Students will be able to apply the Pythagorean Theorem to solve problems in real-life and mathematical situations (MATH.8.2A, MATH.8.2B). The student should be able to demonstrate and verify the following:
• I can identify types of numbers: whole numbers, naturual numbers, integers, rational numbers, and irrational numbers.
• I can explain how types of rational numbers are related to each other.
• I can explain the difference between rational numbers and irrational numbers.
• I can explain how squared numbers and square roots are related.
• I can define a perfect square.
• I can calculate the square root of a number using the number line.
Monday – Rational Numbers
In grade 8 mathematics, we have reached a point in our year that we can review all of the content in preparation for algebra 1 next year. First, we will begin with reviewing the different types of numbers that we have encountered. You and your friends decide to go to the movies together. When deciding what to watch, you and your friends might talk about the TYPE of movies you enjoy – comedy, science-fiction, romance, drama, or horror. In math, we talk about the TYPE of numbers we are working with, just as we talk about the TYPE of movies we like to watch. We do this with many things in our lives: music, animals, food, clothing style, and so much more! Let’s talk about different TYPES of numbers. First, when we talk about the TYPE, we are talking about the SET that a number belongs to. From now on, that is the word we will use: SET.
Natural Numbers: Sometimes we call natural numbers, counting numbers. We do this because the Natural Number SET includes all the numbers we learned when we first learned to count: 1, 2, 3, 4, 5, 6, . . . and so on. The Natural Number SET does NOT include 0. Let’s Give It A Try: Identify all the numbers that are included in the Natural Number Set. Put a check under the correct column (either Natural Number or Not a Natural Number).
Whole Numbers: Whole Numbers INCLUDE all the Natural Numbers, but also INCLUDE 0. This means that Whole Numbers start at 0 and include all the positive counting numbers that we listed earlier: 0, 1, 2, 3, 4, 5, 6, . . . and so on. Notice that Whole Numbers do NOT include negative numbers or numbers that can only be written as a fraction (we’ll get into this later). Let’s Give It A Try: Identify the set that a number belongs to. It can belong to one set, two sets or neither set. Put a check under the correct column showing which set(s) a number belongs to.
Integers: Integers INLUDE all Natural Numbers and Whole Numbers, but also INCLUDE Whole Number’s opposites. When you look in the mirror, you see your reflected image. Look at the number line below. Can you see how negative numbers are a reflection of whole numbers?
Number Lines by HISD Secondary Mathematics using HandyGraph
Let’s Give It A Try: Identify the set that a number belongs to. It can belong to one set, two sets, three sets or none of the sets. Put a check under the correct column showing to which set(s) a number belongs.
If you said, 2, then you are CORRECT! And we know from earlier that 2 is a Natural Number, a Whole Number, an Integer and a now we know it is also a Rational Number. The DIFFERENCE is that it is NOT only a Rational Number. Let’s Give It A Try: Identify the set that a number belongs to. It can belong to one set, two sets, three sets, four sets or none of the sets. Put a check under the correct column showing to which set(s) a number belongs.
Summary: Did you notice that Natural Numbers were included in all the sets that you learned about after you first learned about Natural Numbers? Sometimes classifying numbers can make our brain CONFUSED! That is okay, it just takes practice. Here is another diagram to help you think about the set(s) to which a number might belong.
Before moving on, let’s talk about squares and square roots. In order to understand the relationship between squares and square roots, let’s first look at inverse operations. Answer the following questions:
1. 3 + 9 = __________ 2. 2 + 8 = __________
3. 6 + 7 = __________ 4. 4 + 5 = __________
As you solve the next 4 questions, think about what you NOTICE about the first set of problems and the second.
5. 12 - 9 = __________ 6. 10 - 8 = __________
7. 13 - 7 = __________ 8. 9 - 5 = __________
Did you notice that the answer to #5 was the same number we started with on #1? Is this true of all the problems you completed? Addition and Subtraction “undo” each other. In mathematics, we call these two operations, “inverse operations.” Can you think of two other inverse operations? Try a few more problems to help you answer this last question.
As you solve the next 4 questions, think about what you NOTICE about the first set of problems and the second.
1. 45 ÷ 5 = __________ 2. 24 ÷ 6 = __________
3. 12 ÷ 8 = __________ 4. 28 ÷ 7 = __________
Did you notice that the answer to #5 was the same number we started with on #1? Is this true of all the problems you completed? In your own words, explain what it means when two operations are inverse operations: ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ Activity 1: Write the length of each side of the squares below and find the area.
Let’s record your findings in the table below and extend the information to include squares with side-lengths up to 15.
Side Length Area
1 1 • 1 =
2 2 • 2 =
3 3 • 3 =
4 4 • 4 =
5 ____ • ____ =
6 ____ • ____ =
7 ____ • ____ =
8 ____ • ____ =
9 ____ • ____ =
10 ____ • ____ =
11 ____ • ____ =
12 ____ • ____ =
13 ____ • ____ =
14 ____ • ____ =
15 ____ • ____ =
This is fantastic work! You just found the “square” of the numbers 1 – 15. We represent this operation with an exponent. Here is what it looks like:
72 = 49
Do you think there is an inverse operation that “undoes” the square of a number? ___________ If you said yes, then you are correct! In order to undo the square of a number, we find the square root. Tomorrow, we’ll investigate square roots in more detail.
Look back to Activity 1. Look at the side lengths of each square, that is the square root! Let’s try another activity. Activity 2: Write the side length of each side of the squares below based on the area.
The answers you found in the middle column are called perfect squares because they are the product (the answer to a multiplication problem) of two whole numbers (You just learned about those!). What if you were trying to find the square root of a number that was not a perfect square? Let’s play mathematician! You are a world-renowned mathematician that has just been given a problem nobody else has been able to solve. Can you figure it out? Problem: How do you find the square root of 78? Can you make a prediction as to what the answer might be? ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ If you guessed a number between 8 and 9, you are spot on! There is a way to find the square root of numbers that are not perfect squares. Here’s how: Step 1: Find the two perfect squares that your number is between.
Step 2: Find the square root of the two perfect squares.
You need to pat yourself on the back, you have done a lot of really GREAT work!
Number Lines by HISD Secondary Mathematics using HandyGraph
Thursday – Irrational Numbers
Think back to lesson 1, what did you learn about? ______________________________________________________________________________ ______________________________________________________________________________
If you said that you learned about the different TYPES of numbers, you are correct? Do you remember another name for TYPES of numbers? SETS of numbers, exactly! You are now going to learn about one more SET of numbers, the irrational set. Irrational Numbers cannot be written as any other TYPE of number. You cannot write them as Natural Numbers. You cannot write them as Whole Numbers. You cannot write them as Integers. You cannot write them as a Fraction (Rational Numbers). If I asked you to write 0.25 as a fraction, what would your answer be? ________________
That’s right, 1
4 .
If I asked you to write 0. 33̅̅̅̅ as a fraction, what would your answer be? ________________
That’s right, 1
3 .
There are two things to notice about the two examples above: (1) the first rational number (0.25) is terminating. That means the number does not continue, it stops. (2) the second rational number (0.3333333333333 . . . ) repeats, it creates a pattern. An Irrational Number does NOT terminate or repeat. Can you think of an irrational number that you used when finding the circumference and area of a circle? __________________________
If you said pi, you are absolute right! Pi is famous because its numbers go on FOREVER! They don’t ever repeat, and they don’t ever end. This is why pi is an Irrational Number. Many square roots are also irrational numbers, but not when they are perfect squares. Let’s take a look at t a few examples:
1. √49 = 7
7 is a Natural Number, Whole Number, Integer and Rational Number. So, √49 is NOT an Irrational Number.
2. √29 = 5.385164807134504 . . .
The answer does not end and does not repeat. So √29 IS an Irrational Number.
Let’s Give It A Try: Identify the set that a number belongs to. It can belong to one set, two sets, three sets, or four sets. Put a check under the correct column showing to which set(s) a number belongs.
Here is a diagram to help you think about the set(s) that a number might belong to. This time, it includes Irrational Numbers. Notice that the 5 sets: Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers are all part of the set of the Real Number System. You’ll learn more about Real Numbers (and Imaginary Numbers!) in high school.
12. Create a Venn Diagram or other type of illustration that explains your understanding of how numbers are classified. The figure must demonstrate the relationships between natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
For instance, the representation must be able to show how a number such as “2” is a member of natural numbers, whole numbers, integers, and rational numbers … but not irrational numbers. This representation must include:
• An illustration, figure, diagram
• Explanation of how the relationships are depicted
• A justification for how two example numbers would be placed into the representation Alternative Option/Extension for #12: Create a game to illustrate the relationship between types of numbers. The same three characteristics must be included in the game.
Student successfully completes the portfolio task that includes a demonstration of their understanding for how to classify types of numbers and an illustration of how the types of numbers are related to each other.
