Topic of the Week:
Ratio Tables and the Coordinate Plane
Expressing ratios in three different ways, Understanding ratios, Rate and Unit Rate, Ratio Tables and Double Number Lines, Ratio Tables, Value of the Ratio Table, Coordinate Plane, Graphing Ratios
Mr. Chin's HR: Module 1, Lesson 19 - Complete Exercises 2 and 3. Study for Mid Unit Assessment next week.
Mr. Andersen's HR: Module 1, Lesson 19 - Complete the Problem Set. Study for Mid Unit Assessment next week.
Basic Notes from our class. Look these over to help remind you of information from our latest lessons. These are basic, you should have more details in your own notes.
Think of them as skeleton notes.
A. A ratio is a comparison between two numbers, quantities, or amounts in which both numbers can not be zero.
There are three ways in which ratios are expressed:
1. Using the word 'to' to separate the numbers.
2. Write as a fraction. Numbers are separated by a vinculum
3. Use a colon to separate the numbers.
B. Proportions are math statements that show how one ratio or fraction is equal in value to a separate ratio or fraction.
Example: one-half is equal to three-sixths.
1/2 = 3/6
C. Associated Ratios - ratios that we can find that are related to the original ratio; from a part-to-part ratio, we know that we can also find ratios by switching the order or by combining and finding the sum of the parts to get a whole and use that in the ratio. In our notes, we have referred to this with the letters below.
From the original ratio of A:B in which we can say that A + B = C, we can also get the following associated ratios: B to A, A to C, B to C, C to A, and C to B
With the ratio 2:3, we can also get the associated ratios: 3:2, 2:5, 3:5, 5:2, and 5:3.
D. There are three main ways that information may be given to us as we find the value of nonzero number c.
1. we may be given the value of one of the ribbons in a tape diagram,
2. we may be given the value of both ribbons in a tape diagram, or
3. we may be given the value of part of a ribbon(the difference) in a tape diagram.
4. we may also be given two tape diagrams in which information may be given to us. If this happens, be sure to double check the information they are giving and match it up to the proper ribbon.
E. Steps for simplifying ratios (fractions)
1. First check to see if the numerator can be divided into the denominator. If it can, then the numerator is automatically a 1. The new denominator is going to be the quotient.
2. If it can not be divided into the denominator, then make a list of factors for the numerator and the denominator.
3. Find all of the common factors and identify the largest or greatest common factor. This is known as the GCF.
4. Divide both the numerator and denominator by the GCF.
5. The quotients for both of these will be the new numerator and new denominator.
F. Using Nonzero Number C to help find Equivalent Ratios
Nonzero number c is a number that is used as an identity for a unit square, but it can also be thought of as a number that helps us to find equivalent ratios.
For example: For every 3 lions, there were 4 bears at the zoo.
We can say that the ratio of the number of lions to the number of bears is 3:4. If we wanted to find equivalent ratios to 3:4, we could multiply the first part and the second part of the ratio by the same number. If use the number 5 as our nonzero number c to find other ratios.
3 x 5 = 15, 4 x 5 = 20.
A new ratio could be 15:20.
G. Proportions are math statements that show a ratio or fraction that is equal to another ratio or fraction.
We can use the length and width of a rectangle to help us with understanding proportions.
Example: The ratio of the length to width of a given rectangle is 3 to 5. If we learn that the length of each square unit is 4m, we can multiply both the original units and get a new ratio of 12 to 20. This new ratio is equivalent to the old one, but it uses different numbers. This is a proportion.
H. To simplify a ratio, you are going to follow the basic steps involved with simplifying a fraction.
1. First: make a list of the factors of the first number and the factors of the second number
2. Identify the GCF or Greatest Common Factor.
3. Divide the first and second number of the ratio by the GCF.
4. The quotients from dividing will be your new (and simplified) ratio.
I. Cross-Multiplication to check for Ratio Equivalence.
Cross Multiplying - a strategy that is used to check to see if fractions/ratios are equal in value; this is done by setting up the ratios in a fraction form right next to each other, only separating them by a single space for an equal sign or not-equal sign.
Steps for Cross-Multiplying
Multiply the denominator of the first fraction by the numerator of the second fraction. Write the product above the second fraction.
Multiply the denominator of the second fraction by the numerator of the first fraction. Write the product above the first fraction.
Compare the products from both multiplication problems. If the products are the same or equal, then the ratios/fractions are equal in value.
J. If you have a proportion with a missing number, look at the relationships between the numbers. If you can find the number that has that relationship (it could be multiplying by a number or dividing by a number), then you can complete the proportion.
For example in 5:6 = 20:___
The matching first numbers are 5 and 20. In order to get a 5 to 20, you could multiply it by 4. If we multiply the second number in the first ratio by 4, we can share the same type of relationship. Multiply 6 by 4, and you will see that the missing number is 24.
K. Ratio tables are charts or graphs that show a ratio and equivalent ratios along with them. They often have an additive and a multiplicative structure to them. Having an additive structure means that to find other numbers, we can use addition of the same number. Having a multiplicative structure means that you can find another part of a ratio by using multiplication.
For example: Jimmy eats a ham and cheese sandwich for lunch every Monday through Friday. On his sandwiches, for every 3 slices of ham, there is 1 slice of cheese.
If we were to set up a chart showing the ham and cheese he has each day, we could see that on Monday, his sandwich has 3 slices of ham and one slice of cheese. By Tuesday, the number of slices of ham would be six, and the number of slices of cheese would be two. On Wednesday, the number of slices of ham would be 9, and the number of slices of cheese would be 3. Following this pattern, by Friday, Jimmy would have eaten 15 slices of ham and 5 slices of cheese.
L. Scaling Up and Scaling Down with Ratio Tables
-Scaling up and scaling down are steps that we take in order to find equivalent ratios for a given ratio.
Steps for Scaling Up
1. Identify the numbers used in the ratio.
2. Using the multiples of each number, count up.
Example: Using the ratio 20:50 By scaling up, we can find equivalent ratios 40:100, 60:150, 80:200, etc...
Steps for Scaling Down
1. Identify the numbers used in the ratio.
2. Calculate or identify the value of the ratio - simplify.
3. Using multiples of the numbers in the value of the ratio, count up.
Example: Using the ratio 20:50 By scaling down, we first need to identify or calculate the value of the ratio as 2:5. Now we can use multiples of the numbers in the value of the ratio to find equivalent ratios: 4:10, 6:15, 8:20, 10:25, etc...
M. Rate is defined as a comparison of two amounts or quantities that are expressed in different kinds of units. For example: Kenny finished 4 math problems in 10 minutes.
The two quantities are the number of math problems and time in minutes.
N. Unit rate is similar to rate, except that one of the quantities has to be one. For example: Speed Racer drove his car at 55 miles per hour.
In this case, the quantities are distance in miles and time in hours. The distance is measured in a time of one hour. The number one is what makes this a unit rate.
O. Ratio Table Comparison.
When you compare ratios in different tables, look for common numbers that happen under corresponding columns. If there are none, then find the value of the ratio or find an equal number by scaling up or scaling down.
P. Drawing Double Number Line Steps
1. Read the passage or scenario and highlight, circle or underline the most important information (ratio, labels, names, questions),
2. Write the ratio that is introduced from the passage and underline,
3. Draw a pair of horizontal parallel lines with arrows on either end, and label the lines on the left,
4. Draw a vertical line segment close to the left where you wrote your labels, making sure that it crosses over both horizontal lines; this will be your zero line,
5. Label your zero line with a zero on the top and on the bottom,
6. Draw another vertical line segment that will be used to show your given ratio, write the numbers of the ratio; be sure that the first number in the ratio is on the top, and the second number in the ratio is on the bottom, and then
7. Draw additional vertical line segments, and label them using multiples (count by numbers) of the ratio.
Q. Writing Equations from Ratios
1. Read the passage or scenario. Underline, highlight, or circle the important information.
2. Identify what it is you want to express and give it a variable designation - think about a letter to use as the variable. Write this letter and an equal sign. On our practice sheet, Z and Y were commonly used.
3. Write your other variable on the other side of the equal sign. You should have something similar to this: Y = Z. This is not how it is going to stay. This is just the beginning.
4. Leave the quantity that you find alone. Go to the other side, and determine what is happening to the other side in order to get the number on the other side. For example, if you wanted to find out how many quarters you could get for a dollar, you need to multiply dollars by 4 to get the same value of quarters.
5. Test your equation using numbers and a ratio table. If Z was quarters and Y was dollars, then a proper equation might be 4 x Y = Z or Z = 4 x Y.
When writing an equation, make sure that you have an equal sign. Also, when you multiply a variable by a number, be sure to write the number first and then the letter (variable). You do not write the multiplication sign.
5 x c = d should be written as 5c = d.
y x 5 = X should be written as 5y = x.
If you have more than one variable on one side of the equal sign, try to rewrite them in alphabetical order.
A. The Coordinate Plane
The concept or idea of the coordinate plane is a large flat surface that goes on in all directions forever. In this world of the coordinate plane, there are four different areas called Quadrants. All points in this universe of the coordinate plane either lay upon an axis (x or y axis) or within one of the quadrants. The x-axis is a horizontal line. The y-axis is a vertical line. Where the two lines intersect (cross over each other) is called the origin.
Each place on the coordinate plane is made up of an ordered pair. This consists of two numbers known as an x-coordinate, and a y-coordinate. Together (between a set of parenthesis) they make up an ordered pair.
The x-coordinate tells us how far to the right (or left) we should travel.
The y-coordinate tells us how far up (or down) we should travel.
B. Comparing Ratio Tables
If you are trying to compare ratio tables, it is made easiest if you are using smaller numbers. Try to find the value of the ratio table (simplify). Then compare the ratios. If you don't have a common second number (denominator), then use multiples to find the least common multiple and then compare. Think of it as comparing fractions.
C. Remember your Options for Rates and Ratios
double number lines
equations (keep in mind, your constant will likely be used in your equation)
coordinate planes and graphing ordered pairs