. Uncover the pennies and ask, “How many pennies are there?” Watch to see if the
child counts by ones, or sees the group as a whole. Repeat with two pennies. Encourage a quick
response. The goal is to tell how many in a one-second glance.
Making twos and threes
The objective is to see and make twos, not to see how many twos. Give each child 10 counters. Have
them break the set into parts by making twos. Then put the whole set of objects back together and repeat
until the twos are made quickly. Next start with 12 counters and make sets of three. Initially it may take
a minute, but with practice children can accomplish these tasks in almost 10 seconds.
Dot Cubes
Dot cubes with only one, two, or three dots on the sides are useful for young children. You can draw
dots on blank wooden cubes or cover regular dot cubes with stick on labels. Once children are able to
subitize these quantities then begin using dot cubes with one through six dots on the sides. As children
roll the cubes, often ask, “How many dots are there?” and “How did you see them?”
Two children can play, “Who has more dots?” or “Who has fewer dots?” by each rolling one cube and
then comparing the number of dots. Observe the strategies used by the children to compare.
Dominoes
Have children to explore with dominoes. Ask them to describe the various arrangements of dots, “How
are the dots arranged?” “What do you notice about the dots?” The dominoes can be used for various
activities, such as finding all dominoes with a total of six dots, seeing how quickly one can tell how
many dots are on each side of a domino, finding the doubles, or making matches of adjacent dominoes.
Observe children to see if they count by ones, use groupings, or are able to identify the quantities.
Dot Plates and Dot Cards
Prepare a set of paper plates, cards, or transparencies with groupings of two through ten dots in many
different arrangements. Use the dot plates/cards for various activities as described on the “Dot Pattern
Activities” sheet. Often ask children, “How many dots do you see?” and “How did you see them?”
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Dot Pattern Activities
Dot plates or dot cards consist of patterned sets using dots. The
dot patterns represent quantities from zero through ten in
different configurations. The patterns show the common dice and
domino patterns, as well as many other variations that are
combinations of two smaller patterns or a pattern with one or two
additional dots. Some of the patterns can also be made with two colors to more clearly show the two
parts. Keep the patterns compact so they are easier to see. The dots plates or cards can easily be made
with paper plates and stick-on-dots.
Flash Math
Hold up a dot pattern for about 3 seconds. The children should visualize the dots and try to figure out
how many dots were on in the pattern. Show the plate again for another 3 seconds. Then ask, "How
many dots did you see?" The students can respond in unison. Set up some type of signal so that children
do not call out the number of dots until all everyone has had a chance to visualize and think about the
pattern. Then ask a student to explain, "How did you see it?" Have several students explain how they
saw the dots. There is no right or wrong way to see a pattern and often students see them in different
ways. Begin with easy patterns and then add a few new patterns each day.
Make the Pattern
Flash a dot pattern for a few seconds. Then tell the students to make the pattern they saw using counters
on their work mats. It is helpful to have a work mat so that you can easily see the pattern made by the
student. Spend some time discussing the configuration of the pattern and the number of dots.
Hold Up
Flash a dot pattern. Then tell students to hold up that many fingers. Next discuss how the students saw
the dots, e.g. as a three and a two or as a four and a one.
Show the Numeral
Flash a dot plate. Tell the students to hold up the numeral card that tells how many dots. Then discuss
the configuration of the pattern.
One More Than or One Less Than
Flash a dot pattern and have the students tell the number that is one more than (or one less than) the
number of dots. Have the students explain their reasoning.
Write the equation
Flash a dot pattern. Each student writes an equation that reflects how they show the dots.
Compare
Working in pairs, each student displays a dot pattern. Then they discuss, "Which plate has more dots?"
Combine
Working in pairs, each student displays a dot pattern. They they figure out, "How many dots all together
on both cards (plates)?"
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Dot Patterns
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Part-Whole Activities
Part-whole mats
Children can place a pencil or piece of string down the middle
of a work mat or be given a mat that clearly shows two parts.
Tell the students to place a specific number of counters,
e.g. six, on their mat. They should place part of the counters
on the left side and part of the counters on the right side.
This leads to an interesting discussion as to whether zero is
considered a part of the number. This is something your class can decide. The children can be asked to
represent their combinations in some way on paper. The different combinations can also be recorded in a
table on the board for discussion.
Story Boards
Children can design their own story boards, such as an ocean scene, school yard, playground, or any
place that is special to them. Then using a designated number, such as six, of counters, teddy bears, fish,
or other shapes, have the children tell part-whole stories. For example, “There are five fish swimming in
the ocean. Three of the fish are swimming at the top of the ocean and two fish are swimming at the
bottom of the ocean.” The stories can be recorded in a whole class journal, dictated to older students or
an adult to be recorded, or just shared orally.
Shake and Spill
Tell the students to place a specific number of two–color counters, e.g. five, in a cup or in their hands.
Next they shake up the counters and then spill them onto a work mat (felt works well) on their desks.
Children report how many counters are red and how many are yellow. The different combinations can
be recorded in a table for discussion. Children can also fold a paper into fourths and record a
combination from each toss in each section. These can be cut apart and graphed. This activity can lead to
an interesting discussion on the likelihood of certain combinations, as well as to a discussion on all the
possible combinations.
How Many of Each Problems—Seven Peas and Carrots
Tell students that you have seven things on your plate. Some of them are peas and some of them are
carrots. Explain that there is more than one correct solution to the problem. The students may use
counters to help them solve the problem. When students have found a solution, they record it on blank
paper using pictures, numbers, and/or words. The number of objects and the kinds of objects can be
varied as this type of problem is revisited throughout the school year.
Source #1: Schulman, L, & Eston, R. (1998, October). A problem worth revisiting, Teaching Children
Mathematics, 5, 72-77.
Source #2: Kilman, M., Russell, S. J., Wright, T., & Mokros, J. (1998). Mathematical thinking at grade
1. White Plains, NY: Dale Seymour Publications, p. 43-53.
Problems About 10
Have children make up their own problem about ten things. The problem should be about two different
kinds of things, such as apples and bananas or turtles and rabbits. The children first select the two kinds
of things. Then they decide how many of each there are to have ten things all together. The children may
use counters or other materials to help them. Then they record their problem on paper.
Source: Kilman, M., & Russell, S. J. (1998). Building number sense. White Plains, NY: Dale Seymour
Publications, p. 48-52.
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Towers of Six (Seven, Eight, Nine, or Ten)
Working in pairs, children build three towers of six together. Each child uses a different color of
interlocking cubes, e.g. red and brown. The children take turns rolling two dot cubes with one, two, and
three dots on the sides. (If building larger towers, use regular dot cubes.) If the first child rolls a two and
a two, she would use the red cubes to build a tower four cubes high. Then the second child rolls the
cubes and gets, for example, a three and a two. This child would add two brown cubes to make a
complete tower of six and then start a new tower with the remaining three cubes. They continue to play
until they have three complete towers. It is not necessary to roll an exact number to finish the game. The
children then find the number of cubes of each color in each tower and record this information on paper
using pictures, numbers, and/or words.
Source: Kilman, M., Mainhart, Murray, M., & Economopoulo, K. (1998). How many in all? White
Plains, NY: Dale Seymour Publications, p. 42-44.
Hiding Hands
Tell the students to place a specific number of counters, e.g. five, into their hands. Then they should put
their hands with the counters under their desks and secretly put part of the counters in one hand and the
other part of the counters in the other hand. Now the children take turns revealing their hidden parts by
first showing the counters in one hand and then showing the counters in the other hand. For example,
Morgan has three counters and two counters. When children become familiar with the combinations, ask
for predictions before the counters in the second part or hand is revealed. Observe which students need
to see the counters and which do not. In either case, what strategies do children us to determine the
hidden part. Do they know the combination? Do they count by ones? Do they use grouping strategies?
Do they visualize the hidden part?
Under the Cup
In pairs, children take turns hiding counters in a cup. Assign each pair a number to work on or let them
select a number. One child secretly hides part of the counters under the cup and places the remaining
counters on the table. The other child figures out how many counters are hidden and then shares his/her
strategy. The number of counters in each part is recorded. Then the children continue taking turns as
they repeat the activity with the same amount of counters. When they have found many different
combinations, they can select a new number of counters and begin again.
Turn over Five
The object of this Concentration-type game is to find pairs of cards that add to 5. Children play in pairs
or small groups. Use a set of twenty-four numeral cards with four cards for each of the numbers 0
through 5. Lay the cards facedown in four rows of six. Players take turns choosing two cards to turn
over, trying to find a combination that adds to 5. The game can be varied by selecting a different target
sum, such as 10.
Source: Kline, K. (1999, April). Helping at home. Teaching Children Mathematics, 5(8), 456-460.
Fives Go Fish
This game is a variation of the popular game of go fish. The same rules apply, but rather than search for
pairs of the same number, a player searches for pairs of two numbers that add to 5 or to other numbers as
appropriate. Use a set of twenty-four numeral cards with four cards for each of the numbers 0 through 5.
Deal out six cards to each child. Place the rest of the cards face down in the pond or center of the group.
Children take turns asking each other for a specific card, such as "Mary, do you have a three?" If Mary
has a three she must give it to the person asking. If she does not have a three she tells the person to "Go
Fish" and the person draws one card from the pond.
Source: Kline, K. (1999, April). Helping at home. Teaching Children Mathematics, 5(8), 456-460.
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Ten Frame Activities
Numbers are represented on a ten-frame by filling in,
from left to right, the top row and then the bottom row.
Each number is then seen simultaneously in relationship
to five and ten.
Fill Ten frames
Call out numbers between one and ten in random order and have children represent the numbers on their
ten-frame. Discuss the relationships to five and ten.
Flash Ten frames
Flash ten-frame cards to the class to see how fast they can tell how many dots are shown. Frequently
spend three to five minutes with this activity. Encourage children to see how quickly they can name the
amount. An alternative to saying the number of dots is to say the number of empty spaces.
Ten-frame Facts
Flash ten-frame cards to the class. The children should state the ten-fact, for example if seven dots are
shown, they state, "Seven and three." A variation is to have the children write the equation, “7 + 3.”
Five And
Draw a large ten-frame on the chalkboard or use an overhead transparency. Encourage the children to
think in terms of the ten-frame. Call out a number between five and ten. Children respond with "five and
_________" using the appropriate number. For example, the teacher calls "Eight" and the children
respond, "Five and three."
Make Ten
Show a ten-frame. Encourage the children to think in terms of the ten-frame. The teacher calls out a
number between zero and nine. The children respond with the number required to make ten. For
example, if you call "Four," the children respond with "Six."
One or Two More, or Less
Flash ten-frame cards to the class and have the children respond with the number that is one more than
the number shown. When the children are good at this, have them respond with the number that is two
more. For even more of a challenge, have the children respond with the number that is one or two less
than the number shown.
Crazy Mixed-Up Numbers
Each child has a ten-frame and counters. The teacher calls out numbers from zero to ten. The children
figure out what changes are needed to make the new number. The children say "plus ________" or
"minus _________" [whatever is required to change their ten-frame to the new number.] They then add
or remove counters accordingly. Small groups of children can use long strips of paper with a list of
about 15 "crazy mixed-up numbers." A student can read one number at a time to the rest of the players.
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Collect Ten
Children play in pairs or small groups. Each person has a ten frame and ten counters. Children take turns
rolling a dot cube that has one, two, or three dots on a side. If a three is rolled, the child places that many
counters onto his/her ten frame. The goal is to fill the ten frame. The exact amount must be rolled. If the
number is too big, then the child must try again on his/her next turn.
Fill-Up Together
Two or three children play as a group. Place two or three ten frames in the middle of the group. Each
person needs 10 to 20 counters. The goal is for the group to fill all the ten frames by working together.
working together. The players take turns rolling a dot cube and placing that many counters on a ten
frame. Each ten frame must be filled from left to right, top row then bottom row. A number may not be
split up. If the player is unable to fit the entire number in a ten frame, they must start a new ten frame. If
this is not possible the player will lose a turn. The game is over when all the ten frames are filled.
Cover Up
The object of this game is to cover up two 10-frames completely using interlocking cubes. To prepare
for the game, make eight trains of one color that are five cubes long and twenty trains of another color
that are two cubes long. Also have about forty single cubes of yet another color available. The player
rolls a number cube and takes that many cubes to cover the 10-frames on their game board. The player
may take any combination of cubes to make the total rolled but is not allowed to pull apart any of the 2trains or 5-trains. For example, a player who rolls a 6 may take one 5-train and one single cube or may
take two 2-trains and two single cubes. Play continues until one player has completely covered up both
10-frames with interlocking cubes.
Source: Kline, K. (1999, April). Helping at home. Teaching Children Mathematics, 5(8), 456-460.
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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Calculator Activities for Developing Number Relationships
Counting
•
Count by ones.
0 +
•
Count backwards by ones.
10 –
•
Skip count: Count by fives.
0 +
5 =
Count by tens.
0 +
10 =
1 =
•
Counting-on by ones.
5 +
•
Counting-on by tens.
23 +
1 =
=
1 =
=
=
=
=
=
=
=
=
10 =
=
=
=
=
=
=
=
=
Number Relationships
•
•
•
•
•
•
One More Than Machine
Program the calculator:
0 +
1 =
Use it: 3 =
10 =
8 =
32 =
Two More Than Machine
Program the calculator:
0 +
2 =
Use it: 7 =
5 =
10 =
23 =
One Less Than Machine
Program the calculator:
1 –
1 =
Use it: 9 =
20 =
5 =
48 =
Two Less Than Machine
Program the calculator:
2 –
2 =
Use it: 6 =
10 =
5 =
9 =
Part-Whole Machine
Program the calculator:
10 –
Doubler Machine
Program the calculator:
2 x
10 =
1 =
Use it: 3 = 4 = 8 = 7 =
(Ignore the negative sign. )
Use it: 6 =
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
4 =
11 =
25 =
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Basic Fact Practice
•
•
•
•
Practice adding seven.
Program the calculator:
0 + 7=
Use it: 2 =
10 =
8 =
32 =
Practice subtracting three.
Program the calculator:
3 – 3=
Use it: 9 =
15 =
7 =
28 =
Practice multiplying by nine.
Program the calculator:
9 x 1=
Use it: 3 =
5 =
10 =
7 =
Practice dividing by five.
Program the calculator:
Use it: 40 =
15 =
5 ÷ 5=
100 =
37 =
Place Value Activities
•
Wipe Out
Enter a number into the calculator, for example 458.
Challenge your opponent to wipe out the “5” without changing any of the other digits.
Then challenge your opponent to wipe out the “7” and then the “4.”
A point can be awarded for each correct move.
•
Change it
Enter a number into the calculator, for example 28.
Challenge your opponent to change the “2” to a “5” without changing the other digit.
Then challenge your opponent to change the “8” to a “6” without changing the other digit.
A point can be awarded for each correct move. This game can go on forever.
•
Make a New Number
Enter a number into the calculator, for example 359.
Challenge students to make new numbers without pressing the clear key. For example, change 359
to 659 to 679 to 609 to 209 to 239 and finally to 235
For each change, students should record the keys they pressed to modify the number. For example,
to change 359 to 659, students might record “ + 300” or “ + 100 + 100 + 100.”
Silent Races
Select a number for skip counting, such as 2, 5, 10, or 25.
Then select a target: e.g. 50, 100, 75, or 300.
Each student then programs his or her calculator: e.g. 0 + 5 =
On the teacher’s mark, the students begin the skip counting race by repeatedly pushing the equal key
on their calculators. When a student reaches the target number, he or she stands up. If a student goes
past the target number, he or she must start over.
DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000)
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