Time & Base

This week we only had one day of class so I'll only have information to cover for Tuesday's class period. We went over many different time problems as well as base systems when doing a problem like time.

We are used to a base system of 10. When we were taught to subtract, we always had to "borrow" a group of ten from the next number to in a sense "roll over." We practiced that this week in class but changing the base systems to be something completely different like 3, 8, or 6. These type of systems don't exist, at least we don't use them in our math, it's just a way to get us as future teachers to understand how difficult it can be for a kid to learn these concepts as well.

When dealing with time, there are multiple base systems to keep in mind. The first is that there are 60 seconds in one minute, and 60 minutes in one hour. It is also crucial to remember that we work in a 12 hour system. So we start at start at 1, and then continue until 12 and start over again after that at 1 to end up with 24 hours in one day. Now that is confusing! Here is an example of a time problem:

Jessie started baseball practice 3 hours and 45 minutes ago. It is now 5:15. What time did Jessie's baseball practice start at? 

Possible Mistakes

In class we were assigned to think about the possible errors that a kid could make. I came up with at least three different things that could have gone wrong. 

1. a kid could forget that the base system is 60 for time rather than 10 which they are used to

2. a kid could have done 5 hours - 3 hours = 2 hours

3. try to use the traditional method of borrowing and carrying and get confused

Number Line Method:

I first did the problem on a number line because that is the easiest tool for me to understand how time works with the visual of seeing each jump of time in either minutes or numbers.

So I first started at 3:45 and jumped up 15 minutes to get to 4 o'clock because that was the next easier time to deal with. Remember my goal here is to end up at 5:15 because that is the current time. Once I was at 4 o'clock I jumped 1 hour to 5 o'clock. Our goal is 5:15 so then I knew I only needed to jump 15 minutes more to get there. Last, I added up the numbers above each jump which were 15 min + 15 min + 1 hr. = 1 hour and 30 minutes, or 1:30 which is the correct answer for what time Jessie's baseball practice started.

Algorithm Method: 

After completely the problem on the number line, Christina asked us to do it with the algorithm. This is really confusing to explain so try and stay with me! 

I lined up the numbers the old school way and then stared because I could not figure out how to borrow time! The base was in 60 minutes not 10 so how the heck does that work?! 

I took one group of 60 minutes away from 5 hours making it 4 hours. Are you following?

Then naturally we have to add the one to 15 minutes, but thats confusing because its not a base 10 still. So what really happened to the 15 minutes was I added 60 minutes to it because we took 60 minutes away from the 5 hours. So the 15 minutes became 75 minutes by doing 15 min + 60 min = 75 minutes. Confused yet?

Then I rewrote the problem as you can see in the picture above with 4:75 - 3:45. NOW you are able to subtract as we normally would to get 1:30 for the start time of Jessie's practice. I absolutely hated doing it this way! My friend sitting next to me in class has to keep explaining it to me over and over until I fully understood. Think about how frustrated an elementary aged kid would be trying to do this method! I'm going to stick to the number line, what about you?

I have had such a fun time blogging about math these past six weeks.  I hope my blog has been helpful to you and understanding the concepts you learned as a kid. To conclude this lesson on algorithms and confusing base systems, I'll leave you with this cheesy math cartoon :

Hands on Addition & Subtraction

Using Visuals to Solve Problems

Tuesday in class we have been going over different types of alga rhythms to do math problems. Now alga rhythms is a big word, but it basically is just a big math term that explains different ways to solve problems.

We first began using those bright yellow hundreds, tens, and ones cubes from elementary school. Although I was definitely a hands on and visual learner as a kid, when we pulled these out in class I had a little flash back that was not quite pleasant.

One is represented by one little cube. The tens are represented by 10 little cubes put together to equal one stick of 10. Lastly, the hundreds slate is made up of 10 sticks of 10 to equal up to 100 little cubes. Bare with me, I'll have pictures to represent this better later on.

So with these manipulatives, we rolled a dice to represent each place value.
My group rolled a 6 for hundreds place, 11 for the tens, and another 11 which represented the ones. Here is what that looked like when we put all the numbers together and counted out each place value.

As you can see in the picture, we selected the correct amount of slates, sticks, and cubes for each number but it could be simplified more. So our next task was to simplify the manipulatives as much as possible.

Here is what our manipulatives looked like after combining and simplifying. 

We first realized that we had 11 sticks that represent 10. We know that 10 sticks equals 100 which is what a slate is worth, so we traded the 10 sticks for 1 slate equal to 100. Then we were left with 1 stick representing 10, and 11 cubes. We knew that 10 cubes equal 1 stick which is also worth 10. So we traded the cubes in for a stick worth ten. This left us with 7 slates adding up to 700, 2 sticks adding up to 20, and 1 cube. The total was then 721. 

Methods to our Madness 

On Thursday in class we went over all the different ways to solve an addition and subtraction problems. There are a total of six different ways to solve one addition problem, but I am just going to show you four of those ways today. It is important for us as aspiring teachers to understand all of these different ways to do the problem because it represents all different thinking processes that potential students may have when they do this same problem.

So I used the same problem in all four ways to demonstrate what is is going on. 

34 + 28 = 62

Partial Sum 

The first strategy is called the Partial Sum. When we learned how to do this is class it blew my mind. I will never do the "traditional" method we were taught in school ever again after learning this way. The picture shows how you can take 8 + 4 and get 12 for the ones place, and then take 30 + 20 and get 50 for the tens place and then just add 12 and 50 together to get 62. This is so simple and each step is so clear for me. It is also important to understand that you can still do this method if the numbers were listed side by side and not lined up the way I have them. You also may start with the tens place first and do 30 + 20 and then do 8 + 4. Because it is addition, it is interchangeable. 

Compensation

The next method of addition is compensation. We use this to get to more friendly numbers and then have to come back and compensate for what we took away or added later on in the problem. You may be more familiar with term rounding, but that is basically what this method is showing you to do. For this particular problem I decided to round 28 to 30 so that it was easier to work with. I think knew that 3 + 3 was 6,  and 4 + 0 was 4 so I got 64. But since we added 2 to 28 to get 30, now we have to compensate for that step and subtract 2 from 64 to get our final answer of 62. 

 Decomposing

When explaining decomposing in class, Christina asked us what the actual word decomposing means when we aren't talking about math. We all answered that it is the act of something breaking down, so in this case we are breaking down the numbers to make them easier to work with. I also love the method of a number line. It is so visual and easy to see step-by-step what is happening. For this problem I started with 28 on the number line and then added 2 to get to 30 because that is a much more friendly number. Then I added another 30 to get to 60, and then just 2 more got me to 62. What can be confusing about the number line is which numbers to pay attention to. In this case we wanted to add up the 2 + 30 +2 that we created above the number line which equals 34, the other part of the problem. Once the numbers over got to 34, we knew that the number we landed on in the number line, in this case 62, was our final answer. Here's another visual to help you understand better.

Traditional

The last method is the traditional way that we learned to do addition way back when. I don't think it takes much explaining because we all already know how to do it, but I made a point of showing you this method last because we tend to get caught up doing it this way because our teachers only taught us this one way to do it. I don't know about you, but it is hard to break habit so once you see this way you don't want to do the problem any other way. 

Hopefully using these visuals and different ways of solving an addition problem will help you in the future to better understand the thinking strategies that future students may have. 

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! 

Problem Solving

This week in class we have been going through all different types of problem solving. It's so funny because as the problems were presented in class, I found myself thinking the problems were harder than they actually were. This is a common misconception with kids as well when they are presented with a word problem of this sort. Going back to the basics of math concepts can be harder than you think after learning more complex concepts in the more recent years. If you get down to it and break the problem apart step by step, they are not very difficult and hopefully won't scare kids away as much. 

Here are some step by step helpful tips to keep you going if you get stuck in the middle of a word problem! 

http://www.livingpolesapartbipolar.com/2012/07/08/effective-problem-solving-bipolar-patients/

The problem I would like to share is posted below.

In a stock car race, the first five finishers in some order were a Ford, a Pontiac, a Chevy, a Buick, and  a Dodge. What order did the 5 cars finish? Here are a few hints to help you solve the problem.

Ford finishes 7 seconds before Chevy

Pontiac finished 6 seconds after the Buick

Dodge finished 8 seconds after the Buick

The Chevy finishes 2 seconds before the Pontiac

When I first read this problem I began to think in through in my head and try to randomly place the cars in the right order. I realized this was not a very accurate method, so I decided that I needed to draw a picture instead. There is nothing wrong with drawing a picture! In school I was always more of a visual and hands on learner, so these word problems always scared me away and I felt like drawing a picture was silly or meant I was stupid. Most kids will be able to understand what the words mean when a picture is drawn on paper and they will probably end up grasping the concept more fully rather than trying to process words and numbers on a paper. Other kids may need something in their hands to put the cars in order. It would also be a great idea to use toy cars for the kids to be able to manipulate the cars in the right order to reach all different types of learners.

I used the clues to decide what I should do first. I decided to make a number line to start out. I am in love with the idea of using a number line whenever possible! 

• First, I put a bunch of hash marks on the line representing 1 second per hash mark.  The first clue said Ford finished 7 seconds before the Chevy. I put Ford down first and then counted 7 seconds backwards on the number line and wrote down Chevy. 
• The next clue mentioned the Pontiac and the Buick. So far there was nothing written down referring to those two models of cars so I went on to the next clue. 
• The next clue said Dodge finished 8 seconds after the Buick. Again, there was no reference to the Ford or Chevy that were already on the number line so I skipped to the last clue. 
• The last clue stated that the Chevy finishes 2 seconds before the Pontiac. Because the Chevy was already on the number line, I counted back 2 seconds from the Chevy and wrote down Pontiac. Now that Pontiac was on the number line I was able to continue with the previous two clues. 
• The previous clue said that the Pontiac finished 6 seconds after the Buick. This gave me the reference point of the Pontiac. I counted 6 seconds ahead of the Pontiac and wrote down the Buick. 
• That left only one more clue which stated that the Dodge finished 8 seconds after the Buick. Now that the Buick was on the number line I counted 8 seconds backwards and wrote down the Dodge. 

My final answer is this: 

Even though some of the clues were not helpful right away, I kept going through the problem and came to them at the end to be able to complete the problem. It's important not to let confusion or getting stuck in the middle of a word problem discourage you from moving forward and identifying another aspect of the problem that could be solved first. Drawing pictures can make the problem so much more fun and not as painful to get through.

 Happy Problem Solving!

Mental Math

This was our first week of class learning about the fundamentals of elementary math. In class we were assigned a few different subtraction problems to solve in our head. I am not too fond of math on a personal level but I figured this couldn't be too hard. After a few minutes we talked with our table group about the way we solved the problem and then were assigned to form groups with other students who solved the problem the same way. I used the traditional regrouping way to solve the first problem. The problem with this method is that we were taught in school that you can't take 9 away from 6 so you must make the 6 a bigger number. The problem with that statement is that later in math kids will learn negative numbers and have to do 6-9 but won't understand how that is possible. Christina used a really good example in class that kids may not understand the actual term "negative number", but they can understand the concept of owing somebody $3.00. It's really important that we as aspiring teachers learn to the right WORDS to say so that our phrases don't impact the kids idea of math in the future when they discover that you really can have negative numbers. 

66-29 = 37    

I lined the numbers up like we were taught in 2nd grade and crossed out the first 6 to make it a 5 and so on. I ended up getting the answer 37 which was correct. The problem with this method is dealing with 66 as two separate numbers instead of thinking of it as a whole count of 66 crayons for example. It was really interesting to see the other ways that people in the class solved the same problem. Some people changed the 29 to a 30 to make a friendly number. I had never heard that term "friendly number" before. The tens numbers are referred to as "friendly numbers" because it is easier to add or subtract from a number like 30 as opposed to 29. 

After we went through all of the mental subtraction problems, it was brought to our attention that you never have to actually subtract. When Christina first said this I was pretty confused but curious to what she meant by that statement. Instead of understanding the actual concept of subtracting, like many kids, I would just go through the motions of regrouping hoping to get the correct answer. Most people are better at addition than they are at subtraction. With the use of a number line you can actually add to find the correct answer and it makes the problem much more visual for kids to see how you got the answer you got. 

Using the same example of 66-29 = 37 here is the number line method of finding the same answer using addition rather than subtraction. 

1. You can add one to get to 30 which is a friendly number
2. from 30 it is fairly simple to go to 60 so you can then add another 30 (still dealing with friendly numbers)
3. from 60 it is easy to count up 6 more to get a total of 66
4. using proportional arks is important to show the space between numbers on the number line so the child can easily look at each ark to add up 1+30+6 = 37