Addition & Subtraction

We went over a lot of information this week in class! We reviewed a few more things with the venn diagrams from last week and then moved on to some addition and subtraction. I took a lot of notes so bare with me, the information is all very important.

Models for Addition

There are two different models, one is called a set and one is called a measurement or number line. Here is an example of each:

Set: Jake has 7 chocolate chip cookies and 8 oatmeal cookies. 

How many cookies does he have altogether?

The main thing to keep in mind with a set is that it is two distinct sets that don't come together unless we physically put them together. So picture it like two separate plates of cookies, but not combining them. The way this problem is worded makes it difficult for young kids to understand that they should be adding the two types of cookies together without an action or something changing.

Measurement/Number line: Jake has 7 chocolate chips cookies. 

He bakes 8 more, how many does he have now?

The key point to keep in mind with a measurement model is that something is changing throughout the problem. For example, a time sequence if we are referring to a number line. So first Jake had 7 cookies, but then he made 8 more which was the action in the problem. This model makes a lot more sense to young kids in terms of an addition problem. They can see that because Jake made 8 more, they should add 7 + 8. Both the set and measurement models above are essentially getting at the same thing, to add 7 + 8, but the way a problem is worded can affect the outcome of what a kid thinks he/she is supposed to do.

Properties of Addition

There are 4 different properties that we went over in class that will probably ring a bell from your awesome memories of math as a kid. Okay, maybe that was a little bit sarcastic considering most of us hated these properties. I think we can agree that we were taught to strictly memorize what they were rather than learn the meaning behind them and how we were supposed to apply them to a math problem.

Commutative Property: the order doesn't matter, the act of moving numbers around to make them more friendly numbers

a + b = b + a  or with numbers  4 + 8 = 8 + 4  

Associative Property: using parenthesis to move numbers around to make them more friendly numbers

a + (b + c) = ( a + b) + c  or with numbers (8 + 7) +3 = 8 + (7 + 3) 

It is easier to first do 7 + 3 because it adds up to 10 which is then a friendly number for kids to move forward from. Once they have found 10, they can fairly easily add 8 to 10 and get 18.

Identity Property of Zero: any number plus 0 remains itself or keeps its identity

a + 0 = a or with numbers 5 + 0 = 5 

This concept is extremely hard for young kids to understand that you can add "nothing" to another number. As adults, it seems so simple because we were taught this concept when we were young. Now think about how you would explain this to kids. Using manipulatives or blocks to represent the problem can help kids grasp the concept better.

Using Properties of Learned Facts: using other tactics within a problem to help you come to a complete answer

How do you think a child would solve 7 + 9? Neither 7 or 9 are friendly numbers to work with. When I first saw this problem I realized that 9 is just one digit away from 10 which is very easy to work with. Because of that, I took 1 away from 7 which gave me 6, and added that 1 to the 9 to get 10. The problem went from 7 + 9  to 7 - 1 = 6 ; 9 + 1 = 10. Then I took 10 + 6 to get 16.

Another aspect of the Property of Learned Facts is using an addition chart. A lot of times when we're learning our addition facts, a visual with a color coated pattern can help reinforce those facts for kids to eventually have them memorized. It's important as a teacher and as a student to understand that it is okay to have a kid use these charts. 

I know it was a lot of reading and processing in todays post, but hopefully it gave you a better understanding of what some of those funky words actually mean and how to apply them to an actual math problem. I find it great importance that us, being the trained up future teachers of generations to come, educate ourselves the best that we can in order for kids to learn to love math rather than to hate it! 

Christina shared this really cool spectrum of learning concepts in class on Thursday. I really loved it!

There are so many different ways to solve the problem 7 + 9, which I used as an example above to explain the different thought processes that can be done. A kindergartener can do that problem by counting out 7 red blocks and 9 blue blues to count all the way up to 16. A first or second grader may not have all their addition facts memorized so they may either look at a chart like I showed above, or start at 9 and count up to 16 using their fingers to represent the number 7 being added. The next in line is using derived facts to solve the problem which is the way I still solved that problem. I took 1 away from 7 to make the 9 an easier number and then added 6 to 10 from what was left over. Also if the student already knows their double facts, which can come easier to others, than they may know that 9 + 9 = 18; and be able to then subtract 2 from 18 to 16. Lastly, the students will eventually reach sheer memorization of facts by looking at the numbers and being able to visualize them in their head with dots or whatever it may be and know the answer right away. It is important to know that kids get lost when we go from counting straight to facts because they have not developed along the spectrum correctly and they end up missing out and not understand the material fully. 

Thanks for reading,  hopefully this information was helpful!