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!

Grasping the Greatest Common Factor

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It feels like just yesterday I was learning about greatest common factors in elementary school. The only difference from learning GCF’s now is that I actually understand it. The greatest common factor is the largest possible number that can divide into a set of numbers equally. An example of this is find the GCF of (24,36), the answer is 12 because 12×2=24 and 12×3=36. There are no larger numbers that divide equally into 24 and 36.

Although the example GCF(24,36) is simple, there is a trick for the more complex problems. For example find the greatest common factor of (2100,3360) seems almost impossible, but I have learned a trick to help speed up the process. Instead of plugging numbers into your calculator you simply need to draw a factor tree for both numbers. The picture provided is what the factor trees should look like.

Once you have the factor trees drawn, you write out the prime numbers for 2100 and 3360. The prime factors of 2400 are 2x2x3x5x5x7, and the prime factors of 3360 are 2x2x2x2x2x3x5x7. The trick to finding the GCF of the 2100 and 3360 is to look at the intersection (overlap) of the numbers. The numbers in read are the overlap (2x2x3x5x7). After you find the intersection, you need to multiply the numbers together (2x2x3x5x7=420). This means that the greatest common factor of 2400 and 3360 is 420. This may look like a lot of work, but it is actually much less work than using your calculator to guess and check numbers.

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Factors can be intimidating, but there are some tricks that help to understand them. Factors are numbers that are multiplied together to equal the final number. For example in the problem 2×4=8, the factors are 2 and 4. Although these are factors, they are not prime factors. Prime means any number greater than 1, and its only factors are 1 and itself. Some examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39, 41, and 43. If we go back to the problem 2×4=8, clearly 4 is not a prime number; therefore, we have to break it down until it is a prime number. To break down 4 into prime numbers we can start by finding a multiple of 4, which is 2 (2×2 = 4). Now we have all prime numbers: 2x2x2=8; this is the same as saying 2×4=8.

Factor trees help to find the prime factors of a number by first obtaining any two factors and then obtaining their factors. This may seem confusing, but it is actually quite simple. For example if we have the number 40, we can start by obtaining any two factors (8×5). Since 5 is already a prime number, we only need to obtain the factors of 8. These factors could be 4×2. Once again, 2 is a prime number; therefore we only need to break down the number 4. The number 4 breaks down into 2×2. Now we have all the prime factors of 40: 5x2x2x2=40. I have provided a picture of a factor tree to help visualize the problem.

A fun game that involves practicing factors is called Roast Turkey. This game helps to gain further knowledge of factors and helps to practice them. The game begins by letting the player choose what level you want. After you choose your level, it explains that you use your keyboard arrows and space bar to navigate “Professor Super”. Once you click the begin button, it will bring up a page that explains factors. Now it is time to begin the game; a question will pop up like: what is not a factor of 25. There will be three uncooked turkeys floating around your screen with different numbers in each one. You have to use the arrows to navigate “Professor Super” and the space bar to “fire” the correct turkey. This is a fun interactive factoring game!

Magnifying Multiplication

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A very commonly used math concept is multiplication; it can be defined as repeated addition. For example, 2×4=8 can also be written as 2+2+2+2=8. This seems like a very simple definition, but can be confusing for young students. I can remember learning multiplication when I was in elementary school, and it took me a while to fully grasp the idea of it. I practiced my multiplication with flashcards all the time! Now that I am learning how to teach students multiplication, I think it is important to get them involved in math activities.

A great activity to help students understand multiplication is to draw chocolate chip cookies. In the drawing the chocolate chips stand for the number that we are multiplying and the cookies stand for the number that we are multiplying by. In the problem 2×4=8, the 2 is the number we are multiplying and the 4 is the number we are multiplying by; therefore, we would have 2 chocolate chips in each of the 4 cookies. When we add up all of the chocolate chips in the cookies the answer is 8. I think this is a fun and creative way to get students to understand multiplication.

The cookie activity is a great way to learn how to multiply, and Factortris is a great way to practice multiplication. Factortris is an online game that helps students who may struggle with their multiplication tables. It is also great practice for students who know their multiplication tables, and just need more practice. The idea of the game is to answer the problem before the time runs out. For example, it may say 54 and the player has to use the arrows on the keyboard to click the answer. In this case the player could use the down arrow and move it to 9, and use the right arrow and move it to 6. This would show up on the screen as 9×6 which is 54. Factortris is just one online game that focuses on multiplication practice, but I am sure there are many more.

Avoiding Confusion

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There are two different properties of equality. They are the addition or subtraction property of equality and the multiplication or division property of equality. The addition or subtraction property of equality is to add or subtract the same element on both sides of the equal sign. The multiplication or division property of equality is to multiply or divide by the same element on both sides of the equal sign. One important reminder when using the multiplication or division property of equality is that you never divide by zero.

Although the addition or subtraction property of inequality is the same concept as the property of equality, the rule for multiplication or division property of inequality is different. The rule for this is that we can multiply or divide by a positive number on both sides…no problem, but if we multiply or divide by a negative number the inequality must be reversed. These rules are used for problem solving and are fairly easy to remember.

It feels like just yesterday when I was sitting in my 3rd grade classroom learning about inequalities. My teacher taught my class the “alligator trick.” The “alligator trick” is the idea that the alligator eats the larger number, but I have learned that the “alligator trick” is not a very good technique to teach to elementary students. This is because the idea that the alligator eats the larger number can become very confusing. An example of this is -5<-8 (-5 is less than -8) or -5>-8 (.5 is greater than -8). -5 is actually greater than -8 on a number line, but because most students look at the number before the sign it can get confusing.  The picture below is also confusing because the 3 is written smaller than the 2 in size, but the 3 is actually greater than the 2 on a number line. Avoiding this trick is a good idea to avoid any confusion.

“Alligator Trick”

Understanding Problems

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While working on my first set of text questions I realized that confusion truly is a sign of understanding. There were multiple questions where I had to stop, reread, and unclutter the information before I even began the question. This part of my problem solving is what I would call confusion. Although problem solving can take me a while to figure out, I feel relief and happiness when I find the answer. I think throughout my first couple homework problems I was able to understand the questions more because they started off confusing. The article entitled “Confusion is a Sign of Understanding” allows me to not stress when I am confused, but to continue through frustration to really understand the homework questions and feel confident with the end result.

The terms arithmetic sequence, geometric sequence, and finite difference were all intimidating at first glance, but as I began to study them they were quite simple. Arithmetic sequence is a term used to explain a common difference in a set of numbers. The common difference is simply adding or subtracting the same number from each number in the sequence. Geometric sequence is a term used to explain a common ratio in a set of numbers. Common ratio is multiplying or dividing the same number from each number in the sequence. Finite difference is a term used to explain a “hidden” pattern. The “hidden” patterns vary from problem to problem. For example the sequence: 1, 2, 4, 7 do not seem to have a pattern, but it does. Step one: take the difference of the numbers which is 1, 2, 3. Step two: take the difference of the difference which is 1, 1, 1. This was a simple finite problem, but it gives a good idea to the definition of “hidden” pattern.

I used problem solving to help me understand the terms arithmetic sequence, geometric sequence, and finite difference. I wrote down multiple examples of each term and worked them out until I truly understood them. This helped me to look at the differences between the terms. The hands on activity we did in class also helped me to understand the terms on a greater level. The hands on activity consisted of categorizing different sequences into the three terms. Although a couple of the sequences were complex, we were able to place them in the correct category. This activity was a great way to visualize the terms and understand the differences.

Class Activity