Cluster: understand ratio concepts and use ratio reasoning to solve problems

Standard: know the concept of a ratio and also use proportion language to explain a proportion relationship between two quantities. Because that example, “The proportion of wings to beaks in the bird residence at the zoo was 2:1, due to the fact that for every 2 wings there to be 1 beak.” “For every poll candidate A received, candidate C received practically three votes.”

Learning Domain: Ratios and also Proportional Relationships

Standard: recognize ratio concepts and use proportion reasoning to settle problems

Indicator: recognize the ide of a ratio and also use ratio language to describe a proportion relationship between two quantities. Because that example, "The ratio of wings to beaks in the bird residence at the zoo was 2:1, since for every 2 wing there was 1 beak."ť "For every vote candidate A received, candidate C received nearly three votes."ť

Learning Domain: Ratios and also Proportional Relationships

Standard: understand ratio concepts and also use ratio reasoning to solve problems.

You are watching: A ratio is the comparison of two quantities by what operation?

Indicator: recognize the ide of a ratio and use ratio language to explain a proportion relationship between two quantities.

Ratios compare Numbers through Ratios to compare Numbers with Ratios

This lesson formally introduces and defines a proportion as a method of comparing numbers to one another.

Key ConceptsA *ratio* is defined by the adhering to characteristics:

*a*:

*b*).Ratios are used to compare two numbers.The worth of a proportion

*a*:

*b*is the quotient

*a*÷

*b*, or the result of separating

*a*by

*b*.

Other vital features of ratios encompass the following:

A proportion does not always tell girlfriend the worths of amounts being compared.The bespeak of values in a ratio matters.Goals and also Learning ObjectivesIntroduce a formal meaning of ratio.Use the an interpretation of ratio to solve problems related to comparing quantities.Understand the ratios carry out not always tell friend the values of the amounts being compared.Understand that the bespeak of values in a proportion matters.advent to Ratios

lesson Guide

In this lesson, students find out the formal meaning of a ratio and also then use it to fix problems. The start of the lesson introduce the principle of a ratio as a method of comparing numbers of objects making use of division, i beg your pardon is an different to compare numbers using subtraction. Have students look at the photo of stars and also triangles and then check out the text.

Ask:

The ratio of triangle to stars is 3:10. What is the ratio of stars come triangles? (Answer: 10:3)ELL: save in psychic that part students may not feel comfortable reading aloud. Be prepared to support them in this task.

## Opening

arrival to Ratios

Look in ~ the photo of stars and triangles and read the complying with information.

One means that you deserve to compare the variety of stars and also the number of triangles is come say the there are 7 much more stars 보다 there space triangles. This to compare looks in ~ the difference in between two quantities; it provides the operation of subtraction.Another method that you have the right to compare the number of stars and the number of triangles is come say the for every 3 triangle there space 10 stars. You deserve to say the the*ratio*of triangle to stars is 3 come 10 or 3:10. This comparison uses the operation of division.The worth of the proportion of triangles to stars is 310, or 0.3.

ratio of Egginess

class Guide

Have a volunteer check out the definition of a ratio aloud.

Review the definition of ratio: A ratio is a to compare of 2 numbers through division.

The worth of a ratio is the quotient that outcomes from splitting the 2 numbers. Because that example, the worth of the proportion 35:7 is 5, i m sorry you uncover by computer 35 ÷ 7 = 5.

Have students testimonial the video clip of the egginess difficulty from the vault lesson. Comment on the proportion in the egginess problem. Demonstrate how to compose the ratio. Talk around the fact that the ratio could be flour to eggs or egg to flour. The proportion of flour to egg is 3:2; the ratio of egg to flour is 2:3.

ELL: as soon as showing the video, be certain that ELLs are complying with the explanations. Pause the video at vital times to permit ELLs time to process the information. Ask students if they have to watch that a second time. Remind students that they room finding the proportion in the egginess problem.

## Opening

proportion of Egginess

A *ratio* is a compare of two numbers by division.

The *value* of a ratio is the quotient that outcomes from dividing the 2 numbers. Because that example, the value of the proportion 35:7 is 5, i m sorry you uncover by computing 35 ÷ 7 = 5.

In the vault lesson, friend looked at just how to settle the egginess in a mixture. Clock the EgginessPart 2 video.

What is the ratio in the egginess problem?VIDEO: Egginess part 2

MP4

Egginess component 2

Download

math Mission

class Guide

Discuss the mathematics Mission. College student will explain how ratios are offered to compare quantities.

ELL: comment on the ide of ratio. Map out diagrams that the examples so students have the right to make associations with comparison that quantities. Use manipulatives to show the to compare of two amounts using subtraction and also division. Use illustrations or manipulatives come model efficient learning to explain ratios. The is important to emphasize the you can describe ratios by saying “for every” or “per” since these terms will certainly be used interchangeably transparent the unit. Underline the two quantities so the students know specifically which quantities you space comparing.

## Opening

Explain just how ratios are used to to compare quantities.

Ms. Lee’s course

class Guide

Have students work in bag on the problems and also the presentation.

SWD: assist students through disabilities build their math vocabulary by continuous modeling the usage of new terms in the paper definition of class work and also activities.

Mathematical Practices**Mathematical practice 2: reason abstractly and also quantitatively.**

Listen because that students who use the problem cases to aid them make sense of the worths they are working with.

**Mathematical exercise 6: attend to precision.**

Listen likewise for college student who use the ax *ratio* appropriately or who talk about the correct usage of the term as they work together to solve the problems.

**Student thinks the you compose a proportion as a subtraction.**

**Student reverses the boys and also girls in the ratio.**

## Work Time

Ms. Lee's Class

There space 15 boys and 17 girl in Ms. Lee's mathematics class.

What is the difference in between the variety of girls and the number of boys in the class?What is the ratio of guys to girls?What is the ratio of girl to boys?Ask yourself:

When you need to find the difference between two numbers, what procedure do girlfriend use?For the proportion of boys to girls, what must the first number be, ”the number of girls or the variety of boys?For the ratio of girl to boys, what have to the very first number be, the variety of girls or the number of boysA Tennis game

mathematics Practices

**Mathematical practice 2: reason abstractly and quantitatively.**

Listen because that students who usage the problem cases to help them make feeling of the values they are working with.

**Mathematical exercise 6: attend to precision.**

Listen also for college student who usage the ax *ratio* appropriately or who comment on the correct consumption of the term together they work-related together to fix the problems.

**Student thinks there are exactly 3 females and 2 males the town hall the tennis game.**

**Student thinks there is1 more female than male watching the tennis game.**

**Student to trust the number of females, males, or civilization watching the tennis game can be calculated with only the ratio.**

## Work Time

A Tennis Game

The proportion of the number of females watching a tennis video game to the number of males city hall the tennis video game is 3 come 2. You have the right to write the as32, or 3:2.

Can friend tell from this proportion how numerous females room watching the tennis game? Explain.Can friend tell indigenous this ratio just how many world are city hall the tennis game? Explain.Can girlfriend tell indigenous this ratio whether an ext males or much more females space watching the game? Explain.Can you tell native this proportion the difference in between the variety of males and also the variety of females city hall the game? Explain.Could you usage this proportion to create the ratio of the number of males watching the tennis video game to the variety of females the town hall the tennis game? Explain.Ask yourself:

Sketch a diagram showing several possible numbers the females and males the town hall the game. Have the right to you phone call which numbers space correct based upon the ratio?In looking at a ratio, how deserve to you call which number represents the bigger amount?What would you require to know to find the difference between the number of males and also females watching the game?In determining whether the ratio deserve to be rewritten to stand for males to females, think around the an interpretation of a ratio.Prepare a Presentation

prepare for methods of Thinking

Listen and look for the complying with student reasoning to highlight throughout the means of reasoning discussion:

Students that explicitly comment on subtraction and division as ways of comparing numbersStudents who comment on the boundaries of what a ratio tells friend (e.g., “It doesn’t tell us how countless …”)Students who divide 3 through 2 to gain 1.5 and also then talk about the definition of 1.5 with reference to the trouble situationChallenge Problem## Possible Answer

The declare is constantly true. A number separated by chin is equal to 1, so if 2 quantities acquire closer to every other, your ratio’s value is closer to a number split by itself, or 1.## Work Time

Prepare a Presentation

Explain what varieties of conclusions friend can and cannot make based on the tennis video game ratio.

In your own words, define what a proportion is.

Challenge ProblemAs 2 quantities obtain closer to every other, the value of the proportion of the quantities viewpoints 1.

Is the above statement*always true*,

*sometimes true*, or

*never true*? Explain.

Make connections

Mathematics

Have students share their presentations. If over there are any misunderstandings, facilitate a class discussion that concentrates on the an interpretation of a ratio and also what that does and does no tell us.

Ask:

What space some examples of numbers of females and also males that can be watching the tennis game?Compare these examples to the situation around the number of boys and girls in Ms. Lee’s class, in which the actual number of students are offered rather than a ratio between numbers. Highlight correct usage of the ax *ratio*, or provide opportunities because that students to review their usage of the term.

Conclude the discussion with a focus on even if it is the ratio tells friend whether more males space watching the video game or more females are watching the game. If over there is a college student who split 3 by 2 to obtain 1.5, ask them come share your strategy. Asking the course what 1.5 method in this situation:

Does that make feeling for a ratio of numbers of civilization to have a worth of 1.5?Check the all students understand that the bespeak of values in a proportion matters. The ratio of males to females (2:3) is different from the proportion of females to males (3:2).

Mathematical Practices**Mathematical practice 2: reason abstractly and quantitatively.**

Ask students to talk about how they can use the paper definition of a problem instance to assist them make sense of the worths they are working with.

**Mathematical practice 6: attend to precision.**

Call attention to correct provides of the hatchet *ratio*,or ask for clarifications about its use and an interpretation as needed as students existing their work and also ask questions of presenters.

## Performance Task

ways of Thinking: do Connections

Take notes about your classmates' explanations the the conclusions that can and cannot it is in made based on the tennis video game ratio, and also their explanations the what a proportion is.

As your classmates present, ask questions such as:

What possible numbers of females and also males watching the tennis video game did girlfriend find?How did you recognize these numbers?How did you decision what varieties of conclusions you can and also cannot make based on the tennis game ratio?Is over there anything friend can include to her explanation to make it much more specific or precise?What ns Know about Ratios

A possible Summary

Ratios permit you to to compare quantities, but by themselves, they perform not tell friend the actual values of the quantities. Using a ratio to compare amounts is different from utilizing subtraction to find the difference between quantities since a ratio tells friend the worth of one quantity for a provided value that the various other quantity. Because that example, for a ratio of 3:2, you recognize that if the first quantity has actually a worth of 6, the 2nd quantity has a worth of 4.

SWD: create a resource for some students that consists of this explanation in simpler language (perhaps with visual illustrations). Annotate illustrations that show ratios and also what castle represent.

Additional conversation PointsIf over there is time, comment on the following:

A summary of various ways to compare numbersA meaning of a ratioA summary of what a ratio tells you and what that doesn’t tell you## Formative Assessment

review of the Math: What ns Know around Ratios

Write a review of what girlfriend learned around ratios.

Check her summary.

Do you define what a proportion is?Do you comment on what species of conclusions can and also cannot it is in made based on a ratio?Do you describe how making use of ratios come compare two numbers is different from using subtraction to compare two numbers?Reflect ~ above Your occupational

lesson Guide

Have each student compose a brief reflection before the finish of class. Review the reflect to discover out what students wonder about ratios.

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## Work Time

Reflection

Write a reflection around the ideas discussed in course today. Use the sentence starter below if you find it to it is in helpful.