June 19, 2019

Top 5 Things I’ve Learned This Year

Here are what I think are the most important topics I’ve learned so far in Math

#5 Polynomials

Although this may seem like a self-contained topic that doesn’t apply to much, this is still an important topic that applies to things like graphing and equations, you need to know how polynomials work for equations that contain variables and although much less important, graphing and t-charts utilize variables where polynomials could still apply.

#4 Exponents

This is quite an important topic that applies to certain topics like Polynomials and Square roots, alongside certain geometry formulas like solving for surface area and volume. You need to know how they work in order to get the right estimates for square roots, aligning like terms in Polynomials, finding out exponent values for variables, and so much more.

#3 Equations

Equations are what you use when finding the answer to any type of question, even basics like 1+1=2 are an equation, you’ll need them to find out answers in topics like values in t-charts and to solve polynomial questions, and even geometry can apply since finding out the formulas all equate to the final volume or surface area of an object.

#2 Fractions

Fractions are a topic that encompass a large variety of questions and topics, so wide that it’s difficult to pinpoint the topics it can encompass. Fractions can be used to answer scale factors that have reductions, fractions can be used to find out the unknown value in an equation if it’s larger than x (3x for example) fractions can be used to subtract and divide exponents, and quite a lot more. But because fractions are essentially division questions, they’ll show up in almost every other topic

#1 BEDMAS

This is by far one of the most important topics that I’ve learned this year, it applies to everything and is required for so many different topics. BEDMAS are the steps that you need to follow in order for you to get the correct answer, if you mess up in the order then you’re almost guaranteed to get the question wrong. This topic applies in so many different scenarios, from exponents to polynomials to rationals to geometry, this is one of the essential basics that the other topics build off of.

April 29, 2019

What I learned in Grade 9 Patterns and Graphing

What are patterns?

Linear Patterns, as the name suggests, are equations that have a consistent pattern when a number is plugged in. For example, 2x is an example of a linear equation since if you plug in any number, it will be accurate no matter what.

2, 4, 6, 8, 10, 12

2(1) = 2. 2(4) = 8. 2(6) = 12

The next kind of linear equation is something slightly different, if we add a constant to the equation, like 2x+1, we’ll get a different result.

3, 5, 7, 9, ,11, 13, 15, 17, 19, 21

2(1)+1 = 3. 2(4)+1 = 9. 2(6) = 13

It may not be immediate at first, but you can notice that it still goes up by 1. This type of equation is used for patterns that aren’t perfect multiples. Here’s another example, this time involving negatives.

29, 22, 15, 8, 1, -6, 13, -20

It’s clearly descending this time, so that always means the variable is negative, since no constant will stop it from going up, we can also tell that the pattern goes down by 7, so we continue from there.

-7x

Now that we know the variable, we can find out the constant as well by simply adding the variable to the first number in the pattern, in this case 29.

29+7=36

So our full equation is -7x +36, at this point we can plug in any number and it will work just fine

29, 22, 15, 8, 1, -6, 13, -20

-7(1) +36 = 29. -7(3) +36 = 15. -7(6)+36 = -6

Another method for finding the answers to a pattern equation is by using what’s called a T-chart, this chart allows you to look for the pattern if a number is missing. So let’s say our pattern is this:

We can put it into a T-chart, which is a simple chart that splits the amount of multiples and the values themselves like this:

The X value represents the multiples while the Y value represents the values of the numbers themselves, from here on out we can look for the equation by looking at clues. We can clearly tell that it goes up by 6 right off the bat, so we already have the variable 6x, then we can trial and error to look for something that matches our table:

Here we simply look for what matches, 1 to 3 is too low and 5 and above is too high, but 4 matches our chart perfectly, so now we have our equation – 6x+4.

Our next topic is graphing, in graphing you simply put the equation into a graph using dots, so if we had an equation like x+2 it would look like this on a graph

I’ll be covering how to accurately plot the dots on the line to make a correct graph. So here there are two intersecting lines, the Y axis and the X axis, and just like in the T-chart, the X axis controls the multiples while the Y represents the values themselves, the upper boxes represent the positive end and the bottom represents the negative end.

Let’s have a demonstration with 2x+4. The way you put an equation into a graph is like this, first you go up on the Y axis according to the constant which is +4

Then you go up by the variable while moving a tile to the right which is 2x, so you go up by 2 and move to the right

Keep doing this until you achieve the desired amount of dots on the graph.

Just remember to move the dots according to their signs as well. If the equation is something like -x-5 I would have to go down the Y axis like this:

Then go down by 5 and then move down and to the right by 1 like this:

A quick side note is that negative patterns always slant downhill, this is because even if the equation is something like -x+100, it WILL become negative at some point, and sometimes the Y axis is higher than the limit of the graph you have, like x+10

If you can’t go any further, simply move to the left, since this still counts since it’s now multiplying by negatives: x+10 = -1+10

And that’s what I learned in Grade 9 Patterns and Graphing

April 17, 2019

Horizontal and Vertical Line

Here is my horizontal and vertical line activity. My letters are meant to spell out “Mahoutsukai No Yome” which is the Japanese name for my favorite book and show series: The Ancient Magus’ Bride.

Here are the equations used for writing it all out.

March 15, 2019

What I learned in Grade 9 Inequalities

So, what is an inequality? An inequality is an equation wherein the two values could or could not be equal to each other, but we should know what the symbols make an inequality equation and a normal equation.

These are the four symbols that will define how an inequality question shall play out.< (less than) > (greater than) ≤ (less than or equal to) ≥ (greater than or equal to).

Usually in an inequality equation the unknown variable is on the left side of the equation with the other value is on the right, like so:

x > 5

This equation denotes that x is greater than 5, and this can be anything so long as it’s higher than 5, its obviously impossible for us to list out every single individual number that is higher than 5, so we graph it on a number line

The arrow faces wherever the inequality is, so if it’s less than it points to the left, and if it’s greater than it’s to the right. If the inequality has a dash beneath it, denoting an “or equal to” equation, then it will instead turn out like this:

The reason why the normal inequalities are open in are because they are the boundary point between what is a correct value or not, while the “or equal to” is shaded because the number is included in the values that could be the correct answer.

With that out of the way, let’s solve a normal inequality question and check if our answers are correct. Let’s say our equation is x+2.5<-6

We do the standard legal move procedure until we find an equation where one side has the variable and the other has a value.

Afterwards we can graph it on a number line, but what we’re focusing on here is how to check if our answer is correct, so the best thing to do is plug in the answer we got into the x and solve as per usual.

This shows that our dashed inequality is correct, but we should also check if the inequality is facing the right direction, so we pick a number according to the inequality, in this case we will be using -9.

This confirms that the inequality symbol is now facing in the right direction.

That’s what I’ve learned in Grade 9 Inequalities

March 2, 2019

What I learned in Grade 9 Equations

So, what is an equation? We usually associate equations with problems that need to be solved, something like 5+5=10, but equations boiled down are statements wherein both sides are equal, 5+5 is equal to 10. The equations that we’ll be dealing with are comprised of variables, coefficients and constants, for the sake of distinction we’ll call them “linear” equations and our main goal is to find the answer, or the equivalent, of the variable.

So a simple linear equation would be something like 2x-3 = x+4

Our main goal is to find the value of x, to do that we need to place the variable on one side of the equation and the answer on the other. In order to do this we need to make zero pairs, zero pairs are pairs of like terms that are combined to cancel each other out, like terms must only be made with like terms, so -3 can’t be combined with 2x, and whatever we do to one side also has to be done to the other, this is called a legal move.

3x-2=x+2

-x +2 = -x +2

2x = 4

If the variable doesn’t have a value of 1, divide both sides by the coefficient and simplify further if needed.

$\frac{2x}{2}$ = $\frac{4}{2}$ = x-2

And that’s the basics of solving a linear equation, but sometimes questions won’t appear like this, and have fractions, brackets and decimals that make solving much harder than usual, but everything can be made easier if we know what to do. So our tougher problem is this:

We can use a tactic called BFSD (Brackets, Fractions, Sort, Divide)

12( $\frac{m}{4}$ – $\frac{2-m}{3}$ ) = 12( $\frac{1}{4}$ )

So first we have to get rid of the brackets, to do that we have to distribute the constant behind the brackets

12 x $\frac{m}{4}$ – 12( $\frac{2-m}{3}$ ) = 12( $\frac{1}{4}$ )

Then we apply what we did earlier, and combine all like terms and distribute further if needed.

3m-4(2-m)=3

3m-8+4m = 3

7m-8 = 3

Then we do a legal move and add 8 to -8 and 3

7m-8 = 3

+8 +8

7m=11

Afterwards we divide the variable by it’s coefficient, and do the same to 11

$\frac{7m}{7}$ = $\frac{11}{7}$

m= $\frac{11}{7}$

If the final answer is a fraction and can’t be simplified further, simply leave the fraction as is, but if you’d like you can convert it into a mixed fraction if it’s easier for you, and that’s it, that’s the algebraic way of solving linear equations, another way to do this is visually using algebra tiles. This is practically identical to the algebraic way, but personally I prefer the algebraic way since I can display more information more easily without having to cut up my algebra tiles when a fraction comes in.

Let’s say our problem was 2x-5=3

In this case you would still apply the same rules to the algebra tiles as you have done algebraically.

First we add 5 to the -5 on the left side, and add 5 to the right side in order to make a zero pair

Now that the zero pair is in place, we can get rid of the -5

Now we can more easily find out the value of x, we simply have to arrange the tiles so that we can divide more easily. 8 divided by 2 is 4, so our answer would be x = 4

And there you have it, that’s how you use algebra tiles to solve linear equations. Be wary that you can do these equations provided that you have enough tiles, sometimes the problem has more numbers than you have tiles, so know when to use algebra and when to use tiles.

But there are a few things that I’d like to mention that could prove helpful when dealing with certain linear equations. Like this

$\frac{1}{12}$ x +2 = $\frac{3}{4}$

You can make this question easier by looking for the lowest common denominator in the equation and multiplying it to the fractions.

12( $\frac{1}{12}$ +2) = 12( $\frac{3}{4}$ )

x+24 =9

x=-15

You could do this to any linear equation that contains a fraction by simply multiplying the denominator and the fraction itself, as long as you maintain legal moves and multiply the other side.

$\frac{1}{2}$ x +1 = 6

2( $\frac{1}{2}$ x +1) = 6

x + 2 =6

-2 -2

x=4

And that’s what I learned in Grade 9 Equations.

January 16, 2019

What I learned in Grade 9 Surface Area

Let’s start off with Surface Area. Surface area is the combined measurement of all faces in a 3D shape. This usually applies to prisms, prisms are solid objects that have identical ends and flat faces.

Many different prisms have different formulas to solve them, but boiled down it’s simply finding the area of each face and combining them together.

Composite surface area isn’t very different either, composite surface area is when two prisms are glued together to have a sort of “super-prism”, solving the surface area is similar but with a twist.

Let’s say these are the dimensions for our composite shape:

Rectangular Prism 1:

Length : 15mm

Height :8mm

Width : 10mm

Rectangular Prism 2:

Length :6mm

Height: 4mm

Width : 5mm

In order to find the surface area of a composite prism, you first need to find the surface area of both prisms

Rectangular Prism 1: 700mm

Rectangular Prism 2: 148mm

Let’s say Prism 2 is on top of Prism 1, and the side glued to Prism 1 is 30mm, since its already glued to Prism 1, then it’s practically vanished away from both prisms, so we have to subtract 30mm x2, one for the 30mm taken away from Prism 1, and again for Prism 2.

So we add together both surface areas

848mm

And subtract the 60mm

788mm

And there we go, that’s how you solve for composite surface area, the same process applies to different shapes, it doesn’t matter as long as you solve for the surface area and the side that’s attached to the bigger shape

December 13, 2018

What I learned in Grade 9 Polynomials

So, let’s start with the low-hanging fruit. What is a polynomial? For starters, a Polynomial is a sentence for algebraic equations, something like this:

4xy+3x + 2x+2

It’s usually comprised of numbers, variables and exponents. A set of these is called a term, terms can help us identify what kind of algebraic equation it is.

3xy is a Monomial

5ab-3a is a Binomial

4vk-3k+v is a Trinomial

5cd-3x+4-y-12 is a Polynomial

Note that anything that has 4 or more terms is classified as a Polynomial.

Polynomials also have degrees, in simple terms the degree of a Polynomial is the largest exponent the Polynomial has.

( $6x^3$ ) + ( $6xy^2$ ) – ( $3x$ ) + ( $2$ )

3 3 1 0

If there are two different variables, add them together to get the degree

If two exponents are both the largest, then the Degree is still the same.

Operating with Polynomials

Addition :

Adding Polynomials is as simple as it can get, add all like terms together to get the answer.

( $4x^2$ ) + ( $2x^2$ ) = $6x^2$

Subtraction:

Subtracting Polynomials only requires you to flip anything to the right of the minus symbol, all negatives become positives and all positives become negative, afterwards add them as you would normally.

( $3xy^2$ ) – ( $4xy^2$ + 3)

( $3xy^2$ ) + ( $-4xy^2$ -3) =

( $-xy^2$ )

Multiplication:

Multiplying is quite easy, multiply all like terms together, but add the exponents up.

( $3a^3b^3\cdot 5a^2b$ ) =

$15a^5b^4$

Division:

Like multiplication, all you have to do is divide all like terms together, but subtract the exponents instead.

( $10x^3y^2 \div 5x^2y$ )=

$2xy$

A few tips I could share:

If there is a number outside of the bracket, then you need to distribute the number to everything inside the bracket

Likewise, if there’s an operation behind the number closest to the bracket, leave it as is and distribute the number to everything inside the bracket.

If you’re dealing with smaller polynomials, you can use algebra tiles.

Have $x^2$ as a large square

$x$ as a rectangle

and 1 as a small square

Color them in on one side

Flip them over to the clear side when using a negative

Use them as you like.

November 17, 2018

What I learned in Grade 9 Exponents

So, what have I learned about Exponents up until this point? Well, why don’t we start off with the basics.

What is an exponent?

An exponent is a simple version of multiplying a number to itself, the exponent is located at the top right of a variable, like so:

$x^2$

The exponent shows us how many times it has to multiply itself, if a number like 2 is raised to the power of 2, then the equation will look like (2 x 2), if it’s raised to the power of 4, then it’s (2 x 2 x 2 x 2).

There are special terms for certain exponents however, the power of 2 is called a square, the power of 3 is called a cube, the rest are simply “x raised to the power of y”

What is the difference between evaluating and simplifying?

If we had a question like “ $3^2\cdot 3^4$ “, then the end result can be “ $3^6$ ” without any repercussions, that is evaluating.

But if we had a question like “ $4^4\cdot 4^5$ “, then we need to simplify it to it’s lowest form, the end result is then “1280” instead of “ $4^9$ “, that is simplifying.

What is Multiplication Law?

If we are multiplying two exponents that have the same base, we can just add the exponents together.

$5^4\cdot 5^4$ = $5^8$

What is Division Law?

The opposite of multiplication law, if we are dividing two exponents that have the same base, we can just subtract the exponents.

$6^9\div 6^8$ = $6^1$

What is Power of a Power law?

If we have an exponent that has another exponent outside of it, then we need to multiply the exponents together.

( $7^3$ ) $^3$ = ( $7^9$ )

What are Exponents on Variables?

The only difference between exponents on numbers and exponents on variables is that there isn’t a simplified answer for exponents, since the value is unknown, then you can simply just add the exponents like in Multiplication law.

$x^2\cdot x^3$ = $x^5$

October 17, 2018

What I learned in Grade 9 Fractions

Here’s what I learned in Grade 9 fractions so far

For putting negative fractions on a number line, you have to determine the fraction’s value and put it on the number line

$\frac{1}{4}$ $-3\frac{3}{4}$

<5—-4 $-3\frac{3}{4}$ —3—-2—-1—-0 $\frac{1}{4}$ –1—-2—-3—-4—-5>

For $\frac{1}{4}$ , it’s clearly less than 1, but still has a value greater than zero, so it’s placed in between

For $-3\frac{3}{4}$ , since it’s a negative, it goes backwards on the number line, since if going up with positive means going on the right, negative means going on the left.

Comparing fractions is quite easy, it’s as simple as checking which one is bigger than the other, like this :

$\frac{1}{3}$ $\frac{2}{3}$

The line is clear as to which number is bigger, because 2 is clearly larger than 1. But since teachers want to test their students, you’re more likely going to find questions like these :

$\frac{1}{3}$ $\frac{3}{6}$

These questions aren’t as hard as you think either, all you need to do is make them equal, an easy way of doing this is by making sure both denominators are the same, and you can do this by multiplying.

$\frac{2}{6}$ $\frac{3}{6}$

The reason why the 1 turned into 2 for the first number was because whatever happens to the denominator, happens to the numerator, it’s kind of like if 3 got caught stealing food from a grocery store, 1 would go down with them.

So now that both have equal denominators, both can easily be compared, you do this process with all fraction comparison questions that have uneven numerators and denominators.

Some numbers are a combination of a whole number and a fraction, like this :

$1\frac{1}{2}$

In order to make this into an improper fraction for use, we need to multiply the denominator to the whole number, then add the result to the numerator, the end result is something like this:

$\frac{3}{2}$

One important thing to do before comparing any fraction is to make all mixed into improper first for ease of solving.

For adding and subtracting fractions, it’s just as easy, you do the same steps, but with slightly different rules.

$\frac{1}{4}$ + $\frac{3}{12}$

So make both of them equal like so, make sure to multiply the numerator as well:

$\frac{3}{12}$ + $\frac{3}{12}$

= $\frac{6}{12}$

The same applies to subtraction questions as well. But when you put a negative, that’s when the whole thing flips on it’s head

$\frac{-6}{12}$ + $\frac{3}{12}$

$\frac{-8}{10}$ – $\frac{7}{10}$

Both addition and subtraction can catch you off guard if you’re not careful, if it’s an addition question like this, then do a little game that goes like this :

There are two nations, Positive and Negative, they are currently fighting a war of attrition, and every group of reinforcements can be decisive to winning a battle, there are multiple ways clever field commanders can strategize, some skirmishes can go like this:

Battle 1:

-6 negative soldiers vs. 3 positive soldiers

Whatever side is larger wins the battle, so the aftermath will look like this

-3 negative soldiers surviving

This is how some battles will play out, either side must win through sheer force of manpower. Another way the battle could play out is like this:

Battle 2

-8 negative infantry units vs. 7 positive infantry units

Since this is a subtraction question, or as commanders like to call a propaganda attack, all 7 positive units defect and aid the negative side:

-15 negative soldiers

This causes the Positive nation’s defenses to be overrun, if a subtraction question has a negative on the left side, then any positive must turn into a negative as well, negative + negative = negative.

Either nation would keep fighting, one may win the battle, but not the war, so the fight between Positive and Negative will continue until the end of time.

Multiplication and Division

All you have to do for Multiplication is direct and easy, just multiply straight across

$\frac{4}{7}\cdot\frac{6}{14}$

= $\frac{24}{98}$

And you’re done.

For Division, there are multiple ways you can go about dividing, but I’d say you should just use the Reciprocal method :

$\frac{12}{15}\div\frac{7}{9}$

Just flip the right fraction on it’s head then multiply

$\frac{12}{15}\cdot\frac{9}{7}$

= $\frac{108}{105}$

October 1, 2018

Digital Footprint

How might your digital footprint affect your future opportunities? Give at least two examples.

If you attempt to enroll for a college or look for a job, your employers will sleuth the internet in search for any information that will shed light on your personality. They can check if you have any…undesirable traits like aggression, racism, or unprofessional behavior simply from any posts that you made in the past. While at face value these don’t give a 100% accurate reflection on your character, they will carefully scrutinize these since they have little to no information outside of your resume.

Any future relationships that you may have, like potential partners, might also check your profile to see if you have traits they may not want in someone they intend to spend their life with. If they find out that you are aggressive and abusive, they may turn you down and look for better people, besides, there are plenty more fish in the sea.

Describe at least three strategies that you can use to keep your digital footprint appropriate and safe.

Do not show any personal information. Use a username instead of your full name, don’t show your home address, keep your account so that only friends can see them, and overall be secure and keep any confidential information away from prying eyes.

Keep watch over your accounts and passwords. Always keep them in a security app like LastPass or Applock, and to reinforce security even further, make passwords for these security apps even harder than your personal account passwords

Don’t be reckless. By simply exercising caution you can effectively protect your information to a large degree.

What information did you learn that you would pass on to other students? How would you go about telling them?

Add the site name to the end of your password for certain sites, like using passwordInstagram, passwordSteam or passwordGmail

Be careful when dealing with location based services, since certain apps may request permission for services that aren’t even tied to their original use, like a Camera requesting location permission, while it has it’s fair share of positives, always be aware of the potential security threat.

There will always be people who would want to steal your personal information through viruses and spam mail, always make sure that your anti-virus is up to date and to be on the safe side, manually go through your emails and messages to make sure that no files like viruses are present.

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