As we move along with ratios/proportions, we start to look at equivalent ratios. Finding equivalent ratios is exactly the same as equivalent fractions (fractions are ratios in the sense of part to whole). We can find equivalent ratios in a number of different ways.
Lets start with halving and doubling. Any ratio can be halved (if both numbers are even), or doubled. Take a look at this ratio. The easiest way to find the simplest fraction is to divide the denominator (bottom number) and the numerator (top number) by the GCF.
4 boys to 8 girls. This means for every 4 boys in a class, there are 8 girls. We can simplify this by halving both numbers.
2 boys to 4 girls is equivalent. We can even half this again to get a unit rate (we will cover this more later)
1 boy to 2 girls = 2 boys to 4 girls = 4 boys to 8 girls.
Alternatively, we could find equivalent ratios by doubling each number in the ratio.
4 boys to 8 girls is equal to 8 boys to 16 girls.
These two ways are, in my opinion, the easiest way to find equivalent ratios. You can also multiply each number in the ratio by the same number and find larger equivalent ratios.
What about finding equivalent ratios when given a fraction? Let's take a look. In class we have been using $300 to describe a fundraising goal. Let's say over the course of a few days that 3/5 of the goal has been raised. We can set this up like an equation
3
--- = --------
5 300
We have to determine what number 5 has to be multiplied by to get to 300. We can do this by dividing 300 by 5, which gives us 60. So we would know that 5 * 60 equals 300, so 3 * 60 would give use what 3/5 of 300 is.
3 * 60 180
--- = --------
5 * 60 300
We can do this with any fraction and ratio.
Wednesday, September 27, 2017
Ratios
Ratios are comparing two things together. Let's take comparing boys to girls in a classroom. There are 27 students in class. Of the 27 students, 15 are boys and 12 are girls. These ratios may be represented as:
15 boys to 12 girls
For every 15 boys, there are 12 girls (however, only used when comparing larger quantities through scaling)
15 boys : 12 girls
15 boys
12 girls
These ratios all compare part to part.
It is important to keep the same order in which was given in the problem. Notice I said:
"Let's take comparing boys to girls in a classroom".
I kept the same order because that was what was given to me in the problem.
You can compare ratios in terms of part to whole as well.
15 boys out of 27 students
12 girls out of 27 students
In this case, we used part of the classroom (boys or girls) and compared it to the total amount of students. Since we know that there are only 2 genders, if I said 15 boys out of 27 students we could subtract the part FROM THE whole to find the amount of the other part.
27 total students
-15 boys
12 girls.
These are the basics to ratios.
15 boys to 12 girls
For every 15 boys, there are 12 girls (however, only used when comparing larger quantities through scaling)
15 boys : 12 girls
15 boys
12 girls
These ratios all compare part to part.
It is important to keep the same order in which was given in the problem. Notice I said:
"Let's take comparing boys to girls in a classroom".
I kept the same order because that was what was given to me in the problem.
You can compare ratios in terms of part to whole as well.
15 boys out of 27 students
12 girls out of 27 students
In this case, we used part of the classroom (boys or girls) and compared it to the total amount of students. Since we know that there are only 2 genders, if I said 15 boys out of 27 students we could subtract the part FROM THE whole to find the amount of the other part.
27 total students
-15 boys
12 girls.
These are the basics to ratios.
Wednesday, September 20, 2017
Wednesday, September 13, 2017
Order of Operations
We have reviewed Order of Operations in class and I have previously written about it in a blog but I have decided to dedicate one post to it prior to this test.
Order of Operations is the standard, accepted method for evaluating expressions and equations. Order of Operations goes like this:
Parentheses
Exponents
Multiplication or
Division (whichever comes first, left to right)
Addition or
Subtraction (whichever comes first, left to right)
Let's look at an example. (remember that the ^ means an exponent)
2^3 + 9 (6 - 4) ÷ 2
The parentheses would have to be expressed first so we would rewrite the expression changing only what was inside the parentheses.
2^3 + 9 (2) ÷ 2.
Now, the (2) means to multiply, so we would not do that next. We would, however, express the exponent.
2^3 = 2 x 2 x 2 = 8, so when we rewrite the expression it would be
8 + 9 (2) ÷ 2
Multiplication would come next, so we would multiply 9 and 2
8 + 18 ÷ 2
Division would come next and 18 divided by 2 would have use rewrite the expression
8 + 9
With addition left, we add 8 + 9 and our answer would be
17.
Step by step would look like this:
2^3 + 9 (6 - 4) ÷ 2
8 + 9 (2) ÷ 2
8 + 18 ÷ 2
8 + 9
17
I call this in class the upside down pyramid and I prefer students doing it this way because it shows each and every step and it is organized.
Use this link to find order of operation problems.
http://www.math-drills.com/orderofoperations/ooo_integers_foursteps_positive_pemdas_all.html
Order of Operations is the standard, accepted method for evaluating expressions and equations. Order of Operations goes like this:
Parentheses
Exponents
Multiplication or
Division (whichever comes first, left to right)
Addition or
Subtraction (whichever comes first, left to right)
Let's look at an example. (remember that the ^ means an exponent)
2^3 + 9 (6 - 4) ÷ 2
The parentheses would have to be expressed first so we would rewrite the expression changing only what was inside the parentheses.
2^3 + 9 (2) ÷ 2.
Now, the (2) means to multiply, so we would not do that next. We would, however, express the exponent.
2^3 = 2 x 2 x 2 = 8, so when we rewrite the expression it would be
8 + 9 (2) ÷ 2
Multiplication would come next, so we would multiply 9 and 2
8 + 18 ÷ 2
Division would come next and 18 divided by 2 would have use rewrite the expression
8 + 9
With addition left, we add 8 + 9 and our answer would be
17.
Step by step would look like this:
2^3 + 9 (6 - 4) ÷ 2
8 + 9 (2) ÷ 2
8 + 18 ÷ 2
8 + 9
17
I call this in class the upside down pyramid and I prefer students doing it this way because it shows each and every step and it is organized.
Use this link to find order of operation problems.
http://www.math-drills.com/orderofoperations/ooo_integers_foursteps_positive_pemdas_all.html
Monday, September 11, 2017
Distributive Property
In class today we began discovering the distributive property. They have several notes pages that will help you with their homework. We also determined the definition of even and odd numbers, and found some things to be true:
The sum of 2 even numbers will always be EVEN
The sum of 2 odd numbers will always be EVEN
The sum of an even and odd number will always be ODD
The product of 2 even numbers will always be EVEN
The product of 2 odd numbers will always be ODD
The product of an even and odd number will always be EVEN.
We also learned two different ways to right a multiplication problem. Lets look at 16.
8 x 2 = 16
8 · 2 = 16
8(2) = 16
From here on out, we will use the parentheses version to multiply.
Before we continue with the distributive property, lets review order of operations.
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
When doing math problems, we have to look at this to know what order to perform the operations.
In terms of distributive property, we typically do the parentheses in different ways. Lets look at the number 24.
24 can be seen at 3 (8) = 24. However, it can always be seen as
3 ( 2 + 6) =24 (we can read this at 3 times the sum of 2 and 6)
In the second case, we can "distribute" the 3 to both numbers inside the ( ). This would look like
3 (2) + 3 (6) = 24, or
6 + 18 = 24.
Notice we had to multiply AFTER distributing the 6 through the parentheses. Otherwise it would have looked like this:
3 (2) is 6, so 6 + 3 (5), then 6 + 3 is 9, so 9 (5) = 45. That entire process is wrong according to order of operations.
Let's look at one in terms of area of a rectangle.
The sum of 2 even numbers will always be EVEN
The sum of 2 odd numbers will always be EVEN
The sum of an even and odd number will always be ODD
The product of 2 even numbers will always be EVEN
The product of 2 odd numbers will always be ODD
The product of an even and odd number will always be EVEN.
We also learned two different ways to right a multiplication problem. Lets look at 16.
8 x 2 = 16
8 · 2 = 16
8(2) = 16
From here on out, we will use the parentheses version to multiply.
Before we continue with the distributive property, lets review order of operations.
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
When doing math problems, we have to look at this to know what order to perform the operations.
In terms of distributive property, we typically do the parentheses in different ways. Lets look at the number 24.
24 can be seen at 3 (8) = 24. However, it can always be seen as
3 ( 2 + 6) =24 (we can read this at 3 times the sum of 2 and 6)
In the second case, we can "distribute" the 3 to both numbers inside the ( ). This would look like
3 (2) + 3 (6) = 24, or
6 + 18 = 24.
Notice we had to multiply AFTER distributing the 6 through the parentheses. Otherwise it would have looked like this:
3 (2) is 6, so 6 + 3 (5), then 6 + 3 is 9, so 9 (5) = 45. That entire process is wrong according to order of operations.
Let's look at one in terms of area of a rectangle.
The area of the entire rectangle can be found by added together the areas of the two smaller rectangles. Remember, area is equal to length times width ( a = l (w) )
Using the distributive property we can "distribute" the 5 to what is inside the parentheses.
5 (10 + 4) = 5 (10) + 5 (4) = 50 + 20 = 70
We can actually do this another way. We can add the lengths on the bottom (10 + 4) to get 14 then multiply by the width, 5.
5 (14) = 70.
Introduction to the Distributive Property
Here is a video you can watch to help you with the distributive property.
https://youtu.be/R-vxK9CFCR0
Here is a video you can watch to help you with the distributive property.
https://youtu.be/R-vxK9CFCR0
Here is a video you can watch to help you with the distributive property.
Tuesday, September 5, 2017
Prime Factorization part 2
This is an extension of prime factorization.
For any composite number, we can look at ANY factor pair that we know and derive the prime factorization. Lets look at 36. The prime factorization of 36 is 2 x 2 x 3 x 3. Let's look at the picture and see the ways we can derive that from each factor pair.
Let's look at 6 x 6. If you decompose 6 into 2 x 3, you can see how we can get to 36's prime factorization. Now look at 9 x 4. 9 decomposes to 3 x 3 and 4 decomposes to 2 x 2. This can be done with ANY composite number.
Now let's look at how to write prime factorization is a different way. 36's prime factorization is 2 x 2 x 3 x 3. We can rewrite it as 2^2 x 3^2 (read as two squared times three squared OR two to the second power times three to the second power). The picture below explains this as exponential notation.
Here is another look. Exponential notation is on top and expanded notation is at the bottom.
One more time, but as a factor tree.
For any composite number, we can look at ANY factor pair that we know and derive the prime factorization. Lets look at 36. The prime factorization of 36 is 2 x 2 x 3 x 3. Let's look at the picture and see the ways we can derive that from each factor pair.
Let's look at 6 x 6. If you decompose 6 into 2 x 3, you can see how we can get to 36's prime factorization. Now look at 9 x 4. 9 decomposes to 3 x 3 and 4 decomposes to 2 x 2. This can be done with ANY composite number.
Now let's look at how to write prime factorization is a different way. 36's prime factorization is 2 x 2 x 3 x 3. We can rewrite it as 2^2 x 3^2 (read as two squared times three squared OR two to the second power times three to the second power). The picture below explains this as exponential notation.
One more time, but as a factor tree.
81 decomposes to 9 x 9, or 9^2 (read as nine squared or nine to the second power). Each 9 decomposes to 3 x 3, making the prime factorization of 81 at
3 x 3 x 3 x 3 or 3^4 (read as three to the fourth power)
GCF and LCM using Prime Factorization
We have been using techniques to find factors of numbers including the greatest common factor. Previously we have used a Venn Diagram. Below is an example.
Yesterday we used the prime factorization technique. An example can be found below.
Basically, you find the prime factorization of both numbers. We used the factor tree method. The prime factorization of 24 is 2 x 3 x 2 x 2. The prime factorization of 60 is 5 x 2 x 3 x 2. The factor string in bold is what they have in common. By multiplying the common factor string of 2 x 3 x 2, we can find the greatest common factor of 24 and 60, which is 12.
We also can find the least common multiple of a number, or LCM by doing the same thing. First lets look at the previous method. This method is the listing method, which we just list the multiples by multiplying the original number by 1, 2, 3, etc. until we find common multiples. In this case the LCM of 24 and 60 is 120.
By using the factor tree method and finding the prime factorization of both 24 and 60, we can see that 3 x 2 x 2 x 2 is the prime factorization of 24 and 5 x 2 x 3 x 2 is the prime factorization of 60.
When finding the LCM, we still use the factor string that they have in common, 2 x 3 x 2, but we add to it the factors they don't have in common. If we take the new factor string that combines what they have in common and what they don't have in common ,2 x 2 x 2 x 3 x 5, we get 120.
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