Integers

Modeling Integer Operations

Well, we have to model these equations using these chips. (Red= negative, Yellow=positive) So these are our models and explanations for them for them.

(Model and explain)

6 +  -4  = Positive 2

So we start out with 6 and we have to add 4 negatives. When you add one positive and one negative together they cancel each other out. So when you add -4 to +6 then it basically takes away 4 from 6, and in the end you end up with 2.

(Model and explain)

4 -  -3  =

We start out with 4, and we need to take away -3. However 4 has no negatives to take away. SO in order to get those negatives you need to add in some zero pairs (zero pairs equal 0, so you are not actually adding any more value)! Now you have some negatives to take away, but when you take them away you still have the positives left. So because of that it actually adds onto the original number you were taking away from.

-4 * -2  =

SO basically you have to get two negative groups of negative 4 in order to do that you need to first get two groups of -4. Then after that since negatives are the opposite of positives then you just change the answer of -4*2 (-8) into it’s opposite (+8) and that's your answer.

-84 =

Since it starts out -8. You first need to get 8 negative chips. Then, since you are dividing by 4, you need to split the negative 8 into 4 different groups. You are left with negative 2 in each group so the answer is negative 2.

Rules for adding, subtracting, multiplying and dividing integers: (Clearly explain the rules as you understand them here. Use math language.)

Adding & Subtraction - When you add positives, you add on top of each other. Adding positive moves you right on the number line, subtracting positives moves it left. The opposite is for negatives, adding moves you left, subtracting moves you right.

Multiplication - If you love to love you’re a lover + * + = +

If you hate to hate you’re a lover - * - = +

If you hate to love you’re a hater - * + = -

If you love to hate you’re a hater + * - = -

Division- Dividing is basically splitting it up into equal groups. So dividing a positive into another positive will always result in a positive. Dividing a negative into positive groups is also the same, just splitting it into how many groups of the positive. Dividing a negative by a negative, you think, how many groups of this number is in this negative.

Dividing a positive by a negative, you have to rephrase it like a multiplication problem

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It's a little messed up so here is the link:
https://docs.google.com/document/d/1ZjHZ7YwpMxrEql8Vt3lphB-XYu4FfwULTHZuU3eaQl8/view
                    

Height of a Typical 6th Grader

    I think that the mode, which is 61, is the best way to find the height of a typical 6th grader. I think this because the mode is the height of most of the 6th graders making it the most frequent, the most typical. I am 66 inches so I am taller than the typical 6th grader. I am also in the top 25% of 6th graders, which is from 63 to 72. I know this from finding the 3rd quartile, which is 63, to the maximum ,which is 72. I am 5 inches taller than the typical 6th grader and 3 inches taller than the height of quartile 3. I don’t think that the height of 6th graders in Math 6+ has low variability. I say this because the IQR is only 4 inches which I don’t think is that much.

Personal Finance Project

We did a Personal Finance Project in math class where we learned about investing and saving your money and also interest. Before this project, the only thing that I really knew about it was that you can save your money in the bank. I learned a lot from this project, from class and home. For example, your money earns money, and also that there are many different ways to invest like real-estate investing. This will affect me in probably the most simplest way possible, I'm going to start investing and saving, so that I don't have to worry about money that much, or at least as much as I would if I didn't. My favorite part of this project was making the presentation because it was really colorful and simple. My least favorite part was probably doing the research and the worksheets at the start of the project.

My Project: https://magic.piktochart.com/output/4716283-invest

Fraction Division

3/4 ÷ 1/8 =  6

The equation 3/4 ÷ 1/8 simply means how many times 1/8 can fit into 3/4. There are many ways to solve this equation but the way i used get a common denominator between the two numbers. First, I changed 3/4 into 6/8 by multiplying both the numerator and denominator by 2. This way, it is much easier solve. Now all you have to do is figure out how many times 1/8 fits into 6/8 and that's easy, 6!
Another way to solve this is by using the reciprocal. The reciprocal of a number is another number that if multiplied by that number will be one. For example, 1/8's reciprocal is 8 or 8/1 because 8 * 1/8 is 1. When you use the reciprocal strategy you switch out the second number in the problem to its corresponding reciprocal, then you change the operation sign to either division or multiplication depending on which one you started with. In this problem you would change 1/8 into 8/1 and change the division to multiplication. You end up with an easy multiplication problem of 3/4 * 8/1. Now you can multiply the fractions and get 24/4, which when you simplify, comes out with 6! The reason the reciprocal strategy works is because reciprocals are basically opposite and so are multiplication and division.

Math Reflection 3

       I think it is easier to use fractions when you are dividing smaller numbers by bigger ones. For example, 7/4. This would be hard if you tried to do it in decimal notation so instead you use fractions which are so much easier for these kind of questions. You know that 4 fits into 7 once and you are left 3/7, and all you have to do is add 1 and 3/7 which is just 1 3/7. At other times though, it is easier to use decimal notation. It would be easier to use decimals in a problem if it had some benchmark decimals that are a little harder to use in fractions like 0.33 and 0.125, or questions where you are multiplying big numbers with decimals. For example if it was 1.125 x 12, you would do 12 * 1 = 12 and also 12 * 0.125 = 1.5.
       When comparing two positive whole numbers with different numbers of digits, such as 115 and 37, the one with more digits is always greater. This rule does not work when comparing decimals. For example, if you are trying to compare 3.15729 and 55.23, it is obvious that 3.15729 has more digits but 55.23 is obviously the greater number.

Fraction Comparison Reflection

       In class we learned how to use cross products, LCD (least common denominator), and reasoning to compare fractions, but when is it better to use one method over the other?

       I think if either one of the denominators is a multiple of the other one then you should use the LCD method. This is because it is very easy to find the LCD. For example if I had to compare the fractions 2/5 and 7/15, know that 15 is a multiple of 5 so Ido 5 * 3 ad get 15 and also 2 * 3 to get 6 which is smaller than 7 so 7/15 is bigger.

       When you are on relatively low numbers that are easy to multiply for both numerator and denominator then you should use the cross multiplication. If I was comparing 4/5 to 2/3 I could easily do 3 * 4 = 12 and 5 * 2 = 10. Since 12 is bigger than 10 4/5 is bigger.

       When the fractions are close to that are easier to deal with like 5/10 or 75/100. If I was comparing 16/30 to 23/48 I know that 16 is a little over 15, which is half of 30. I also know that 23 is just under 24, which is half of 24. This obviously means that 16/30 is bigger.

1,000 Lockers Explanation

Question :

The problem involves many things that we have learned in class including multiples, factors, and square numbers. The question is that if student 1 opens all 1,000 lockers, student 2 changes the state of all lockers that are multiples of 2, and student 3 changes all multiples of 3, etc., which lockers would be open?

Explanation :

First, we discovered that all prime numbers would be closed and in doing so, that narrowed it down. Then, we tried to find out which ones were open in 30 lockers because we didn’t see how the prime number helped that much, but we decided against it because we knew the pattern wouldn’t repeat. So after that, we went back to the prime numbers because it was the only thing we had. We started to think about it factor-wise. Why were all prime lockers closed? Because they all had an even amount of factors (2). That means all numbers with an odd amount of factors would be open, the problem was finding every number with in 1,000 with odd factors. We then recalled having a lesson about square numbers and them all having odd amounts of factor because one of their factors multiplied by itself was that number. That means every square number would be open. Every square number is open because they have an odd amount of factors, but why? That is because all square numbers have one factor pair where there are 2 of the same number so we only count it as one factor and therefore all square numbers have an odd amount of factors. Of course that is just generalization, we also want to find out how many lockers there were. We needed to find out how many square numbers were in a thousand. Now all square numbers are not in 1,000, so we just needed to find which square number goes over a thousand. I tried 50 * 50 first and it was way over a thousand, so I tried halving it (25 * 25) but it was too small. So I tried 30 * 30 and it was just a little too small. So then I tried 31 * 31 and it still fit 1,000 but I didn’t know if the next square number (32 * 32) fit in 1,000. It turns out that it doesn’t. So there are 31 square numbers in 1,000 and that means 31 of the 1,000 lockers would be open.

Representations :

Out of 30 :       O = OPEN       C = CLOSED       BOLD = SQUARE      UNDERLINE = PRIME

1 - O	2 - OC	3 - OC	4 - OCO	5 - OC	6 - OCOC
7 - OC	8 - OCOC	9 - OCO	10 - OCOC	11 - OC	12 - OCOCOC
13 - OC	14 - OCOC	15 - OCOC	16 - OCOCO	17 - OC	18 - OCOCOC
19 - OC	20 - OCOCOC	21 - OCOC	22 - OCOC	23 - OC	24 - OCOCOCOC
25 - OCO	26 - OCOC	27 - OCOC	28 - OCOCOC	29 - OC	30 - OCOCOCOC

Square Numbers :

50 * 50 = 2,500  Too Large

25 * 25 = 625  Too Small (Could be a Bigger One)

30 * 30 = 900 A Little Too Small (Maybe One or Two More still Bigger)

√√√  31 * 31 = 961 (There could be a bigger one) √√√

32 * 32 = 1024 Too Large  (This means 31 is the biggest and is correct)

Math Reflections: Factors and Multiples

1.
     You can decide if finding common multiples is useful if the question is asking for you to find out the next time multiples 2 numbers will ever be the same. You can decide if finding common factors is useful if the question is asking for you to divide 2 numbers equally. You will know to find the CM or CF because if it is asking you to divide, it is CF because it is going down and factors are smaller than the original number and if it is asking for you to multiply then it is CM because multiples are bigger.

2.
     a.
          To find the common factors of 2 numbers or CF, the easiest way would be to list all the factors of those 2 numbers so you can see which ones are common. To find the greatest common factor, or GCF, will also be very easy, you just take all the common factors and find the greatest one, the one with the highest value.

Example: 24 and 60      Bold = CF      Underline = GCF
     24 - 1, 2, 3, 4, 6, 8, 12, 24
     60 - 1, 2, 3, 5, 6, 10, 12, 20, 30, 60

     b.
          The GCF of 2 numbers will give us a number to divide both numbers by and if we put them into groups the groups would be equal.

3.
     a.
          Finding the CM and LCM of 2 numbers is very much like finding the CF and GCF, all that's different is that you are listing multiples instead of factors. Of course, this list could go on forever, but no one would want to do that. To find the LCM, it is just the first common multiple you come across, if your listing them in order that is. Another way to find the LCM is that if the 2 numbers are both prime, you can just multiply them together.
 
Example #1: 24 and 60      Bold = CM      Underline =LCM
     24 - 24, 48, 72, 96, 120, 144, 168, 192, 226, 240
     60 - 60, 120, 180, 240

Example #2: 5 and 17 (Both Prime)

17 * 5 = 85

 

     b.
          The LCM of 2 numbers will tell us which number those 2 numbers have in common and also when those 2 numbers will meet. Also, even though you probably won't use this information, it gives us 2 factors of the LCM.

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