## Finding areas and squares

November 11, 2009 - Leave a Response

Today, we finished covering chapter 6 with section 6.4. We covered two concepts: finding the area of a shape and finding the square root of a number.

To learn the concept of finding the area, we looked at pictures of shapes on a geoboard. Every square that is within a shape is one square unit. So, to find the total area, you just have to count the number of squares within the shape. This square on a geoboard consists of 9 small squares, so the area is 9.

Some shapes are not as simple to find the area of, however. If half of a square is included, then you add a half to the total area. If the shape is a triangle, then you can find the area of two of those triangles, which would be a rectangle. Divide that area by 2 to get the area of the triangle. The area of the rectangle is 2. Since the rectangle is twice the amount of the triangle, the area of one triangle is half of the rectangle. The area is 1.

If the shape is a square that is tilted, then you will have to use this same rectangle method to find the total area. To do this, first find the number of complete squares in the shape. Then make rectangles for all the triangles left on the shape. This shape does not have any complete squares in it. So, you just have to add the areas of the triangles in it. There are 4 triangles. The area for each one can be found by finding the area of the square that surrounds it. The area of the squares are 1 each, so each triangle’s area is 1/2. The total area of the shape would be 2.

Next, we learned that finding squares of numbers can be done by doing factor trees. So, if I wanted to find the square root of 24, I would make a factor tree for 24 first. It would look like this: So, the prime factorization for 24 is 2•3•2•2.

Next you would replace the 24 under the square root sign with this prime factorization. Any number that has a pair under the square root sign would be moved outside of it. Then you would combine the terms outside and inside the symbol. So it would look like this:

√24 = √2•3•2•2 = 2√3•2 = 2√6

To do the cubed root, you would still replace the number with its prime factorization. Instead of moving pairs of numbers out, you would move triples out from the symbol.

Watch this video for more examples of this.

Well, it looks like that’s it for my blogging project! I enjoyed this project, and it turned out to be a good way to review the material covered in class. Thanks for reading!!

~Ashley

## When decimals and fractions meet

November 9, 2009 - Leave a Response

Hi all! Hope your weekend was great! Today, we learned how to convert fractions to decimals and decimals to fractions. We also went over how to set up algorithms for operactions with decimals.

To convert fractions to decimals, you divide the numerator by the denominator. So, the fraction a/b would be a divided by b. Here’s an example of how to write a fraction as a decimal: To solve for this decimal, you would have to add a decimal point and a zero in the tenths place because 10 does not go into 3. Then, you would divide as if it were 30 divided by 10. Carry the decimal point under the division sign up to the answer.

The decimal may not always come out nicely like that. If the numbers after the decimal point form a repetition of a pattern of numbers, then you can stop divided and add a line over the repeating portion of the decimal. Here’s what that would look like. The answers for both of these examples are rational numbers. A rational number is a fraction of two integers that is either a terminating decimal or a repeating decimal. An irrational number would be a decimal that does not end but has no repetition of any patterns. Ex. 0.12657893462…..

There are two ways to convert a decimal to a fraction. If the decimal is a terminating decimal, you would just put all the numbers after the decimal point over the appropriate number. If the decimal is written in tenths, it would go over 10. If it were written in hundredths, it would go over  100, and so on. Then reduce to find the final answer. This decimal is written in tenths, so 8 would be written over 10. 8/10 reduces down to 4/5 by dividing 2.

If the decimal is a repeating decimal, then you will have a slightly more complicated process to go through to convert it to a decmial. First, the decimal would represent the variable x. So, if the decimal is .3333…, x would equal .3333…. Then, look at one set of the repeating group. In this case it would be 3. Since that group has only 1 number in the pattern, you would multiply x by 10. So, you would have 10x=3.3333… To get the fration, you would subtract the 10x and x then solve for x. Here’s what that would look like:

10x=3.333…-   x=0.333…=9x=3.00

x= 3/9= 1/3

Solving algorithms for decimals are not that tricky once you know how to set them up. I’ll show you an example for each operation. For addition, you have to make sure that the decimal point in each number is lined up. To solve, just add and carry like normal. The answer for this problem is 5.23.

Subtraction: Set up the algorithm for subtracting decimals just like addition. Solve like normal. The answer to this problem is 2.35.

Multiplication: To multiply decimals, the place that the decimals are in does not matter. Just multiply like normal, then count the total number of numbers that are after a decimal point in the algorithm. Place that many places in the decimal in the final answer. The answer to this problem is 11.088.

Divison:

0.1 ÷ 0.8 would look like: To divide, the number that is outside the brackets cannot be a decimal. So, you move the decimal point to the right until it is a whole number. Move the decimal under the bracket the same amout of spaces. Place the decimal point in the answer above the number under the brackets. Divide like normal.

## Mastering decimals

November 4, 2009 - Leave a Response

Hi all! Today in class, we finished up with lesson 6.2 and started 6.3.

For 6.2, we learned more methods for doing operations with decimals. The all can be done by either drawing decimal units and then use different colored highlighters to shade.

For addition, all you would have to do is shade the first number on the unit and then the second on the unit. The answer would be the total amount shaded.

For subtracting decimals with decimal squares, the first number would be shaded, and then the amount being subtracted would be shaded with another color OVER the first. The answer will be the amount that is not overlapped. This shows the problem 1-.5.

1 is shaded on the decimal square in green, and then .5 is covered in white. The answer is .95 because is is the amount in green remaining.

Multiplying on decimal squares is similar to the overlapping model for fractions I told you about a few posts ago. To do this, you would have to shade one amount in one color, and then shade the second amount in a different color perpendicularly to the first.  An example would look like this: This example shows the problem .4 • .2. The overlap of the two numbers is the answer, which is .08.

Division can be done two different ways using these squares. If you are dividing a decimal by a whole number, then you would shade the decimal on the square and divide it into the an equal number of sections. The answer would be the amount in each section. Here’s what that would look like: .45 is shaded on the square, and is divided into 3 equal sections. Each section has 15 hundredths shaded, so the answer would be .15.

If you divide a decimal by another decimal, then you will have to shade the first amount on the square and make sections out of that containing the amount it is being divided by. The answer will be the total number of sections. This would look like: This decimal square represents the problem .40 ÷ .20. The amount of 40 hundredths is shaded in blue, and then divided into sections of 20 hundredths. There are 2 sections, so the answer is 2.

6.3 involves percentages. To start us off in this section, we had a student presentation. She had us split into groups of 3 and play a game that was similar to Connect Four. We would spin to get a number between 1 and 10, and then take that percentage from an amount of our choosing. We then would cover a square on the board that she made up (i.e. 6% of 100=6. A square containing the number 6 would be covered). The first player to get 4 of their pieces in a row would win. I liked this game because it was fun, and it also helped us to recognize patterns of percentages.

After this presentation, we did a worksheet that involved finding percentages of prices and amounts just by finding 10% of it. For example, we would have to find 40% of \$250.00. To do this in your head, you could just find 10% of \$250 first. Taking 10% is just moving the decimal place two places to the left. So, in this case, 10% of \$250 would be \$2.50. Then, you would multiply this amount by 4 to find 40%. \$2.50 • 4= \$10.oo.

By knowing how to find percentages with taking 10%, you can easily calculate discounts and tips amounts in your head. So, next time you see a sale at a store, instead of calculating it on your cell phone, try to do it in your head!

That’s if for today! Next post will be after Monday’s class.

Hope the rest of your week is full of smiles!

~Ashley

## Working with decimals

November 3, 2009 - Leave a Response

Hope your Halloween weekend was great! Yesterday in class, we summed up 6.1 (introduction to decimals) to review for this section first.

As a continuation of 6.1, we covered that a decimal, such as 4 tenths (.4), can be converted from tenths to hundredths by just adding a zero after the 4. So, 4 tenths is the same as 40 hundredths because .4=.40. Then, another zero can be added to convert the decimal to thousandths, and so on.

Here is a visual of this information: As you can see in the above picture, 6 tenths has the same amount of the unit shaded as 60 hundredths.

After this quick lesson on 6.1, we moved onto 6.2, which involved operations with decimals. For this, we used only one method to visually see how decimals can be added, subtracted, multiplied, and divided. It was really easy to understand. Each person took a scrap piece of paper and split it into 3 sections: tenths, hundredths, and thousandths. We then were given beans. For this, we had to create the problem gived by placing beans in each section to represent the number of tenths, hundredths, and thousandths. For addition, we just had to combine the beans under each section.

So, if the problem were 0.123 + 0.456, we would combine the beans in the thousandths place (9 total), then the hundredths place (7 total), and finally the tenths place (5 total).  The answer would then be 0.579.

If the sum of one section is 10 or more, however, then we would have to carry over into the next highest section. So, if the problem were .123 + .457, the thousandths section would add up to be 10. This would carry 1 bean to the hundredths place. Then there would be 5 beans in the tenths place and 8 beans in the hundredths place. The answer is then 0.58.

The same method would be used for subtraction, except we would take the second amount of beans away from the first instead of combining the two. So, if the problem were 0.782 – 0.213, you would originally put 7 beans in the tenths place, 8 beans in the hundredths place, and then 2 in the thousandths place. Next, you would take away 3 beans from the thousandths place, 1 from the hundredths place, and lastly 2 from the tenths place. If you cannot subtract one place, like the thousandths place, then you will have to borrow from the next place up, hundredths. So, 2 thousandths would become 12 thousandths after 1 hundreth is borrowed, and the problem can be completed. The answer would be 0.569.

This method only works when a decimal is being multiplied by a whole number. For example, .326 x 2. You would just have to set up the decimal twice to get the final answer. So, there would be 6 beans in the tenths place, 4 beans in the hundredths place, and 12 beans in the thousandths place. The same rule with addition applies to multiplication: if there are more than 10 beans in one place, then carry over to the next highest place. So, there would be 2 thousandths and then 5 hundredths. The final answer would be 0.652.
Division also works best when dividing by whole numbers. For this, you would have set up the first number, the decimal, and then split it into the number of groups indicated. For example, if the problem were 0.46 ÷ 2, then you would place 4 beans in the tenths column and 6 in the thousandths column. Then, split those beans into two equal groups. The answer is the total in one of these groups, 0.23.
We also had two IBL presentations in class today. The first, which was done by Ashley, involved addition and subtraction with decimals. She came up with a neat activity that involved finding prices for multiple items. Prices are great for applying decimals to real-life scenarios. Cory presented next. He did a bingo game, but we had to convert fractions to decimals and vice versa to find the corresponding square. This would be a fun game to do with kids to practice fractions and decimals.
For practice with decimals, try the games on this site.
Next post will be Wednesday!
Hope your November is off to a great start!
~Ashley

## It’s important to understand fractions!

October 28, 2009 - Leave a Response

We went over sections 5.3 and 6.1 in class today. Let me warn you ahead of time: there was a lot of information and examples in these sections, so  sorry if this post gets a little long!

Section 5.3 was on operations with fractions. We used very visual methods to help us understand how to add, subtract, multiply, and divide with fractions.

To add and subtract with fractions, the main way method we used was to visually add them. In our activity books that we have, the problem would be given with a fraction bar for each addend. Seeing the fractions on the page of our books, we could see how they needed to be added up. This is how the problem would be presented. This would show us that the two fractions need common denominators to be added up. What we would do is divide the fraction bars into equal sections to find a common denominator. By dividing each bar into 6 equal sections, the two fractions are now in sixths and can be added up. To do this, we just add the 3 sections shaded in the first bar with the 2 sections shaded in the second.

Visually, the addition would look like this: Subtraction used the same method, but we would take the numerator of the second bar from the numerator of the first once the denominators were equal.

3/4-2/4 For this problem, you would take the two-fourths from the three-fourths.

Addition and subtraction with fractions didn’t seem to be too hard to understand, but multiplication is where it got a little tricky.

When you multiply fractions, the answer will be smaller. This is because you are taking a fraction of another fraction. We used fraction bars to understand this. The first fraction would be shown, and you would overlap the other fraction on top. It was confusing, but it ended up looking like this: So, the fraction 1/4 was shown, and then that fourth was split into thirds. The overlap of 1/3 and 1/4 would be the answer, which is 1/12.

Don’t worry if this is confusing, my whole class was also pretty confused. So, we found an easier method to do this. This model shows 2/3•2/4. Each side of the entire square represents 1. The answer can be found by finding the overlap of the two shaded fractions. In this problem, the answer is 4/12, which can be reduced to 1/3.

We did this method with a square piece of paper. Our teacher gave us each a piece of paper and told to use it to multiply 1/2 by 2/3. To do this with the paper, we folded the paper into halves one way and shaded 1/2. Then, we folded it into thirds the other way, so the folds were perpendicular to each other. 2/3 was shaded on that side. It’s best to used two colors to shade both fractions. To find the answer, you would count the number of overlapping shaded squares (this will be the numerator) and the number of total squares (the denominator).

To multiply fractions on paper, you simply multiply the numerators together and the denominators together. So, to multiply 1/2 by 1/2, you would first multiply 1•1 (1) and then 2•2 (4). The final answer is 1/4.

Lastly, there is division. This is the most complicated operation to understand with fractions. Dividing fractions really is finding the number of times the second fraction can fit into the first. Visually, this looks like: So, to see how many times 1/3 will go into 4/6, you will have to visualize picking the third up and comparing it to the 4/6 bar. In this example, it turns out nicely and equals 2. This example is 5/12 ÷ 1/4. This does not turn out evenly like the last example, so you will get a mixed fraction as an answer. The answer to this problem is 1 and 2/3. You can see that 1/4 = 3/12, and since 1/4 has to go into 5/12, there will be 2/3 added to the one.

To do division on paper, you will have to rewrite the problem a little. Since division is the opposite of multiplication, you can take the reciprocal the second fraction and then change from division to multiplication. This is because division is the opposite of multiplication.

Here’s an example:

Section 6.1 involved decimals. We didn’t go into too much on decimals since fractions took most of our class time today. The main concept that we learned was decimal places. We know the label to number places- like ones, tens, hundreds, etc.- but decimals have a different way to label its places. The first decimal place is directly after a decimal. This is called the tenths place because it represents 10 to the -1 power, or 1/10. The pattern continues in the reverse pattern of numbers before a decimal place. So, after the tenths place is the hundredths place, then the thousandths, and so on.

Next, we looked at units that represent a decimal. Here is how these would look: This is a single unit (left) compared to a tenth unit (right). The tenth unit is the sam size as a single unit, but is split into 10 equal sections. The left unit represents hundredths and the right unit represents thousandths.

To show a specific decimal on a unit, you will have to shade part of the unit. This represents two-tenths, or .2 This represents 6-hundredths, or .06 This represents 1-hundredth and 5-thousandths, or .015

Sorry again for the very long post today, but there was a lot to cover!

Have a great Halloween!

~Ashley

## Learning something non-math related in math class?!?!

October 27, 2009 - Leave a Response

Instead of learning a new lesson today in class, we went over the last test that we took. This wasn’t the typical test review that most classes do, though. We had to analyze the test to see how the study guides that we get on the class’s blackboard (aka webpage) matched up to the test itself.

Maria had us split into groups and gave us each a copy of the test. What each group had to do was write down what topics from the study guide were used for each question of the test. Then, we could check off each topic on the study guide. Almost every study topic was used on the test. This gave us a visual idea that the test was made for us to demonstrate our understanding of what we went over in class and what we should have studied outside of class.

The next task that we had to do with our test was to calculate the amount of points from each book section that were on the test. To do this, my group decided the main topics and types of problems that were used for each question. We then placed the question in the section or sections that we thought that it related to the most. For example, if question 3 on the test fit into both sections 3.1 & 3.2 and was worth 4 points total, we assigned 2 points to section 3.1 and 2 points to section 3.2. This was done for each question, and the points were all added up. Each group had a slight difference in the total amount of points for each section, but we could all agree that the points for each section reflected the amount of time we spent in class learning that section.

Then we got the tests that we took last week back. We were supposed to review them and write what we thought we needed to work on more and what we thought we did well.

You might be wondering what the point of all this detailed test analysis was. I was too at first. But Maria told us that this would help us with our studying for the future. To understand how a test is constructed-based on the study guide, the homework questions, and the amount of time taught in class-we can learn what is most important to study and not to skip over any topics.

I also think that this was a helpful exercise to do in this class because it taught us how much detail and planning is put into test-making. Since the class is directed towards students interested in elementary education, we got to learn some methods behind making a test. So, a few years down the road when I am a teacher myself, I will know how to construct a test along with a study guide that will reflect the knowledge of my students best. So, today I didn’t learn very much math, but some important lessons to use now and in the future.

Next post will be next Wednesday when we return to math! Get ready for more fractions!

Stay warm,

~Ashley

## Integers and fractions- a kid’s worst nightmare

October 21, 2009 - One Response

We had a substitute teacher in class today, and I think she did pretty well with teaching the material. It made me realize that the same lesson can be taught in a very different way according to the instructor.

For the first half of the class, we went over lesson 5.1, which is integers. It involved the addition, subtraction, multiplication, and division of positive and negative numbers. We were taught two methods to use for this lesson.

The first was to use different colored tiles representing positive and negative numbers: In this example the red tile represent negative numbers, and the black tiles represent positive numbers. We can visually solve the problem given above this way. In this problem, the answer is -3, because the 2 positive tiles cancel out with two negative tiles.

The second method was to use a number line: This is a visual representation of 3×5. On a number line, you would go up three groups of 5 or up 5 groups of 3 to get the answer, which is 15.

I found that colored tiles were more helpful for addition and subtraction, while a number line was more useful for multiplication and subtraction. When multiplying and dividing negative numbers, there are a few rules:

A•(-B)=(-K) and (-A)•B=(-K)

A÷(-B)=(-K) and (-A)÷B=(-K)

(-A)•(-B)=K and (-A)÷(-B)=K

This site provides a unique way to explain these multiplication an division rules.

The second half of the class focused on section 5.2, introduction to fractions. This lesson was focused mostly on the understanding of fractions in general and finding equivalent fractions. We used our fraction pieces to visually see how fractions can be equal. These pie graphs show that 1/2, 2/4, and 4/8 are all equal fractions because the same amount of each graph is shaded red.

To know if one fraction is less than, greater than, or equal to another fraction, you can either visually compare the two fractions like this: In this illustration, we can see that 2/4 is greater than 1/3. That would be written as 2/4>1/3.

To figure this out on your own, you will have to find the a common denominator. For 2/4 and 1/3, the least common denominator would be 12. So, you will have to multiply 2/4 by 3/3 and 1/3 by 4/4. You would get: 2/4=6/12 and 1/3=4/12. Now, you can find which fraction is larger- 6/12, or 2/4.

To practice this, go to this link.

The lesson for Monday covers operations with fractions.

Until then, have a great weekend!

~Ashley

## What will I post in my blog?

October 19, 2009 - Leave a Response

This blog is for a project in my Math 105 class. My teacher is Maria Andersen.

I will be updating this blog frequently with an overview of what we went over in class, tips and hints to understand the material, and links to websites that I found helpful.

So, this is just a introductory blog post. My blog will cover unit 3, chapters 5 and 6.

Check back on Wednesday for my next post! We will be covering section 5.1, integers, in class that day.

Enjoy the beautiful fall weather!

~Ashley