Understanding Proportions

Understanding Proportions

Proportions is a way of comparing 2 similar things. You do them every day. Have you ever figured out how many miles your car can get per gallons of gasoline? Or how much something is per ounce? Or How how long it will take you to finish a project? Or how much money you can make per day? If you have you have just used the concept of proportions.

Before we look at problems and solutions using proportions, let’s look at and example together.

Question: Al painted 500 square feet in 6 hours. At that rate how many hours will it take him to paint 1500 square feet?

What is the question? At that rate how many hours will it take him to paint 1500 square feet?

What is the necessary information? 500 square feet, 4 hours, and 1500 square feet. The words at this rate are a clue that we will be using proportions as a setup to solve this problem.

What is the operation? They have given us three pieces of information and we will need to solve for the fourth. To do this we are going to put like things across from each other, cross multiply, and then divide.

So we have our

500 sq ft   =   1500 square feet

6 hours                      x

Remember we are going to place like things across from each other. And we are solving for x, the number of hours. And we need to cross multiply and then divide.

Solve it. So we are going to multiply 1500 x 6 = 9000 and then we need to divide 9000 by 500.

That is going to give us 18.

Does this answer make sense? Do you see that 1500 is 3 times larger than 500? So is 18 three times larger than 6? Yes, so this makes sense and is the correct choice. If you see patterns that double or triple, you can just multiply to get the answer.

 

500 x 3 = 1500 and

   6 x 3 = 18

So the answer is it will take 18 hours.

 

Understanding Fractions

Understanding Fractions

Don’t be afraid of fractions.  Fractions are nothing more than a piece of a whole number. For example, if you baked a birthday cake and you were expecting 10 people to come, but an 11th person showed up would you back an entirely new cake? No, you would take that cake and cut it into smaller pieces. A fraction is a piece of the whole number, each piece of the cake when added together creates the whole cake, the whole number.

Fractions are a piece of cake!

Let’s look at an example.

So now I want to show you how to use a dollar to turn fractions into decimals to make it easier. So let’s look. A dollar is represented as 1.00. Do we need the zeros? No, the zeros just it means there is no change. How much of a percentage is one dollar? Well if we move the spaces over by two places to the right we get 100%.

Now, let me show you how I am going to break this dollar into factions and how I am going to turn those fractions into percents.

I am going to break the dollar into 4ths. Because we know that there are four 25-cent pieces in a dollar or in other words, 4 quarters make up a dollar. Right? so,  we have…

Each one is also represented as a fourth because there are four pieces.

If you had just one piece, you’d have one-fourth or .25. If you have 2-pieces, you’d have .25 and/plus .25 or .5 which is half a dollar, or 50%.

No, if you have three-fourths, then you would have .25 added three times, which is .75 which as a percent moved over three times that would be 75%. And the same with the whole dollar.

Now let’s say you had three friends and you wanted to divide the dollar evenly among you and your friends. So, each would be .33 cents. Also written as ⅓. What happened to that extra penny?  I will show you where that extra penny goes.

Shows you division of 1 divided by 3. The three is repeated… Remember the rule for rounding. Fi it is less that 5 we round down. This is why we get .33.

Next, 2 divided by 3.

Now let’s take all of your money and add them all up.

Addition and Subtraction (Lesson 2)

Let’s focus on whole numbers.

Now we will look at how what we’ve learned in the previous lesson relates to addition and subtraction. For now we will only focus on whole numbers.

You are a painter and you are walking down the road. You painted 2 houses on the right side of the street and 5 houses on the left side of the street. How many houses have you painted? Yes, houses. There are terms that use when adding individual parts.

The parts we add together are called addends and the answer is called the sum. You can sue the dot method or finger method to find the solutions, but you should dedicate time to learn these numbers with speed because as math gets harder, not having an immediate recall of addition will slow you down.

Quiz: Single Digit Speed Test

Part A: Large Numbers

If I ask you to give me the immediate answer for:

You may need a moment to find the sum. Chances are that to get the solution you must do something in your head. Most likely you will rearrange the numbers because when adding or subtracting large numbers the math is not as easy as with single digit numbers.

To start we typically set up an addition of large numbers vertically.

Which is correct? Why?

As we learned before, place values are important! WE must align each digit based on place value.

We are actually thinking of numbers in expanded form and adding each little part, all at once. Be sure to add place values to the same place value. Having neat handwriting helps so that these numbers are not aligned incorrectly.

We must align our numbers by place value or it will not make any sense and we will get the problem wrong. You can do it in you head, but we will write these numbers out.

Steps:

  1. Align vertically.
  2. Only one digit fits in each column so if we have a two digit sum, we must carry the tens digit to the next column. We continue this process until all values are added.

Gives us the following:

In addition we will see how order does not matter. We will go over properties. Just realize that for addition, you will always get the same answer no matter the order.

Part B: Properties

There are some rules you must be aware of when dealing with addition.

What happens if I take 7 and add zero to it?

What happens if we switch the order? This applies to any number added to zero. This is called the addition property of zero or additive identity. We call it an identity because it reflects the number value added where any number added to zero gives us that number back, like a mirror reflection, hence the term ‘identity’.

If I add:

or

This ability to switch order is called the commutative property. Like if you are driving a car, you commute and can switch lanes to get to your destination. Same thing applies here. We can switch the order and still get the same answer for addition.

Does it matter if I rearrange addition?

As you can see, order does not matter. I can rearrange the these numbers in any order, each time I will get a solution of 11. Let’s see what happens if we use parenthesis. Will the order of parenthesis matter? Will we get a different answer? Let’s find out.

The answer was the same in both, so we can say that with parenthesis in addition, order doesn’t matter because addition is commutative. When we use parentheses we now label addition as associative property. Meaning the parenthetical order is associated with the other. With associative property you must realize that parenthesis are being used.

Note: parentheses means “do this section first”

Do we have to add these numbers 2 at a time or is there another way? Yes, we must write addition vertically and keep all digits in the correct position.

Quiz: Can you define the following additional properties?

Now that you understand the properties to addition let’s move on to see the differences of subtraction.

Part C: Subtraction

green applesWe will learn that addition and subtraction can be defined as one another.

Let’s say you have 9 green apples and your friend asks you for 4 apples. If you give away four apples, how many are left?

You need to commit to memory subtraction of single digits of whole numbers with speed and accuracy. As matha advances this will be needed.

Like addition, there are names we give to each part of subtraction math problems.

The first number is called the minuend, the part being taken away is called the subtrahend. We don’t tend to call out these parts often in math problems, but knowing them is good to know.

The answer or solution is called the difference. The difference between two numbers suggests that we are subtracting.

Now let’s look at the properties for subtraction.

Part D: Properties

Subtraction properties have significant differences. Let’s analyze them.

There is not identity property for subtraction

As you can see, if we switch the order of subtraction problems we get different answers. Let’s look at bit further.

Unlike addition, we can say that subtraction is not commutative. Because the order gives us different solutions.

Now lets see what happens with parenthesis.

Associative properties do not work either. We get different answers.If we use parenthesis as a the associative properties rules dictate the regrouping changes our answer. Example:

Like addition, zero taken from a number gives us just that number. Example:

A number subtracted from itself gives us zero, called the property of zero for subtraction.

No other properties apply. Notice that for subtraction, order matters.

Large Numbers

Subtracting large numbers is similar to adding large numbers in that we need to rearrange in vertical form.

Step 1: Align by placing the values vertically. Remember order matters.

Step 2: subtract digit by digit according to place value.

Sometimes the subtraction gets messy and doesn’t come out as we’d like.

When there is not enough to cover, we must borrow a ten and overload the ones place. Then we subtract.

E. Perimeter

If you watch the show Perimeter, you’d hear actors say, “Secure the perimeter.”

What does perimeter mean? Perimeter is the distance around something, called a polygon. Now we need to know what is A polygon? We define polygons as a closed figure with straight sides. Perimeter is not the area, but the outside distance by length.

Can you find the perimeter of this polygon?

Add up all of the sides and use units if given.

What if you were a contractor and decided to visit Home Depot to purchase supplies. You decide that you need 1000 feet of baseboards. You would request it by the perimeter of the room.

What about rectangles? Do you need to measure the the other sides? Why not?

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Resources:

ASVAB Practice Exams

Numbers in Expanded Form (Lesson 1 Part B)

Part B: Expanded Form

In the last lesson, we learned the fundamentals of number place values. Now we are going to look at expanded form. In expanded form, we are writing a number it terms of a sum of its parts. Expanded form gives us the basis for addition and subtraction which we’ll review in the next lesson. To put in expanded form, all we first need to know the place values. Which we just reviewed. Now we will use the place values and expand a number.

Let’s do an example:

We must be sure to keep all of the place values lined up, otherwise, this will not make sense or we’ll get the wrong answer (see Fig. 1).

Fig. 1 Expanded form example

If we expand each place value and then add all of these numbers up we are basically separating and recombining each digit to get our final answer in the standard form. We call our answer, 4382, the standard form because this is normal way we would write values and numbers.

Why do you think the expanded form so important? Expanded form is important because it is how our minds add numbers. We don’t talk about this or even think about this process when we are adding, but this is in fact what we are doing. All addition is the expanding and contracting of numbers to get them in standard  form.

Expanded Form Exercise

Next, we will look at addition and subtraction in lesson 2.

Lesson 1 Part A: Place Value

 photo scale_zpsvqcm94bl.png
Fig. 1 scale.

Introduction: There are some things you should know about place values, but first you should already know about how to draw a number line (Fig. 1), how to putting numbers in ascending and descending order, and how to plot numbers on a scale using whole numbers, fractions, and decimals (see Fig. 2).

Fig. 2 plot scale.

For now, we will only deal with whole numbers to review the basics. Later we will look at decimals and fractions.

Part A: Place Value

Let’s first look at place value. Here is an example: 

 photo place value

We can think of this number as money. As you can see all numbers are in a particular location. This is called place value, meaning each value has a position and a role to play as a member of this number group, similar to a family.

In English, we read from left to right. So the numbers further to the left are larger numbers. So in order to understand place value, we must work backwards and define each position so that we can read a number correctly. Let’s dissect this number to see what is meant.

Starting from the far right, let’s look at each place value (see Figure 3).

 photo place value example
Fig. 3. Understanding place values of numbers.

The first position is called the ones place and in our example above that number is 9.

Move one position to the left of 9: 

  • The second position is called the tens place, and that number value is 5.

Move one position to the left of 5:

  • The third position is called the hundreds place, and that number is 4.

And we continue. Take a moment to locate each place value for our example number above. Be sure to commit the place values to memory. When we get into higher math this will be an important foundational step.

<PLACE VALUE QUIZ>

We could continue naming place value positions because numbers don’t end. They continue for infinity, or forever. Some examples of numbers that go beyond trillion include, quadrillion, quintillion, and dectrillion.

The point is, each position has a specific name. The reason why knowing place value is important is because the place values tell you how much we have. Knowing your place value for numbers is fundamental working addition problems. Another importance to place values is that they teach us how to read values and write numbers in word form.

Now let’s look at a few examples: 

How would you say this value?

You’d say “one trillion, eight hundred seventy-four billion, three hundred four million, thirty-nine thousand, two hundred seventy.”

Notice that when writing numbers we only hyphenate numbers between twenty and ninety-nine. Also, notice that there is no “and” in this statement. Using the word “and” means we have two parts – which is incorrect. This is only one value. This is a big number, but it is a whole number. You’d use the word and to denote bringing two whole numbers together or a part of another number, like in decimals. We will discuss more of this in a later lesson.

Practice

  1. What place value is the 3 in  the number 93,205?
  2. What place value is the 7 in  the number 679,430,125?

Exercise

<Examples or Quiz Here>

In Lesson 1 Part B we will learn about expanded form, how this relates to addition, and why it is important.

 

Additional Resources:

Big Numbers: What Comes after Trillion?

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Congruence

Today we are going to go over Congruency.

This lesson will be broken up into four sections. We will start off by defining congruency and experiment with transforming shapes in any plane, understand congruence and how they relate to rigid motions, prove geometric theorems, and how to make geometric constructions with ease.

At the end of today’s lesson you will be able to make sense of geometric theorem problems and be able to solve them.

Defining Congruence

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

 

geometry_congruence_definition

Now, let’s take a look at planes, perpendicular lines, parallel lines, circles and arcs.

geometry_congruence_definition 2

 

As you can see above, in geometry we use notation to show congruency among shapes. Now lets represents these transformations and compare their distance and angles.