MCTM Games

© 2012 Joan A. Cotter, Ph.D. • [email protected] • www.rightstartmath.com

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Addition StrategiesStrategies are a way to learn a fact or to recall a forgotten fact. These strategies are not to betaught as rules, but should be thought of a powerful visual tool. Counting should be discou-raged because it is slow, often inaccurate, and unmindful of place value. Some facts can belearned with more than one strategy. Also teach the names of the strategies.

Number plus 1. Adding 1 to a number is the next number.

Even number plus 2. Adding 2 to an even number is the next even number.

Odd number plus 2. Adding 2 to an odd number is the next odd number.

Adding 5 plus numbers 1 to 4. Adding 5 to a number is obvious on fingers or the abacus.

What makes 10. Enter 10 on the abacus. Separate one quantity and see what’s left.

Adding 9. Enter 9 on the first wire and the other number, for example, 4 on the second wire.Take 1 from the 4 and give it to the 9 to make a ten. The sum is 10 plus 3, or 13.

Adding 8. This strategy is similar to adding 9. Two beads are moved back, which is similar tocounting by 2s backward.

Two-fives. Both numbers need to be between 5 and 10. For example, enter 8 and 7 on two wiresof the abacus. The sum is 10 plus the “leftovers,” 3 and 2.

Doubles. New facts with the doubles are 3 + 3 and 4 + 4. Beyond 4 + 4 they can be seen as theTwo Fives strategy.

Near Doubles. New facts with the doubles are 3 + 4 and 4 + 5. Beyond 4 + 5 they can be seen asthe Two Fives strategy.

Relating facts. There are four facts not covered with these strategies: 6 + 3, 3 + 6, 4 + 7, and7 + 4. They can be seen in relation to the 10's facts.

Even + 2. Odd + 2.

Two 5s: 8 + 7 = 10 plus 3 and 2.

Making 10: 8 + 2 = 10.

Adding 9: 9 + 4 becomes 10 + 3.

Relating facts. If 6 + 4 = 10, then6 + 3 = 9.

Doubles: 4 + 4.

Near doubles: 5 + 4.


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Subtraction StrategiesA child should know all the addition facts before attempting to master the subtraction factsto avoid confusing the two. Some addition strategies require subtracting and some subtrac-tion strategies require adding. The whole point of strategies is that they be efficient and ac-curate. Visual strategies are very powerful. Counting is not an efficient strategy.

Some facts can be learned with more than one strategy. Teach the names of the strategies.

Part/whole circles. Part/whole circles, as shown below,help children see the correlation between addition andsubtraction. The whole is written in the large circleand the parts in the smaller circles. They also helpchildren solve word problems.

Subtracting 1. Subtracting 1 from a number is the previousnumber.

Subtracting 2. Subtracting 2 from an even number is the previous even number. Subtracting 2from an odd number is the previous odd number.

Subtracting consecutive numbers. Explain the meaning of consecutive. The result is 1.

Subtracting from 10. These were learned first with the Go to the Dump game.

Subtracting from 9 and 11. Do these by comparing them to subtracting from 10.

The facts ≤ 10. The above strategies include all the facts with the following exceptions: 6 – 3,7 – 3, 7 – 4, 8 – 3, 8 – 4, and 8 – 5. Two 8 facts easily can be seen with 5 as one part. Theremaining four facts can be found with the following Going Up strategy.

Going Up < 10. The Going Up strategy works for any fact. To subtract 9 – 6, start with 6 and re-member how much is needed to get to 9. [3] If the number being subtracted is less than 5,first find how much is needed to go to 5 and then add the amount over 5.

Going Up > 10. Use the same procedure for subtracting from numbers over 10. For 13 – 9, ittakes 1 to get to 10 and 3 to get to 13; so the answer is 1 + 3 = 4. See the left figure below.

Subtracting from a 10. Subtracting 15 – 9 can be also thought of as subtracting 9 from the 10,getting 1, and adding the result to 5 to get 6. See right figure above.

Derived strategies. Of course, there are also derived strategies. For example, if you know 12 – 6= 6, then 13 – 6 = 7.

Doubles and near doubles. The doubles and near doubles are not very obvious in subtraction,and therefore, not good strategies.

Going up: 13 – 9, start at 9, go upto 10, then go up to 13. The an-swer is 4, the total added.

Going up: 9 – 6, start at 6 and goup to 9 by adding 3, the answer.

Going up: 7 – 3, start at 3, go upto 5, then go up to 7. The answeris 4, the total added.

Subtracting from 10: 15 – 9, sub-tract 9 from the 10, then 1 + 5 = 6.

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4 6

Part/whole circlesshowing 10 = 4 + 6,10 – 4 = 6, and10 – 6 = 4.


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GO TO THE DUMP(From Math Card Games: Over 300 Games for Learning and Enjoying Math. Fifth editionby Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.)

Objective To learn the combinations that total 10

Number of players 2 to 4

Cards 4 or 6 of each basic number card 1 to 9

Deal Each player takes five cards; the remaining cards face down form the dump, or stack.

Object of the game To collect the most pairs that equal 10

Materials Beginners need an abacus or at least a list of the facts.

Preparation Before starting, the players check over their hands for pairs that total 10.To do this, they look at each card in turn, determine what is needed tomake 10 and look for that number among their other cards. (Somechildren may need to spread the cards out on the playing surface.)Store paired cards face up on two piles. (This allows verification andkeeps the cards shuffled for the next game.)

6 is needed with 4 to make 10.4

4

6

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8

8

2

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1 + 92 + 83 + 74 + 65 + 5

Play When all are ready, the first player asks the player on her left for anumber needed to complete a pair. If he has it, he must give it to her,whereupon she receives another turn. If he does not have it, he says, “Goto the Dump,” which is also the signal for him to begin his turn. He takesa turn by asking the player on his left and so forth.Meanwhile, the firstplayer concludes her turn by picking up the top card from the dump.She does not receive an additional turn even if she picks up a neededcard. However, she may put a new pair on top of her other pairs.A player running out of cards takes five more cards, but the turn isended. When the dump is exhausted, players may ask any player (notonly the players on their left) for a card.At the end of the game, players combine their two stacks and comparethe heights. (Counting the cards is too time consuming.) No shuffling isnecessary for subsequent games.

Player 1. Player 2.


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ROWS AND COLUMNS(From Math Card Games: Over 300 Games for Learning and Enjoying Math. Fifth editionby Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.)

Objective To practice adding three numbers

Number of players 2 or 3. (with more than three, the game may become too long.)

Cards 12 of each basic number card 1 to 9

Layout Sixteen cards are laid face up in a 4 × 4array. The remaining cards form the stock.

Object of the game To collect the most cards

Play During a turn, the player checks each rowand column for two or more cards that total15. The same card cannot be used for both arow and a column.In the figure, the 7 and 8 can be collectedfrom the first row. Also the 5, 1, and 9 fromthe first column and the 6, 7, and 2 from thelast column can be collected. Alternately, the5, 3, and 7 from the second row and the 9and 6 from the last row could be picked up.After a turn, fill in the array for the next player. If a player cannot play,she skips her turn and replaces the four corners.

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7

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8

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7

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9

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ROWS AND COLUMNS SOLITAIRE(From Math Card Games: Over 300 Games for Learning and Enjoying Math. Fifth editionby Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.)

Objective To practice adding three numbers

Number of players 1 or more. (Explain that in a solitaire, they want to “beat the cards.”)

Cards 6 of each basic number card 1 to 9

Layout Sixteen cards are laid face up in a 4 × 4 array. The remaining cards formthe stock.

Object of the game To collect all the cards

Play From each row or column, collect the facts of cards that total 15. Then fillin the gaps from the stock and again collect the facts. Continue until thestock is exhausted, at which time the cards may be combined regardlessof rows or columns. If no mistakes have been made, the cards will comeout in complete groups.


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Multiplication StrategiesMultiples.

• Needed for multiplication facts, fractions, and algebra.• Start as soon as 1-100 is understood; use patterns.

Multiples pattern explanations.• Twos. The second row is 10 plus the first row. They are the

even numbers.• Threes. Consider the ones: they increase starting at the low-

er left with 0 (30) and continue up the first column andover to bottom of the second column and to the thirdcolumn. Next consider the tens: 0, 1, 2 in each column.Sum of the digits: 3 in the first column (1 + 2, 2 + 1, 3 +0), 6 in the second column, and 9 in the third.

• Fours. The second row is 20 more than the first row, everyother even number.

• Fives. They have an obvious singsong pattern.• Sixes. The first row is the even 3s. Second row is 30 more

than the first row.• Sevens. Within each row the tens increase by 1. The ones in-

crease by 1 starting at the upper right (21) and continu-ing down the column and over to the next column.

• Eights. In each row the ones are the decreasing even num-bers. The second row is 40 more than the first row, alsoevery other 4.

• Nines. The sum of the digits in all cases is 9. The ones de-crease while the tens increase. The second row has thedigits of the first row reversed, as shown by the arrow.

Ditties. • Rhymes and songs are stored in the language, not math,

part of the brain.• A child who learns “6 ¥ 1 = 6, 6 ¥ 2 = 2, . . . ” often cannot

recognize multiples, which is necessary for simplifyingfractions and algebra.

Multiplication seen visually is shown below.

2 4 6 8 1012 14 16 18 20

4 8 12 16 2024 28 32 36 40

6 12 18 24 3036 42 48 54 60

8 16 24 32 4048 56 64 72 80

5 1015 2025 3035 4045 50

7 14 2128 35 4249 56 6370

3 6 912 15 1821 24 2730

Skip counting patterns.

9 18 27 36 4590 81 72 63 54

7 × 7 = 25 + 10 + 10 + 4 = 49. 9 × 8: 10 × 8 – 8 = 72.4 × 4 = 8 + 8 = 16. (See the twodot patterns for 8.)

6 × 3 = 5 × 3 + 3 = 18.


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A game in progress: Theplayer on the left collectsthe 2s while the player onthe right collects the 5s.

SKIP COUNTING MEMORYObjective To learn the skip counting patterns on previous page.

Preparation To prepare the envelopes, see page 13. The players use the envelopes forreference during the game to memorize the patterns.

Number of players 2 or 2 teamsCards Each player or team chooses an envelope and removes the cards. Mix the

cards together and shuffle lightly. Lay the cards out face down in a 5 by 4array.

Object of the game To be the first player to collect in order the complete set of cardsPlay The first player turns over one card so both players can see it. If it is the

needed card, the player collects the card and receives another turn. If it isnot the needed card, the card is returned. Next the second player takes aturn. Turns alternate until one player has picked up all ten cards.Stress the importance of returning the cards to the correct envelopes following a game.

2 4 6 8 1012 14 16 18 20

5 1015 2025 3035 4045 50

MULTIPLICATION MEMORYObjective To help the players master the multiplication facts.

Cards 10 basic number cards with numbers 1 to 10 and one set of product cards.Also a sticky note with the set number and “×” and another note with “=.”

Number of players Two. Beginners should sit on the same side of the cards.Object of the game To collect the most cards by matching the multiplier with the product.

Layout Lay the basic number cards face down in two rows. To the right in separaterows lay the product cards.

Play The first player turns over a basic number card and states the fact. Forexample, if the card is 4, the player says, “Three taken four times is 12.” Hethen decides where it could be among the product cards. If he is correct, hecollects both cards and takes another turn. If it is not a match, both cardsare returned face down in their original places, and the other player takes aturn.

12

12

4

4

=3 ×

MCTM Games

Education

Transcript of MCTM Games