Rate

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Understand quantity as a number used with a unit of measurement.
Solve problems involving quantities such as distances, intervals of time, liquid volumes, masses of objects, and money, and with the units of measurement for these quantities.
Understand that a ratio is a comparison of two quantities.
Write ratios for problem situations.
Make and interpret tables, graphs, and diagrams.
Write and solve equations to represent problem situations.

Lesson Flow

In this unit, students will explore the concept of rate in a variety of contexts: beats per minute, unit prices, fuel efficiency of a car, population density, speed, and conversion factors. Students will write and refine their own definition for rate and then use it to recognize rates in different situations. Students will learn that every rate is paired with an inverse rate that is a measure of the same relationship. Students will figure out the logic of how units are used with rates. Then students will represent quantitative relationships involving rates, using tables, graphs, double number lines, and formulas, and they will see how to create one such representation when given another.

In this lesson, students define rate. After coming up with a preliminary definition on their own, students identify situations that describe rates and situations that do not.Students determine what is common among rate situations and then revise their definitions of rate based on these observations. Students present and discuss their work and together create a class definition. They compare the class definition of rate with the Glossary definition and revise the class definition as needed.Key ConceptsA good definition of rate has to be precise, yet general enough to be useful in a variety of situations. For example, the statement “a rate compares two quantities” is true, but it is so general that it is not helpful. The statement “speed is a rate” is true, but it is not useful in determining whether unit price or population density are rates.A good definition of rate needs to state that a rate is a single quantity, expressed with a unit of the form A per B, and derived from a comparison by division of two measures of a single situation.Goals and Learning ObjectivesGain a deeper understanding of rate by developing, refining, testing, and then refining again a definition of rate.Use a definition of rate to determine the kinds of situations that are rate situations and to recognize rates in new and different situations.Understand the importance of precision in communicating mathematical concepts.

In this lesson, students are introduced to rate in the context of music. They will explore beats per minute and compare rates using mathematical representations including graphs and double number lines.Key ConceptsBeats per minute is a rate. Musicians often count the number of beats per measure to determine the tempo of a song. A fast tempo produces music that seems to be racing, whereas a slow tempo results in music that is more relaxing. When graphed, sets with more beats per minute have smaller intervals on the double number line and steeper lines on the graph.Goals and Learning ObjectivesInvestigate rate in music.Find beats per minute by counting beats in music.Represent beats per minute on a double number line and a graph.

In this lesson, students explore rate in the context of grocery shopping. Students use the unit price, or price per egg, to find the price of any number of eggs.Key ConceptsA unit price is a rate. The unit price tells the price of one unit of something (for example, one pound of cheese, one quart of milk, one box of paper clips, one package of cereal, and so on).The unit price can be found by dividing the price in dollars by the number of units.The unit price can be used to find the price of any quantity of something by multiplying the unit price by the quantity.Goals and Learning ObjectivesInvestigate rate as a unit price.Find a unit price by dividing the price in dollars by the number of units.Find the price of any quantity of something by multiplying that quantity by the unit price.

In this lesson, students use an interactive map to compare the crowdedness of three countries of their choice. They learn that to compare countries with different areas and populations, they need to calculate population density—a rate that compares the population of a region to its area.Key ConceptsA ratio is a comparison of two quantities by division. It can be expressed in the forms a to b, a:b, or ab, where b ≠ 0. The value of a ratio is found by dividing the two quantities. A ratio provides a relative comparison of two quantities. A rate is a ratio that compares two quantities measured in different units. Population density is a rate that compares the population of a region to its area. The value is given in number of people per unit of area.ELL: Identifying key words are crucial for students. Spend some time discussing the key vocabulary in this unit.Goals and Learning ObjectivesExplore rate in the context of population density.Compare three countries to see which is most crowded—that is, which has the greatest population density.

In this lesson, students use a ruler that measures both inches and centimeters to find conversion factors for converting inches to centimeters and centimeters to inches.Key ConceptsRates can be used to convert a measurement in one unit to a corresponding measurement in another unit. We call rates that are used for such purposes conversion factors.The conversion factor 2.54 centimeters per inch is used to convert a measurement in inches to a measurement in centimeters (or, from the English system to the metric system).The conversion factor 0.3937 inches per centimeter is used to convert a measurement in centimeters to a measurement in inches (or, from the metric system to the English system).In the calculation, the inch units cancel out and the remaining centimeter units are the units of the answer, or vice versa.Goals and Learning ObjectivesExplore rate in the context of finding and using conversion factors.Understand that there are two conversion factors that translate a measurement in one unit to a corresponding measurement in another unit, and that these two conversion factors are inverses of one another.

In this lesson, students focus on the units used with rates. Students are given calculations without units and must determine the correct units to use.Key ConceptsWhen dividing quantity A by quantity B to find a rate, the unit of the quotient is expressed in the form A per B.When multiplying a B quantity by an A per B rate, you get an A quantity.Some rates, while mathematically correct, are physically impossible in the real world.Goals and Learning ObjectivesUnderstand the units that result from rate calculations.

In this lesson, students write formulas to represent different rate relationships.Key ConceptsA formula is a mathematical way of writing a rule for computing a value.Formulas, like c = 2.50w or d = 20g, describe the relationship between quantities.The formula c = 2.50w describes the relationship between a cost and a quantity that costs $2.50 per unit of weight. Here, w stands for any weight, and c stands for the cost of w pounds at $2.50 per pound.The formula d = 20g describes the relationship between the distance, d, and the number of gallons of gas, g, for a car that gets 20 miles per gallon.Goals and Learning ObjectivesUse equations with two variables to express relationships between quantities that vary together.

In this lesson, students first watch three racers racing against each other. The race is shown on a track and represented on a graph. Students then change the speed, distance, and time to create a race with different results. They graph the new race and compare their graph to the original race graph.Key ConceptsA rate situation can be represented by a graph. Each point on a graph represents a pair of values. In today's situation, each point represents an amount of time and the distance a racer traveled in that amount of time. Time is usually plotted on the horizontal axis. The farther right a point is from the origin, the more time has passed from the start. Distance is usually plotted on the vertical axis. The higher up a point is from the origin, the farther the snail has traveled from the start. A graph of a constant speed is a straight line. Steeper lines show faster speeds.Goals and Learning ObjectivesUnderstand that a graph can be a visual representation of an actual rate situation.Plot pairs of related values on a graph.Use graphs to develop an understanding of rates.

In this lesson, students watch a video of a runner and express his speed as a rate in meters per second. Students then use the rate to determine how long it takes the runner to go any distance.Key ConceptsSpeed is a rate that is expressed as distance traveled per unit of time. Miles per hour, laps per minute, and meters per second are all examples of units for speed. The measures of speed, distance, and time are all related. The relationship can be expressed in three ways: d = rt, r = dt, t = dr.Goals and Learning ObjectivesExplore speed as a rate that measures the relationship between two aspects of a situation: distance and time.In comparing distance, speed, and time, understand how to use any two of these measures to find the third measure.

Students use their knowledge of rates to solve problems.Key ConceptsGiven any two values in a rate situation, you can find the third value.These three equations are equivalent, and they all describe rate relationships:y = rx,  r = yx,  x = yrAt the beginning of this lesson (or for homework), students will revise their work on the pre-assessment Self Check. Their revised work will provide data that you and your students can use to reassess students' understanding of rate. You can use this information to clear up any remaining misconceptions and to help students integrate their learning from the past several days into a deeper and more coherent whole.The work students do in this lesson and in revising their pre-assessments will help you and your students decide how to help them during the Gallery. In this lesson, students will reveal the depth and clarity of their understanding of rate.Students whose understanding of rate is still delicate should get extra help during the Gallery.Students who feel that they have a robust understanding of rate may choose from any of the problem-solving or deeper mathematics problems in the Gallery.Goals and Learning ObjectivesUncover any partial understandings and misconceptions about rate.Develop a more robust understanding of rate.Identify which Gallery problems to work on.

Students watch a video in which two students discuss the problem of how to compare fuel efficiency. Students then analyze the work of the two students as they use rates to determine fuel efficiency in two different ways.Key ConceptsFuel efficiency is a rate. Fuel efficiency can be expressed in miles per gallon (mpg). This rate is useful for determining how far a vehicle can travel using any number of gallons of gas. Fuel efficiency can also be expressed in gallons per mile (gpm). This rate is useful for determining how many gallons of gas a vehicle uses to travel any number of miles.The rates miles per gallon and gallons per mile are inverse rates—they both describe the same relationship. For example, the rates 20 miles per gallon and 0.05 gallon per mile both describe the relationship between 300 miles and 15 gallons. The greater the rate in miles per gallon, the better the fuel efficiency. The smaller the rate in gallons per mile, the better the fuel efficiency.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. This will help to highlight for students the critical features and/or concepts and will help them to pay close attention to salient information.Goals and Learning ObjectivesExplore rate in the context of fuel efficiency.Express fuel efficiency as the rate miles per gallon (mpg) and as its inverse, gallons per mile (gpm).Use the rate miles per gallon to find the number of miles a car can travel on a number of gallons of gas.Use the rate gallons per mile to find the number of gallons of gas used for a number of miles driven.

This lesson teaches about Rates, Ratios, and Unit Rates, using visually appealing art and real-life examples that will engage teenagers.

The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards.  Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.

This lesson unit is intended to help sixth grade teachers assess how well students are able to: Analyze a realistic situation mathematically; construct sight lines to decide which areas of a room are visible or hidden from a camera; find and compare areas of triangles and quadrilaterals; and calculate and compare percentages and/or fractions of areas.

Two besotted rulers must embrace proportional units in order to unite their lands. It takes mathematical reasoning to identify the problem, and solution, when engineers from Queentopia and Kingopolis build a bridge to meet in the middle of the river.

The purpose of this task is to help students see that when you have a context that can be modeled with a ratio and associated unit rate, there is almost always another ratio with its associated unit rate (the only exception is when one of the quantities is zero), and to encourage students to flexibly choose either unit rate depending on the question at hand.

Learn about the dynamic relationships between a jet engine's heat loss, surface area, and volume in this video adapted from Annenberg Learner's Learning Math: Patterns, Functions, and Algebra.

The battle is on in this game where you build your own potions! Check your ratios to win this mixture mix-off. Ratio Rumble guides students in: identifying ratios when used in a variety of contextual situations; providing visual representations of ratios; solving common problems or communicating by using rate, particularly unit rates; and explaining why ratios and rates naturally relate to fractions and decimals.