CHAPTER 14

Addition, Subtraction, Multiplication, and Division

Once your child has a good grasp of how to recognize and create patterns in the natural world, he’s ready to make the connection that there are patterns in the way we work with numbers, as well. Addition, subtraction, multiplication, and division (known, to some degree, as operational math), are all based on patterns between numbers and groups of numbers. The activities in this chapter help your child work with numbers in ways that make those patterns visually apparent.

Build Fact Family Houses

One of the more common themes in operational math is fact families. Every fact family has three core numbers that are related to each other. Just as in a human family, the family grows as connections between these numbers are created. The original three numbers can always be used to make at least four math facts. Your child will begin learning about fact families in relation to addition and subtraction, then later move on to multiplication and division.

Recognizing fact families, especially the ones that make up the “tens facts” in the following example, are key in being able to add by rote. Putting those families in “houses” extends the analogy of a family, giving your child a fun, visual way to see the relationships.

Skills Being Practiced

What You Need

Get Ready to Play

Here’s some information you need to know before you begin.

How to Play

  1. Use a piece of lined paper and a pencil to help your child make a list of the tens facts. She may need a little help, but once you get going, together you should be able to figure out all the combinations of numbers that add up to 10. To get started, write the number 1 on the first line of the paper and ask your child, “What do you need to add to the number 1 to get 10?” Continue this process, remembering to list the inverse facts, too. That means your child will list the problem 9 + 1 as well as the problem 1 + 9.
  2. On a piece of card stock or construction paper, have your child draw a basic house. The house should be simple: just a basic square with a triangle on top for the roof. On the roof, your child should draw three square windows, one in the gable (the peak) of the roof, and the other two somewhere near the bottom corners of the triangle.
  3. On the body of the house, ask your child to draw four long, horizontal rectangular bay windows, one on top of the other, each one going across the whole front of the house (these are not typical of the square windows in most children’s drawings, so you may need to show her how to do this).
  4. Write two unfinished addition problems (_____ + _____ = ____) on two of the long windows and two unfinished subtraction problems (_____ - _____ = ____) on the two other long windows.
  5. On the windows of the roof, ask your child to write three numbers that make up one of her tens facts families. The number 10 should always be in the top window in the gable. Since it is the number that all the other numbers are related to, it is sort of the head of the fact family.
  6. Ask your child to use the numbers from the roof to fill in the addition problems. You can give her a hint that 10 will always be the number after the equal sign in these addition problems.
  7. Then have her complete the subtraction problems using the same numbers. Remind her that subtraction is just the reverse of addition, prompting her with questions like, “You added 7 to 3 to get 10, so what number do you think you’ll be left with if you take that 3 away from 10?”
  8. When she’s done, it’s time to decorate the house!

EXTEND THE LEARNING

Have your child make one house for each of her tens facts. She can cut them all out and create an entire neighborhood of fact families.

Multiplication Magic: Tricks to Make Multiplication Easy

Being able to pull up the answers to multiplication facts is a very important skill your child will need to master in order to move on to more complicated types of math. This is why so much time is spent in school trying to make sure that she can recall the answers as quickly as possible. Not all kids are able to learn their multiplication facts using rote memorization, and if your child is one of them, don’t worry too much about it. There are Multiplication Magic Tricks to help and amaze her.

Research has shown that rote memorization doesn’t help kids to learn the connections between numbers or understand the rules of multiplication. Studies have found that practically-based math, or finding ways to help kids relate math to real life, is more effective than just teaching the facts.

Here are some practically-based multiplication strategies you can do easily:

How to Play: Magical Multiplication Tricks

While it’s easy to see through the magic of some of the more practical multiplication tricks, there are other ways to get the answers that aren’t as easy to see through. Once your child understands how to do the trick, though, he’ll be able to amaze his friends and teachers with his talent!

The Magically Appearing Zero

  1. Write out the tens times table for your child to see and ask if he notices a pattern. What he should be able to see is that when multiplied by the number 10, a number looks like itself with a zero on the end.
  2. Let him try it out on a calculator using large numbers, too. He’ll see that that every time he multiplies by 10, a zero “magically” appears on the end!

Multiplying by Zero

Multiplying by zero doesn’t lend it itself easily to magic. It’s not easy to understand that when you multiply a number by zero the answer is zero, not the number you started with. Once your child understands that question really is “How much is zero groups of something?” he’ll realize the answer is “Nothing,” and understand how the number disappeared.

Seeing Double

The magic of the elevens times tables may only work with single digits, but that’s probably all your child is working with right now anyway. Demonstrate how multiplying by 11 always makes your child see double of the number he’s multiplying. For instance, 11 × 7 = 77 and 11 × 3 = 33.

Doubling the “Doubles”

  1. If your child has figured out the trick to his twos, then he’ll be able to make magic with the fours. Ask him to fold a piece of paper in half lengthwise, then unfold it to make two columns separated by the fold. Have him write his twos tables in one column.
  2. Have him write his fours table in the next column.
  3. The magic that he should see is that the answers are the doubles doubled. That is, if 4 × 2 = 8 (the double), then 4 × 4 = 16. The double has doubled!

Magic Fives

  1. This trick is a little odd, as it only works with odd numbers. Write down all the fives multiplication facts that use an odd number, and have your child find the magical oddity.
  2. He may see that if he subtracts one from the multiplier, “cuts” it in half, and puts a five after it, that’s the answer to the problem. If that went too fast for you to see, look at it like this: 5 × 9 = 45, which is actually 9 minus 1 (8), cut in half (4) with a 5 on the end (45).

Even More Magic Fives

  1. If your child doesn’t like skip-counting as an option to make the fives tables appear, there is another trick to try. Ask your child to write down all the fives facts that involve even numbers, and ask him to look for a pattern.
  2. What should appear before his eyes is that each answer is simply half of the number he’s multiplying by five, with a zero on the end. Not a believer? Check out these examples: 5 × 6 = 30, and 5 × 8 = 40.

Magical Finger Math

  1. To learn the nines tables, all your child really needs is his hands. Have him place his hands in front of him and explain that the fingers on his left hand represent the numbers 1 through 5. The fingers on his right hand represent the numbers 6 through 10.
  2. For his first trick, ask him to fold down the middle finger on his left hand, or finger number 3.
  3. Remind him that 9 × 3 = 27, and then have him look at his hands. To the left of his bent finger there are 2 fingers. To the right are his remaining 7 fingers.
  4. The magic to this trick is that the number given to the finger that he folds down × 9 is equal to the number of fingers to the left of the bent finger (in the tens place) and the fingers to the right (in the ones place.)

Multiplication Card Games

For children who aren’t taken in by rote memorization or “magic” tricks, another way to make those multiplication skills become second nature is to just make it part of everyday fun. Here are a few card games that help you incorporate multiplication without making it a big deal.

Skills Being Practiced

What You Need

How to Play: Multiplication War 1 (Two Players)

  1. Shuffle the cards and deal them facedown equally between the players.
  2. On the count of three, each player turns over a card. The first player to say the product of the two cards wins the hand. For example, if Player One turns over a 6 and Player Two turns over a 5, the first player to say 30 gets to keep both cards.
  3. The game is over when all the cards have been turned over. The player with the most cards is the winner.

How to Play: Multiplication War 2 (Multiplayer)

  1. Shuffle the cards, and place them in a facedown stack in the middle of the table. Each player draws two cards.
  2. Go around the table and ask each player to give the product of his two cards. Once everyone has had a turn, all players must show their cards so the other players can check the multiplication. The player with the highest product gets to take all the cards.
  3. This continues until all the cards are gone. The player who has the most cards at the end of the game is the winner.

How to Play: Multiplication Go Fish

  1. Shuffle the cards and deal six cards to each player. Place the remaining cards in a pile in the middle of the table. This pile is to be used as the draw pile.
  2. The game is played very much like typical Go Fish, but when a player asks another player if he has a card, he must come up with a multiplication fact to represent the card’s value instead of just saying the card. For example, instead of asking, “Do you have any 6s?” a player may ask, “Do you have any 2 times 3s?”
  3. If the player asked has a card of that value, he must relinquish it to the player who asked, who then can put the matched pair aside. If the player asked does not have that card, he says “Go fish,” and the player who asked takes a card from the draw pile.
  4. The game continues this way until one of the players runs out of cards. The player with the most matches is the winner.

How to Play: Ladder Multiplication Game

  1. This is a more complex card game than any of the other multiplication games. It should only be played with children who have a good grasp of multiplication facts and don’t get frustrated easily. It’s also a game that can go on for a long time, so having a fairly long attention span is also important.
  2. Designate a recorder to keep track of all the players’ points. Give the recorder a piece of paper and a pencil and ask him to write everybody’s name on the paper. Place another piece of paper in the middle of the table and draw a ladder on it.
  3. Deal six cards to each player. These cards remain as each player’s hand for the entire game and can be displayed faceup or hidden from view. They are never discarded or lost.
  4. Place the rest of the cards to the side. This is the draw pile.
  5. Turn over the first card in the draw pile, and place it in the middle of the table. This card serves as the first “rung” in the ladder, so the recorder writes the card’s number on the first rung of the ladder (on the paper in the middle of the table).
  6. The first player takes a card from the draw pile. He then looks at the cards in his hand and tries to find one that can be multiplied with the card he drew to create a product larger than the number on the first rung of the ladder. If he is able to do it, then this product becomes the next rung in the ladder. The recorder writes that number on the second rung, the first player is awarded a point, and his draw card is placed at the bottom of the draw pile.
  7. The next player turns over a card from the draw pile and tries to find a card in his hand that can create a product larger than that of the second rung of the ladder. If he is unable to do so, he places his draw card on the bottom of the draw pile and it is the next player’s turn. If the third player is able to come up with a product larger than the number on the second rung, that number is written on the third rung, and that player is awarded a point.
  8. Game play continues until no player is able to create a product larger than the number on the highest rung of the ladder. The player with the most points is the winner.

Division Card Games

Card games are a good way to practice multiplication, but they can also be modified to practice division, too. It takes a little more creativity, but you and your child will both have fun and end up with a better understanding of division.

Skills Being Practiced

What You Need

How to Play: Ready, Set, Divide!

  1. Assign numbers to the face cards, writing down the key or placing a small piece of tape on each card with the number written on it. For the purpose of this game Ace = 1, King = 12, Queen = 12, and Jack = 11.
  2. Shuffle the cards, then deal them facedown equally between the two players.
  3. On the count of three, each player must turn two cards faceup.
  4. Looking at all four cards, the players must try to find three cards that can be put in sequence to create a division problem. For instance, if one player reveals a 5 and a 3 and the other player reveals a King and a 4, one of the players could take the 4, the 3, and the King to make the problem King (12) ÷ 4 = 3.
  5. The first player to recognize a potential division problem, and set it out with all the cards, wins the hand. All the cards not used go back into the players’ hands for the next round.
  6. The game is over when there are no more cards left, or the players are unable to make any more division problems.

How to Play: Division Go Fish

  1. This game is played almost exactly like Multiplication Go Fish, except that instead of thinking of a multiplication problem to represent the card value she is looking for, a player must come up with a division problem. So, if a player is looking for a match for her 6, she might ask, “Do you have any 18 divided by 3s? or “Do you have any 12 divided by 2s?”
  2. If the player being asked has that card, she must give it to the asking player. If not, she must say “Go fish,” and the other player takes a card from the draw pile.

Operation Words Brainstorm

Your child might know his number facts, but when it comes to word problems, it’s likely that he is a little confused. For many kids it’s not the math that confuses them, it’s figuring out which operation they need to use. There are always key words or phrases in the problems that can help, but your child needs to learn which words go with which operations. That’s what the Operation Words Brainstorm activity is all about.

Skills Being Practiced

What You Need

COMMON WORDS FOR OPERATIONAL MATH

Addition Subtraction Multiplication Division
plus minus times divided by
add fewer than multiplied by goes into
and difference between product of into
combine take away multiply divide
added to from by half of
together less than groups of (fraction) of
greater than decreased by doubled by per
in addition to reduced by tripled quotient
more than subtract how many each groups of
increased by how many left twice as much every
total dropped by

How to Play

  1. Talk to your child about what he thinks about math word problems, reassuring him that you’re not going to be solving any problems. Ask him what he finds the most difficult about word problems and, if he doesn’t bring it up, mention to him that a lot of people find it hard to understand what the problem is asking them to do.
  2. Explain that if you know the “magic words” to listen for, word problems can actually be really easy to solve. Let him know that every problem has phrases or words that are clues to what type of math he needs to be doing. Tell him that what you’re going to do is help him learn as many of those clue words as possible. Most word problems include more information than is necessary to solve the problem. Once your child is able to recognize the key words and phrases, it will be much easier to figure out what other information is also important to the problem.
  3. Tape four large pieces of paper somewhere where they are easy to see and easy to reach. Label them: Addition, Subtraction, Multiplication, and Division.
  4. Ask your child if he can tell you one word or phrase that he knows is used to mean “add these numbers” (the previous table provides a number of examples for you to use).
  5. Write his words on the Addition paper, and add a word or phrase of your own. Then tell him you’re going to set the timer for 5 minutes, and the two of you are going to see how many other phrases you can come up with that mean the same thing. Write them down.
  6. Repeat this process with each of the other mathematical operations.
  7. When you are finished, demonstrate how some of the phrases can be used as clue words. For example, if you have the phrase “in addition to” on the Addition paper, you might say: “I bought three apples. In addition to those, I have four apples in the kitchen. How many apples do I have?”

EXTEND THE LEARNING

Ask your child to create word problems using some of the clue words.

Toothpick Subtraction

Your child may understand the concept of place value in that she knows which is the ones place and which is the tens place, but understanding that there’s a connection between the two is a little harder to understand. One way to help her see this is to help her actually see it. The Toothpick Regrouping activity helps your child understand how ones become tens, and how it’s possible to borrow from a ten when subtracting.

Many parents remember learning the concept of “borrowing” in subtraction and “carrying over” in addition. For the most part, these terms have been replaced with the singular term “regrouping.” Regrouping is used to describe both processes, since they both involve “trading” or making changes to place value “groups.”

Skills Being Practiced

What You Need

How to Play

  1. Label two sets of index cards with the numbers 0 through 9. Then fold a piece of paper in half lengthwise, open the page, and trace the fold with a marker. Label the top of the right-hand column “Ones” and the top of the left-hand column “Tens.” Set these materials aside.
  2. Ask your child to help you make five or six “tens bundles” with the toothpicks or craft sticks. Explain to her that ten bundles consist of ten sticks held together by an elastic band. Set the tens bundles aside as well.
  3. Lay out one set of the numbered index cards in a vertical row (up and down) in numerical order. That means the 0 card will be at the top, and the 9 at the bottom of your set. Ask your child to place the correct number of toothpicks on each card. Explain that since these numbers are less than ten, they remain single sticks and cannot be bundled up.
  4. Open up the sheet of paper on which you have identified the Tens and Ones columns, and ask your child to tell you in which column these cards would belong. She should be able to tell you they go in the Ones column because they are single sticks and not tens bundles. If she is unable to explain this, then remind her that only tens bundles go in the Tens column.

Check for Understanding

  1. Your child may say he understands the concept of “tens bundles,” but to really test his understanding, place one tens bundle in the Tens column. Ask him how much that is. He should say “ten.” Place an index card with the number 1 on it under the ten bundle, and ask him if that is how he writes ten.
  2. When he says no, explain to him that what he has written is the equivalent of “one Ten,” but in order to make the number complete, he’ll need to add a 0 in the Ones column. That’s because the number really says “one Ten and zero Ones.” Place an index card with the number 0 in the Ones column to show him how the two cards together make the number 10.
  3. Ask him to use the index cards to show you “1 Ten and 1 One,” then ask him to identify what number he has made (11). Now ask him to put the correct number of tens bundles and single sticks down on the cards, and ask him what number he gets by adding one tens bundle and one single stick (11).

How to Play: Begin with Addition

  1. Write the plus sign (+) on an index card. Use the sheet of paper with the Tens and Ones columns again. In the Ones column, use the index cards to set up a vertical addition problem that adds up to more than ten (for example, 9 + 2, but you’ll stack it instead of writing it from left to right).
  2. Have your child put the correct number of sticks on each card, and count the total number of sticks to get the answer. Ask her if there are enough sticks to trade some of them in for a tens bundle. When your child is adding and is able to trade in single sticks for a tens bundle, make sure to use the word “trade.” It’s the language she’s likely to use at school, and it verbalizes exchanging one thing for another of equal value.
  3. Have her trade in ten sticks, and then place the newly acquired tens bundle and the remaining single sticks in the correct columns below the addition problem. Ask: What is the answer (1 Ten and 1 One)? Can you show the answer with the index cards? What number have you made?

How to Play: Subtraction with Regrouping

  1. Clear your Tens and Ones columns and create a minus sign (–) on an index card.
  2. Set up a new vertical problem, using both the Tens and Ones columns. Start with something that doesn’t require any regrouping (formerly known as “borrowing”). For example, try the problem 15 - 4.
  3. Again, have your child create the problem using tens bundles and single sticks, placing the correct index cards and number of sticks in the columns below the problem. In this example, he should have 1 Ten and 1 One left. Do a few more subtraction problems without regrouping before moving on to the more difficult problems.
  4. Set up the problem 11 - 2. Ask your child to put out the right number of sticks and to try to do the problem. Ask: Can you do the problem with the sticks you have handy? Why not?
  5. Remind your child that when he was adding, he was able to trade in to make new tens bundles. Ask: Is there a way to make more single sticks? If your child doesn’t see the solution, explain to him that all he has to do is “break” one of the tens bundles and regroup them. Once a tens bundle is broken, it’s just ten single sticks (or ones) that can be placed in the Ones column with the rest of the singles.
  6. Open a tens bundle and add it to the Ones column, putting a 0 card in the Tens column to remind your child there are no more tens. Have your child complete the problem 11 - 2. Ask: Can you do it now? How did you do it?

EXTEND THE LEARNING

Create a problem that asks your child to regroup, but also to leave some Tens behind. For example, try the problem 21 - 9. See if your child can figure out how to complete the problem, reminding her that once she breaks open a tens bundle, she needs to make sure the Tens column has the number cards that matches how many tens bundles are left.