# Tricky Significant Digits

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Significant digits can be a frustrating concept if you do not understand the rules. Significant digits are the digits needed to have a precise answer. If you were to take a chemistry class, then you will be familiar with them. Recently, I began working with significant digits in my math class. I think the more you work on a concept the more you understand it. There are different rules when dealing with significant digits. Adding and subtracting numbers have different rules compared to multiplying and dividing numbers.

Adding and subtracting numbers focuses on the accuracy of the answer. The number with the least amount of decimal places is the “winning” number. For example if we are adding 5.36+2.2, the “winning” number is 2.2 because it has the least amount of decimal places. The answer to this problem is 7.56, but since we need to have 1 decimal place the answer is 7.6. In the picture below I have provided an example of a subtraction problem.

Multiplying and dividing numbers focuses on the least number of significant digits. Instead of looking at the amount of decimal places, we look at the whole number and count the amount of significant numbers; therefore, the “winning” number is the number with the least amount of significant digits. The solution to the problem should be rounded to the same amount of numbers as the “winning” number. For example if we are multiplying 82.1×3.2, the “winning” number is 3.2 because it only has 2 significant digits. In the picture above I have included a division problem and some tricky problems dealing with zeroes. I think with effort and determination anyone can understand the rules and differences between adding and subtracting numbers compared to multiplying and dividing!

# Refreshing Real Numbers and Decimals

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Understanding the different types of numbers is great way to learn more about math in general. Real numbers are the main type of numbers that elementary students learn, but there are imaginary numbers that are learned during higher grade levels. Although real numbers are the main type of numbers, there are five different types of real numbers. Making a diagram of these different types is a great way to understand the differences.

There are also irrational numbers which are not in the diagram above. They are kind of in their own circle. These numbers are nonrepeating decimals that do not terminate and have no pattern. An example of an irrational number is the square root of 2. This number does not come to an end (not terminating) and has no pattern. If you type the square root of 2 into a calculator you will get something like 1.414213562 and it continues on and on. Clearly, you can see that this is an irrational number.

Adding and subtracting decimals are very similar concepts. In order to add and subtract numbers you need to line up the decimals in the problem. An example of adding two decimal numbers is 1.2+2.4. If we line up the decimal numbers it will be 3.6. Subtracting two decimal numbers like 3.6-2.1 uses the same method. You need to line up the decimals and subtract the numbers; therefore, the answer is 1.5. Lining up the decimal numbers can be hard if the numbers are super long. A great way to eliminate this problem is by lining up your decimal numbers on graph paper. I think this is a great tip that may create less confusion!

# Dealing with Decimals

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I think a great way to understand what decimals are is to put decimals into a table. As you can see from the table below, decimals can be written in many ways. One example of this is the number 0.1 can be written as 1 tenth, 10 raised to the -1, or 1/10. Decimals are base-ten units; therefore, we use the unit ten in the table.

The United States separates numbers with commas every third number:  5,853,567.75; however, many other places around the world separate their numbers with spaces and use a comma the way the United States uses the decimal point like this 5 853 597,75. I think this is a very interesting fact! Another thing I learned is that the number 0.0000000009 is easier to read if we put spaces every third number: 0.000 000 000 9. If we put spaces it does not mean anything different; it is just to help us count the zeros or read the number easier. These are great tips to keep in mind while working with decimals.

As you can see from the decimal table, decimals can be written in many ways; essentially a decimal is a fraction. I say this because 0.732 is 732/1000 and can be written as 732 thousandths.  Another example of this is 0.20 is 20/100 and can be written as 20 hundredths. No matter the decimal number it can always be written as a fraction which may be easier for some people. You can also look at a decimal by expanding it. If we look back at the decimal 0.732 we can break it down into three parts 0.732=0.7+0.03+0.002. This can also be written as 732/1000=7/10+30/100+200/1000. Decimals are very versatile which allows for many different strategies to understand them better!

# Digging Deeper into Fractions

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Did you know fractions are easier to picture in your head if you estimate? I have learned from experience that fractions truly are easier if you use estimation. An example of this is 7/8 + 1/3, this may look tricky if we actually were to work out the problem and find the same denominator. If we were to attempt this problem using estimation, we could say that 7/8 is almost equivalent to 8/8 (1), and 1/3 is a little larger than 1/4. If we add 1 +1/4, the answer is 1 1/4. Although this may not be the exact answer, it gives us a good idea of what the fraction will be. This is an easy way to compare fractions and gain a better understanding of them!

Another great way to help gain an understanding of fractions is by using fraction bars. Fraction bars are rectangular shaped pieces of paper that are divided up into different amounts. The different amounts are the fractions that the shape equals. For example, I have 5 different fraction bars: twelfths, sixths, fourths, thirds, and halves. The rectangles that represent twelfths are divided into 12 sections; the rectangles that represent sixths are divided into 6 sections, and so on. These are great tools to work out fraction problems. Finding a fraction between 2/6 and 3/6 is easier with fraction bars. This may sound easy, but it is hard to figure out if you do not have a good understanding of fractions. Let’s use fraction bars to help us out; for starters we will need a 2/6 fraction bar and a 3/6 fraction bar. If we set these fractions bars side by side, we can easily see that 5/12 is between both 2/6 and 3/6. I found this by looking for a fraction bar that was in between 2/6 and 3/6.

When dealing with whole numbers and fractions, it is easier to set them up into improper fractions. To set up an improper fraction you need to multiply the denominator by the whole number and add the numerator. We do this because we are essentially looking for a common denominator. An example of this is 4 1/2, we start with 2(denominator) x 4(whole number) = 8, then we add 1 (numerator) = 9/2. 9/2 is an improper fraction because the numerator is larger than the denominator. We can also find the improper fraction by finding the common denominator. 4 1/2 can also be written as 4/1 + 1/2; in order to find the common denominator we need to multiply the 4 by 2. Once this is done we can add the two fractions together which gives us the sum of 9/2. This idea is helpful when adding and subtracting fractions: 1 2/3 = 5/3 – 1/3= 4/3. Take your time to learn fractions because I think it is a great tool that we can use in our daily lives.

# Learning Fractions

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Fractions can be confusing and hard to understand. It can be easy to misread or do a problem wrong that is dealing with fractions. For example, in the problem 5/3 +3/3 many may get confused and say the answer is 8/6. This is wrong because you cannot add two like denominators in the problem. Denominators are the numbers on the bottom of the fraction; in this case the denominator in 5/3 and 3/3 is 3. It is less likely to get confused if 5/3 is written as 5 thirds and 3/3 is written as 3 thirds. Clearly, you can see that the denominator is going to stay 3 or a third. You do not add the denominators (3+3=6) because that is not correct. This is a nice tip to follow especially if you are teaching younger students who have never dealt with fractions before.

Another great tip to help learn more about fractions is having the same unit. For example, if you have 2 yards + 9 feet you cannot add them because yards and feet are not the same unit. In this case you would have to convert one of the units in order to make them both have the same unit. You can do this two different ways with the example provided: yards can convert to feet and feet can convert to yards. Let’s convert 2 yards into feet. In order to do this we must know how many feet are in a yard, which is 3. So if there are 3 feet in a yard and we have 2 yards that means there are 6 feet in 2 yards. Now we can do the problem! 6 feet +9 feet = 15 feet.

I found that drawing measuring cups helps to understand the conversions between halfs and fourths.  For example, if we have 3 halfs + 1 fourth it may be confusing to solve the problem if you don’t have something to look at. First of all we need to convert either halfs to fourths or fourths to halfs. I am going to solve the problem by converting halfs to fourths because a fourth is smaller than a half. I have included a picture of measuring cups to work out the problem. The end result is 6 fourths + 1 fourth= 7 fourths.

# Investigating Integers

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The word integer may seem a bit simple, but as I have learned it can actually be tricky to understand. The first thing that one must know about integers is that the number can be either positive or negative. For example, a set of integers looks like this: {…-3, -2, -1, 0, 1, 2, 3…}. This shows that the negative numbers will continue infinitely as well as the positive numbers. Once the idea of integers is understood, adding and subtracting begins.

A simple way to understand the meaning of an integer is to name the number n. So if n=the number, -n=the opposite of the number, and -(-n)=the opposite of the opposite of n. The opposite of the opposite of n would simply mean the number is positive. An example of n+n is 2+2=4. In this example there are two positive numbers which result in a positive answer. An example of -n+n is -2+2=0. This shows that 2 negatives plus 2 positives cancel each other out resulting in zero. An example of -(-n)+n is -(-2)+2=4. In this example the -(-2) is equivalent to 2, so when we add 2+2 it equals 4.

A great hands on activity that helped me to understand integers was using color counters. Color counters are little square shapes; one side of the square is red and the other side is black. The red side stands for the negative integer and the black side stands for the positive integer. For example to show 2+2=4, you would have 4 black color counters. The example -2+2=0 you would have 2 blacks and 2 reds, which cancel each other out. To show -(-2)+2=4 you would simply have 4 black color counters like in the example 2+2=4. The picture provided is a great representation of the examples I have explained.

# Gaining Knowledge

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Adding and subtracting are math concepts most people learn at a young age. Although we start by learning simple math problems, we still develop an understanding of the concepts. For example, if there are 5 sheep on a farm and 2 get lost there are now 3 sheep on the farm; if there are 2 cows on a farm and the farmer buys 2 more cows, then there are now 4 cows on the farm. These simple math problems help students to understand the idea of adding and subtracting. As we get older we begin to learn more complex problems like 159+253=412. This may be easy to understand for most people, but may be very confusing for a student who has no idea what they are doing. How does 9+3=2? This is a question a student may ask because clearly 9+3 does not equal 2; it equals 12. In this case we use the idea of “carrying” numbers, and if it were a subtraction problem we may have to use the idea of “borrowing” numbers. What is a good way to teach a student to “carry” and “borrow” numbers? How can we help a student understand how to add and subtract numbers? I have learned great ways to answer these questions and gained a better understanding for adding and subtracting numbers.

A great way to teach a student to “carry” and “borrow” numbers is to use manipulatives. Using a manipulative is a hands-on-way to get students involved with math. For the standard adding and subtracting problems we use base-ten units; therefore, we will use base-ten pieces for a manipulative. The base-ten pieces include: flats (100 units), longs (10 units), and units (1 unit). For the problem 159+253=412, we will use 1 flat, 5 longs, and 9 units to show the number 159; we will use 2 flats, 5 longs, and 3 units to show the number 253. After we decide what base-ten pieces we need for the problem, we then begin “carrying” units. In this manipulative we show “carrying” by exchanging units for longs and longs for flats. The first exchange in this problem is to exchange 10 units for 1 long; the second and final exchange is to exchange 10 longs for 1 flat. Once the exchanges are done, we are left with 4 flats, 1 long, and 2 units. 4 flats is equivalent to 400, 1 long is equivalent to 10, and 2 units is equivalent to 2; therefore, if we add 400+10+2 we get 412. This is time consuming, but it is a great way to explain addition and subtraction. Although this is an example of addition, the same idea is used when using manipulatives to show subtraction. The pictures provided may help to clear up any confusion about adding with base-ten pieces.

Hands-on-activities are great ways to get students involved in the math problems, but there are also other options. Technology is growing very quickly; therefore, there are dozens of online websites that have interactive games and manipulatives to get students involved in math problems. This is a great way to stay updated with new opportunities available for teaching math concepts. National Library of Virtual Manipulatives is a great website that has a bunch of activities. This website allows you to download different manipulatives for free! This is a wonderful way to enhance any classroom.