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.
For this problem, you would take the two-fourths from the three-fourths.
The answer would be 1/4.
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:
For practice with fraction and operations, try this website.
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!