By now our students are likely beginning to understand and feel comfortable with multiplication. The good news is that division is a natural extension of multiplication...if we remember our multiplication tables, we can generally come up with the answer to a division problem using the same numbers. But what does it really mean to “divide” numbers? Once again we have the opportunity to teach a deeper concept, as well as help our students understand why this is important.

If we let Merriam-Webster’s Dictionary define division for us, we see that in mathematics division is, “the process of finding out how many times one number is contained in another”. For a more visual definition, we can begin to think about dividing a number of apples into a number of baskets or shopping bags.

Let’s say we are selling apples at a local farmers’ market. We have 50 apples to sell, and all at once we get five customers. We sell the same number of apples to all five customers, and when we are finished we have only ten apples left. How many apples did we sell to each customer? This is a great example of a real-world division problem! We can use our 9 Apples cards to demonstrate this idea visually, and make it more clear to our students.

Let’s begin by our cards, face down, to represent the apples. We begin with 50 apples at our apple stand:

We know we are left with 10 apples at the end, so we know we have sold 50 - 10 = 40 apples. So our first task is to remove 10 apples, leaving the 40 sold apples.

We also know that each of five customers purchased the same number of apples. This means we must divide 40 apples equally among our five customers. This becomes the equation 40 ÷ 5.

We know from our multiplication tables that 8 x 5= 40 (this may be a good time to remind our students of their multiplication tables, and ask what times 5 equals 40). Now we can ask our students to group their 40 apples into equal piles for each customer. There should be 8 apples in each of 5 piles.

We know from our multiplication tables that 8 x 5= 40 (this may be a good time to remind our students of their multiplication tables, and ask what times 5 equals 40). Now we can ask our students to group their 40 apples into equal piles for each customer. There should be 8 apples in each of 5 piles.

Let’s try another problem: What if we have 80 apples at our stand, and we have 9 customers. What is the most apples can we sell, and still give each customer the same number of apples? How many apples will we have left?

We may begin with 80 cards face down (this is a good opportunity to show your students how important knowing division is...80 cards will get quite cumbersome! You wouldn't want to use this method with very large numbers.). We now must divide this number of apples into 9 equal piles, giving each customer the maximum number of apples while still giving each the same number of apples. We’ll begin by asking what is 80 ÷ 9? If we actually calculate this equation, we find that 80 ÷ 9 = 8.88. This means we do not have enough apples to divide them all equally between 9 customers. Since this number is between 8 and 9, we can sell each customer 8 apples, and will have some left over. If we divide our apples into 9 equal piles of 8, how many are we left with? We know from our multiplication tables that 8 x 9 = 72, therefore we will have 8 apples left (80 - 72 = 8).

Now that your students understand what division is, and why it’s so important, help them practice their skills in dividing numbers by playing the 9 Apples Base Game with Booster Pack! See a tutorial on how to play by clicking here.

]]>We may begin with 80 cards face down (this is a good opportunity to show your students how important knowing division is...80 cards will get quite cumbersome! You wouldn't want to use this method with very large numbers.). We now must divide this number of apples into 9 equal piles, giving each customer the maximum number of apples while still giving each the same number of apples. We’ll begin by asking what is 80 ÷ 9? If we actually calculate this equation, we find that 80 ÷ 9 = 8.88. This means we do not have enough apples to divide them all equally between 9 customers. Since this number is between 8 and 9, we can sell each customer 8 apples, and will have some left over. If we divide our apples into 9 equal piles of 8, how many are we left with? We know from our multiplication tables that 8 x 9 = 72, therefore we will have 8 apples left (80 - 72 = 8).

Now that your students understand what division is, and why it’s so important, help them practice their skills in dividing numbers by playing the 9 Apples Base Game with Booster Pack! See a tutorial on how to play by clicking here.

What does it mean to “multiply”? As teachers, most of us have done it for many years; we may even take it for granted. But we must be able to teach our students not only the standard multiplication tables, but also an understanding of what multiplication really is, and why it is important.

So how do we define multiplication? Merriam-Webster’s Dictionary defines multiplication as follows:

process of adding an integer to itself a specified number of times

and that is extended to other numbers in accordance with laws

that are valid for integers”

To rephrase, multiplying two numbers is just shorthand for repetitive addition--for adding one number to itself over and over again, the number of times depending on what number you have multiplied it by. We can use our cards to demonstrate this idea visually, and make it more clear to our students.

For example, let’s say we want to explore the problem 2 x 3:

For example, let’s say we want to explore the problem 2 x 3:

We may think of this as two of the number 3 being added. Using our cards to visualize this problem, we may think of the first number in the equation as the number of cards used, and the second number as the number on the cards:

This translates our multiplication problem into an addition problem: 3 + 3. Now we may use our cards again:

We know 3 + 3 = 6, and this is the same as 2 x 3. To satisfy ourselves that the problem will work the same way if we reverse the 3 and the 2, we may use the same visual test. The new problem:

Applying our rule as before, three is now the number of cards, and two is the number on the cards:

Translating our equation into an addition problem, we have 2 + 2 + 2:

We know 2 + 2 + 2 = 6, the same as our previous answer. We can therefore see that 3 x 2 = 2 x 3.

Let’s try another equation: 6 x 5:

As per our rule, there are six cards needed, each with the number 5. This addition problem looks like this:

If we add or count the apples, we know that 5 + 5 + 5 + 5 + 5 + 5 = 30. This is the same as 6 x 5 (or 5 x 6). We are also beginning to see something important--how much more work it is to add 5 six times! We are starting to see how much faster multiplication is, and therefore how important it is.

Let’s do one more, 8 x 9:

As per our rule, there are eight cards used, each with the number 9. This addition problem looks like this:

Adding or counting the apples, we know that 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 = 72, the same as 8 x 9 (or 9 x 8). It should also by now be very clear how much easier it is to remember that 8 x 9 = 72, instead of having to add 9 to itself so many times! While our cards only go up to the number 9, it should be easy to imagine how difficult trying to add very big numbers would be (11 x 15 for instance, or 250 x 373)! Using multiplication offers us a fast and easy solution!

Now that your students understand what multiplication is, and why it’s so important, help them practice their skills in multiplying numbers by playing the 9 Apples Base Game with Booster Pack! See a tutorial on how to play by clicking here.

Experts in the field of education, as well as common core standards, state that students should be introduced to negative numbers in the sixth grade (Ryan). By this time, they should have a thorough understanding of the positive number line, how numbers change when added, subtracted, multiplied, or divided with each other. Once they comprehend this, they may be introduced to numbers less than zero, a more abstract concept. With standards in some states becoming stricter, this skill may soon be taught in fifth grade (Ryan).

Whether they realize it or not, most children do have experience with negative numbers. One excellent way to introduce your students to the idea of negative numbers is using the thermometer. Children living in colder climates will recognize the idea of negative temperatures, and based on experience they will understand that a temperature below zero is colder than zero, and thus lower. The thermometer on the right clearly shows temperatures above and below zero. |

Negative numbers may also be illustrated using images of large buildings. The floors of a building that are above ground are counted up as on the positive number line (the first floor is floor 1, the second floor is floor 2, etc). If the building has multiple levels below ground, these are denoted with negative numbers (the first floor below ground is -1, the second floor below ground is -2, etc.). The ground may be denoted as zero.

We can use the 9 Apples Base Game to illustrate how negative numbers are formed. One way they may be formed is by subtracting a larger number from a smaller number. (Many children may have been told this is not possible. You may need to clarify that this is not possible only if you wish the answer to the problem to be a positive number, which is what was meant in the previous situation.) For example, let’s take the example of a farmer’s market:

*You are running the apple stand at the farmer’s market, and you have 5 apples left. Mrs. Johnson comes to the stand, and says she wants to buy seven apples. How many more apples would you need to give Mrs. Johnson all the apples she wants?*

Mathematically, this story has the following formula: 5 - 7 = -2

You have five apples, but Mrs. Johnson wants to take away 7. This means that you are two apples short for Mrs. Johnson.

We may begin the illustration of this problem using a number line. Create a number line using the playing cards numbered -5 through 5, using the back of a card for zero (there are no zero cards in the deck). Alternatively, you may ask the student to draw the number line on paper.

Mathematically, this story has the following formula: 5 - 7 = -2

You have five apples, but Mrs. Johnson wants to take away 7. This means that you are two apples short for Mrs. Johnson.

We may begin the illustration of this problem using a number line. Create a number line using the playing cards numbered -5 through 5, using the back of a card for zero (there are no zero cards in the deck). Alternatively, you may ask the student to draw the number line on paper.

Remind the student that we have five apples, and seven should be taken away. Begin at the number five, and count out seven cards going down the number line. The student should land on -2, therefore we are two apples short.

For more advanced testing, we may present the student with an equation, and ask them to find the answer. For example, if we present the equation 6 - 9 = ?, we may ask the students to construct a number line to find the answer, or we may offer a few cards as possible solutions and ask them to choose the correct one.

We must also address the way negative numbers affect multiplication and division. One of the most difficult things to grasp is why multiplying a negative number and a positive number should equal a negative number. For an excellent explanation of why this principle makes sense, please watch this video from Khan Academy.

We may present students with equations using the cards, as well as presenting them with possible answers. For example, we may present the equations 3 x -2 = ? or -8 ÷ 4 = ?.

We may present students with equations using the cards, as well as presenting them with possible answers. For example, we may present the equations 3 x -2 = ? or -8 ÷ 4 = ?.

For additional practice, play the game of 9 Apples (with or without addition of the booster pack, depending on whether you wish to exercise multiplication and division skills) and require that all the numbers formed be negative! Instead of starting at +9, going to +8, and so forth, begin at -9, then -8, etc. Please see our tutorial on playing 9 Apples, or see the game rules for more information.

Citations:

Ryan, A.J. “When do kids learn negative numbers?” Global Post. http://everydaylife.globalpost.com/kids-learn-negative-numbers-3458.html

]]>Citations:

Ryan, A.J. “When do kids learn negative numbers?” Global Post. http://everydaylife.globalpost.com/kids-learn-negative-numbers-3458.html

In this tutorial we will show two players playing 9 Apples using the Base Game and the Booster Pack. We will show you typical hands of cards during game play, and how those hands may be used to create equations equaling the numbers 9 and 8 (the first two numbers that each player must create equations to equal). We will also show you all of the final equations created by the player who first got to the number 0, and thus ended the game.

First hand, Player 1:

Following shuffling and dealing 7 cards to each player, the first hand of Player 1 looks like this:

As per the rules, **the first equation must be equal to +9**. To generate an equation with the greatest number of cards, and thus the greatest number of points, Player 1 creates the following equation:

(-9 x -5) / 5.

**NOTE: While there are no parentheses cards in the deck, the player may imply their use by placing the cards in groups, which will make it clear to the other players that these functions are being performed first. **

The wild card (which may take any number value 1 through 9, or -1 through -9) stands for the number 5. The remaining cards in this player's hand are a sign change card, and a multiply card.

(-9 x -5) / 5.

The wild card (which may take any number value 1 through 9, or -1 through -9) stands for the number 5. The remaining cards in this player's hand are a sign change card, and a multiply card.

Player 1 is unable to make another equation which equals +8, and she now chooses to discard the multiply card. Play then moves to Player 2.

The hand dealt to Player 2 looks like this:

This player's first equation must also equal 9. She uses these cards to create the following equation:

(9 / 3) + 6. The remaining cards in this player's hand are -9, -8, and -6.

(9 / 3) + 6. The remaining cards in this player's hand are -9, -8, and -6.

Player 2 is unable to make another equation which equals +8, and she now chooses to discard the -9. Play then moves to Player 1.

On her last turn, Player 1 had one card in her hand. She must now draw 6 cards to begin her turn with the required 7 cards. Below is the resulting hand:

She must now use these cards to create an equation which equals 8. She uses these cards to create the following equation: 2 + 5 + (1 / 1). She does this using a sign change card to turn the -2 card to a +2, and another sign change card to turn the -5 card to a +5. She has no cards remaining in her hand, and is therefore not required to discard.

Player 2 had two cards remaining in her hand after her previous turn; she must therefore draw 5 cards. Below is the resulting hand:

She must now use these cards to create an equation which equals 8. She uses these cards to create the following equation: 6 + 2 - 4 + 4. The remaining cards in her hand are -8, -6, and -1.

Player 2 must now choose a card to discard, and she selects -8. Play then proceeds to the left.

Play continues with each player creating equations in reverse numerical order, from 8 to 7 to 6 and so on until one player creates a final equation which equals 0. In this game Player 1 was first to create this equation. Below is a picture of each equation she used from 9 to 0.

For more information, please see the rules for 9 Apples Base Game and the rules for 9 Apples Booster Pack.

]]>In this tutorial we will show two players playing 9 Apples using the Base Game only. We will show you typical hands of cards during game play, and how those hands may be used to create equations equaling the numbers 9 and 8 (the first two numbers that each player must create equations to equal). We will also show you all of the final equations created by the player who first got to the number 0, and thus ended the game.

First hand, Player 1:

Following shuffling and dealing 7 cards to each player, the first hand of Player 1 looks like this:

As per the rules, **the first equation must be equal to +9**. To generate an equation with the greatest number of cards, and thus the greatest number of points, Player 1 creates the following equation:

9 - 7 + 8 - 1. The remaining cards in this player's hand are +3, -4, and -7.

9 - 7 + 8 - 1. The remaining cards in this player's hand are +3, -4, and -7.

Player 1 is unable to make another equation which equals +8, and she now chooses to discard the -7 card. Play then moves to Player 2.

The hand dealt to Player 2 looks like this:

This player's first equation must also equal 9. She uses these cards to create the following equation:

5 + 2 + 3 - 1. The numbers 5, 2, and 3 have been changed from negative numbers to positive numbers by the use of the sign change cards. Player 2 has no more cards in her hand, therefore there is no need to discard. Play moves to the left.

5 + 2 + 3 - 1. The numbers 5, 2, and 3 have been changed from negative numbers to positive numbers by the use of the sign change cards. Player 2 has no more cards in her hand, therefore there is no need to discard. Play moves to the left.

On her last turn, Player 1 had two cards in her hand. She must now draw 5 cards to begin her turn with the required 7 cards. Below is the resulting hand:

She must now use these cards to create an equation which equals 8. She uses these cards to create the following equation: 3 + 1 + 4 + 4 - 4. She does this using a sign change card to turn one of the -4 cards into a +4 card. The remaining card in her hand is -8.

Since Player 1 has one card remaining in her hand, she must discard this card. Play then moves to the left.

Player 2 had no cards remaining in her hand after her previous turn; she must therefore draw 7 cards. Below is the resulting hand:

She must now use these cards to create an equation which equals 8. She uses these cards to create the following equation: 8 - 8 + 7 + 5 - 4. The remaining cards in her hand are -2 and -9.

Player 2 must now choose a card to discard, and she selects -2. Play then proceeds to the left.

Play continues with each player creating equations in reverse numerical order, from 8 to 7 to 6 and so on until one player creates a final equation which equals 0. In this game Player 2 was first to create this equation. Below is a picture of each equation she used from 9 to 0.

For more information, please see the rules for 9 Apples Base Game.

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