Week 3  (Chip Models)

Note: While using the chip model in math, the black (colored circle) chip is positive and red (open circle) is negative.

Addition on Integers
4 + (-7) = ?

Step 1: The first step to do fill the first circle with the first number.  In this case you would fill in the first circle with a positive 4.  You would then fill the second circle with a negative 7 since that is the second number in the equation.

Step 2: You would then combine the two circle together into one, which mixes all the chips together.

Step 3: Lastly, you need to pair off the positive chips with the negative chips.  In this case there are 3 negative chips left which means the answer is -3.

Subtraction of Integers:
5 - (-2) =?

Step 1:  For subtraction, you first need to fill the first circle with the first number in the set.  

Step 2:  Since there is two negative chips, pair them with two positive chips to make the zeros.

Step 3:  You then continue to the second number that is in the equation that says to take away -2 chips.  After you take those away, count how much you have left which is going to be a positive 7.

Multiplication of Integers

4 x (-2) = ?

Step 1:  For multiplication, you first need to know how many sets of what number you are doing.  In this case, you are doing 4 sets of -2.  

Step 2:  You then combine the chips together into one, and that's your answer.

Division of Integers

-8 / 2 = ?

Step 1:  In division, you first need to put the first number of chips into the circle.

Step 2:  Then divide that number by your second number so there is an equal amount of chips into each circle.  

Step 3:  How much chips is in each circle is your answer.  In this case the answer is 4.

Expanded Notation

(When breaking up a number, we usually break it up by each digit.  Any number after the first digit should be zero(s).  You can check your work by adding all the numbers together and you should end up with your original number.)

(We can also break up multiplication problems)

(Scientific Notation can also be a different way to show the result of the broken down numbers)

Number Bonds - Division

(Decomposing a number to make it more friendly)

During this week of class we learned the models for multiplication and division.
Multiplication Models

1. Repeated Addition Model
Example:
Ryan has 4 plates of cookies with 7 cookies on each plate.  How many cookies does Ryan have altogether?

For this problem I drew 4 plates (circles) and drew 7 cookies in them (green dots).  After doing so, I counted all the dots together in every plate.  I could have also multiplied 4 by 7 or add 7 together 4 times.  Just be aware of what facts the question gives you and draw it out to help you visualize it.  Then look at the question which usually involved the "total" of something.  

2. Array/Area

Example:

In the classroom there are desks lined up in 4 rows with 7 desks in each row.  How many total desks are in the classroom?

                                   

For this problem I started with labeling 4 rows with numbers.  According to the question, there are 7 desks in each row, so I counted 7 blocks and put it into each row.  In order to find the total amount I counted all the blocks that I drew out.  *For this type of problem, you usually have to use column and rows.

3. Tree/Combinations

Adam has 4 pairs of pants and 7 different shirts.  How many different outfits can he make?

To solve this problem I drew 7 shirts in a row at the top and 4 pants in a row on the bottom.  Then I drew lines connecting every shirt to every shorts.  As I was drawing the lines, I counted each line which led me to my answer.

Properties of Multiplication

1. Commutative Property

2. Associative Property

3. Identity Property

4. Distributive Property

Division Models

1. Repeated Subtraction (? * members = total)

Example: 

Meghan has $48.  She wants to give $6 to needy kids for school supplies.  How many kids can she give money to?

When I first did this problem, I drew out 48 dots in pairs of 10 and then circled groups of 6 dots.  It looks very complicated but it made sense in my mind.  I'm not sure if that is the best way to solve it, but it worked for me.

2. Partition (group * ? = total)

Example:

Meghan has $48. She wants to give an equal amount of money to 6 needy kids for school supplies. How much can she give each kid?

I solved this problem by drawing 6 circles that represents each kid.  Then I drew a line in every circle and counted up to 48 lines.  To check my work I made sure that each circle had the same amount of lines.

Area Model Practice

1. Partial Products

After learning this Partial Products Technique, I found that multiplying big numbers is actually not hard at all.  I found this way of doing multiplication a lot easier than the traditional way.  

2. Area Model

This area model is just like an Open Array model.  

3. Ratio Model

This is the one technique I found difficult.  I won't be using this technique unless I really had to.