Math - 2018-19
6.1 - Ratios
6.1 The student will represent relationships between quantities using ratios, and will use appropriate notations, such as a/b , a to b, and a:b.
BIG IDEAS
- I can figure miles per gallon, cost per unit, and estimate the size of an animal population when the entire population cannot be counted.
- I will be
able to solve practical problems with proportional reasoning.
UNDERSTANDING THE STANDARD
· A ratio is a comparison of any two quantities. A ratio is used to represent relationships within a quantity and between quantities. Ratios are used in practical situations when there is a need to compare quantities.
· In the elementary grades, students are taught that fractions represent a part-to-whole relationship. However, fractions may also express a measurement, an operator (multiplication), a quotient, or a ratio. Examples of fraction interpretations include:
- Fractions as parts of wholes: represents three parts of a whole, where the whole is separated into four equal parts.
- Fractions as measurement: the notation can be interpreted as three one-fourths of a unit.
- Fractions as an operator: represents a multiplier of three-fourths of the original magnitude.
- Fractions as a quotient: represents the result obtained when three is divided by four.
- Fractions as a ratio: is a comparison of 3 of a quantity to the whole quantity of 4.
· A ratio may be written using a colon (a:b), the word to (a to b), or fraction notation .
· The order of the values in a ratio is directly related to the order in which the quantities are compared.
- Example: In a certain class, there is a ratio of 3 girls to 4 boys (3:4).
Another comparison that could represent the relationship between these quantities is the ratio of 4 boys to 3 girls (4:3). Both ratios give the same information about the number of girls and boys in the class, but they are distinct ratios. When you switch the order of comparison (girls to boys vs. boys to girls), there are different ratios being expressed.
· Fractions may be used when determining equivalent ratios.
- Example: The ratio of girls to boys in a class is 3:4, this can be interpreted as:
number of girls = ∙ number of boys.
In a class with 16 boys, number of girls = ∙ (16) = 12 girls.
- Example: A similar comparison could compare the ratio of boys to girls in the class as being 4:3, which can be interpreted as:
number of boys = ∙ number of girls.
In a class with 12 girls, number of boys = ∙ (12) = 16 boys.
· A ratio can compare two real-world quantities (e.g., miles per gallon, unit rate, and circumference to diameter of a circle).
· Ratios may or may not be written in simplest form.
· A ratio can represent different comparisons within the same quantity or between different quantities.
Ratio |
Comparison |
part-to-whole (within the same quantity) |
compare part of a whole to the entire whole |
part-to-part (within the same quantity) |
compare part of a whole to another part of the same whole |
whole-to-whole (different quantities) |
compare all of one whole to all of another whole |
part-to-part (different quantities) |
compare part of one whole to part of another whole |
- Examples: Given Quantity A and Quantity B, the following comparisons could be expressed.
Ratio |
Example |
Ratio Notation(s) |
part-to-whole (within the same quantity) |
compare the number of unfilled stars to the total number of stars in Quantity A |
3:8; 3 to 8; or |
part-to-part^{ 1} (within the same quantity) |
compare the number of unfilled stars to the number of filled stars in Quantity A |
3:5 or 3 to 5 |
whole-to-whole ^{1} (different quantities) |
compare the number of stars in Quantity A to the number of stars in Quantity B |
8:5 or 8 to 5 |
part-to-part^{ 1} |
compare the number of unfilled stars in Quantity A to the number of unfilled stars in Quantity B |
3:2 or 3 to 2 |
^{1}Part-to-part comparisons and whole-to-whole comparisons are ratios that are not typically represented in fraction notation except in certain contexts, such as determining whether two different ratios are equivalent.
ESSENTIALS
- What is a ratio?
A ratio is a comparison of any two quantities. A ratio is used to represent relationships within a set and between two sets. A ratio can be written using fraction form (2/3), a colon (2:3), or the word to (2 to 3).
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
· 6.1_{1} Represent a relationship between two quantities using ratios.
· 6.1_{2} Represent a relationship in words that makes a comparison by using the notations, a:b, and a to b.
· 6.1_{3} Create a relationship in words for a given ratio expressed symbolically.
KEY VOCABULARY
ratio, notation, relationship, corresponding parts, simplest form,
comparison, symbolically, fraction, numerator, denominator, rate, equivalent