# Research and Background What are the "basic facts"?

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Research and Background

What are the “basic facts”?

Most research agrees that basic facts for addition refer to combinations where both addends are less than 10 and basic facts for subtraction refers to the corresponding addition fact; for example, 17 – 9 = 8 is a basic fact for subtraction while

17 – 4 = 13 is not a basic fact. Likewise, basic facts for multiplication have both factors less than 10 and division facts are those that refer to corresponding multiplication facts.
What does it mean to “know” a fact?

The rule of “3” – if a student can consistently give a quick response (in about 3 seconds) to a fact without resorting to an non-efficient method, such as counting, then they have mastered that fact.

Why is knowing the basic facts important?

“Fluency with basic facts allows for ease of computation, especially mental computations, and therefore, aids in the ability to reason numerically in every number-related area.” Van de Walle, John A. (2007). Helping Children Master the Basic Facts. Elementary and Middle School Mathematics, Teaching Developmentally, 165

Which of our students should we expect to know their basic facts?

“All children are able to master the basic facts – including children with learning disabilities. All children simply need to construct efficient mental tools that will help them.” Van de Walle, John A. (2007). Helping Children Master the Basic Facts. Elementary and Middle School Mathematics, Teaching Developmentally, 165

I learned my facts by just memorizing them; we drilled until we knew them; what’s wrong with that?

“Students who memorize facts or procedures without understanding often are not sure when or how to use what they know, and such learning is often quite fragile. Learning with understanding also makes subsequent learning easier. Mathematics makes more sense and is easier to remember and to apply when students connect new knowledge to existing knowledge in meaningful ways. Well-connected, conceptually grounded ideas are more readily accessed for use in new situations.” National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics, 20

Drill is by far the most popular method for working on basic facts in our school, but, if that worked for all students, we would not have students in the next grade level still struggling with the facts or students in middle school and beyond that do not know their facts.
Drill is appropriate and beneficial for students that have an efficient strategy that they understand, like and know how to use - but have just not become proficient with it.
How can I help students learn the basic facts?

Children learn math the same way they learn everything else - by constructing their own knowledge. Another way of saying this is, even if you teach or ‘feed’ a child math facts, they will have to experiment with, explore and think about numbers before they really learn about and understand numbers and how they work. Research has proven that math facts can, in fact, be learned by young children using "two kinds of activities: situations in daily living … and games.” Kamii, Constance (1985). Young Children Reinvent Arithmetic

So, what is One-to-One Math?

One-to-One Math was developed based on the following beliefs:

1. All students can and should learn math.

2. If children like math and feel successful at math, they will learn math.

1. For children to like math, it must be fun! Students like to play games, so One-to-One Math is a game based program.

2. For children to feel successful at math they must be successful. One-to-One Math starts students with concepts they can quickly master and then builds on that success.

3. Students need to go through three stages as they learn math: 1. concrete or manipulative, 2. mental representational, 3. abstract or symbolic

One-to-One Math was developed to help teachers, parents, para-professionals, community members, etc. work with students on math facts and critical concepts.

One-to-One Math is a non-profit program – a CD is provided with all of the materials that we use in the training. Please share these activities, materials, and information with others who are involved in helping our children become life-long learners of mathematics!
Suggested Questions/Prompts

To make sense of mathematics:

• Tell me what you are thinking.

• How would you explain this to a student who doesn’t understand?

To foster predicting, inventing, and problem solving:

• What would happen if…?

• Is there a pattern? What is it? Why not?

• What decisions can you make from this pattern?

• Can you do it a different way?

• What is the same or different about your two ways of doing this?

• Will it be the same if we use different numbers? Why or why not?

To rely more on themselves:

• Does it make sense to you? Why or why not?

• What do you think?

• What would seem more reasonable to you? Why?

• How can you check to see for yourself?

• What do you want to do next?

• Can you draw a picture or build a model to illustrate the problem?

To foster reasoning:

• Will what you did always work this way? How do you know?

• Do you see a pattern in this? What is it?

• How could it be done a different way?

• Can you explain your reasoning?

• Could you explain this in another way?

• What other numbers will work?

• Are there some numbers for which it will not work? How do you know?
• Write a new problem that is different in some ways but the same in others.

To help connect and apply mathematics:

• Have you ever solved a problem like this before?

• Tell (or write) a story problem that uses this kind of mathematics.

Researchers have separated addition and subtraction problems into three categories: join problems, separate problems, and part-part-whole problems.

These categories are based on the different types of relationships involved.

Each category can then be divided into sub-categories depending upon which of the three quantities in the problem is unknown.
In most mathematics curricula, the major emphasis is on the easier “join” and “separate” problems with the result as the unknown part. This leads to the definitions of addition as “put together” and subtract as “take away”. These definitions are limited and if these are the only exposure students have, they will have difficulty when the situation calls for something other than “put together” or “take away”. Take for example, the following problem: Bob has 3 nickels and Bill has 7 nickels. How many more nickels does Bill have than Bob?
Students need exposure to all the different types of addition and subtraction problems.
Examples of Join Problems

Join Problem: the result is unknown

Katie has 8 baseball cards. Mason gave her 4 more. How many baseball cards does Katie have altogether?

Join Problem: the amount of change is unknown

Katie has 8 baseball cards. Mason gave her some more. Now Katie has 12 baseball cards. How many baseball cards did Mason give her?

Join Problem: the initial amount is unknown

Katie has some baseball cards. Mason gave her 4 more. Now Katie has 12 baseball cards. How many baseball cards did Katie have to begin with?

Examples of Separate Problems

Separate Problem: the result is unknown

Katie had 12 baseball cards. She gave 4 baseball cards to Mason. How many baseball cards does Katie have now?

Separate Problem: the amount of change is unknown

Katie had 12 baseball cards. She gave some to Mason. Now she has 8 baseball cards. How many baseball cards did she give to Mason?

Separate Problem: the initial amount is unknown

Katie had some baseball cards. She gave 4 to Mason. Now she has 8 baseball cards left. How many baseball cards did Katie have to begin with?

Examples of Part-Part-Whole Problems

Part-Part-Whole Problem: the whole is unknown

Mason has 4 baseball cards and 8 basketball cards. How many cards does he have?

Mason has 4 baseball cards and Katie has 8 baseball cards. They put their baseball cards together in a notebook. How many baseball cards did they put into the notebook?
Part-Part-Whole Problem: one of the parts is unknown

Mason has 12 cards. Eight of his cards are baseball cards, and the rest are basketball cards. How many basketball cards does Mason have?

Mason and Katie put 12 baseball cards into a notebook. Mason put in 4 baseball cards. How many baseball cards did Katie put in?
Examples of Compare Problems

Compare Problem: the difference is unknown

Mason has 12 baseball cards and Katie has 8 baseball cards. How many more baseball cards does Mason have than Katie?

Mason has 12 baseball cards and Katie has 8 baseball cards. How many fewer baseball cards does Katie have than Mason?
Compare Problem: the larger amount is unknown

Mason has 4 more baseball cards than Katie. Katie has 8 baseball cards. How many baseball cards does Mason have?

Katie has 4 fewer baseball cards than Mason. Katie has 8 baseball cards. How many baseball cards does Mason have?
Compare Problem: the smaller amount is unknown

Mason has 4 more baseball cards than Katie. Mason has 12 baseball cards. How many baseball cards does Katie have?

Katie has 4 fewer baseball cards than Mason. Mason has 12 baseball cards. How many baseball cards does Katie have?
Using 5 and 10 as Benchmark Numbers
5 and 10 are powerful numbers that can be used as anchors to “build” other numbers. We want children to be able to be able to recognize the combinations that make these numbers.
Start the process with a simple game called I Wish

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I have 3 cars in my parking lot; I wish I had 5 cars. How many more cars do I need? Be sure to ask them how they know they are correct. You may want to model your thinking by using the five-frame to build and draw the problem.

5-frame
This is a very good formative assessment of your student. Watch to see how they find the missing added – do they know the amount or do they have to unit-count to find the missing number.
A second game to play with the student is Make Five Go Fish. This game uses the (0-10) number cards. Pull out all of the cards above 5and the wild cards from the deck before you begin play. On the CD there is a deck of cards that have 5-frames on them instead of 10-frames if you would like to use them instead.
Make Five “Go Fish”

Materials: number cards 0, 1, 2, 3, 4, 5 (four of each) from the number card deck

Objective: make sets of 2 cards with a sum of 5.

1. Each player is dealt five cards. The rest of the cards are placed down in the center of the table.

2. If you have any pairs of cards that total 5, put them down in front of you and replace those cards with cards from the deck.

3. Take turns. On your turn, ask the other player for a card that will go with a card in your hand to make 5.

4. If you get a card that makes 5, put the pair of cards down. Your turn is over.

If you do not get a card that makes 5, take the top card from the deck. Your turn is over.

If the card you take from the deck makes 5 with a card in your hand, put the pair down. Your turn is over.

1. If there are no cards left in your hand but still cards in the deck, you take two cards from the deck.

2. The game is over when there are no more cards.
3. At the end of the game make a list of the number pairs you made.

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When you feel that the student is comfortable with all of the combinations for 5, move on to the combinations for 10. Again, start with the

10-frame and the I Wish game: I have 3 necklaces in my jewelry box; I wish I had 10.

Change the rules to where they are now looking for combinations that make 10 and play Go Fish. These cards are designed so that the student can count the missing part for 10. You may want to model out loud how to use them: I have a 7; what I should ask you for; oh, I’m missing 3. Do you have a 3?
When you notice that they are getting a fairly good grasp on the combinations that make 10, you may want to try the 10-beads with them. Notice that the beads are designed with 5 of each color – this is to help them “quick count” numbers such as 8 by seeing that 8 is 5 and 3 more. Hide the beads behind your back, move some of them into the palm of your hand, show the student the rest of the beads and ask them, “How many are hiding in my hand?”

For variety, you can use the 10-frame cards or the 10-beads and one of the board games. Take turns; turn over one of the cards; the missing addend is the number of spaces that the player gets to move.

Combinations for 10 are crucial for other strategies that we will use with the rest of the addition and subtraction facts. Keeping that in mind, we have tried to give you a variety of games to play with your students to practice that concept. Two additional games to play with your students are Snappo and Rummy.
SNAPPO

Materials: 0-10 number cards

Objective: to recognize pairs of addends with a sum of 10 and to “capture” those cards.

1. Deal out all the number cards (0-10) face down into 2 stacks.

2. Player #1 lays the top card from his/her stack face up on the table.

3. Player #2 lays the top card down from his/her stack face up on the table. If that card makes the sum of 10 with the other card that is already down on the table, he/she should place it face up beside the other card and call out SNAPPO. He/she has captured the two cards and should place them in his/her collection of captured cards. If the card does not make a SNAPPO, it is still placed face up in the center of the table.

4. As play continues, the new card that is turned over can be matched with any card that is already on the table that makes a sum of 10.

5. Any person recognizing a match may call SNAPPO and capture the cards.

6. The game continues until there are no matching cards remaining.

Make 10” RUMMY

Materials: 0-10 number cards

Players: 2 or more

Objective: Make sets of two cards that add up to 10.

1. Deal out 7 cards to each player.

2. Turn over the next card and start a discard pile - lay it down on the table so everyone can see the number.

3. Place the remaining deck of cards face down in the center of the table.

4. Check to see if you have any sets – a set is 2 cards that have a sum of 10. If you have a set, you may play it (lay it down) when it is your turn.

5. Each player in turn can either draw a card from the remaining deck or pick up the top card from the discard pile. If you wish to pick up more than the top card from the discard pile, you must be able to make a set with the last card in the stack that you picked up.

6. The game ends when a player “goes out” (has no more cards) or when there are no more cards left in the deck. The winner is the player with the most cards in his or her “captured” pile.

7. Calculate you score by adding 5 points per card for any sets that you have laid down and subtracting 5 points per card for any cards that remain in your hand when the game is over.

Don’t feel like you have to play all of these games an equal amount of time. We have found that some students will like a certain game and ask to play it over and over – that’s fine! As long as they are having fun, feeling successful, and learning math, we are accomplishing our goal!

Feel free to change the rules of any of these games to make them better suited for your students!

As you do the above games, don’t forget to model your thinking, have the students build and draw the problems, and always ask the student to say the addition problem two ways:

Example: 2 + 3 = 5 and 3 + 2 + 5 or 3 and 2 make 5 and 2 and 3 make 5

Make Five and Make 10 are games where the student circles pairs of addends that have a sum of 5 or 10. On the CD, there are two additional versions of each game. You may want to time them and see if they can improve on their time each week.

 Make Five Make Ten 1 4 2 3 0 1 9 2 3 6 4 5 3 2 4 5 5 4 8 7 9 7 0 4 1 1 1 5 6 0 6 1 3 3 2 5 3 4 3 7 10 4 5 5 4 1 0 2 0 2 2 8 0 1 9 2 3 1 4 5 8 3 7 0 6 4

It will probably take longer that you had thought (or planned) for the student to become proficient at the combinations for 5 and 10 and you might be tempted to move on before they have mastered these concepts. PLEASE DON’T! Combinations for 5 and 10 are critical concepts. We will build other strategies and concepts around these as we work through the rest of the addition and subtraction facts.

Make It Fun!!

Mastery of a basic fact means that a child can give a quick response (in about 3 seconds) without having to resort to an inefficient method such as counting.
An efficient strategy is one that can be done mentally and quickly. Our goal is to help the student develop an efficient strategy and then provide practice of that strategy through games.
Adding and Subtracting 0, 1, and 2

Adding and subtracting zero does not require any strategy; just a good understanding of the meaning of zero and addition/subtraction. Even though it does not require a strategy for adding and subtracting zero – don’t neglect it! Be sure to give them several story problems to model for adding and subtracting zero.

For adding one and two, focus on the strategy of “more than” rather than “counting on”; for example: 7 is 1 more than 6; 2 more than 7 is 9. If a student uses the “counting on” strategy for these two addends, don’t try to stop that use, but do discourage that strategy for larger addends. Likewise, for subtracting one and two, focus on “less than” rather than “take away”.
Start with working on the relationship of numbers that are 1 more or less than a given number before you do 2 more or less than a number. The number line is a great tool to help students see this relationship of numbers. You can cover all of the numbers on the number line except for one number and ask the student, “What is 1 more than 8”? or “What is 1 less than 8? and “What is 2 more than 8”? or “What is 2 less than 8?”

Some students will have to unit-count from 1 to 9 to answer the question about one more than 8 and won’t be able to quickly tell you what number is one less than 8 – don’t panic, that just lets you know where the student is developmentally – with practice, they will develop that understanding.

After using the number line for these questions you can try different ways to practice this concept – turn over one of the 10-frame cards and ask the same type questions, or let the student turn over a number card and then spin the less than/more than spinner or roll the less than/more than die to generate the problem. There is also bingo games to practice one/two more than and one/two less than.

Adding and subtracting 10 to any number without having to unit count is an extremely important concept – not only in learning the basic facts but later when we work with different strategies to add and subtract 2-digit numbers.

 Number Ten More Than The Number
To work on adding ten, have the student use the double 10-frame and build problems such as 10 + 3 and 4 + 10. Ask questions such as, “What’s ten more than 3?” We want the student to see the pattern of what happens when we add 10 – please don’t tell them the pattern, just do enough problems and keep asking them if they see a pattern. There is a recording chart that should be helpful for the student to see the pattern for adding ten.

The 0-10 number cards, 0-9 die, 0-9 spinner, 0-10 spinner can all be used to help generate problems. Turn over one of the cards, spin the spinner, or roll the die and ask the student, “What is 10 more than ??”.

To work on subtracting 10, ask the student to build a number such as 17 on the double 10-frame and then ask them to subtract 10. There is a recording chart for ten less than a number and number cards from 11 through 20 made with double 10-frames on the cards to help with the visualization of subtracting 10.
A hundreds chart is a great tool to use to look for patterns.
You can use Math Basketball, Math Baseball, Math Race, or Catch Me If You Can as a game to practice these concepts.

Ask the student to build 9 + 5 on the double 10-frame.

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