Unit Overview

Content Area:      Mathematics
Unit Title: Ratio Concepts and Using Ratio Reasoning to Solve Problems
Name:    Suzanne Brummitt
School: Lakeside Middle School
Date:      1/2/11

Unit Summary
The unit introduces the concepts of ratio and rate and compares ratios to fractions.  It builds on students’ understanding of equivalent fractions.  If students have not mastered equivalent fractions in earlier grades, per the standards, then it is necessary to re-teach this content.

By the end of the unit, students should be able to solve problems by reasoning about equivalent ratios.  Such problems include unit rate problems, percent problems, and measurement conversion problems.

Primary interdisciplinary connections
SCIENCE:  Many scientific terms are defined in terms of ratios, for example, mass per unit volume (density) and pound force per square inch (pressure).  Rate of change is an essential concept in scientific inquiry.  Without the ability to interpret and manipulate ratios, students will be lost in the mathematics of science.

WORLD GEOGRAPHY:  The study of geography also references ratios, such as population density (people per square mile).  Ratios are also used to compare demographic groups to each other (men to women, native born to immigrant, children to adults).  Finally, of course, ratios are implicit in map scales; and interpreting map scales is a basic geography skill.

LIFE SKILLS:  The ability to manipulate ratios and rates is essential in many life skills, such as the following:

• adjusting recipes to serve more or fewer people
• computing miles per hour for travel planning
• computing miles per gallon to track fuel efficiency
• computing unit price for comparison shopping
• calculating simple interest, tax and tips
• calculating discounts and savings
• calculating mark-ups when pricing merchandise

THE ARTS:

• A photographer uses lighting ratios to plan the lighting of his subjects, in order to achieve the desired effect.
• An understanding of proportions is important to the skill of drawing the human body.

HEALTH PROFESSIONS:
A chemist gives the following examples of how ratios and proportions are used in two health professions:
In order to determine how long someone can work in an area with a lot of radiation, first we take a measurement of the radiation field. In general, we have an idea how long specific jobs will take. We can use ratios and proportions to scale up or down the time they are allowed to work in a

Primary interdisciplinary connections:

particular area. Health physics uses a lot of proportions to give an "on the fly" estimation of radiation dosage.
Another example would be in the field of medicine. Ratios and proportions are used to determine proper medication dosage for a patient if you have to change it for body mass, age, etc.
21st century themes
Global Awareness, Financial, Economic, Business and Entrepreneurial Literacy

Unit Rationale

Learning Targets

Standards
Ratios and Proportional Relationships (6.SP)
Number and Operations—Fractions (4.NF)

Content Statements

Understand ratio concepts and use ratio reasoning to solve problems.
Extend understanding of fractional equivalence.

6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and 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.” “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.”

6.RP.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.
a.     Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b.     Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c.     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.
d.         Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n*a)/(n*b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.  Use this principle to recognize and generate equivalent fractions.

Unit Essential Questions
The following questions come from Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning, Grades 6-8 (p. 14):

• How does ratio reasoning differ from other types of reasoning?
• What is a ratio?
• What is a ratio as a measure of an attribute in a real-world setting?
• How are ratios related to fractions?
• How are ratios related to division?
• What is a proportion?
• What are the key aspects of proportional reasoning?
• What is a rate, and how is it related to proportional reasoning?
• What is the relationship between the cross-multiplication algorithm and proportional reasoning?
• When is it appropriate to reason proportionally?

Unit Enduring Understandings
The following “Essential Understandings” come from Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning, Grades 6-8 (pp. 12 – 13):

• Reasoning with ratios involves attending to and coordinating two quantities.
• A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit.
• Forming a ratio as a measure of a real-world attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest.
• A number of mathematical connections link ratios and fractions:
•           Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning.
•           Ratios are often used to make “part-part” comparisons, but fractions are not.
•           Ratios and fractions can be thought of as overlapping sets.
•           Ratios can often be meaningfully reinterpreted as fractions.
• Ratios can be meaningfully reinterpreted as quotients.
• A proportion is a relationship of equality between two ratios.  In a proportion, the ratio of the two quantities remains constant as the corresponding values of the quantities changes.
• Proportional reasoning is complex and involves understanding that—
•           Equivalent ratios can be created by iterating and/or partitioning a composed unit.
•           If one quantity in a ratio is multiplied or divided by a particular factor then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship; and
•           The two types of ratios—composed units and multiplicative comparisons — are related.
• A rate is a set of infinitely many equivalent ratios.
• Several ways of reasoning, all grounded in sense-making, can be generalized into algorithms for solving proportion problems.

Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities

Unit Learning Targets
Students will ...

• Practice using ratio language and ratio notations and to express ratios.
• Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane.
• Use tables to compare ratios.
• Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
• 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.

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Evidence of Learning

Summative Assessment (X days)
Students will take the online quiz at Skillwise: http://www.bbc.co.uk/skillswise/numbers/wholenumbers/ratioandproportion/ratio/quiz.shtml .  Advanced students will take Quiz C.  Less advanced students will take Quiz A.  (Quiz B is too British-centric, and it is almost as hard as Quiz C.)

Equipment needed
Technology:  SMART board; Document Camera; Computers for students, with Internet Access and Windows PowerPoint; Computer for teacher, with Internet Access, Windows PowerPoint or Glogster , and SMART Notebook software.

Teacher Resources
Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning, Grades 6-8; JoAnne Lobato, Amy B. Ellis, Randall I Charles, Rose Mary Zbiek; National Council of Teachers of Mathematics. Inc.; 2010.

Elementary and Middle School Mathematics:  Teaching Developmentally; John A. Van de Walle et al; Allyn & Bacon; 2010.

Internet sources:

Video, Lighting Ratios

Lesson, Bean Counting (Comparing Ratios)

Presentation:  Using Tape Diagrams for Problem Solving

Eight Approaches to Proportions

Direct Proportion Graphs

Graphing Direct Proportions

SMART board lesson on Ratios and Proportions

SMART board lesson, Direct Variation

Shodor.org Graphit! to graph coordinate pairs

Video from Wood Magazine:  Designing proportional projects

Purchase a Golden Mean (Fibonacci) Gauge

Site containing a free plan for a Fibonacci Gauge

Purchase Plans from Amazon for How to Make a Fibonacci Gauge

Make Your Own Golden Section (Fibonacci) Gauge Using Heavy Cardboard or Plastic and Brads

Video:  Ratios, Proportions, and the Wright Brothers

Making Comparisons with Ratios and Proportions

Video:  The Golden Ratio in the Human Body

Video:  Fibonacci and the Golden Mean

Video:  Proof of Why We Can Cross Multiply

Formative Assessments

• Pre-assessment
• Daily exit card
• Homework
• Observing execution of class work
•  Daily games/exercises on the SMART Board.
• Results of exercises on the Choice Board.

Lesson Plans

Lesson 1

Pre-assessment

1 40-minute period

Lesson 2

Review/Reteach 4th Grade Content:  Equivalent Fractions
1 80-minute block

Lesson 3
Ratios, Ratio Language
1 80-minute block
Lesson 4

Equivalent ratios:  Using Tables to Compare Ratios
1 80-minute block
Lesson 5
Equivalent ratios:  Plot Pairs of Values on the Coordinate Plane
1 80-minute block

Lesson 6
Unit Rates
1 80-minute block

Lesson 7
Unit Rate Problems
1 80-minute block

Lesson 8
Percent as Rate per 100
1 80-minute block

Lesson 9
Convert Measurements Using Ratios
1 80-minute block

Lesson 10
Summative Assessment:  Quiz
1 40-minute block

Teacher Notes

Connecting to past learning:  It is very difficult to predict how long the review lessons will take in a resource-room class.  Because of delays, or because of changes in curriculum, students may never have been exposed to any of the prerequisite concepts or skills.  Therefore, although the lessons in this unit plan assume review of familiar material, the teacher should be prepared to introduce and cover 4th grade content, if required.  A pre-assessment of this material is essential.   However, in order to get to the 6th grade content in a timely fashion, the teacher may use discretion in how much time to devote to 4th grade content.

Connecting to future learning:  By the end of this unit, students should be ready to proceed to the 7th grade required content.  It is especially important that the groundwork be laid for understanding that proportions can be represented on a graph by a straight line with positive slope passing through the origin, as students will have to take this to the next level in 7th grade, using equations to describe the graph and identifying the constant of proportionality.  It is also important for 6th graders to master solving simple proportion problems so that they are ready for multi-step problems in 7th grade.

Curriculum Development Resources

Click the links below to access additional resources used to design this unit:http://www.udlcenter.org/
http://www.corestandards.org/the-standards/mathematics
http://www.p21.org/route21/index.php?option=com_content&view=article&id=5&Itemid=2
http://daretodifferentiate.wikispaces.com/

Lesson Plan 1
Content Area: Mathematics
Lesson Title: Equivalent Ratios
Timeframe: 80-min. block

Lesson Components

21st Century Themes
Financial, Economic, Business,  and Entrepreneurial Literacy

See Explanation.

Critical Thinking and Problem Solving
See Exploration Activity, See Elaboration Questions.
Communication and Collaboration
Life and Career Skills

See Explanation.
Interdisciplinary Connections:
Indirectly, all interdisciplinary connections for the unit apply to this lesson.  Most directly and explicitly, in this lesson, connections are made to the arts and engineering, including drawing and architecture.  Also, practical problems are related to life skills, such as comparison shopping and adjusting recipes.
Integration of Technology:  Lesson plans are developed using SmartNotes and supported through the use of smart board.  The document camera, connected to the SmartBoard, is used to model procedures using manipulatives and to display student work.  In preparing and presenting their projects, students use computers and access web sites
Equipment & materials needed:

• SMART board
• Fibonacci gauges for each student, pair of students, or group of students (made of cardboard and brads).
• One purchased gauge for the teacher to demonstrate with maximum accuracy.
• One computer with Google’s Sketchup program installed.
• Other computers with Internet access (at least one for every four students)
• Networked printer available to all computers.
• Exit slip

Homework

Goals/Objectives/CPIs
CPI:
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios.
a.     Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables. Use tables to compare ratios.

Objectives:

• Given a ratio, SWBAT create a table of equivalent ratios, with at least 3 more entries.
• SWBAT create and recognize equivalent ratios that are the result of iterating (scaling up) a ratio.
• SWBAT create and recognize equivalent ratios that are the result of partitioning (scaling down).

SWBAT create and recognize equivalent ratios that represent PART:PART relationships as well as ratios that represent PART:WHOLE relationships.

Learning Activities/Instructional Strategies

Engagement
Students watch an engaging, interdisciplinary video showing how to design furniture with pleasing proportions using the Golden Ratio or Golden Section.

Exploration

Students use Fibonacci Gauges, as shown in the video, to draw pairs of line segments that are in the ratio of 1:1.618.  Here is a picture of a Fibonacci Gauge:

Students should ensure that the smaller segment (FG) is an integer, a multiple of 1cm that is at least 5cm long.  Students measure pairs of segments (FG and FH), to the nearest 0.1 cm and record them in a table like the one below.

Some expected values are:

 FG 5 cm 6 cm 7 cm 8 cm 9 cm 10 cm FH 8.09 ≈ 8.1 cm 9.708 ≈ 9.7 cm 11.326 ≈ 11.3 cm 12.944 ≈ 12.9 14.562 ≈ 14.6 cm 16.18 ≈ 16.1 cm

On the SMART Board, using a table prepared by the teacher, student reporters from each group record the data from their investigations in a table of equivalent ratios.

On their calculators, students compute multiples of 1.618, rounding to the nearest .1 cm, to determine whether the observed values are the expected values, in proportion to 1:1.618.

Differentiated Strategy:  Students explore independently, in greater depth, the use of tables to represent equivalent ratios.  Students choose enough of the activities on the choice board to earn at least the minimum number of points (20).  Typically, it will entail four or five activities. Students record their points on a hard copy of the choice board.

Explanation
The class follows this presentation step-by-step to ensure that students have learned how to use tables to represent equivalent ratios (skipping the warm-up and problem-of the day):

Students submit the answers to Quiz Part I and Quiz Part II (the last two slides) as exit tickets (formative assessment).

Elaboration
A common shortcut that is used for establishing whether two ratios are equivalent is the cross-multiply method.  Now that students understand how to derive equivalent ratios in a meaningful way, by multiplying and dividing the numerator and denominator by the same number, it may be a reasonable time to teach the shortcut, as long as students understand the reasoning behind the shortcut.  (Note:  It is also very possible that they have already been acquainted with cross-multiplication in working with equivalent fractions and will be eager to apply it to equivalent ratios.)

Students watch a video that shows the use of cross product to determine whether two ratios are a proportion.

Video:  Three problems solved by cross product

Discuss four ways to confirm that 2/3 = 8/12 is a proportion:

1. Multiply one side of the equation by 1 in the form 4/4:  (2 * 4)/(3 * 4) = 8/12 à           8/12 = 8/12.
2. Divide one side of the equation by 1 in the form 4/4:  2/3 = (8÷4)/(12÷4) à 2/3 = 2/3 (which is the same as putting 8/12 in its simplest form).
3. Multiply both sides of the equation by the same value:  12 (2/3) = 12 (8/12) à              24/3 = 96/12  à 8 = 8.
4. Cross multiply:  2 * 12 = 3 * 8 à 24 = 24.

The following are some questions that can be used to push the levels of thinking up the pyramid during the class discussion:

REMEMBERING:  Given a proportion, state two methods for showing whether it expresses equivalent ratios. (Bloom’s)
UNDERSTANDING:  Explain how each of the methods is performed.  (Bloom’s)
APPLYING:  Apply each of the methods to particular problems. (Bloom’s)
ANALYZING:  Compare the method of cross-multiply to other methods.  Which one is it most like?   How is it alike?  How is it different? (Bloom’s)
EVALUATING:  Which of the methods is easiest?  Why?  Which method most effectively teaches you, the student, the concept of equivalent ratios? (Bloom’s)

CREATING:  Design a real-life problem of equivalent ratios.  State it as a word problem.  Translate it to an equation.  Explain how to solve it using the three methods and tell which one is the best from your point of view.  Tell why it is the best for you. (Bloom’s)

Evaluation:

Exit Ticket:  Respond to the quick quizzes at the end of the PowerPoint Lesson.
Homework:    Print and assign the worksheet found at this link :  Exercises, equivalent ratio tables

• Teacher observation of independent and group activities
• Teacher observations during class discussion, including questioning at all levels of Bloom’s taxonomy.
• Class work (results of Choice Board activities)
• Exit ticket
• Homework

Universal Design for Learning Options

Multiple Means of Representation

1. Perception

When displaying tables of ratios, use effective color contrast to help students with partial sight or color deficiencies to distinguish values in rows from values in columns and/or to distinguish columns from one another. Applies during:  Exploration and Explanation.

1. Language & symbols

For students whose native language is not English or for students with language-based disabilities, use the National Library of Virual Manipulatives to illustrate key concepts (Fibonacci Sequence and Golden Ratio) non-linguistically.  Applies during:  Engagement and Exploration.

1. Comprehension

For students with limited background knowledge, due to native language other than English or specific learning disabilities:  Activiate background knowledge using Windows to the Universe.  Applies during:  Exploration

Multiple Means of Action and Expression

1. Physical activity

For students with limited manual activity:  No Keys Virtual Keyboard provides an on-screen key selection screen for students who have trouble using a regular keyboard.    Accommodation on choice board and exit ticket, if required.  Applies during:  Exploration and Evaluation.

1. Expressive skills and fluency

For students who want to participate in the artistic activities but cannot draw: Type a picture with Kerpoof! Applies during:  Exploration.

1. Executive functions

For students with attention deficit (ADD or ADHD):  Create-a-Graph allows students to track progress toward goals (number of points on the choice board).  Applies during: Exploration.

Multiple Means of Engagement

1. Recruiting interest

For all students:   Extensive use of manipulatives, especially the Fibonacci Gauge.  Applies during:  Exploration.

1. Sustaining effort and persistence

Use of SMART board.  Applies during:  Explanation.

1. Self-regulation :

For all students:  Through the use of choice boards, students choose which activities to engage in, according to their learning styles, interests, and capabilities. Students track their own progress through annotations on the Choice Board.  Applies during:  Exploration.

Resources

• See Unit Resources.

Universal Design for Learning Options: Teacher Toolbox for Entire Unit

Multiple Means of Representation

1. Perception

For students with limited vision:  Use AIM Explorer for text access features such as magnification, custom text and background colors, text-to-speech (synthetic and human), text highlighting, and layout options.  Implement if necessary as an accommodation for homework.

1. Language & symbols

For students with specific learning disabilities:  Use Illuminations to allow students to model key mathematical concepts (ratio, rate, proportion) non-linguistically.

1. Comprehension

To assist all students in seeing the relationship among the concepts of fractions, ratios, rates, proportions, and percentages, use options that guide information processing, for example, graphic representations using WebSpiration
To assist students in seeing all of the components of a proportion, and their relationships to each other, use a tic-tac-toe organizer as described in the video Solving Ratio Proportion Problems the EASY Way.

Multiple Means of Action and Expression

1. Physical activity

For students with limited manual activity:  No Keys Virtual Keyboard provides an on-screen key selection screen for students who have trouble using a regular keyboard.    Accommodation on exit ticket, if required.  Applies during:  Evaluation.

1. Expressive skills and fluency

For students with limited writing skills:  use VoiceThread for students to respond to videos when working independently.

1. Executive functions

For students with attention deficit (ADD or ADHD):  Teacher should meet with students to set realistic goals for independent work that provides the student a large variety of choice among differentiated activities.  The student will need help choosing the right activities and a reasonable number of  them, with a realistic plan to achieve.

Multiple Means of Engagement

1. Recruiting interest

For all students:   Extensive use of manipulatives, videos, and technology.

1. Sustaining effort and persistence

Use of SMART board and on-line games.

1. Self-regulation :

For all students:  Learner diaries not only allow the teacher to get to know students one-on-one but also encourage the student to gain understanding of his or her learning style.