End-of-Year Reflection for Math

My new learning in Math 8+ this year included factoring and quadratics. More specifically, the different methods to factoring quadratics (such as the diamond problem method, the rectangle method, and factor by parts), and the different methods to solving quadratics (such as using Desmos, square root, factoring, and the quadratic equation). Before this school year, I didn't know anything about quadratics; however, now I am very comfortable with this concept.

I really understand the different types of functions, ranging from linear relationships to exponential relationships, to quadratic relationships. I know how to recognize which relationships are which and how to find the equation from a graph and/or table. I can use this knowledge to solve a variety of applied problems.

Over summer break, I should learn how to solve systems of equations with exponents and how to solve three equation systems of equations. I should review and practice quadratics, especially the special cases and the ways to find the solution because I feel that practicing these things will help me become less careless and a better learner in 9th grade.

Looking at all my tests and my mistakes from those worked well in preparing for the End-of-Year Test test. I felt that this method worked because I would know what to focus on so that I wouldn't make the same mistakes again. If I got a question correct on a previous test, I know that I understand that concept, which means that I wouldn't have to focus on it as much. I also just looked at all the notes I took throughout the whole year and compiled them into one sheet which made it easier to review. Overall, I feel that I did pretty well studying for the EOY test.

This year, I learned that taking notes is very very important. On top of studying the concepts I'm unfamiliar with more, to better prepare for tests in high school, I should continue taking notes on new information and continue analyzing my formative and summative tests, keeping in mind the concepts from the questions I got wrong. This would help me with EOY exams just like the one I took this year (8th grade).

The Learning Behaviors I will focus on improving is reflection because I feel that reflecting on my learning is a key to success. I will continue to ask questions and get help whenever I need help understanding a concept.

Designing A Spinner

Math Normal 7th Grader Project

Statistics Homework 3: Data Analysis

How alike or different are the data values from each other? 
They are pretty similar, they all have quite similar values of central tendency. The biggest difference in these three data sets is that the thrid set's data is more spread out that the first and second's data.

Which data values occur more frequently or less frequently? 
Some data values that occur more frequently than others are 7, 7.5, and 8, and this is the cause for the mean to be 7.5. There are some data values that occur less frequently, such as 1.5 and 4, and they are outliers.

How spread out or close together are the data values in relation to each other? 
I would say that the first set and the second set of data values are pretty close together, with the exception of the 1 on the second set, and the third set of data values are very spread out. The reason the thrid set is more spread out is that the thrid set has an outlier, 4, that messes up the data. There is also another outlier, 1.5, in the second data set, which made the data seem more spread out.

How spread out or close together are the data values in relation to a measure of center?
The measure of center for all the three data sets are about 7.5 and 8. However, the data isn't that close to it. Yes, there are a lot of students with the shoe sizes of 7.5 and 8, but there are way more students with shoe sizes of 6 and 9.

What do you infer about the shoe sizes of 7th graders?
I infer that the average shoe size in 7th grade is 7.5

Statistics Homework 1: Video Facts

1. Identifying a Random Sample: Instead of conducting a census, which is difficult as you have to survey each individual of the population, you can survey a sample of the population and make a generalization of the population from the sample.

2. Generate a Representative Sample: Since we don't know what the whole population looks like, we can't handpick the people for the sample, so we have three different unbiased sampling methods: simple random sample, systematic random sample, and stratified random sample.

3. Understanding Biased Samples: Often students think that biased sampling methods are obvious and easy to avoid. Just because you've selected a part of the population that seems like a good sample population, like the basketball team, it doesn't mean that your sample is unbiased.

Cellphone Plans

You are a representative for a handphone company and it is your job to promote different phone plans. 

• Plan A costs a basic fee of $29.95 per month and 10 cents per text message 

• Plan B costs a basic fee of $90.20 per month and has unlimited text messages

• Plan C costs a basic fee of $49.95 per month and 5 cents per text message 

• All plans offer unlimited calling
• Calling on nights and weekends are free
• Long distance calls are included

1. Determine the cost of each plan if you send 500 text messages in one month. Show your working out. 
Plan A cost $79.95 because on the graph, on plan A's line, the y coordinate for the x coordinate 500 is 79.95. The x coordinate is the number of text and the y coordinate is the cost.

Plan B cost $90.20 because the cost doesn't change from the number of text since it has no rate of change.

Plan C cost $74.95 because on the graph, on plan C's line, the y coordinate for the x coordinate 500 is 79.95. The x coordinate is the number of text and the y coordinate is the cost.

Plan A

Plan B

Plan C

2. Your boss asks you to visually display three plans and compare them so you can point out the advantages of each plan to your customers.

The Graph

3. A customer wants to know how to decide which plan will save her the most money. Determine which plan has the lowest cost given the number of text messages a customer is likely to send.
If you text 0 to 400 times in a month, then Plan A is the cheapest plan for you.
If you text 400 to 805 times in a month, then Plan B is the cheapest plan for you.

If you text more than 805 times in a month, then Plan C is the cheapest plan for you.

Proving Similarity

I think that Triangle B is similar to Triangle C because of two reasons: the angles are the same and the sides are proportional. 

Since Triangle B is similar to Triangle A, the angles must be the same even though the side lengths are different. Same with Triangle C, since Triangle C is similar to Triangle A, the angles must be similar too. So Triangle A, B, and C's angles are all the same. I checked this by making up angles. I looked at the corresponding angles for all three shapes and found that they were all the same.

For the side lengths, I made some up myself. The side lengths for the triangles are shown below in the image. I found that Triangle B's side lengths were proportional to Triangle A's side lengths and Triangle C's side lengths were proportional to Triangle A's lengths. This means that Triangle B's length is proportional to Triangle C's length, making them similar. I made sure that this was correct by comparing corresponding sides from all three sides. I found that all of the corresponding sides were proportional to each other. I also found that there was a scale factor for all the triangle relationships providing more evidence that Triangle B is similar to Triangles C.

Mathematical Reflections on Polygons

1. The common properties of polygons are... 
Polygons are multisided, meaning that they have more than one side. The sides must be straight and there can be no curves. Polygons also are closed, meaning that there are no gaps on the edges. There cannot be intersections, which means that there can't be anything inside the polygon. Also, the number of lines should be equal to the number of vertices. Finally, polygons are 2D/planar and the vertices must be on the same plane as the shape.



2. The measure in degrees in an angle tells me that the angle... Some of the common angles are... 
Some common angles are 90°, 180° , 270°, 360°.

3. Some strategies to estimate angle measures are... To find accurate measurements with tools, I should...
A strategy is to look at benchmark angles like 45° and 90° and see how close the angle is to the benchmark angle. Then you can change the angle by 5 or 10 to get a good estimation. To find accurate measurements, you should use a protractor. A protractor measures angles.

Modeling Integer Operations

6 + -4  = 2

4 - -3  = 7

-4 * -2  = 8

-8/4 = -2

Math Comparison: Time Spent on Homework

1. How was the data similar?
They both had outliers. The data for both blocks were skewed to the right. One more similarity these data sets had is that they both have the same maximum. The maximum for F and B block is 180.

2. What was different in the 2 sets of data? 
The median for F block was higher than the median of B block. Also, the minimum and maximum were different for both data sets.

3. Were there any outliers?
Yes, there were more outliers in B block than in F block. The outliers are the dots on the graphs.

4. Which class spends longer on their homework? PROVE IT!

F block spends more homework than B block because first of all, the median for F block is 80 while the median for B block is 60. Secondly, although the maximum for both graphs are 180, the whisker for F block is higher which means that there is data in that area. Finally, the box in F block is higher than the box for B block this means the data is for F block is more higher.

F Block

B Block

How Tall is a Typical 6th Grader?

What is the height of a typical 6th grader at SAS?
The height of a typical 6th grader at SAS is 61 inches. I know this because of the measures of center, the median, mode and mean. The median is 61, the mode is 61, and the mean is about 60.9577. If you round the mean to the ones place, you get 61, the same number as the median and mode. Therefore the typical height of a 6th grader at SAS is 61 inches.

How does you own height compare?
My height is 5 inches over the average height of 61. I'm 66 inches tall therefore I'm taller than average. Also, the IQR is 4 and ranges from 59 inches to 63 inches. My height still doesn't fit in there.

Does the height of 6th graders have a lot of variability? How do you know?
Yes, the height of 6th graders have a lot of variability, because there are no big clusters and the data is spread out really well. It also has a range of 20 inches which is a big range. Therefore, 6th graders have a lot of variability.

Data:

Mean: 60.95774648

Median: 61

Mode: 61

Q1: 59

Q2: 61

Q3: 63

Range: 20

IQR: 4

Personal Finance Project

1) What did you already know? I already knew that when people put their money in the bank, they get interest. Interest gives the banker more money.  2) What did you learn? (from the project? from your parents?) To become a millionaire, you need lots of time. Without banks, becoming a millionaire will be really hard. During the project, I saw that a lot of the money that makes you a millionaire is interest. I also learned that 5% interest makes a big difference. 3) How might this affect what you do in the future? This projects might affect me in the future because I am now aware of how saving in a bank gets you interest. Interest gives you more money. 4) What was your favorite part about the project? Least favorite part? My favorite part was when I didn't have to do the math and the spreadsheet would do it for me. Like when you drag the numbers down. It saves a lot of time. The least favorite part of the project was when the spreadsheet wasn't working right. Links Worksheet Spreadsheet Presentation

Fraction Division (3/4 ÷ 1/8)

3/4 ÷ 1/8

First, understand the question. 3/4 ÷ 1/8 is another way of saying how many groups of 1/8 are in 3/4.
Second, start solving it. You should start by finding the reciprocal of 1/8 which is 8/1. Remember, you are finding the reciprocal of 1/8 because it is the divisor. When dividing fractions, you always find the reciprocal of the divisor. The answer is 24/4
Finally, simplify your answer.  24/4 = 6.

Using a number line to solve this problem!

Student VS Beast

Math Reflection, Fractions and Decimals

When is it more useful to use fraction or decimal notation?

Fractions and decimals are both really useful, but there are times where one way is better than the other. For example, fractions are really easy to use when you are dividing numbers because fractions are division. But, when fractions get too big, they're hard simplify, add, etc. Decimals are easy to use when you do anything, from division to addition. The way you solve decimals is the same way you solve natural numbers. But there is a huge con, it's when decimals go on forever and ever.

Examples:
1) 5÷3
For this question, using fractions is more easier because all you have to do is take 5 and make it the numerator and take 3 and make it the denominator. So the fraction your going to end up with is 5/3 which is equivalent to 

1⅔. This way is way easier than using decimals because the way your going to solve the problem using decimals is long division, and with problems like this, long division will take forever.

2) 0.25+0.80
For this question, using decimals is the fastest way because all you have to do is add the two numbers like how you add natural numbers. You could use fractions but you would have to do two steps to get to your answer. Step one is to convert the decimals to fractions and step 2 is to solve.

When comparing two positive whole numbers with different numbers of digits, such as 115 and 37, the one with more digits is greater. Does this rule work for comparing decimals?

No it doesn't because with positive numbers, the more digits the bigger but with negative numbers (decimals), the more digits the smaller. You can see how it works on the place value chart.

Examples:
1) 2100 or 0.1232523
2100 is bigger because it is positive, even though 0.1232523 had more digits. 2100 is bigger because 0.1232523 is a negative number.

Fraction Comparison Reflection

You know how to compare fractions using cross products, LCD, and logical reasoning. When is it better to use one method over another?

Methods - Pros and Cons

Cross Products
Pro: It's a quick way to compare fractions, you only need to solve 2 equations.
Con: When the numbers get higher, it's harder to solve the equations.

LCD
Pro: Once you find the LCD, you just need to compare the numerators.
Con: If the LCD is really high, it's harder to find the new numerator.

Logical Reasoning
Pro: It requires no solving of equations!
Con: If the numerators are about the same, it's really hard to compare.


Examples

1) 5/7 or 6/8
For this question, I would use cross products because it's the quickest way to solve it. The numbers aren't huge and massive. Logical reasoning would not be a good method to use for this problem because the numerators are really close to each other. You could use LCD to solve this problem but that would take longer.

2) 21/35 or 46/70
For this question, I would use LCD because it's the easiest way to solve it. The LCD is pretty easy to find because 70 is a multiple of 35. Cross products would be another way to solve this problem but the numbers are really high which makes multiplying them harder. Logical reasoning would be hard to use because if you simplify the fractions, they both are more than 1/2.

3)3/82 or 32/35
For this question, I would use Logical reasoning because it's easy to use logical reasoning for this question. Also, you don't have to waste time doing unnecessary math. LCD and cross products would not be good methods because the numbers are really high.

My Favorite Number Project

To summarize Unit 1, we chose a number and studied it. It was a fun thing to do and a great way to end a unit. My number was 16. I hope you like it!

"Gran, How Old Are You?

How does prime factorization help you to answer this problem more efficiently?

Prime factorizations would help you solve this problem because they are numbers that are divisible by 111,111. So you keep on dividing the numbers in the prime factorization of 111,111 until you reach 0.

Blog the Reflection for Investigation 2

Question: How can you decide whether finding common multiples or common factors is helpful in solving a problem?

Answer:

Common Multiples:

When to find common multiples: If you want to find when 2 or more numbers reach the same number again.

Question that requires you to find common multiples: Billy Bob and Billy Bob Jr swim. Billy Bob swims every 2nd day and Billy Bob Jr swims every 3rd day. If they both start at the 1st day of the month, when will the swim together again?


Common Factors:

When to find common factors: When you want to equally share 2 or more items.

Question that requires you to find common factors: Johnny has 50 cans of apple juice and 2 packs of cheese. He wants to make packs of apple juice and cheese. All packs have the same amount of apple juice and cheese. What is the maximum number of packs Johnny can make?

Quicktime Player Test