Math 11

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Week 6 – Foundations 11

This week we started on scale factor and how to find them with 2 different shapes.

In this image, the dimensions of the first rectangle is 3x5cm. To find the total perimeter of the rectangle we can do 3(2) + 5(2) = 16cm. And to find the area we use base x height which will be 3×5= 15cm².

It’s pretty obvious that the rectangle has doubled in size based on the fact that the sides are multiplied by 2. We can use the steps previously to find information such as the total perimeter and area or we can see that if entire rectangle is doubled, then to find the total perimeter, all we need to do is multiply our final answer. So 16(2) = 32cm is our perimeter. When we double the perimeter, we can see that the area has been multiplied by 4, so the area would be 60cm². Or we can just use base times height to find it aswell.

Remember that the trick I used to find the area for the larger shape only works if the scale factor is 2.

And if the scale factor is less than 1, the shape will become smaller.

Ex. SF = 0.99 = smaller

Ex. SF = 1.3 = bigger

Ex. SF = 1.01 = bigger

 

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Week 5 – Foundations 10

This week we learned everything about angles and how to find them without using a protractor.

When dealing with parallel lines connected with a transversal, there are a lot of ways be can connect angles and there also a lot of vocabulary included.

Supplementary angles: These angles always add to 180. Ex. 1 and 2 or 5 and 7.

Also: every angle around a point equals 360 degrees. For example the angles 1, 2, 3, 4 all equal 360.

Vertically opposite: angles that are vertically opposite are always equal. Ex. 1 and 4 or 6 and 7.

Corresponding angles: these are angles that correspond with each other and are equal as well. Ex. 1 and 5 or 4 and 8.

Interior angles: these are angles that are both found inside the two parallel lines and are on the same side. These angles when added always equal 180. Ex. 4 and 6 or 3 and 5.

Exterior angles: the same thing as interior angles but on the outside of the parallel lines. These angles also always when added equal 180. Ex. 1 and 7 or 2 and 8.

Alternate interior angles: These are the same as interior angles but on opposite sides of the transversal. They are always equal. Ex. 4 and 5 or 3 and 6.

Alternate Exterior angles: Same as exterior angles but on opposite sides of the transversal. They are also always equal. Ex. 1 and 8 or 2 and 7.

Knowing all these different terms, you can find every single angle on the diagram without using a protractor with only one given angle.

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Week 4 – Foundations 11

This unit we learned about inductive vs. deductive reasoning and the differences between them.

People often say inductive reasoning is the weaker of the two because it’s an observation —> a rule which can’t always be correct. For example if I take 6 blue marbles from a bag of 10 marbles I can’t be certain that all 10 marbles are blue. The observation that I only got blue marbles made the rule that there are only blue marbles in the bag, which can’t be 100% accurate.

Deductive reasoning is quite the opposite. It’s a rule —> an observation. So if we know for a fact that there are all 10 blue marbles inside of the bag because it’s a rule then we can make the observation and predict that there will be all blue marbles. Another example is gravity. Gravity is a constant “rule” that if something goes up, it must come back down. If you throw an apple up a million times, every single time it will come back down. The concept of gravity would be deductive because it’s been proved that it’s a rule and not a coincidence that the apple comes back down every time.

These methods of reasoning can be used for pretty much any concept.

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Week 3 – Foundations 11

This week we learned about the differents forms of equations and how to identify them. There are 3 that we became familiar with and how to differentiate them.

1. The general form: y = ax2 + bx + c. This equation is good because we can find the y-intercept, which is c. The a signifies the amount of curve the parabola has. For example if a = 1 then the spacing between points will be 1-3-5. If it’s 2 then you multiply all the numbers by 2 so it will be 2-6-10 and so on.

2. The vertex form: y = a(x + b) + c. Like the name says we can find the vertex, which is the most important point. For example if the equation is y = (x – 5) + 4 then the vertex will be on (5,4). Whatever is inside the brackets must be flipped. Is the equation is y = -(x – 5) + 4 then the whole parabola will be flipped upside-down because of the negative. And like general form, a signifies the curve of the parabola.

3. The factored form: y = a(x + b)(x – b). We can use this equation to find the x-intercepts. And like what we did before, we can find the vertex using these intercepts. If the equation is y = (x – 5)(x – 3) then we know the x-intercepts are +5 and +3. Remember to flip the numbers since they are in brackets. To find the vertex we first add both the numbers and divide by 2. So 5 + 3 = 8/2 = 4. So now we know the line of symmetry will be x = 4. Then to find the vertex we take 4 and plug it into our equation to get our y-coordinate. y = (4 – 5)(1 – 3) -> y = (-1)(-2) = 2. So the coordinates of our vertex will be (4,2).

 

 

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Week 2 – Foundations 11

This Week we learned about parabolas and what they are. First we learned the proper vocabulary for important parts of the parabola and then we learned how to properly graph them. Before we start, when you see “x²” in an equation you should automatically know that it’s going to be a parabola.

Let’s start off with the vocabulary, the highest or lowest point of the parabola (the curved part) is called the vertex. I say the highest or lowest point because depending on the equation the parabola can be flipped upside-down making the vertex the lowest point.

The line of symmetry or the axis of symmetry is the line that goes down the center of the parabola. For example, if we cut the parabola in half it’s the line where both sides will look the same but mirrored. There are many ways to find the line of symmetry, we can just use our eyes and find the middle of the parabola. But sometimes it isn’t that simple so we can find the average of the two point to find the midpoint (the middle).

To graph a simple parabola, we can use the 1-3-5 rule. These numbers represent the spacing we need to use when graphing the parabola. For example the equation: y = x² + 6 we would start with 6 which is the y-intercept. Next we use the 1-3-5 rule. We start by going up 1 and then 1 out, then we go up 3 and 1 out, and finally 5 up and 1 out, making a sort of exponential curve.

Now we know the important vocabulary for parabolas and how to graph them.

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Week 1 – Foundations

Inequalities

I good amount of the stuff we learned during this unit was review from previous years. In grade 9 we learned how to re-arrange the equations correctly and last year, in grade 10 we learned how to make the slope formula and how to use it on a graph.

This year, we learned about inequalities and how to graph them.

When looking at this equation: y = 5x + 4 it isn’t anything new, because it follows the y = mx + b format. We should be able to graph this equation easily starting off with the y-intercept which is 4. Then recognizing 5 is the same as the fraction 5/1 then using rise over run to make our line.

But what if I told you this equation: y > 5x + 4 means almost the same thing, the only difference is what you do after drawing the line. You would still do the same steps as the previous equation but this time the line you would draw wouldn’t be a full line, it would be a dashed-line.

It would be a dashed-line because the inequality says that it is greater than so the solution would include the line. If the symbol in the equation is ≥ (greater or equal to) we would use a full line because it shows us that the solution doesn’t include the line. Sort of like the number line when we shade in the circle to show if the number is included or not.

Now with this information we move onto the next step, locating the region in which our solution may be in. We have the dashed line and now what we can do is use any coordinates of our choice to find the side of the line the solution is. One of the easiest ways to do this is to use the coordinates (0,0), because all we need to do is plug the zeros into the equation in place of the variables. Now our equation will become: 0 > 5(0) + 4. Once we solve it we will get: 0 > 4.

If the statement is true, then the region where our solution is includes the coordinates we used (0,0). If it’s false then we use the side of the line where we didn’t use the coordinates. In this case the statement is false so we wouldn’t shade the side where (0,0) is.

When we find the region where we think the solution is in, we can pick a random coordinate in this region. For example: (-2,-2). Like we did before we will plug the numbers into our equation and if the statement is correct it is safe to say we found our region.

-2 > 5 (-2) +4

-2 > -6 is true so we know we got the correct answer.