Arithmetic: Ratios and Proportional Reasoning

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Arithmetic: Ratios and Proportional Reasoning. -Ratio (& Proportions) -Scale Drawings: Increase/Decrease -Percent -Rate. Overview.

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### Arithmetic:Ratios and Proportional Reasoning

-Ratio (& Proportions)

-Scale Drawings: Increase/Decrease

-Percent

-Rate

Overview

HINT: Read the brief text in each slide BEFORE going to any website or looking at a file. There will be instructions on HOW best to use the resource/website.

This is the first modular, and serves as a precursor to the algebra and beyond courses. In this module you’ll be learning about Ratio and Proportional Reasoning.

A ratio describes a relationship between two quantities.

Some distinguish ratios from rates, using the term “ratio” when units are the same and “rate” when units are different; others use ratio to encompass both situations.

Relationships of two quantities may be described in terms of ratios, rates, percents, or proportional relationships.

Rates can be indicated in terms such as “for each 1,” “for each,” and “per.” The unit rate is the numerical part of the rate; the “unit” in “unit rate” is often used to highlight the 1 in “for each 1” or “for every 1.” ” Notation for ratios can include the use of a colon, as in 3 : 2.

Ratios are used in many applications.

Topics
• Ratio Foundation
• Determining and expressing a ratio
• Creating Ratio Proportional Relationships
• Ratio: Scale Drawings
• Ratio: Percent
• Ratio: Rates
RatioDetermining and Expressing a Ratio
• A ratio expresses the relationship of two quantities.
• Ratios can be indicated in words as “5 to 6” and “3 for every 5” and “5 out of every 7” and “5 parts to 6 parts.” This use might include units, e.g., “5 cups of flour for every 6 eggs” or “3 meters in 2 seconds.” Notation for ratios can include the use of a colon, as in 3 : 2.
• This video will show you how to determine a ratio as well as how to express a ratio. The video is 14 minutes long; however, you may only need to watch the first 4 minutes demonstrating HOW TO DETERMINE and WRITE A RATIO of horses to dogs. There are thee more examples that follow horses : dogs ratio. You might want to use these to test your skills. After the problem is introduced, stop the video and try to write the ratio, restart the video to check your answer. Continue through the video until you feel you can suffessfully write a ratio. http://www.khanacademy.org/math/arithmetic/rates-and-ratios/ratios_and_proportions/v/introduction-to-ratios--new-hd-version
RatioDetermining and Expressing a Ratio
• Often you might be asked to use a ratio to determine missing information such as: If the ratio of girls to boys is 5 to 8, and there are 65 students in the class, how many of those students are girls?
• This 2 minute step-by-step video demonstrates the above problem. You should stop the video after the problem is introduced to determine the ratio before going on to check your answer with the one demonstrated on the video. https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-ratios-prop-topic/cc-6th-ratio-word-problems/v/ratio-word-problem-exercise-example-1
• There is a second problem you can try. To get to the second problem click on “Ratio word problem exercise 2” in the left side bar.
RatioCreating Ratio Proportional Relationships
• A ratio expresses the relationship of two quantities.
• Proportions are determined by using a given ratio to create multiplicative relationships of the same two given numbers used in the ratio, thus when two ratios are the same, they are proportional. For example:
• This 5 minute video will demonstrate the use of multiplication to find proportions equal to a given ratio. Note that one factor is used to multiply both numbers in the ratio to find a proportion equal to the given ratio. http://www.khanacademy.org/math/algebra/ratio-proportion-topic/ratios_algebra/v/ratio-problem-with-basic-algebra--new-hd
RatioCreating Ratio Proportional Relationships
• This 5 minute step-by-step video will show you how to determine and express a ratio proportionally. Use this video to practice practical applications problems using ratios. Begin the video, and once the problem is established; stop the video and determine the ratio, before going on to check your answer with the one demonstrated on the video.
• Note on the left side bar of this web site that there are several examples you can work through. You can determine how many problems you need to do. Continue doing problems until you are successful. https://www.khanacademy.org/math/algebra/ratio-proportion-topic/ratios_algebra/v/writing-proportions
Ratio: Scale DrawingsScale Drawings ~ Increase
• Scale drawings use proportional reasoning to create a smaller or larger drawing of the given drawing.
• Scale drawings are created by using multiplication to increase the size of the drawing; or division to decrease the size of the drawing.
• This 4 minute video will show you how to use a scale factor used in creating a floor plan to determine the actual size of the house. Watch this video and note the multiplication used to determine the actual size of the house.
• Note that one factor is multiplied by all sides to maintain the ratio. http://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-geometry/cc-7th-scale-drawings/v/scale-drawings-example
Ratio: Scale DrawingsScale Drawings ~ Decrease
• Scale drawings use proportional reasoning to create a smaller or larger drawing of the given drawing.
• Scale drawings are created by using multiplication to increase the size of the drawing; or division to decrease the size of the drawing.
• This 3 minute video will show you how to use a scale factor to create a floor plan. Watch this video and note the division used to create the scale drawing.
• Note that the same number is used to divideall sides to maintain the ratio. https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-geometry/cc-7th-scale-drawings/v/Constructing%20scale%20drawings
Ratio:Percent
• A ratio can be expressed as a percentage.
• In expressing the percent keep in mind the two quantities in the ratio combined to make the whole. Thus one can compare one quantity to the other; or compare one quantity to the whole of the set.
• This 3 minute video explains what a percent is, and how to find a percent. http://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-ratios-prop-topic/cc-6th-percentages/v/finding-percentages-example
• Now that you have watched the video, think about all the ratio problems you have done and how a ratio can be expressed as a fraction.
• Use the next two web resources to test your understanding of percent. Each website has percentage problems . Work the problem, then click on ‘answer’ to see if you are correct.

http://www.mathgoodies.com/lessons/vol4/meaning_percent.html

Ratio: RateA quantity measured with respect to another measured quantity (sometimes called a unit rate)

Example: a rate of speed

• This rate compares the miles traveled to the time it took to drive those miles. This is sometimes given as a formula rate equals distance divided by time.
• Jose is driving 55 miles per hour.
• The rate of speed is 55 miles per hour.
• This is sometimes written as
Ratio: RateA quantity measured with respect to another measured quantity (sometimes called a unit rate)

Rates are often found in respect to an time. To see how a rate per hour is determined, click on the link below. After you view this two and one-half minute video, click the left sidebar to continue.

To skip the video, advance to the next slide.

Ratio: RateA measure of a part with respect to a whole

Example: a sales tax rate.

• This rate compares the taxes to be paid to the state based on the total dollar amount being spent.
• Ohio’s state sales tax is 5.75%.
• To see how to calculate sales tax, watch a one minute video at http://www.ehow.com/video_4909013_calculate-sales-tax.html
• After you view this one minute video, click the left sidebar to continue. To skip the video, advance to the next slide.
Ratio: RateThe cost per unit of a commodity or service

Example: a postal rate.

• This rate compares amount of money to be paid to the post office based on the weight of the mail being spent.
• The cost to mail a letter weighing one ounce or less is \$0.46
• The postage rate for first class letters of one ounce is \$0.46
• This is sometimes written as
Ratio: RateThe cost per unit of a commodity or service

Click on the link below to find the cost of cleaning one office when the cost of cleaning eight offices is known. After you view this two minute video, click the left sidebar to continue.

To skip the video, advance to the next slide.

Ratio: RatesIf you would like to read more about rates and unit rates, click on the link below.
• http://www.eduplace.com/math/mathsteps/6/e/
• If you do not need to read further information, advance to the next slide.
Ratio:RateUnit Price Practice
• If you would like to practice finding unit rates click on the link below to access 10 multiple choice problems. When you are finished, click the left sidebar to continue. To skip the practice, advance to the next slide.
• http://www.mathopolis.com/questions/q.php?id=1049&site=1&ref=/measure/unit-price.html&qs=1049_1050_1937_1938_2210_2211_3745_3746_3747_3748
Ratio: RateRate of Change

We often talk about how something changes. A graph is a picture of the rate of change of one thing to another.

• Some examples for rate can be found in the 13 minute video at the link below. You will learn about Usain Bolt’s average speed.

After viewing the video, click the left sidebar to continue on with examples of finding unit rates. To skip the video, advance to the next slide.

Ratio: RateGame for practice
• Build your own potions based on what you have learned about unit rates. Click on the link below to access the game.
• http://mathsnacks.com/ratiorumble.php
• Once you are finished playing the game, click the left sidebar to continue.

To skip the game, advance to the next slide.

Ratio: RateGame for practice
• In this Ratio Stadium game, play against the computer or with up to three other people. Click on the link below to access the game.
• Once you are finished playing the game, click the left sidebar to continue.
• To skip the game, advance to the next slide.
Ratio: RateSlope
• The word slope is used to define the incline or steepness of a hill. It is comparing the change in the vertical distance (sometimes called the rise) to the change in the horizontal distance (sometimes called the run.)
• Algebraically we can say
Ratio: RateSlope – Mathematical Definition
• The slope of a straight line is defined by the change in y divided by the change in x
• where x2 does not equal x1
Ratio: RateSlope – Problem

The slope of a line that passes through the points (4,7) and (2,3).

For our purposes (x1,y1) = (4,7) and (x2,y2)=(2,3)

so the slope of the line is 2.

That means that for every two units there is a change in the vertical distance, there is a change of one in the horizontal distance.

The slope of the line is positive.

Ratio: RateSlope – Problem

The slope of a line that passes through the points (1,2) and (-1,4).

For our purposes (x1,y1) = (1,2) and (x2,y2)=(-1,4)

so the slope of the line is -1.

That means that for every one unit there is a change in the negative vertical distance, there is a change of one in the horizontal distance.

The slope of the line is negative.

Ratio: RateLinear Equations
• See what happens when the values of “x” and “y” change using a virtual manipulative. Click belowto access a place where you can see how the slope of the line changes. See what happens when you enter a value of zero for slope and a value of 5 for the y intercept. Try other values for the slope and the y intercept to see what happens to the line. Once you have finished exploring, click the left sidebar to continue.
• http://plaza.ufl.edu/youngdj/applets/graphing_tool.html
• To skip the exploration, advance to the next slide.

If you want to remember the formula for slope, watching this YouTube video may help. Click on the link below to watch the 2 ¼ minute video.

• If you wish to skip the song, advance to the next slide.
Ratio: RateHow to calculate the slope on a ramp
• There are many practical applications of using the slope formula. Calculating the slope of a ramp is one such application. Click on the link below to try it! After you have visited the website, click the left sidebar to continue.
• http://www.ehow.com/how_4522492_calculate-slope-ramp.html#ixzz2nbf9VQj7
• To skip practice, advance to the next slide.
Ratio: RateRate of Zero

If you are driving at 70 mph on the highway and then are stopped for 20 minutes because of an accident, at first your rate is 70 mph. When are stopped, what is the rate of change? Time is passing but you are not traveling any distance so the rate is zero.

• Using the formula
• The graph for a rate of zero would be a straight line. There is no slope.
Ratio: RateSlope – Problem – Rate of Zero

The slope of a line that passes through the points

(0,2) and (6, 2).

For our purposes (x1,y1) = (0,2) and (x2,y2)=(6,2)

so the slope of the line is 0.

That means that there is no change in the vertical distance. The slope of the line is zero.

Ratio: RateUndefined slope

Can you travela distance and no time passes. Is this possible? Not in our universe!

• Remember that the rate is defined by distance over time. If the time is zero, then the rate is undefined.
• The graph for this situation would be a vertical line. The slope is undefined. We may also say vertical lines have no slope.
Ratio: RateSlope – Problem

The slope of a line that passes through the points

(4,7) and (4, -3).

For our purposes (x1,y1) = (4,7) and (x2,y2)=(4,3)

so the slope of the line is undefined.

That means that for any change in the vertical distance, there is no change in the horizontal distance.

The slope of the line is undefined.