Day 1/Lesson 1: Ratio and Proportion
Goals:
1. Understand Ratios.
2. Use ratios correctly to solve problems.
3. Learn and understand proportions
4. Use proportions to solve problems.
Introduction:
Ratios can be used to compare quantities. Example: If you have 2 dogs and 3 cats, then your ratio of dogs to cats is 2/3, or 2:3. If you also have 4 fish, the ratio of dogs to cats to fish is 2:3:4.
Definition 1:
1. If a and b are two quantities that are measure in the same units, then the ratio of a to b is a/b. The ratio of a to b can also be written as a:b.
Main Lesson:
What are ratios?
A ratio between two numbers is a certain relationship between the two numbers.
Example: A certain recipe for muffin mix requires 7 cups of flour for every 2 cups of milk. The ratio of flour to milk is 7:2 or 7/2. What about the ratio of milk to flour? Answer: 2:7.
Why do we equate ratios with fractions and fractions with ratios?
In our previous example, think of the 7 cups of flour getting evenly divided up between the 2 cups of milk, which naturally implies division. Therefore, each cup of milk goes with 7/2=3.5 cups of flour. So in our recipe there are 3.5 cups of flour for every cup of milk.
Definition 2:
2. An equation that equates two ratios is a proportion.
Main Lesson:
Back to the muffin mix problem. If you wanted to make a batch of muffin mix using 3 cups of milk, how much flour would you need to keep the same relationship we had before of 7c. of flour: 2c. of milk. (In other words, we want to increase the recipe, but we want the muffins to taste the same as before).
Use proportions to solve this problem:
To solve for x here, we use the property of cross products described below.
Properties of Proportions:
1. Cross Product Property: The product of the extremes equals the product of the means.
2. Reciprocal Property: If two ratios are equal, then their reciprocals are also equal.
Now, let's finish our problem from above using the cross product property:
7*3=2*x
21=2x
x=number of cups of flour=21/2
Applications Using Ratios and Proportions:
1. John mixed 4 tablespoons of lemonade concentrate with 1 cup of water to make lemonade. In the following table, use ratios and proportions to fill in the correct number of tablespoons of lemonade concentrate and water.
a. Mix ____ tablespoons concentrate with 3 cups of water.
b. Mix 3 tablespoons concentrate with ____ cups of water.
c. Mix ____ tablespoons concentrate with 2.75 cups of water.
d. Mix 6 tablespoons concentrate with _____cups of water.
Answers:
2. The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle.
3. The perimeter of the isosceles triangle shown below is 56 in. The ratio of LM:MN is 5:4. Find the lengths of the sides and the base of the triangle. m<LNM=m<LMN.
Hint: What is the relationship between the sides of an isosceles triangle in terms of perimeter.
4. Solve the Proportions:
Using a Proportion to solve a real world example:
The photo shows Bev Dolittle's painting Music in the Wind. Her actual painting is 12 inches high. How wide is it?
