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.