Weekly Overview

Weekly Topics

The focus of this week’s instruction is to deepen students’ understanding of:

  • From Ratios to Rates
  • From Rates to Ratios
  • Finding a Rate by Dividing Two Quantities
  • Percent and Rates per 100
  • Percent of a Quantity

Materials Needed

  • Tape Diagrams (See example below.)
  • Double Number Line Diagrams (See example below.)
  • Ratio Tables (See example below.)
  • Colored pencils or pens

Standard(s) Covered

6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0. Use rate language in the context of a ratio relationship.

For example, this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. Also, we paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.

(Expectations for unit rates in 6th grade are limited to non-complex fractions).

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).

  1. Solve unit rate problems including those involving unit pricing and constant speed. For example, if a runner ran 10 miles in 90 minutes, running at that speed, how long will it take him to run 6 miles? How fast is he running in miles per hour?
  2. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
  1. Use ratio reasoning to convert customary and metric measurement units (within the same system); manipulate and transform units appropriately when multiplying or dividing quantities.

Representations

Rising grade 7 example

Additional Terms and Symbols

  • Equivalent Ratios (Two ratios A:Band C:Dare equivalent ratios if there is a nonzero number c such that C=cAand D=cB.  For example, two ratios are equivalent if they both have values that are equal.)
  • Percent (One percent is the number \({1 \over 100}\) and is written 1%.  Percentages can be used as rates.  For example, 30% of a quantity means \({ 30\over 100}\)  times the quantity.)
  • Quantity (illustration) (Examples of a quantity include a length, an area, a volume, a mass, a weight, a length of time, or a speed.  It is an instance of a type of quantity.)

All quantities of the same type have the properties that (1) two quantities can be compared, (2) two quantities can be combined to get a new quantity of that same type, and (3) there .always exists a quantity that is a multiple of any given quantity.  These properties help define ways to measure quantities using a standard quantity called a unit of measurement.)

  • Rate (illustration) (A rate is a quantity that describes a ratio relationship between two types of quantities.  For example, 1.25  \({miles \over hour}\) is a rate that describes a ratio relationship between hours and miles:  If an object is traveling at a constant 1.25 \({miles \over hour}\), then after 1 hour it has gone 1.25 miles, after 2 hours it has gone 2.50 miles, after 3 hours it has gone 3.75 miles, and so on.  Rates differ from ratios in how they describe ratio relationships—rates are quantities and have the properties of quantities.  For example, rates of the same type can be added together to get a new rate, as in 30 \({miles \over hour}\)+20 \({miles \over hour}\)=50 \({miles \over hour}\), whereas ratios cannot be added together.)
  • Ratio (A ratio is an ordered pair of numbers which are not zero.  A ratio is denoted A:B to indicate the order of the numbers—the number A is first and the number B is second.)
  • Ratio Relationship (A ratio relationship is the set of all ratios that are equivalent ratios.  A ratio such as 5:4 can be used to describe the ratio relationship {1:\({4 \over 5}\), \({5 \over 4}\):1,  5:4,  10:8,  15:12, …}.  Ratio language such as “5 miles for every 4 hours” can also be used to describe a ratio relationship.  Ratio relationships are often represented by ratio tables, double number lines diagrams, and by equations and their graphs.) 
  • Unit Rate (When a rate is written as a measurement (i.e., a number times a unit), the unit rate is the measure (i.e., the numerical part of the measurement).  For example, when the rate of speed of an object is written as the measurement 1.25 mph, the number 1.25 is the unit rate.)
  • Value of a Ratio (The value of the ratio A:B is the quotient \({A\over B}\) as long as B is not zero.)