Objective 7.11.1: Convert units of length, weight, or capacity within the same system (customary or metric) Objective 7.11.2: Solve problems involving decimals, percents, fractions, and proportions.

Counting with the String Clock
You have two identical pieces of string. You know that if you light one of these pieces of string at one end, it will take exactly an hour for the string to burn completely. However, neither piece of string is evenly made, so each might burn faster at one point and slower at another.
With just these two pieces of string and a lighter, how can you measure exactly 45 minutes?
Answer: The Daily Spark Critical Thinking, p. 31.

What's on Your Radar Screen?
Distance-Rate-Time Problems from NASA Mathematics Teaching in the Middle School, Aug. 2006, pp. 6-12.

CMP unit "Comparing and Scaling" has some interesting real-life problems.

Collaborative Project

Chris Harbeck and I are planning to have our students work together online to learn about ratios and proportions. This page has my ideas for the project.

Hands-on

Cool Kool-Aid Experiment
Seventh grade students have had wide-ranging experiences with fractions and operations on fractions, but ratios and rates may be new concepts for them. This activity introduces students to the idea of rates via a familiar and tangible product: Kool-Aid. Students will add varying amounts of Kool-Aid powder to a set amount of water, thereby producing a sensory experience of rates. The Kool-Aid example also provides a base experience upon which other examples can build.

Video

Designing Toy CarsNew!!
As a professional toy car designer for Mattel's Hot Wheels, Larry Wood uses basic math concepts such as fractions, measurement and scale to create accurate replicas of the coolest cars on the road. The Futures Channel, running time 2:42 minutes.

Technology

What's Your Benchmark?
In this TI-73 activity from Texas Instruments, students explore the importance of benchmarks when estimating measurements. They will also use the Convert menu to change from one unit to another.

Smart Shoppers
In this TI-73 activity from Texas Instruments, students will compare ratios with different denominators, and the with common denominators. Students will also find unit rates and see how they can be used as a way to compare numbers. They will also discuss when each way is the best to compare prices of food.

Math Homework Hotline
Video and auditory tutorial from the Tampa Educational Channel. One of over 30 topics specifically covering problem areas of middle school math. (You may need to set up a free user account to access this site and see all topics). Designed as a homework help site, this makes an excellent link for teachers to include on their own web pages. Students can pause and replay the videos as needed. Requires Flash 6.

Exploring Rate/Ratio/Proportion
this multimedia resource from Junior High Interactives (learnalberta.ca) examines rate, ratio and proporition and shows how they play a role in photography. Using the interactive components, students explore ratio equivalencies by enlargening and reducing images to compare an original ratio and a target ratio. A print activity is included.

PBS Mathline: The Math of Bicycles
Lesson plan/offline activity. Students explore ratios by comparing bicycles to tricycles. They describe the ratio of the relationship between one turn of the pedal and one turn of the wheel, and define the ratio between number of teeth to sizes of the gears.

SpyGuys: Percents Select Lesson 4. These video and auditory tutorials from LearnAlberta.CA allow teachers and students to explore math concepts, pause and replay. Includes interactive exercises that provide immediate feedback. Printable worksheets, parent notes, and a glossary are included for each lesson. In addition to the lessons, there are sections on problem-solving strategies.

SpyGuys: Ratios Select Lesson 3. These video and auditory tutorials from LearnAlberta.CA allow teachers and students to explore math concepts, pause and replay. Includes interactive exercises that provide immediate feedback. Printable worksheets, parent notes, and a glossary are included for each lesson. In addition to the lessons, there are sections on problem-solving strategies.

Body Ratios
Modeling Middle School Mathematics project (MMSM) is a professional development program using video lessons and Web-based Internet materials to examine mathematical concepts. Video clips and full transcripts are provided of actual classroom situations in which the lesson is taught. In this lesson on ratios and scaled drawings, educators use the story of Gulliver's Travels to pose the problem: how to make a shirt for someone who is much bigger.

Ratios and Equivalent Ratios
From Math.com. In-depth tutorial with java applets, visuals, examples and workouts. Could be used at home or in class for review or reteaching.

Proportions
From Math.com. In-depth tutorial with java applets, visuals, examples and workouts. Could be used at home or in class for review or reteaching

Using Scale Drawings - Floorplan your Classroom
The activities presented here allow students to study mathematics by doing real-life architecture to walk through the floor plan design process. Look for measurement and drawing tips as you create a scale drawing on graph paper or using a computer drawing program.

Scale Factors Figure This! Math Challenge #61 : Statue of Liberty. One of a series of NCTM standards-aligned math challenges designed for students to work out with their families. Available in English and Spanish. Learn about similarity and scaling by measuring the length of your nose and your arm in the same units. Then apply this knowledge to find out if the Statue of Liberty's nose is too long.

Ratio and Proportions Figure This! Math Challenge #47 : VCR Recording Tape . One of a series of NCTM standards-aligned math challenges designed for students to work out with their families. Available in English and Spanish. Develop proportional reasoning by solving this problem related to VCR Tape recording settings.

Great Cartoon Blow-up Activity
Students practice drawing to scale and create blow-ups of their favorite Flinestone characters while learning about simultude.

Proportional Reasoning: The Mixture Blues
Session 4, Part B. Annenberg Media Lesson for teachers to improve their own teaching practices. Includes excellent teacher training videos at the beginning of each session. In "Mixture Blues," teachers compare mixtures of blue liquid and clear water and predict which set would be bluer. The video includes reflection on what has been learned.

Proportional Reasoning: Quad Person
Session 4, Part C. Annenberg Media Lesson for teachers to improve their own teaching practices. Includes excellent teacher training videos at the beginning of each session. The lesson includes a scaling exercise with "Quadperson" that could be used with students and the video shows teachers using it in a learning situation. Participants learn the significance of relative vs. absolute change.

Proportioner
A free resource from the Concord Consortium's "Seeing Math" Project. Proportioner allows students to specify image dimensions graphically, numerically or using a scale factor. They can duplicate images and modify comparisons or use one image to "paint" another. Interactives can be downloaded for offline use. There is a user's guide, warm-up and sample activities.

The Ratio of Rest: Proportional Reasoning
In NASA CONNECT™: The Right Ratio of Rest: Proportional Reasoning, students learn about the human circadian clock and how it affects peoples’ daily lives. Students see how NASA scientists are studying the circadian timing system to improve astronaut’s physical and mental tasks while working in space. Conducting research to help astronauts sleep better in space also helps people on Earth with similar problems. The hands-on activity in this guide allows students to demonstrate how fractions, decimals, and percents are related and further develop their proportional reasoning skills. Includes a 28-minute video that can be paused at appropriate intervals.

Ratio and Proportion Game
Find out how to use ratio and proportion in everyday life. Students can view tutorials on 6 related topic areas or play a shockwave game from the BBC to practice writing ratios.

Understanding Rational Numbers and Proportions
students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving. Illuminations/NCTM classroom activities.

Population Ratios
An interdisciplinary project (mathematics, social studies) where students choose a county in the United States and using ratios convert the statistics into meaningful numbers. Illuminations/NCTM classroom activity.

Rags to Riches
"Millionaire" style game from Quia that quizzes students on proportions.

Solving Proportions
Glencoe Online Study Tool with hints for solving proportion problems.

Solve Proportions
Harcourt School Publishers tutorial on solving proportions. Includes auditory reinforcement and the ability to replay screens.

Scales, Ratios, Proportion Jeopardy Game
Use this ready-made PowerPoint Jeopardy game from Taft Middle School teachers to test student knowledge on ratios and unit rates, proportions, similar figures,scale drawings and probability.

In whats can ratios be used to analyze an athlete's performance?

When are some times in your everyday life that you have used rate to determine the better buy? Describe the situation.

In what ways are fractions and ratios alike? How are they different?

The aspect ratio of a film is the relationship between the width of the film's image and the height. What issues do aspect ratios cause for TV's, movie screens, and computer monitors?

A ratio can be expressed as 2/6, 2 to 6, or 2:6. Do you think different expressions of the same ratio are more appropriate in different contexts? Describe when to use each.

A rate is a "ratio that creates a relationship between two quantities with different kinds of units" like miles per hour (miles/hour). What are some other kinds of relationships between quantities you've seen and what characteristics do they share?

What are some ways a builder can use what he or she knows from the blueprint to determine the appropriate amounts of materials to puchase?

Population density is a "rate that compares the population per square mile." Would you rather live in an area with an unusually high or an unusually low population density? Explain your reasoning.

A proportion is "an equation stating that two ratios are equivalent." Other than using cross products, what are some strategies for determining whether or not two ratios form a proportion? Use examples to explain.

The rate of change shows how one quantity changes in relation to another quantity. How would you interpret a situation in which there was a decreasing rate of change? An increasing rate of change?

Proportions are used to determine medicine dosage amounts. What are some conditions that cause the dosage amounts to differ among people?

What are some times that proportions, ratios, or rates must be exact? What are some times that these relationships can be estimated?

Maps are "scale drawings that show distances between two locations." What might help you decide whether to purchase a map with a relatively small scale or a relatively large scale? Explain your reasoning.

Many countries use the metric system and use kilometers instead of miles to measure large distances. How would you convert the rate 65 miles per hour into kilometers per hour?

In ratios, order matters. What is the difference between 3:5 and 5:3?

Two differently priced items are reduced by the same amount. Would you predict that the percent of change for both items is the same? If not, which has the greater percentage of change? Explain.

>

MMI Workshop Notes

Nov. 2, 2006
Don’t teach cross-multiplication!!!!!!!!!!!!!! Let students discover it!
Important prior knowledge for solving proportions

Equivalent fractions

Patterns

Ratios

CMP Comparing and Scaling is a good unit on proportional reasoning. Follows Stretching and Shrinking.

Sue and Julie were running around a track equally fast. Sue started before Julie. When
Sue had run 9 laps, Julie had run 3 laps. How far had Sue run when Julie had run 15 laps? (21) This is not a proportional reasoning problem, but most people want to solve it using proportions and get 5 or 45.

ARMT: 7th grade tests proportions through converting measurements.
8th grade tests proportions through similar figures

3 US dollars equals 2 British pounds.
Y = X
How many British pounds can you get for $21?

A proportion can be viewed as a multiplicative relationship between the quantities in two measure spaces. The quantities across measure spaces are related by multiplication.

Six men can build a house in 3 days. Assuming that all of the workmen work at the same rate, how many men would it take to build the house in one day.

Eighty candies will be divided between two boys in the ratio 2:3. How many will each boy receive?

If 5 chocolates cost $.75, how much do 13 cost?

Between them, John and Mark have 32 marbles. John has 3 times as many marbles. How many marbles does each boy have?

Jane loves to read. She can read a chapter in about 30 minutes. Assuming chapters are all about the same length, how long will it rake her to read a book with 14 chapters?

Six students were given 20 minutes to clean up the classroom after an eraser fight. They were angry and named 3 other accomplices. The principal added their friends to the clean-up crew and changed the time limit. How much time did she give them to complete the job?

If 1 football player weighs 280 pounds, what is the total weight of the 11 starters?

Sandra wants to buy a boom box costing $210. Her mother agreed to pay $5 for every $2 Sandra saved. How much did each pay?

A company usually sends 9 men to install a security system in an office building, and they do it in about building, and they do it in about 96 minutes. Today, they have only three men to do the same size job. How much time should be scheduled to complete the job?

A motor bike can run for 10 minutes on $.30 worth of fuel. How long could it run on $1.05 worth of fuel?

Proportional Reasoning Strategies
Level O Nonproportional reasoning
· Guesses or uses visual clues (“It looks like…”)
· Unable to recognize multiplicative relationships
· Randomly uses numbers, operations, or strategies
· Unable to link the two measures

Level 1 Informal reasoning about proportional situations
· Uses pictures, models, or manipulatives to make sense of situation
· Makes qualitative comparisons

Level 2 Quantitative reasoning(strive to have 7th grade at this level)
· Unitizes or uses composite units
· Finds and uses unit rate
· Identifies or uses scalar factor or table
· Uses equivalent fractions
· Builds up both measures

Level 3 Formal proportional reasoning (strive to have 8th grade at this level)
· Sets up proportions using variables and solves using cross-products or equivalent fractions
· Fully understands the invariant and covariant relationships

## Table of Contents

## ACOS Objective 11

Solve problems involving ratios or rates. using proportional reasoning.ARMT Possible Points = 6 (MC, OE)

Sample problems from Item Specs

## Curriculum Guide

Objective 7.11.1:Convert units of length, weight, or capacity within the same system (customary or metric)Objective 7.11.2:Solve problems involving decimals, percents, fractions, and proportions.## International Baccalaureate

Activity and assessment created at 1-3-06 Vertical Meeting (CR, BH, AHS)

## Cool Problems

Counting with the String ClockYou have two identical pieces of string. You know that if you light one of these pieces of string at one end, it will take exactly an hour for the string to burn completely. However, neither piece of string is evenly made, so each might burn faster at one point and slower at another.

With just these two pieces of string and a lighter, how can you measure exactly 45 minutes?

Answer:

The Daily Spark Critical Thinking, p. 31.What's onYourRadar Screen?Distance-Rate-Time Problems from NASA

Mathematics Teaching in the Middle School,Aug. 2006, pp. 6-12.CMP unit "Comparing and Scaling" has some interesting real-life problems.

## Collaborative Project

Chris Harbeck and I are planning to have our students work together online to learn about ratios and proportions. This page has my ideas for the project.

## Hands-on

Cool Kool-Aid Experiment

Seventh grade students have had wide-ranging experiences with fractions and operations on fractions, but ratios and rates may be new concepts for them. This activity introduces students to the idea of rates via a familiar and tangible product: Kool-Aid. Students will add varying amounts of Kool-Aid powder to a set amount of water, thereby producing a sensory experience of rates. The Kool-Aid example also provides a base experience upon which other examples can build.

## Video

Designing Toy Cars New!!

As a professional toy car designer for Mattel's Hot Wheels, Larry Wood uses basic math concepts such as fractions, measurement and scale to create accurate replicas of the coolest cars on the road.

The Futures Channel, running time 2:42 minutes.## Technology

What's Your Benchmark?

In this TI-73 activity from Texas Instruments, students explore the importance of benchmarks when estimating measurements. They will also use the

Convertmenu to change from one unit to another.Smart Shoppers

In this TI-73 activity from Texas Instruments, students will compare ratios with different denominators, and the with common denominators. Students will also find unit rates and see how they can be used as a way to compare numbers. They will also discuss when each way is the best to compare prices of food.

Using Brain Pop to Address Writing Proportions

Math Homework Hotline

Video and auditory tutorial from the Tampa Educational Channel. One of over 30 topics specifically covering problem areas of middle school math. (You may need to set up a free user account to access this site and see all topics). Designed as a homework help site, this makes an excellent link for teachers to include on their own web pages. Students can pause and replay the videos as needed. Requires Flash 6.

Exploring Rate/Ratio/Proportion

this multimedia resource from Junior High Interactives (learnalberta.ca) examines rate, ratio and proporition and shows how they play a role in photography. Using the interactive components, students explore ratio equivalencies by enlargening and reducing images to compare an original ratio and a target ratio. A print activity is included.

PBS Mathline: The Math of Bicycles

Lesson plan/offline activity. Students explore ratios by comparing bicycles to tricycles. They describe the ratio of the relationship between one turn of the pedal and one turn of the wheel, and define the ratio between number of teeth to sizes of the gears.

SpyGuys: Percents

Select Lesson 4. These video and auditory tutorials from LearnAlberta.CA allow teachers and students to explore math concepts, pause and replay. Includes interactive exercises that provide immediate feedback. Printable worksheets, parent notes, and a glossary are included for each lesson. In addition to the lessons, there are sections on problem-solving strategies.SpyGuys: Ratios

Select Lesson 3. These video and auditory tutorials from LearnAlberta.CA allow teachers and students to explore math concepts, pause and replay. Includes interactive exercises that provide immediate feedback. Printable worksheets, parent notes, and a glossary are included for each lesson. In addition to the lessons, there are sections on problem-solving strategies.Body Ratios

Modeling Middle School Mathematics project (MMSM) is a professional development program using video lessons and Web-based Internet materials to examine mathematical concepts. Video clips and full transcripts are provided of actual classroom situations in which the lesson is taught. In this lesson on ratios and scaled drawings, educators use the story of Gulliver's Travels to pose the problem: how to make a shirt for someone who is much bigger.

Ratios and Equivalent Ratios

From Math.com. In-depth tutorial with java applets, visuals, examples and workouts. Could be used at home or in class for review or reteaching.

Proportions

From Math.com. In-depth tutorial with java applets, visuals, examples and workouts. Could be used at home or in class for review or reteaching

Using Scale Drawings - Floorplan your Classroom

The activities presented here allow students to study mathematics by doing real-life architecture to walk through the floor plan design process. Look for measurement and drawing tips as you create a scale drawing on graph paper or using a computer drawing program.

Scale Factors

Figure This! Math Challenge #61 : Statue of Liberty. One of a series of NCTM standards-aligned math challenges designed for students to work out with their families. Available in English and Spanish. Learn about similarity and scaling by measuring the length of your nose and your arm in the same units. Then apply this knowledge to find out if the Statue of Liberty's nose is too long.Ratio and Proportions

Figure This! Math Challenge #47 : VCR Recording Tape. One of a series of NCTM standards-aligned math challenges designed for students to work out with their families. Available in English and Spanish. Develop proportional reasoning by solving this problem related to VCR Tape recording settings.Great Cartoon Blow-up Activity

Students practice drawing to scale and create blow-ups of their favorite Flinestone characters while learning about simultude.

Proportional Reasoning: The Mixture Blues

Session 4, Part B. Annenberg Media Lesson for teachers to improve their own teaching practices. Includes excellent teacher training videos at the beginning of each session. In "Mixture Blues," teachers compare mixtures of blue liquid and clear water and predict which set would be bluer. The video includes reflection on what has been learned.

Proportional Reasoning: Quad Person

Session 4, Part C. Annenberg Media Lesson for teachers to improve their own teaching practices. Includes excellent teacher training videos at the beginning of each session. The lesson includes a scaling exercise with "Quadperson" that could be used with students and the video shows teachers using it in a learning situation. Participants learn the significance of relative vs. absolute change.

Proportioner

A free resource from the Concord Consortium's "Seeing Math" Project. Proportioner allows students to specify image dimensions graphically, numerically or using a scale factor. They can duplicate images and modify comparisons or use one image to "paint" another. Interactives can be downloaded for offline use. There is a user's guide, warm-up and sample activities.

The Ratio of Rest: Proportional Reasoning

In NASA CONNECT™: The Right Ratio of Rest: Proportional Reasoning, students learn about the human circadian clock and how it affects peoples’ daily lives. Students see how NASA scientists are studying the circadian timing system to improve astronaut’s physical and mental tasks while working in space. Conducting research to help astronauts sleep better in space also helps people on Earth with similar problems. The hands-on activity in this guide allows students to demonstrate how fractions, decimals, and percents are related and further develop their proportional reasoning skills. Includes a 28-minute video that can be paused at appropriate intervals.

Ratio and Proportion Game

Find out how to use ratio and proportion in everyday life. Students can view tutorials on 6 related topic areas or play a shockwave game from the BBC to practice writing ratios.

Understanding Rational Numbers and Proportions

students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving. Illuminations/NCTM classroom activities.

Population Ratios

An interdisciplinary project (mathematics, social studies) where students choose a county in the United States and using ratios convert the statistics into meaningful numbers. Illuminations/NCTM classroom activity.

Rags to Riches

"Millionaire" style game from Quia that quizzes students on proportions.

Solving Proportions

Glencoe Online Study Tool with hints for solving proportion problems.

Solve Proportions

Harcourt School Publishers tutorial on solving proportions. Includes auditory reinforcement and the ability to replay screens.

Scales, Ratios, Proportion Jeopardy Game

Use this ready-made PowerPoint Jeopardy game from Taft Middle School teachers to test student knowledge on ratios and unit rates, proportions, similar figures,scale drawings and probability.

## Questioning

From

Higher-Level Thinking Questions: Secondary Mathematics, by Robin Silbey, Kagan Publishing, 2005.

- In whats can ratios be used to analyze an athlete's performance?
- When are some times in your everyday life that you have used rate to determine the better buy? Describe the situation.
- In what ways are fractions and ratios alike? How are they different?
- The aspect ratio of a film is the relationship between the width of the film's image and the height. What issues do aspect ratios cause for TV's, movie screens, and computer monitors?
- A ratio can be expressed as 2/6, 2 to 6, or 2:6. Do you think different expressions of the same ratio are more appropriate in different contexts? Describe when to use each.
- A
- What are some ways a builder can use what he or she knows from the blueprint to determine the appropriate amounts of materials to puchase?
- A
- The rate of change shows how one quantity changes in relation to another quantity. How would you interpret a situation in which there was a decreasing rate of change? An increasing rate of change?
- Proportions are used to determine medicine dosage amounts. What are some conditions that cause the dosage amounts to differ among people?
- What are some times that proportions, ratios, or rates must be exact? What are some times that these relationships can be estimated?
- Many countries use the metric system and use kilometers instead of miles to measure large distances. How would you convert the rate 65 miles per hour into kilometers per hour?
- In ratios, order matters. What is the difference between 3:5 and 5:3?
- Two differently priced items are reduced by the same amount. Would you predict that the percent of change for both items is the same? If not, which has the greater percentage of change? Explain.

>rateis a "ratio that creates a relationship between two quantities with different kinds of units" like miles per hour (miles/hour). What are some other kinds of relationships between quantities you've seen and what characteristics do they share?Population densityis a "rate that compares the population per square mile." Would you rather live in an area with an unusually high or an unusually low population density? Explain your reasoning.proportionis "an equation stating that two ratios are equivalent." Other than using cross products, what are some strategies for determining whether or not two ratios form a proportion? Use examples to explain.Mapsare "scale drawings that show distances between two locations." What might help you decide whether to purchase a map with a relatively small scale or a relatively large scale? Explain your reasoning.## MMI Workshop Notes

Nov. 2, 2006

Don’t teach cross-multiplication!!!!!!!!!!!!!! Let students discover it!

Important prior knowledge for solving proportions

CMP

Comparing and Scalingis a good unit on proportional reasoning. FollowsStretching and Shrinking.Sue and Julie were running around a track equally fast. Sue started before Julie. When

Sue had run 9 laps, Julie had run 3 laps. How far had Sue run when Julie had run 15 laps? (21)

This is not a proportional reasoning problem, but most people want to solve it using proportions and get 5 or 45.ARMT: 7th grade tests proportions through converting measurements.

8th grade tests proportions through similar figures

3 US dollars equals 2 British pounds.

Y = X

How many British pounds can you get for $21?

A proportion can be viewed as a multiplicative relationship between the quantities in two measure spaces. The quantities across measure spaces are related by multiplication.

M1 M2Dollars pounds

3 2

6 4

9 6 These “covary”

12 8

18 10

21 12

Six men can build a house in 3 days. Assuming that all of the workmen work at the same rate, how many men would it take to build the house in one day.

Eighty candies will be divided between two boys in the ratio 2:3. How many will each boy receive?

If 5 chocolates cost $.75, how much do 13 cost?

Between them, John and Mark have 32 marbles. John has 3 times as many marbles. How many marbles does each boy have?

Jane loves to read. She can read a chapter in about 30 minutes. Assuming chapters are all about the same length, how long will it rake her to read a book with 14 chapters?

Six students were given 20 minutes to clean up the classroom after an eraser fight. They were angry and named 3 other accomplices. The principal added their friends to the clean-up crew and changed the time limit. How much time did she give them to complete the job?

If 1 football player weighs 280 pounds, what is the total weight of the 11 starters?

Sandra wants to buy a boom box costing $210. Her mother agreed to pay $5 for every $2 Sandra saved. How much did each pay?

A company usually sends 9 men to install a security system in an office building, and they do it in about building, and they do it in about 96 minutes. Today, they have only three men to do the same size job. How much time should be scheduled to complete the job?

A motor bike can run for 10 minutes on $.30 worth of fuel. How long could it run on $1.05 worth of fuel?

Proportional Reasoning StrategiesLevel O

Nonproportional reasoning· Guesses or uses visual clues (“It looks like…”)

· Unable to recognize multiplicative relationships

· Randomly uses numbers, operations, or strategies

· Unable to link the two measures

Level 1

Informal reasoning about proportional situations· Uses pictures, models, or manipulatives to make sense of situation

· Makes qualitative comparisons

Level 2

Quantitative reasoning(strive to have 7th grade at this level)· Unitizes or uses composite units

· Finds and uses unit rate

· Identifies or uses scalar factor or table

· Uses equivalent fractions

· Builds up both measures

Level 3

(Formal proportional reasoningstrive to have 8th grade at this level)· Sets up proportions using variables and solves using cross-products or equivalent fractions

· Fully understands the invariant and covariant relationships

Covariant and invariant