Math 8 Summer School

Before we start plotting points, we need to build a standard structure for plotting them.

Steps for building a Coordinate Plane:

1. Start by thinking about the horizon, then draw our horizontal axis.
2. Mark an X on our horizontal axis so we know that it's our X-axis.
3. Draw our vertical axis, perpendicular to the X-axis.
4. Mark a Y on the vertical axis so we know that it's our Y-axis.
5. Confirm Y has a little V in it. Use this to ensure you have the X and Y correct.
6. The origin is the intersection point in the middle (where both axes are at zero).
7. Add a scale to the X-axis. Positive to the right, Negative to the left.
8. Add a scale to the Y-axis. Positive going up, Negative going down.
9. You're ready to graph!

Once we have our Coordinate Plane built, we can start plotting points.

Plotting points:

• The location of a point is determined by its coordinates.
• We need an x-coordinate and a y-coordinate.
• The coordinates are often presented like this (1, -2).
• The set of coordinates can also be called an ordered pair.
• The first number is the X-value (left or right).
• The second number is the Y-value (up or down).

PERIMETER:

We use perimeter to measure the distance around objects. You can think of wrapping a string around a triangle. The length of this string would be the perimeter of the triangle. If you walked around the outside of a park, you walk the distance of the park’s perimeter.

AREA:

Introducing Average!! Consider this situation where employers and employees need to think about data and averages:

You'll find that many conversations you'll have in the future will involve the topic of averages. Being knowledgeable about what "average" means will help you make sense and contribute knowledgeably to these conversations:

• average wages
• average speed
• average size
• average grade
• average age
• average house prices
• average height

Pretty much anywhere you find numbers, you'll find someone talking about the average of those numbers!

There are 3 common ways to determine an average.

The average is a term which is used often to try and best represent the "central tendency" (or "typical value") of a set of numbers. What number best represents the group of numbers? If I could throw away my data and replace it with only one “average” value, what would it be?

Thus, average is a fairly simple term, but it has several meanings.

• Mean Average = Add up all the numbers, then divide by how many numbers there are. (Most common.)
• Median Average = Places the numbers in numerical order and find the middle number. (Deals nicely with extreme values.)
• Mode Average = Count the frequency of each number and pick the most common one. (Just like voting.)

MEAN:

The most common type of average that we think of is called mean average..

The mean is the number you get when you add up all the values and then divide by the number of values.

The advantage of using the mean is that it uses every single number. But the disadvantage is that if any of those numbers are extreme (i.e. an "outlier") then the mean will be stretched (less accurate as an average).

RANGE:

Another common term when dealing with data is the range of data.

The range is the difference between the maximum number and the minimum number.

MEDIAN:

The next most common method for determining average is called median.

The median is the middle number, when they're arranged in order.

Advantage of Median: It doesn't get swayed by outliers. Outliers are data point that differs significantly from the other data points.

With an 'odd' number in the set it is easy to find the middle number...but if you have an even number in the set then there are two numbers in the middle....you must take the 'mean' of those two numbers to calculate the median:

MODE:

The third (and final) way that we'll consider to describe an average is mode.

The mode is the number that occurs most frequently.

Sometimes we can have more than 1 mode...as long as they are tied for the 'most'... we can also have NO mode if there are no numbers that occur more than once...

Three Types of Problems:

You hear the term percent and percentage used often. Percentages are used for taxes, discounts, calculating tips and test results. Percent means 'per hundred' .

Percent to Decimal

Often you need to convert a percent to a decimal, in order to do a calculation.

Decimal to Percent

Knowing how to convert percentages to decimals and back again is a valuable math skill.

Percents and Fractions

Fractions to Decimals to Percents...

A ratio is a comparison of two things. We might compare how many hockey games your team won to how many your team played. We might compare the number of green candies in the bowl to the number of red candies in the bowl. A ratio allows us to compare amounts and make decisions on these results.

Ratios are the mathematical numbers used to compare two things which are similar to each other in terms of units. For example, you can compare the length of a pencil to length of a pen.

You can compare the distance to school to the distance to the store. You can’t compare two things that are not similar to each other. For example, you can't compare the height of a person to the weight of another person.

Many ratios can be written with smaller numbers. This is called writing ratios in their simplest form, reducing to lowest terms or simplifying ratios.

Simplifying ratios makes them easier to work with. To simplify ratios, you can use the same technique that you use to simplify fractions.

Simplify ratios by dividing the number on each side by their greatest common factor.

Ratios as Fractions

Three Term Ratios

A three-term ratio will compare three amounts.

Example: Kalani has a backpack with 3 pens, 4 marbles, 7 comic books, and 1 apple. You could compare marbles to comic books to pens: four to seven to three or 4:7:3.

What would this three-term ratio be comparing with the ration 7:1:3 ?

comic books to apples to pens

Graphs are a great tool for sharing data with people in an easy-to-understand format.

The purpose of a graph is to present data that are too numerous or complicated to be described adequately in text and in less space.

Line graphs are a set of graphed points joined by line segments.

A scatter plot is a way to show collected data and recognize a general trend.

A pie graph (or pie chart) shows the breakdown of something at a given point in time.

Bar graphs are like pie graphs, but they allow the reader to see the details a bit more easily.

A pictograph uses graphics to make the graph more fun or appealing.

The Theorem of Pythagoras is used in many branches of math and science. The following video gives an explanation of this very useful theorem.

Pythagoras (puh thag or us) was a brilliant Greek philosopher and mathematician, born in the sixth century, B.C. He and his followers tried to explain everything with numbers.

He was able to demonstrate that there is a special relationship among the sides of a right triangle. A right triangle is one that has two sides that form a right angle. The side opposite the right angle is called the hypotenuse and the two shorter sides are called the legs. Here's a right triangle:

Notice that the sides of the triangle in the middle are 3, 4, and 5 units. The square formed by the leg of length 3 is 9. You could also say 32 = 9. The square formed by the leg of length 4 is 16. You could also say 42 = 16.

The sum of these two squares, 32 and 42, is equal to the square of the hypotenuse which is 25 or 52. Or, 33 + 42 = 52.

Don't worry if you have to reread the previous few sentences to understand!

This relationship is only true for right triangles.

Here's what the Pythagorean (puh thag or ee un) Theorem says:

The sum of the squares of the lengths of the legs of any right triangle is equal to the square of the length of the hypotenuse.

This is often stated as a2 + b2 = c2

where a and b are legs and c is the hypotenuse.

Therefore, if you know the length of any two sides of a right triangle you can determine the 3rd side.

Remember: the hypotenuse is always the LONGEST side in a right triangle.

One of the uses of the Theorem of Pythagoras is to calculate the length of the hypotenuse when you know the length of the other two sides of a right triangle.

Sometimes you want to find the length of one of the legs of a right triangle given the length of the hypotenuse and one leg. This video gives you a solution.

Exponents are all about saving time. Rather than writing out a repeated multiplication as 3 x 3 x 3 x 3 x 3 x 3, you can simply write 36.

What does it mean when something is squared or cubed?

What does it mean when we call a term a perfect square?

Let's return to BEDMAS. Now that you know what exponents are, we can bring the "E" into our problems.

Finding the square root of a number is the inverse (opposite) operation of squaring that number.

Multiplying & Dividing Fractions:

• The next step in multiplying fractions is to multiply a fraction by a fraction.

In the illustration above, the blue part of the pie on the left represents 1/3. The blue section in the circle in the middle shows the 1/3 piece separated into 2 equal parts (or multiplied by 1/2 ). Choosing one of these parts gives us the last section which is . Having a fraction is like having a part of, so multiplication is like saying what part of (the whole thing).

Finding 1/2 of 1/3 results in 1/6. When multiplying a fraction by a fraction multiply the two numerators (1 x 1), then the two denominators (2 x 3).

Numbers often don't come to us as nice rounded numbers. We'll frequently run into numbers in the form of fractions.

What is a fraction?

A fraction represents part of a whole. When something is broken up into a number of parts, the fraction shows how many of those parts you have.

A fraction consists of a numerator and a denominator.

Fractions are often shown as part of a circle. They can be part of anything.

Factoring is the opposite of multiplying. For example:

• 6 x 2 = 12 is a multiplication.
• 12 = 6 x 2 is breaking 12 down into factors.

Factors (or any numbers) can be categorized as:

• Prime → Only have 2 factors, the number itself and 1.
• Composite → Has more than 2 factors.
• Neither → Only 0 or 1.

In this video, you'll not only learn about adding and subtracting integers, but also see how this relates to working with temperatures.

The rules to remember are below. Be sure you can use an example like temperatures to explain why they make sense.

Remember, always rewrite integer subtraction and addition like the examples below:

3 + (+2) = 3 + 2

3 − (−2) = 3 + 2

3 + (−2) = 3 − 2

3 − (+2) = 3 − 2