Unit Overview
Content Area: Mathematics
Unit Title: Ratio Concepts and Using Ratio Reasoning to Solve Problems
Target Course/Grade Level: 6th Grade, Special Ed (Resource Room or Inclusion)
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:
THE ARTS:
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.
Source: http://www.madsci.org/posts/archives/2005-05/1116616537.Ch.r.html (12/5/10)
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):
Unit Enduring Understandings
The following “Essential Understandings” come from Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning, Grades 6-8 (pp. 12 – 13):
Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities
Unit Learning Targets
Students will ...
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:
Exercises, equivalent ratio tables
PowerPoint Lesson, Using Tables to Explore Equivalent Ratios and Rates
Equivalent Ratio Practice, Tables and Graphs, plus how ratios are used in everyday life (pdf)
Video, Sample Problems, equivalent ratios using tables
Video, Tic-Tac-Toe Strategy for Solving Proportion Problems
Interactive Quiz, Equivalent Ratios
Math.com lesson on proportions
Lesson, Bean Counting (Comparing Ratios)
Presentation: Using Tape Diagrams for Problem Solving
Yourteacher.com, Finding Equivalent Ratios by reducing fractions to lowest terms:
Eight Approaches to Proportions
SMART board lesson on Ratios and Proportions
SMART board lesson, Direct Variation
Direct Variation Discovery (Geometer Sketchpad)
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
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
Grade: 6 (Spec Ed)
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:
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:
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.
Video from Wood Magazine: Designing proportional projects
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 ≈ |
9.708 ≈ |
11.326 ≈ |
12.944 ≈ |
14.562 ≈ |
16.18 ≈ |
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):
PowerPoint Lesson, Using Tables to Explore Equivalent Ratios and Rates
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:
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
Formative Assessment Tasks
Universal Design for Learning Options
Multiple Means of Representation
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.
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.
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
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.
For students who want to participate in the artistic activities but cannot draw: Type a picture with Kerpoof! Applies during: Exploration.
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
For all students: Extensive use of manipulatives, especially the Fibonacci Gauge. Applies during: Exploration.
Use of SMART board. Applies during: Explanation.
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
Universal Design for Learning Options: Teacher Toolbox for Entire Unit
Multiple Means of Representation
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.
For students with specific learning disabilities: Use Illuminations to allow students to model key mathematical concepts (ratio, rate, proportion) non-linguistically.
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
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.
For students with limited writing skills: use VoiceThread for students to respond to videos when working independently.
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
For all students: Extensive use of manipulatives, videos, and technology.
Use of SMART board and on-line games.
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.