# MiddleSchoolPortal/Ratios For All Occasions

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<keywords content="middle school mathematics mental challenge rates in real-world problems, percents

fractions scale factors building models comparisons lengths geometry games mapmaking" />

<metadescription content="This free, standards-based, online publication, developed for middle school mathematics teachers, supports teaching ratios and proportions by linking to and describing inquiry-based lessons and activities." />

### Ratios for All Occasions - Introduction

In Ratios for All Occasions, we feature resources on the concept of the ratio as encountered in middle school: as rates in real-world problems, percents in relation to fractions, scale factors in building models, and comparisons of lengths in geometry. Most of these digital resources are activities that can serve as supplementary or motivational material.

A central theme in the middle school mathematics curriculum, proportional reasoning is based on making sense of ratios in a variety of contexts. The resources chosen for this unit provide practice in solving problems, often informally, in the format of games, hands-on modeling, mapmaking, and questions selected for their interest for students. As students work through the activities, they will exercise reasoning about basic proportions as well as further develop their knowledge of the relationship between fractions and percents.

The section titled Background Resources for Teachers contains links to workshop sessions, developed for teachers, on the mathematical content of the unit. Ratios in Children's Books identifies three picture books that entertain while they explore scale and proportion. In the final section, we look at the coverage of proportionality at the middle level in the NCTM Principles and Standards for School Mathematics.

### Background Resources for Teachers

Ratios, whether simple comparisons or rates or percents or scale factors, are old friends of the middle school teacher. Every year you deal with them in your classroom. However, you may like to explore a particular topic, such as the golden mean or indirect measurement. These online workshop sessions, created for teachers, make use of applets and video to enable deeper investigation of a topic. You may find yourself fascinated enough with a topic to import the workshop idea directly into your classroom.

**Rational Numbers and Proportional Reasoning**
How do ratios relate to our usual idea of fractions? In this session, part of a free online course for K-8 teachers, you can look at ways to interpret, model and work with rational numbers and to explore the basics of proportional reasoning. You can investigate these ideas through interactive applets, problem sets, and a video of teachers solving one of the problems. This session is part of the online course Learning Math: Number and Operations.

**Fractions, Percents, and Ratios**
In this set of lessons created for K-8 teachers, you can examine graphical and geometric representations of these topics, as well as some of their applications in the physical world. A review of percents in terms of ratio and proportion is followed by an investigation of Fibonacci numbers and the golden mean. Why do we study the golden rectangle? In a video segment, an architect explains the place of the golden rectangle as an architectural element throughout history. This set of lessons is from Learning Math: Number and Operations.

**Similarity**
Explore scale drawing, similar triangles, and trigonometry in terms of ratios and proportion in this series of lessons developed for teachers. Besides explanations and real-world problems, the unit includes video segments that show teachers investigating problems of similarity. To understand the ratios that underlie trigonometry, you can use an interactive activity provided online. This session is part of the course Learning Math: Geometry.

**Indirect Measurement and Trigonometry**
For practical experience in the use of trigonometry, look at these examples of measuring impossible distances and inaccessible heights. These lessons show proportional reasoning in action! This unit is part of the online course Learning Math: Measurement.

### Ratios as Fractions and Rates

It is at the middle school level that students move from understanding fractions to working with ratios and setting up proportions. You will find here problem-solving activities that you can use to introduce the concept of ratio as a rate that can be expressed as a fraction—miles per hour, drops per minute, for example. And you will find real-world problems that can be set up as proportions. Each activity was selected with student appeal in mind.

**All About Ratios**
Designed to introduce the concept of ratio at the most basic level, this activity could open the idea to younger middle school students. Each multiple-choice problem shows sets of colorful elements and asks students to choose the one that matches the given ratio. The activity is from the collection titled Mathematics Lessons that are Fun! Fun! Fun!

**Which Tastes Juicier?**
Students are challenged to decide which of four cans of grape juice concentrate requiring different amounts of water would have the strongest grape juice taste. A hint suggests forming ratios that are fractions to compare quantities. Two solutions are given, each fully illustrated with tables. Students are then offered further mixture-related questions.

**Tern Turn: Are We There Yet?**
If you know an arctic tern's rate of flight and hours per day in flight, can you calculate how many days would be required to fly the 18,000-mile roundtrip from the Arctic Circle to Antarctica? A hint suggests that students first calculate how many miles the tern flies in one day. Similar questions follow, offering more opportunities to practice distance-rate-time problems.

**Drip Drops: How Much Water Do You Waste?**
A leaky faucet is dripping at the rate of one drop every two seconds. Students are asked to decide if the water lost in one week would fill a drinking glass, a sink, or a bathtub. The only hint is that a teaspoon holds about 20 drops. The full solution demonstrates how to convert the drops to gallons using an equation or a table. Students then consider, "How much water is lost in one year by a single leaky faucet? By two million leaky faucets?"

**How Far Can You Go on a Tank of Gas?**
Which car will go the farthest on a single tank of gas? Students are given the mileage and gasoline tank capacity of three models of automobiles and are encouraged to begin the problem by calculating how far each car could go in the city and on the highway. In follow-up problems, students compare the fuel efficiency of different sports cars and calculate how often a commuter would need to refuel.

**Capture Recapture: How Many Fish in a Pond?**
A real application of the ideas of proportion! To estimate the number of fish in a pond, scientists tag a number of them and return them to the pond. The next day, they catch fish from the pond and count the number of tagged fish recaptured. From this, they can set up a proportion to make their estimation. Hints on getting started are given, if needed, and the solution explains the setup of the proportion.

**Neighborhood Math**
This site contains four activities in a neighborhood setting: Math at the Mall, Math in the Park or City, Wheel Figure This Out, and Gearing Up. Students calculate the amount of floor space occupied by various stores, find the height of objects, and take a mathematical look at bicycles. The third and fourth activities involve both geometry and ratios. Answers and explanations of the four activities are included.

**Understanding Rational Numbers and Proportions**
To work well with ratios, learners need a solid basis in the idea of rational number. This complete lesson includes three well-developed activities that investigate fractions, proportion, and unit rates—all through real-world problems students encounter at a bakery.

### Ratios as Percentages

In teaching ratio, percentage is where the rubber meets the road! Students need to understand the concept of percent thoroughly, which is the objective of the first five resources here. Students also need practice in converting from fractions to decimals to percents, and in finding percentages. The last four resources offer practice in various scenarios, generally through a game format.

**Grid and Percent It**
This lesson begins with a basic visual used in many textbooks: a 10 × 10 grid as a model for demonstrating percent as "parts per hundred." It goes on to extend the model to solve various percentage problems. Especially valuable are the illustration of each problem and the thorough explanation that accompanies it. This is an exceptional lesson plan!

**Percentages**
In this interactive activity, students can enter any two of these three numbers: the whole, the part, and the percentage. The missing number is not only calculated but the relationship among the three is illustrated as a colored section of both a circle and a rectangle. The exercise is an excellent help to understanding the meaning of percentage.

**Majority Vote: What Percentage Does It Take to Win a Vote?**
This problem challenges students' understanding of percentage. Two solutions are available, plus hints for getting started. Clicking on "Try these" leads to different but similar problems on percentage. Questions under "Did you know?" include "Can you have a percentage over 100?" and "When can you add, subtract, multiply, or divide percentages?" These questions can lead to interesting math conversations.

**Fraction Model III**
Using this applet, students create a fraction for which the denominator is 100 and then make the numerator any value they choose. A visual of the fraction is shown—either as a circle, a rectangle, or a model with the decimal and percent equivalents of the fraction. An excellent aid in understanding the basics of percentages!

**Tight Weave: Geometry**
This is a fractal that can be used to give a visual of percentages. At each stage in the creation of the fractal, the middle one-ninth of each purple square area is transformed to gold. This gives progressively smaller similar patterns of gold and purple. At any stage of iteration, the percentage of gold is given. Interesting questions that your class might consider: At what stage will more than 50% of the area be gold? Or you could pick a stage, show it visually, and ask the students to estimate the percentage of the original purple square that has turned to gold.

**Dice Table**
This activity shows the student the possible results of rolling two dice. It can become a game between several students who select various combinations of results, which appear on an interactive table. The players then figure the probability of winning the roll, giving the probabilities as fractions, decimals, and percentages. Good practice in converting from fractions to percents.

**Fraction Four**
A game for two players, this activity requires students to convert from fractions to percents, find percentages of a number, and more. Links go to game ideas and a brief discussion of the connection between fractions and percentages, presented as a talk between a student and a mentor.

**Snap Saloon**
In this interactive online game, students practice matching fractions with decimals and percentages. Three levels of difficulty are available. This is one of 12 games from The Maths File Game Show.

### Ratios in Building Scale Models

This is the hands-on area of ratios! These activities are for students who like to get in there and get dirty—in other words, all middle schoolers. Here they can make models, maps, floor plans, and pyramids, or consider the length of the Statue of Liberty's nose. All the problems deal with the idea of scale, the application of a scale factor, and the central question: What changes when an object is enlarged or shrunk to scale?

**Floor Plan Your Classroom: Make an Architectural Plan in 3 Steps**
This resource guides the learner step-by-step in creating a floor plan of a classroom. The directions include drawings of student work. The three parts of the activity are: sketching a map of the classroom, making a scale drawing from the sketch, and drafting a CAD (computer-aided design) floor plan from the drawing.

**Statue of Liberty: Is the Statue of Liberty's Nose Too Long?**
The full question is: "The arm of the Statue of Liberty is 42 feet. How long is her nose?" To answer the question, students first find the ratio of their own arm length to nose length and then apply their findings to the statue's proportions. The solution sets out different approaches to the problem, including the mathematics involved in determining proportion. Extension problems deal with shrinking a T-shirt and the length-to-width ratios of cereal boxes.

**Scaling Away**
For this one-period lesson, students bring to class either a cylinder or a rectangular prism, and their knowledge of how to find surface area and volume. They apply a scale factor to these dimensions and investigate how the scaled-up model has changed from the original. Activity sheets and overheads are included, as well as a complete step-by-step procedure and questions for class discussion.

This activity provides instructions for making a scale model of the solar system, including an interactive tool to calculate the distances between the planets. The student selects a measurement to represent the diameter of the Sun, and the other scaled measurements are automatically calculated. Students can experiment with various numbers for the Sun's diameter and see how the interplanetary distances adjust to the scale size.

**Mathematics of Cartography: Mathematics Topics**
This web page looks at scale in relation to making maps. It discusses coordinate systems as well as the distortions created when projecting three -- dimensional space onto a two-dimensional paper. One activity here has students use an online site to create a map of their neighborhoods-to scale, of course!

**Size and Scale**
This is a challenging and thorough activity on the physics of size and scale. Again, the product is a scale model of the Earth-moon system, but the main objective is understanding the relative sizes of bodies in our solar system and the problem of making a scale model of the entire solar system. The site contains a complete lesson plan, including motivating questions for discussion and extension problems.

**Scaling the Pyramids**
Students working on this activity will compare the Great Pyramid to such modern structures as the Statue of Liberty and the Eiffel Tower. The site contains all the information needed, including a template, to construct a scale model of the Great Pyramid. Heights of other tall structures are given. A beautifully illustrated site!

### Ratios in Geometry

Geometry offers a challenging arena in which to wrestle with ideas of ratio. Except for the first resource, the work below is more appropriate for the upper end of middle school than for the younger students. All of the resources include activities that will involve your students in working with visual, geometric figures that they can draw or manipulate online. You will notice the absence of a favorite and most significant ratio: p. You will find several interesting resources on the circumference to diameter ratio in Going in Circles!

**Constant Dimensions**
In this carefully developed lesson, students measure the length and width of a rectangle using standard units of measure as well as nonstandard units such as pennies, beads, and paper clips. When students mark their results on a length-versus-width graph, they find that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, not only is the discovery surprising but also opens up the whole meaning of ratio.

**Parallel Lines and Ratio**
Three parallel lines are intersected by two straight lines. The classic problem is: If we know the ratio of the segments created by one of the straight lines, what can we know about the ratio of the segments along the other line? An applet allows students to clearly see the geometric reasoning involved. The activity is part of the Manipula Math site.

**Figure and Ratio of Area**
A page shows two side-by-side grids, each with a blue rectangle inside. Students can change the height and width of these blue rectangles and then see how their ratios compare--not only of height and width but also, most important, of area. The exercise becomes most impressive visually when a tulip is placed inside the rectangles. As the rectangles' dimensions are changed, the tulips grow tall and widen or shrink and flatten. An excellent visual! The activity is part of the Manipula Math site.

**Cylinders and Scale Activity**
Using a film canister as a pattern, students create a paper cylinder. They measure its height, circumference, and surface area, then scale up by doubling and even tripling the linear dimensions. They can track the effect on these measurements, on the area, and finally on the amount of sand that fits into each module (volume). The lesson is carefully described and includes handouts.

**The Fibonacci Numbers and the Golden Section**
Here students can explore the properties of the Fibonacci numbers, find out where they occur in nature, and learn about the golden ratio. Illustrations, diagrams, and graphs are included.

**The Golden Ratio**
Another site that introduces the golden ratio, this resource offers seven activities that guide students in constructing a golden rectangle and spiral. Although designed for ninth and tenth graders, the explorations are appropriate for middle school students as well.

### Ratios in Children's Books

Middle schoolers may be surprised and pleased to find ratios treated as the subject of these three picture books. You can find the books in school or public libraries. They are also available from online booksellers.

**Cut Down to Size at High Noon**
by Scott Sundby and illustrated by Wayne Geehan

This parody of classic western movies teaches scale and proportion. The story takes place in Cowlick, a town filled with people with intricate western-themed hairstyles that the town's one and only barber creates with the help of scale drawings. Enter a second barber, and the town does not seem big enough for both of them! The story reaches its high point of suspense when the two barbers face off with scissors at high noon. The duel ends in a draw of equally magnificent haircuts, one in the shape of a grasshopper and the other in the shape of a train engine, and the reader learns that scale drawings can be used to scale up as well as down.

**If You Hopped Like a Frog**
by David M. Schwartz and illustrated by James Warhola

Imagine, with the help of ratio and proportion, what you could accomplish if you could hop like a frog or eat like a shrew. You would certainly be a shoo-in for the Guinness World Records. The book first shows what a person could do if he or she could hop proportionately as far as a frog or were proportionately as powerful as an ant. At the back of the book, the author explains each example and poses questions at the end of the explanations.

**If the World Were a Village: A Book about the World's People**
by David J. Smith and illustrated by Shelagh Armstrong

How can you comprehend statistics about a world brimming with more than 6.2 billion people (the population in January 2002)? One answer to understanding large numbers is to create a scale where 100 people represent the total world population and change the other numbers proportionally. In a world of 100 people, how many people (approximately) would come from China? (21) From India? (17) From the United States? (5) In the same way, the book presents statistics about the different languages spoken in the world, age distributions, religions, air and water quality, and much more.

### SMARTR: Virtual Learning Experiences for Students

Visit our student site **SMARTR** to find related virtual learning experiences for your students! The **SMARTR** learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities. Visit the virtual learning experience on **Ratios.**

### Careers

**The FunWorks**
Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

### NCTM Standards

Within the NCTM Principles and Standards, the concept of ratio falls under the Number and Operations Standard. The document states that one curricular focus at this level is "the proposed emphasis on proportionality as an integrative theme in the middle-grades mathematics program. Facility with proportionality develops through work in many areas of the curriculum, including ratio and proportion, percent, similarity, scaling," and more. Another focus identified for middle school is rational numbers, including conceptual understanding, computation, and learning to "think flexibly about relationships among fractions, decimals, and percents" (NCTM, 2000, p. 212).

Characteristically, the document emphasizes the deep understandings that underlie the coursework. For example, to work proficiently with fractions, decimals, and percents, a solid concept of rational number is needed. Many students hold serious misconceptions about what a fraction is and how it relates to a decimal or a percent. They can develop a clearer, more intuitive understanding through "experiences with a variety of models" that "offer students concrete representations of abstract ideas" (pp. 215-216).

The online resources in this unit offer several models for hands-on encounters with ratio under some of its many guises: a rate, a scale factor, a percent, a comparison of geometric dimensions. We hope that your students will enjoy their encounters with ratios and deepen their understanding of this useful concept.

## Author and Copyright

Terese Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

Please email any comments to msp@msteacher.org.

Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org.

Copyright June 2006 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.