We have learned how to find the area of rectangles and triangles.  To review those click the links below:

Rectangles
Triangles

This lesson combines both of those ideas into finding the area of irregular shapes. Take a look at the shape below:

To find the area of a shape like that, you simply need to decompose it into shapes we know how to find the area of. This particular shape would be divided into two rectangles; more specifically, a square and a rectangle.

From there we are able to easily find the area of the two shapes.

Square: 3 x 3 = 9
Rectangle: 9 x 2 = 18

At that point, you simply add them together and you have the area of your irregular shape.

9 + 18 = 27 in²

It doesn’t matter how complex the irregular shape gets, the concept is the exact same.

Start be separating the complex shape into simple shapes that we know.

When we decompose a shape like this, we may end up having to transpose some of the distances so we have all the information we need.  notice the bottom rectangle only has about half it’s length measure at 6.  This is because we have 4 for the left tower, and 5 for the right tower.  This tells us that the bottom rectangle is actually 15 long.  In this irregular shape, then, we have four shapes

Triangle – 4 x 4 ÷ 2 = 16 ÷ 2 = 8
Rectangle – 12 x 4 = 48
Rectangle – 6 x 15 = 90
Rectangle – 5 x 4 = 20

We then add them together and we have the area of our irregular shape.

8 + 48 + 90 + 20 = 166 un²

The other idea I want to bring up is the idea of using shapes that we know to find the area of shapes we don’t know.  For example, this pentagon:

We know how to find the area of the triangle: 5 x 5.6 ÷ 2 = 14.  There are five triangles that make up this pentagon, so we can then take the area of one triangle, and times it by 5 to get the area of the pentagon.

14 x 5 = 70 cm²

In our Understanding Integers lesson we learned how to put all types of integers onto a number line.  We also learned about absolute value being a number’s distance from zero, thus the absolute value of -6 would be 6 because it’s 6 numbers away from 0. The next step for us is to be able to figure out two number’s distance from each other.b  Typically to do this we would subtract.  How far are apart are 9 and 4? Well, 9-4 is 5.  It gets a little harder with integers as we haven’t learned subtraction with negative numbers yet.

Let’s take -8 and 3.  We want to figure out how far apart these two numbers are.  We will start by plotting these two numbers on a number line.
Our first thought could be to just count the number of spaces between our two points. There are 11 spaces between them, therefore the distance between these two points is 11.  If they are opposite signs like this, one positive and one negative, you’re basically adding the absolute values together. |-8| is 8, |3| is 3 and 8+3=11.

It’s a little different if the signs are the same. Take -11 and -2.  Start again with a number line.

Again, you could just count the number of spaces between the two of them. There are 9 spaces between -11 and -2.  Think about our positive numbers though.  The distance between 9 and 4 was 5, which is 9-4.  I can do the same thing with negative numbers, but remember, because we’re dealing with a distance, we’ll work with their absolute value, because that is their distance to zero.  |-11| is 11 and |-2| is 2.  We then subtract those. The distance between -11 and -2 is 11-2, which is 9.  Knowing this comes in handy when dealing with bigger numbers or numbers that are further apart.

What is the distance between -62 and -21?  Well, |-62| is 62, and |-21| is 21,.  62-21=41, therefore the distance between -62 and -21 is 42.

The next step in this lesson is the movement of a number on a number line.  If I start with -31 and I want to move a positive 9, I simply want to picture this on a number line.  Remember, moving a positive number moves us to the right.

Notice after moving 9 to the right, we end up at -22.

Our final step in all of this all of this, is to be able to put these ideas into real world scenarios.  Here’s an example of one for distance:

Johnny was parasailing 23 feet above the water.  His brother Mark was SCUBA diving 15 below the water.  What is the distance between the two brothers?

To solve these kinds of things, I find it easiest to visualize it by drawing a picture.

This allows us to easily see that finding the distance between the two of them is just a matter of adding the two distances together. Just don’t forget that when talking about a distance below sea level, you’re referring to a negative number, so technically what you are doing here is |23| + |-15| which gives us our 38 feet.

The final type of problem would be real life scenarios for movement. Dealing with only positive numbers of this would be easy considering we should have been doing this for years now.  Kaylyn was the flyer for her cheer stunt.  The cheer leaders hold her 5 feet in the air.  When they release her, they toss her an additional 4 feet.  How high did Kaylyn fly?  It’s a simple addition problem.  She was 5 feet.  She flew 4 more. 5+4=9.  It’s a little more difficult when dealing with negative numbers.  Let’s go back to Mark who was SCUBA diving.

Mark is SCUBA diving 15 feet below sea level. He dives down an additional 12 feet.  He then raises by 6 feet.  What depth does Mark end at?

Again, I recommend drawing a picture.  Basically, what you’re working with is a vertical number line.

The picture allows you see, if he started at 15 feet deep, and swam 12 further, he was 27 feet down which would represented by an integer of -27.  If then swam up 6 from there he would move up to 21 feet below sea level, which puts him at -21 feet.

At this point in time, we’ve dealt quite a bit with equivalent ratios in both making them larger, and simplifying them.  Also remember, that proportions are simply a pair of equal ratios.  We’ve also learned several different way to solve proportions which have an unknown variable in them.

This lesson simply takes everything we’ve known, and forces us to look at it in a new format.  Below is a table filled with equivalent ratios.  Notice, all I have done is multiplied the top and bottom by the same number to create the chart.

We may also see this chart vertically.  It means the exact same thing.

In this lesson we will be expected to fill in a chart with missing pieces.

Now remember, a proportion is two equal ratios. To fill in this chart, we are basically solving a proportion.

We could solve this using either method.

• Using a common multiplier.  5 x 4 = 20, so 1 x 4 =4; x=4
• Cross multiplying. 1×20÷5=4; x = 4

We would do this same thing for all three missing numbers.

1 x 6 = 6, so 5 x 6 = 30
1 x 10 = 10, so 5 x 10 = 50

Our finished chart would look as follows:

This concept taken into a real world scenarios through the use of story problems.

There is ax extra step in this problem where we have to come up with the starter ratio in the first row. Peaches would be 5:2 and apples would be 8:3.  From there we would simply multiply them to create larger equal ratios.

To answer the question, which orchard has more beehives per acre, we would want to orchard when they’re on the same number of acres. Peaches are 15:6 and apples are 16:6, which means that apples have more beehives per acre.

We now know what a ratio is.  In a class, there are 3 girls for every boy. If I were to ask what the ratio of boys to girls is, the answer would be 1:3.  This ratio is in simplest form, so it doesn’t tell us exactly how many boys or girls there are.  Let’s say we have 24 students in the class, and the ratio of boys is 1:3, how would we figure out exactly how many boys and girls there are?  We need to change our ratio to an equivalent ratio where the parts of it (the boys and girls) equal the total of 24 students.

Right now, we have 1:3, which equals 4.  One way we could do this is to just continue to increase the ratio until it equals 24. Multiplied by 2, it would be 2:6, which equals 8.  By 3 is 3:9, which is 12. By 4 is 4:12, which is 16.  By 5 is 5:15, which is 20.  Finally, by 6 is 6:18, which is 24.  This process could get very long depending on how far we need to multiply.  There is an easier way.

Follow these steps:

These steps would work in any scenario.  Lets say there are 144 animals in a zoo. The ratio of birds to mammals is 1:2.  How many of each animals are there?

Step 1: 1+2=3

Step 2: 144 ÷ 3 = 48

Step 3: 1 x 48 = 48, 2 x 48 = 96

There are 48 birds and 96 mammals.

In Algebra “Substitution” means putting numbers where the letters are:

If you have:  x – 2
And you know:   x=6
Then you can substitute 6 for x
6 – 2 = 4

Simple, right?  It gets a little more complicated.  Take a look at the expression:

6a

We’ll say that a=3.  It is very important to remember when there is a number right next to variable, it symbolizes multiplication.  One of the most common mistakes make is this:

If a = 3 then 6a = 63.

This is not the case.  Please, please, please, do not forget that a number next to a variable, such as 6a is the same as saying ’6 times a’.  So,

If you have:  6a
And you know:   a=3
Then you can substitute 6 for a
6(3) = 6 ∙ 3 = 18

Another example:

If x=5 then what is 10/x + 4 ?
Remember that ‘/’ represents division.
Put “5″ where “x” is:
10/5 + 4 = 2 + 4 = 6

This same concept applies even when there is more than variable.

If x=3 and y=4, then what is x² + xy ?
Put “3″ where “x” is, and “4″ where “y” is:
3² + 3×4 = 3×3 + 12 = 21

Temporarily taking math out of the picture, the word deviation simply means a change from the normal.  In mathematical terms, deviation is how far a number is away from the norm, in this case, the mean, or average.  The first step in this process is to find the mean of a set of data.  Let’s use the set of data below.

It’s a list of Runs-Batted-In for 15 players on the Seattle Mariners.  To find the average of these numbers, we have to start by adding them all together.

15+51+35+25+58+33+64+43+37+29+14+13+11+14+10 = 456

To find the average we then have to take that total and divide it by the amount of numbers, 15.

456 ÷ 15 = 30.4

Now that we have our average, we are able to calculate the deviation for each of the numbers.  Deviation is dimply the difference between the number and the  average. If a number is higher than the average, than its deviation will be positive.  If a number is below the average, then its deviation will be negative. The one problem we will come across is that we have not learned how to add and subtract with negative decimals.  For this reason, we are going to round our mean to the nearest whole number.

30.2 would round to 30

Below is a chart showing the deviation for each of the numbers in the data set.

The final calculation which will be expected is known as the Mean Absolute Deviation, or MAD for short. This is also known as the Standard Deviation. Simply put, this is the average of all of the deviations we just calculated.  This will require that we add positive and negative integers.  For a review on this topic, click here.

To find the MAD, we have to start by adding all of the deviations together.

-11 + 21 + 25 + -5 + 28 + 3 + 34 + 13 +
17 + -1 + -16 + -17 + -19 + -16 + -20 = 36

The next step in finding the MAD is to divide by the total amount of numbers, 15.

36 ÷ 15 = 2.4

This single number can tell us a lot about the data. The fact that it is positive tells us that there are more numbers above the mean than below.  The closer it is to zero, the more evenly spread out the data is.

Similar to a bar graph.  Similar to a line graph. Similar to a line plot. Yet, a different graph all of its own.

A histogram is a graphical display of data using vars of different heights.  Yes, I realize this sounds and looks very similar to a bar graph.  Notice that there are numbers on both the x and y axis.  This is more typical of what you might see on a line graph rather than a bar graph.  Also notice that each bar seems to represent an interval.  This is more typical of what you might see on a line plot.

YAY for smashing three different types of graphs into one!

Let’s display a graph that actually has a bit of meaning to it.  The histogram below represents the height of orange trees in an orchard.  The heights of the trees vary from 100 cm to 340 cm.

Take a careful look at the x axis.  Notice that it counts by 50.  The bar between 100 and 150 represents all the trees that are in between those two heights.  If you look at how high the bar goes, it tells you, of all the trees in the orchard, there are four of them that are between 100 cm and 150 cm.

Some types of questions you might see for reading a histogram:
* Which interval has the most trees in it?
Ans: 200-250
* How many trees are there between 300 and 350?
Ans: 11

We’re going to use the following data set to help us create and understand Box-and-Whisker plots:

14, 21, 19, 12, 13, 24, 26, 19, 15, 25, 19

Step 1: Find the median of the data.
To find the median, remember, we have to put the numbers in order and find the middle number.

12, 13, 14, 15, 19, 19, 19, 21, 24, 25, 26

The median of this set of data is 19.

Step 2: Find the upper and lower quartiles. If you look at the root of the word, quart.  A quart is a fourth of a gallon.  A quarter is a fourth of a dollar.  The root tells us to divide the data into fourths.  The median split the data in half.  to find the quartiles, you find the median of each of the halves.

12, 13, 14, 15, 19 —————— 19, 21, 24, 25, 26

The median of the lower half is 14.  This number represents our lower quartile. The median of the upper half is 24.  This number represents the upper quartile.

Step 3: Find the extremes.  These are exactly as they sound.  The extremes are the greatest and least values in the set: 12 and 26.

Step 4: Draw a line that includes all of the numbers in the data set.  This should include all numbers from 12 to 26.  It is drawn in green in the plot below.

Step 5: Above the line, draw a point for each of the numbers we’ve previously calculated: The median, 19, the quartiles, 14 and 24, and the extremes 12 and 26. This step is pictured in red below.

Step 6: Draw a box that begins at the lower quartile and ends at the upper quartile. Draw a line in the box at the median.  This is pictured in blue below.

Step 7: Draw the whiskers.  This is lines that come from both ends of the box and end at the extremes.  This is purple in the picture below.

This type of graph is an easy way for us to represent those hot spots (upper and lower extremes, upper and lower quartiles, and the median) all in one quick and easy to read graph.

If you were asked, “How many kilos of potatoes were consumed on Thursday?”  You would follow the x-axis to Thursday, find the dot above it and see that 10 kilos were consumed.

Double line graphs works the same way, with one exception.  Notice the key in the top right corner.  It tells you what each line represents.

The title tells us we are looking at a graph that represents Car Sales.  The x-axis shows us that we’re looking at four different years.  the y-axis represents the numbers.  The key tells us that we need to look at the dots.  The line with the circular dots represents the number of blue cars.  The line with the square dots represents the number of red cars.

Questions for these kinds of graphs will be more comparative.  Which color car sold better in the year 2001?  The circular dot is the higher dot on the graph which tells us that blue cars sold better in 2001.

Above is a visual representation of a bar graph.  It is one of the simpler types of graphs we’ll work with this year. Coincidentally, it’s also one of the more common types of graphs we’ll continue to see throughout life.  To a point, it is similar to what we’ve worked with using the regular coordinate plane.  Notice the y-axis works exactly the same, where the numbers get larger the higher up they get.  There will always be a label next to the y-axis telling what those numbers represent.  In this case, the y-axis represents number of students.  The title, “Our Favorite Sports” tells us that it represents the number of students who like a certain sport.  The specific sports in question move along the bottom of the graph.

We could read this graph to know that 9 students chose soccer, 4 chose softball, 6 chose basketball, and 3 chose something different.

This second graph is a double bar graph.  A double bar graph works the same way, but takes it one step further.  Notice the key in the top right corner.  It tells us that the yellow bar stands for the girls and the red bar stands for the boy.  From this graph we can now tell that of the 9 students who chose soccer, 6 of them were girls and 3 of them were boys.  You would be able to see this same information for softball, basketball and other as well.