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Kids and Addition and Subtraction

If you are a first or second grade teacher or the parent of a first or second grader, you have undoubtedly observed that children find addition easier and more natural than subtraction. Children struggle with subtraction even when they learn “fact families” (1+3=4, 3+1=4, 4-1=3, 4-3=1) that ostensibly help them understand the relationship between addition and subtraction.

Given that children continue to find subtraction difficult despite the use of time-honored practices, I suggest that teachers and parents de-emphasize fluency in subtraction until their children become fluent in addition. Once children’s knowledge of a sum is solid, the related subtraction is easy for them. In other words, fluency in subtraction is dependent on fluency in addition.

The educational implication is that teachers and parents must de-emphasize fluency in subtraction in grades one and two and heavily emphasize addition. Permit children to learn sums first and then deduce differences from their knowledge of sums.

It is imperative that children have, in long-term memory, all the combinations of numbers up to and including 10.

Example: Students need to know all the combinations of 9 – 0+9, 1+8, 2+7, 3+6, 4+5, 5+4, 4+5, 3+6, 7+2, 8+1, 9+0

There is an easy and fun way to get children fluent in addition – math games! Children are intrinsically motivated to play games and to play them well. If they learn arithmetic in the process, they learn it for their own use. When teachers or parents instruct children to complete worksheets or pressure them to do well on timed tests, the children’s motivation to learn comes from external sources, and workbook pages, dittos, and timed tests aren’t nearly as much fun!

Here’s one of my favorite math games for first graders and second graders:

Add-em Up
What you need:
2 players
2 dice
counters
Add-em Up game board for each player – take a piece of 8 1/2 x 11 paper and cut it in half horizontally. Write the numbers 1 through 12 at the bottom of each paper.

Players place a counter above each number.

Player #1 rolls the dice and adds the 2 numbers. He/she may then remove the counter over the sum from the game board or the counters over any 2 numbers that add up to that same sum.

Example: Player #1 rolls a 3 and a 4. He/she may remove the
counter above the 7 or the counters above any
combination for 7, such as 1 & 6, or 2 & 5, or 3 & 4.

Players take turns rolling the dice and removing counters. When a player cannot remove counters that match the sum rolled or a combination, he/she loses that turn.

Play continues until neither player can remove counters. The player with the most counters removed wins.

A Math Game for Third Graders

Math games are a highly effective and engaging way to get students involved in practicing basic math skills. The following double-digit addition math game is great for second graders, third graders, and fourth graders. It not only addresses addition but forces students to look at the importance of place value.

Get Close to 100

What you need:
2 – 4 players
deck of cards, 10s and face cards removed
paper and pencils for each player

The object of the game is to make a two-digit addition problem that comes as close to 100 as possible.

Shuffle cards and place them face down in a pile.

Player #1 turns over 4 cards and moves the cards around until he/she has created a problem whose sum will be as close to 100 as he/she can make it. Player #1 records this problem on his/her paper.

Player #2 checks for addition accuracy.

Example: Player #1 draws a 4, a 7, a 2, and a 5. He/she moves the cards around until she/he decides that 47 + 52 = 99 is the closest that he/she can get.

Player # 2 draws four cards and does the same.

The points for each round are the difference between their sum and 100.
Example: A sum of 95 scores 5 points and so does a sum of 105.

Players compare scores at the end of this first round. They put their four cards in a discard pile and player #2 begins first and turns over four more cards for the second round.

After six rounds, players total their points and the player with the lowest score wins.

Variation: Make this a triple-digit addition game called Get Close to 1000! by drawing 6 cards and creating two triple-digit numbers which when added together, get as close to 1000 as possible.

Real-Life Math in Elementary School and Beyond

Elementary school students in three of Kingsport, Tennessee’s four high school zones took some weekend time this school year to learn practical, hands-on applications of math in the “real world”.

My question is, why isn’t their regular, everyday math curriculum talking about math in the “real world”?

Many educators contend that children must go beyond memorizing rules—they need to know when and how to apply the rules in real-world situations. Many also argue that realistic problems can serve as a powerful motivator in the mathematics classroom. They go on to conclude that the curriculum should consist of real-world problems because students will naturally learn mathematics by solving such problems.

The basics are changing. Arithmetic skills, although important, are no longer enough. To succeed in tomorrow’s world, students must understand algebra, geometry, statistics, and probability. Business and industry demand workers who can-

solve real world problems

explain their thinking to others

identify and analyze trends from data, and

use modern technology.

The mathematics students do in school should prepare them for the new basic skills necessary for their futures.

Instead of problems done with no context using worksheets, dittos, and workbook pages, students should be working on problems to investigate that are related to real life, such as investigating salaries, life expectancy, and fair decisions, for example.

Giving students opportunities to learn real math maximizes their future options.

Using money, counting change, etc. is a real-life skill that children need to learn. Play the following game with your second graders, third graders, and fourth graders.

Money Race

What you need:
2 players
1 die
pennies, nickels, dimes, and quarters
sturdy paper plate for “bank”

The following coins (which equal $1.00) are placed in the “Bank” between the two players. A paper plate makes a great bank.

10 pennies, 5 nickels, 4 dimes, and 1 quarter

Each player also takes the same combination of coins for a total of $1.00.

Money Legend:
1 – subtract a penny and put it in the bank
2 – subtract a nickel or 5 pennies and put it in the bank
3 – subtract a dime or a combination of coins that equals 10 cents
and put it in the bank
4 – subtract a quarter or a combination of coins that equals
25 cents and put it in the bank
5 & 6 – choose any one coin from the bank

Player #1 rolls the die and either adds or subtracts the appropriate coins.

Player #2 does the same.

Play continues in this manner until both players have completed 10 rolls. Players total their own coins. The player with the greatest amount wins.

Teaching Young Children About Money

There are many math games and activities that help children learn about money.

Helping your child learn the value of coins is a real-life skill that can be taught and learned easily if you use the following activities which are educational and fun:

Free Exploration

Give your child a small tub of real coins and allow him/her time to explore. This might be a good time for you to watch your child and note what is happening. Does he/she already know the names of each coin? Does he/she know the values? Do they notice likenesses and differences? Do they sort the coins? Make patterns (i.e. penny, nickel, penny, nickel, or dime, dime, quarter, dime, dime, quarter)?

Alike and Different with a Magnifying Lens

Children need to be able to identify coins before they can learn their values. This activity gives children the opportunity to examine pennies, nickels, dimes, and quarters closely and think about what things are the same and different among them. Noticing likenesses and differences is important in math and reading for young learners.

You will need a magnifying lens and 1 penny, 1 nickel, 1 dime, and 1 quarter.

Allow your child to experiment with the magnifying lens first.

Begin with the penny. Have your child look at it closely and tell you what he/she notices. I usually start with the heads side. Identify the year and place the coin was minted, the other words on the coin, and so on. Then look at the tails side. Don’t forget to examine the edges. You might want to have them cut out a large circle and draw pictures of both sides of the penny.

Look closely at each coin in turn, noting how they are alike and different. You might take a blank piece of paper and draw a vertical line down the center, dividing the paper into two columns. List Alike at the top of the first column and Different at the top of the second column. Begin to write about what you discover. Some things appear on every coin; some do not.

Talk about size and value. This can be confusing for young children because the nickel is larger than the dime but worth less.

What Are the Coins?

You’ll need some coins for your child to use to solve the problems.

Ask your child the following questions:

I have three coins in my pocket. They are worth 7 cents. What do I have? (a nickel and 2 pennies)

I have three coins in my pocket. They are worth 16 cents. What do I have? (a dime, a nickel, a penny)

I have three coins in my pocket. They are worth 11 cents. What do I have? (2 nickels and 1 penny)

I have three coins in my pockets. They are worth 30 cents. What do I have? (3 dimes)

I have six coins in my pocket. They are worth 30 cents. What could I have? (1 quarter and 5 pennies or 6 nickels). This problem has more than one answer. It is challenging for children to experience problems like this.

I have coins in my pocket, which have a value of 11 cents. How many coins could I have?

Teachers – these activities can be used successfully in the classroom, and I think the secret to their success is using real coins.

Place Value Activities and Games

Once children have developed a basic number sense for numbers up to ten, a strong “sense of ten” needs to be developed as a foundation for both place value and mental calculations.

Ten is, of course, the building block of our Base Ten numeration system. Young children can usually “read” two-digit numbers long before they understand the effect the placement of each digit has on its numerical value. For example, a five-year-old might be able to correctly read 62 as sixty-two and 26 as twenty-six, and even know which number is larger, without understanding why the numbers are of differing values.

Place value is vitally important to all later mathematics. Without it, keeping track of greater numbers rapidly becomes impossible. (Can you imagine trying to write 999 with only ones?) A thorough mastery of place value is essential to learning the operations with greater numbers. It is the foundation for regrouping in addition, subtraction, multiplication, and division.

Developing students’ understanding of numbers, ways of representing numbers, and the relationships among numbers are focus areas for elementary mathematics. To help children understand these very important concepts and give them the opportunities to explore numbers, the following very effective and engaging game has been developed for First Grade, Second Grade, and Third grade.

Two-Digit War

What you need:
2 players
deck of cards – 10’s removed
Tens and Ones board for each player. Hold an 8 1/2 x 11 piece of paper with long sides on the top and bottom. Fold it in half vertically. At the bottom of the left half write Tens. At the bottom of the right half write Ones.

Shuffle cards and place them face-down in a pile. Player #1 turns over one card and decides whether to put it on the Tens or Ones. Once it has been laid down, it cannot be moved.

Player #2 does the same.

Player #1 turns over one more card and puts it on the remaining space. Again, it cannot be moved once it is laid down.

Player #2 does the same.

Players read their numbers out loud to each other. The player with the biggest two-digit number wins and takes all four cards. When all the cards have been used, players count their cards, and the player with the most cards wins the game.

Variation: The person with the smallest two-digit number wins.
Variation: Play Three-Digit War with Hundreds, Tens, and Ones

There are many really good games that emphasize a thorough understanding of place value

Taking Time to Understand Time

For years, teachers have observed students’ frustration as they grappled with learning to read an analog clock (as opposed to a digital clock). I remember being frustrated myself and not fully understanding why learning to tell time was so difficult for my students and wondering how to help.

I began to realize that there are two aspects of time that have to be distinguished in teaching time: firstly, one must try to develop a concept of time in a child, and secondly, one must teach the child to “tell the time” (teaching clock time).

Teachers of young children generally concur that their students learn mathematical concepts best when they construct understanding through concrete experiences. When we remember that time can be neither seen nor touched but experienced and measured only indirectly with such tools as clock, we can begin to understand why time-related concepts are difficult for our students to learn.

From the body of research available, as well as from our own firsthand teaching experiences, we know that everything to do with understanding and using time concepts develops rather late. I will go so far as to say that most children do not really fully understand the intricacies of telling time until about the third or fourth grade.

What usually happens in the classroom is that developing the concepts of time is skipped. In keeping with the admonition that children must actively develop concepts of time, I have included a few of the math activities I began to use in my classroom:

Time Intervals

Just How Long Is a Minute?

Have your students close their eyes and you time one minute. Have them keep their eyes closed and put up their hand when they think one minute has passed. Call time at the end of the minute. Now try it again. The more you do it, the better sense of a minute they will have.

Also try some of the following activities:

How many times in one minute do you think (make an estimate) you can:

1. Sing “Happy Birthday”? Estimate ____ Actual _____

2. Touch your toes? Estimate ______ Actual ______

3. Hop on one foot? Estimate ______ Actual ___

4. Do jumping jacks? Estimate ______ Actual ______

5. Write your first name? Estimate ______ Actual ______

6. Run around the basketball court? Estimate ______ Actual ______

7. Draw stars? Estimate ______ Actual _______

8. Recite the alphabet? Estimate ______ Actual _______

9. Snap your fingers? Estimate ______ Actual _______

10. How high do you think you can count in one minute?
Estimate _______ How high did you go? ________

More or Less Than a Minute? Homework

Here are some things you do everyday. For each one, guess whether you think it will take more than one minute or less than a minute to do it. Now try each thing while someone keeps time.

1. Put on your socks and shoes.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

2. Brush your teeth.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

3. Eat a banana.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

4. Read a page from your favorite story.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

5. Pledge Allegiance to the Flag.
Guess ______________ It really took _____________
more or less than 1 minute / more or less than 1 minute

Now make up a short list of things you think will take about one minute, and give them a try.

There are some great games which help children understand time!

Number of the Day

One of my favorite math activities for any age child is Number of the Day.

This is a great activity for anyplace you happen to be! It will give your child lots of computation practice, be a good deal of fun, and everyone (even you) will be forced to “prove” that they are correct!

Let’s say that our “number of the day” is 6. Everyone has to think up one way to make 6. Young children will probably begin with simple addition.

Example: 4 + 2 = 6

Ask your child to “convince you” (prove) that 4 + 2 = 6.

Everyone has to come up with an equation that equals 6, and each one has to be different.

After gaining in confidence, encourage your child to think of 2 different things that equal 6.
Example: 3 + 3 and 5 + 1

Then ask them to find 3 things that equal 6
Example: 1 + 2 + 3 = 6

See how many different ways everyone can find to make the number of the day. Write it all down if pencil and paper are handy.

Depending on your child’s age begin to encourage the use of other operations such as:
• subtraction 9 – 3 = 6
• addition & subtraction 8 – 4 + 2 = 6
• multiplication 3 x 2 = 6
• multiplication & addition 2×2+2 = 6
• division 24 ÷ 4 = 6
• all 4 operations in one equation
(50 ÷ 2) x 3 – 70 + 1 = 6
• coin values – 1 nickel and 1 penny =
6 cents
• fractions 4 ½ + 1 ½ + 6
• decimals 2.4 + 3.6 = 6 or 12 x .5 = 6
• integers – positive 10+negative 4 = 6

Family members can take turns choosing the number of the day. What about the day of the month, someone’s age or weight, number of windows in your home, the sum of your telephone number, etc. Try a variety of numbers, including large ones (such as 555 or 62,437) and small ones (they can be just as challenging as large ones).

Well, you get the idea! Dad might be coming up with 4 x 25 – 80 – 14 = 6!!
Does he have to prove it??!! Absolutely!

Using Math Games at Home

Games offer a pleasant way for parents to get involved in their children’s education. Parents don’t have to be math geniuses to play a game. They don’t have to worry about pushing or pressuring their children. All that parents have to do is propose a game to their child and start to play.

Math games for kids and families are the perfect way to reinforce and extend the skills children learn at school. They are one of the most effective ways that parents can develop their child’s math skills without lecturing or applying pressure. When studying math, there’s an element of repetition that’s an important part of learning new concepts and developing automatic recall of math facts. Number facts can be boring and tedious to learn and practice. A game can generate an enormous amount of practice – practice that does not have kids complaining about how much work they are having to do. What better way can there be than an interesting game as a way of mastering them?

All right, you’ve chosen a math game to play with your child. Now what? How can parents effectively help their child while playing a game?

Parent Responsibilities

Too often the parent is willing to give the child the answer, thus making it possible for him/her to do no thinking whatsoever. Not good! Your primary responsibility is to ask your child questions – questions that will force him/her to think and verbalize what he/she is doing and why.

Sometimes children don’t know what to do. Here are a few good questions to help them begin to help themselves, not just rely on you, the parent, to give them the answer:

What can you do to help yourself?
• Use your fingers to count?
• Count the dots on the dice or cards?
• Use counters (such as beans, paper clips, pennies, etc.) to figure
it out?
• Draw a picture?
• Start with something you already know?
Example 1: If you know that 5+5 =10, how can that help
you know what 5+6 equals?

Example 2: If you know that 5×6 = 30, how can that help you
know what 6×6 equals?

The power of questioning is in the answering. As parents, we not only need to ask good questions to get good answers but need to ask good questions to promote the thinking required to give good answers.

Here are a few more great questions to ask your child when playing a game:
• What card do you need?
• Which cards would not be helpful?
• Prove to me that a ____ is what you need.
• Why do you think that?
• How did you know to try that strategy?
• How do you know you have the right answer?
• Will this work with every number? Every similar situation?
• When will this strategy not work? Can you give a counter-example?
• Convince me that you are right.

Parents who observe and interact with their child while they are playing math games can find out a great deal about what their child knows and can do in math. While playing a game, what do you notice – what are your child’s strengths and weaknesses?

Finally, games provide children with a powerful way of assessing their own mathematical abilities. The immediate feedback children receive from their parents while playing games can help them evaluate their mathematical concepts. Good games evaluate children’s progress. They provide feedback so that parents, and the child know what they have done well and what they need to practice.

Parent Response to Game

As you play a game with your child, ask yourself the following questions:

• What did I think of this game? Did I like it? Why or why not?

• Was this game too easy, too hard, or just right? How did I change it to meet the needs of my child?
• What do I think my child learned from playing this game?

• What did I learn about my child while playing this game? What are his/her strengths? What does he/she need to practice?

Keep in Mind While Playing Math Games…

Inventing, Creating, and Changing the Games

Give your child opportunities to invent and create. The rules and instructions for all games are meant to be flexible. Allow your child to think of ways to change the equipment or rules. Encourage them to make a game easier or harder or to invent new games.

You can easily vary the games within this CD to suit the needs of your child. Some variations have been described within many of the games:

• The operations used within the games can be changed. If it’s an addition game, it might also make a great subtraction or multiplication game.
• The types of numbers used with the games can be smaller or bigger. If it’s a two-digit addition game, can it be made into a three-digit game?
• The rules of the games can be altered.

Please be creative in transforming the games into new forms, and please allow your child to do likewise.

Play the games many times. Children begin to build and practice strategies (plan their moves in advance) only when the game is repeated often. Playing it just once or twice is not very helpful, unless the game is too easy for your child.

Provide repeated opportunities for your child to play the game, and let the mathematical ideas emerge as they notice new patterns, relationships, and strategies. Allow the mathematical ideas to develop over time. This empowers children to independently explore mathematical ideas and create conceptual understanding that they will not forget.

Don’t hesitate to go back to a skill and play a game if you know your child needs to practice it.

Have FUN together!!!!!

Fractions Activity and Game

In 2006, the National Math Panel reported that knowledge of fractions is the most important foundational skill for algebra that is not developed among American students.

Research shows that fractions are one of the most difficult topics for students to understand in elementary school. I think the problem lies in the fact that children are expected to be passive receivers of information rather than be actively involved with the subject matter.

CGI (Cognitively Guided Instruction) has been stressing for many years that the best way to help children really understand fractions is to begin with “fair shares”.

Start with situations of 2 or 4 children, as children’s earliest partitions are based on halving:
4 children share 4 cookies so that each child gets the same amount.
4 children want to share 10 brownies so that each child gets the
same amount.
4 child want to share 22 apples so that each child gets the same
amount.

Move to situations with more sharers:
3 children want to share 7 candy bars.
6 children have ordered blueberry pancakes at a restaurant. The
waiter brings 8 pancakes to their table. If the children share the
pancakes evenly, how much can each child have?
Matthew has 13 licorice sticks. He wants to share them with 8
friends.
20 friends are sharing eight cakes.

Ask your child or your students to solve the problems using a strategy that makes sense to them. Strategy is the primary dimension of development because student-generated strategies can (and I believe should) serve as the foundation for mathematics instruction. A focus on student-generated strategies allows a teacher or a parent to begin with, and build on, what children already know, and it allows children to participate in instruction by making contributions that are personally meaningful.

Give children pencils and paper and access to any kind of manipulative they find helpful and allow them to work out the problem by themselves.

Once the task is completed, children need to be able to demonstrate to each other what they did and the answer that was found. The more students are encouraged to contribute the intact products of their own thinking to class discussions, the more likely they are to identify themselves as understanding math – no matter the level of the thinking.

The key in fraction instruction is to pose tasks that will elicit a variety of strategies and representations. Equal-sharing tasks are not the only problems that can do that, but many teachers, like myself, have found them to be a definite source of variety in thinking. Children learn from each other, and the teacher begins to get a picture of what each child knows.

Another great way to help your child or your students to understand fractions is to play a fraction game. I have found that Fraction War can be highly effective. The first level begins simply, and it is probably best to start here, even with older children. Once you are sure they understand this concept, move to the next concept level.

Fraction War

Materials:
 One deck of cards
 Fraction War Game Board (following)

Game:
Players draw cards and create a fraction. The player with the fraction with the greatest value wins a point for that round. The player with the most points when all the cards have been used is the winner.

Variations:

Concept 1:
Each player finds and places a one in the numerator position on his/her game board. This card remains in place until the end of the game. Each player draws a card and places it in the denominator position. The player with the greatest fraction wins the point. Play continues until all cards have been used.

Concept 2:
Place a one in the denominator position and play as above.

Concept 3:
Decide on a number between 2 and 10. Each player places that number in the denominator position. Play as above.

Concept 4:
Place the same number in the numerator position. Play as above.

Concept 5:
Each player draws 2 cards. The first is the denominator, the second is the numerator. Play as above.

Fraction War Game Board
Player #1 Player #2

_______________________ ______________________

Multiplication Games

More and more in my teaching career, I see that children struggle to memorize their multiplication tables. I’ve even worked in many 6th grade classrooms where it was perfectly evident that a majority of the 6th graders had not yet fully memorized their multiplication tables.

Simple multiplication is usually introduced into the math curriculum in 2nd grade, so it is important that 2nd and 3rd graders begin to get a strong handle on their multiplication facts. If they don’t, understanding division becomes ever more difficult.

I’ve found that multiplication games are wonderfully useful. Multiplication games put children in exactly the right frame of mind for learning. Children are normally very eager to play games. They relax when they play, and they concentrate. They don’t mind repeating the multiplication facts over and over.

Teachers and parents are partners in this process, and both can offer greater opportunities for their students/child to succeed in memorizing the multiplication tables. Multiplication games fit the bill wonderfully!

One of my favorite multiplication games is Multiplication Fact Feud. It’s a great way target and practice certain facts.

Multiplication Fact Feud

What you need:
2 players
deck of cards

Teacher or parent decides the particular multiplication fact to practice
(i.e. x7, x4, x8, etc.) Once the constant factor is determined, that card is placed between the two players. Players then divide the remaining cards evenly between themselves.

Each player turns over one card and multiplies that card by the constant in the middle. Players must verbalize their math sentence. The player with the highest product collects both cards.

Example: Player #1 Player #2
4 5 7
“4 x 5 = 20” “7 x 5 = 35”

Player #2 would collect both cards.

In the event of a tie (i.e. both players have the same product), each player turns over one more card and multiplies that by the constant factor. The player with the highest product wins all four cards.

When the cards are all used up, the player with the most cards wins the game.

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