Week 1: Models for multiplication & Division

This week in math we jumped right into the second unit! On Tuesday, we learned several different methods to model multiplication. On Thursday, we learned various division models. I know it's a lot of information to take in and go over so just bear with me!

I'll start off by going over the seven different ways to model multiplication:

1. Repeated addition
2. Array/Area
3. Tree (combinations)
4. Commutative property
5. Associative property
6. Identity property
7. Distributive property

The models that I personally picked up on quickly were Repeated addition, Array/Area models, Tree combinations, commutative property, and identity property.

Example 1: Repeated Addition-
Nate is going on a adventure walk and sees three bushes, with two bunnies under each bush. How many bunnies does Nate see altogether?

2+2+2 =6 

By repeatedly adding two, three times. I figured out that Nate saw a total of 6 bunnies on his adventure walk. This problem can also be easily turned into a multiplication problem 2*3 = 6. 

Example 2: Array/ Area models-
In Natalie's garden there are four rows with five daisies in each row. How many total daisies are in Natalie's garden?

4*5 = 20

I approached this problem by first drawing out four flowers vertically next to each other to resemble a "row". Then I made sure each row had five daisies. By drawing out four different rows and adding five daisies into each row I can visually see and count that Natalie has a total of 20 daisies. Although, you can solve this problem in a more complex way! By multiplying 4 by 5. Since you have four rows of the same amount (5). 

Example 3: Tree combinations-
Kelley went shopping and bought four pairs of skirts and six different tops. How many different outfits can Kelley make?

4*6 = 24 

When kids are given pictures they can physically solve it by drawing lines to each article of clothing. By connecting lines and drawing arrows this then creates the model of a tree. A simpler way to solve this problem is to multiply the two numbers (4*6).  By multiplying the two numbers, I figured that Kelley can make up to 24 different outfits.

Example 4: Commutative property-
You can switch the numbers around and the numbers will still have the same meaning.

                                                         A*B = B*A 
Example 5: Identity property- 
Anything multiplied by 1, is itself.

                                                         A*1 = A 

The next set of models are the ones we spent more time on in class. Such as, Associative property, and distributive property! 

Example 6: Associative property-
 Associate numbers to make them "friendlier". Such as multiples of five, ten, or doubles.
                                                       A*(B*C) = (A*B)*C

Example 7: Distributive property-
Distributive property reminds me of the FOIL method I was taught in middle school. You take the number that is not in the parenthesis and multiply it by the numbers that are in the parenthesis. Then add the two sums. (It doesn't matter which number you multiply first). Also, to help me remember which numbers I am multiplying I draw arrows.

Now For Division! 

As a kid I was not a fan of division. Because the process to do a long division problem was extremely tedious and self draining. By going over strings, models of division, partial products, area models, and ratio tables. I became more comfortable with dividing. Class on Thursday did not start off with a division problem. Instead class began with a slide saying, "Using Strings" with problems such as:

Example 1. 10*3 = 30
                    9*3 = 27

Now, I'll admit multiples of nine are not exactly my cup of tea. I do not find them friendly. Looking at these numbers at first glance of course I would assume they're multiples of three. Although there is a strategy behind this problem. The relation between the solutions 30 and 27 is that they have a difference of 3. By knowing my multiples of ten I can relate it back to my multiples of nine. Christina made a valid point through teaching us "strings"

"Teach a strategy instead of a trick. So it'll work for all numbers."

Another strategy we learned recently was rounding. We round to compensate to make a "nice" number. By building connections and relations. For example:

Example 2. 39*7 = ? 

                                      (I can compensate the 39 to a 40, to make it into a nice number)  

                   40*7 = 280 

                                      (Subtract 7 from 280. Since we compensated a unit of 7 earlier)                

                   280-7 = 273    

                                                             

There are two different kinds of subtraction problems. Repeated subtraction and partition subtraction. 

Example 3. Repeated Subtraction:

Nohelani has $20 she wants to give $4 to each of her friends to buy mothers day presents. How many friends can she give money to? 

               What we know: -Total amount = $20

                                         -Number of members = $4 

Solution: You can solve this problem by drawing out groups of 4 to find out that Nohelani can give $4 to 5 of her friends. Because 4*5 = 20. 

Example 4. Partition Subtraction:
Maria has $20. She wants to give an equal amount of money to 5 friends. How many friends can Maria give money to?
                 What we know: -Total amount = $20 
                                           -Number of groups = 5 friends 
Solution: You can solve this problem by starting with the 5 friends, and figure that each friend gets $4. Since 20/5 = 4.

Next, we learned about Partial products, Area models, and Ratio tables!

Partial products is when you take a traditional multiplication problem (two numbers stacked on top of each other) and multiply each place value like shown in my picture:

An area model is when you use a grid and split the numbers into friendly numbers (by place value). Once I divide the numbers into friendlier numbers. I write the multiplication problem into each box so I can visually see the numbers I will need to add. After just three simple steps I can see each of my steps, and understand how I got my answer. For example, the problem 123*26 = ?

A ratio table for 26*123 is as follows: