# Learning Modules for Ratios and Proportions

Our friends at the Ohio Resource Center have developed two new learning modules on the topics of ratio and proportion. These two stand-alone learning modules for ratios and proportions are online tutorials that provide engaging interactive problems, immediate feedback to answers, and real-world applications. Mathematics educators who are teaching or reviewing the often hard-to-learn topics can encourage their students to use the modules to support their learning. Each module comes in two formats. In the Student version, the user must correctly answer check-for-understanding questions before proceeding. In the Teacher version, the user can proceed without answering questions. Educators can direct students to a particular chapter within a module, or they can incorporate an entire module into their lesson plans.

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# Ratios as Seen in Scale Factors

Ratio underpins so much mathematics in our real world that it deserves occasional return visits. These sites deal mainly with making and building and constructing; mathematically, they concentrate on scale factor, a topic chosen by NCTM as a Focal Point for Grade 7. The very last site is just for teachers who may want a refresher at the professional level on basic but essential concepts. Please let us know any of your favorite sites for exploration!

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!

Statue of Liberty
This activity asks students to determine if the statue’s nose is out of proportion to her body size. It carefully describes the mathematics involved in determining proportion, then goes on to pose problems on  enlarging a picture, designing HO gauge model train layouts, and analyzing the size of characters in Gulliver’s Travels. The page features links to a solution hint, the solution, related math questions, and model building resources. Other ratio problems in the Figure This! Series include Tern Turn, Capture Re-Capture, Drip Drops, and Which Tastes Juicier?

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.

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.

Size and Scale
A more challenging and thorough activity on the physics of size and scale! The final 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.

Golden Rectangle (grades 6-8)
This virtual manipulative can help students visualize the golden rectangle. It shows how a golden rectangle is generated by using the golden ratio (the ratio of the longer side to the shorter side of a golden rectangle) to create smaller and smaller golden rectangles within an initial rectangle. Instructions for using this online manipulative are included on the site.

Similarity
In this workshop session, elementary and middle school teachers explore scale drawing, similar triangles, and trigonometry in terms of ratios and proportion. 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, participants use an interactive activity provided online. This is session 8 of Learning Math: Geometry, a free online course.

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# Investigating Cell Size

How big are cells? Ask most middle school students that question, and they’ll tell you that cells are very small – even microscopic! But what if you asked them to compare the size of cells in a whale and a shrew?

That’s the engaging scenario laid out in “Whale and Shrew,” a formative assessment probe from Uncovering Student Ideas in Science, Volume 2: 25 More Formative Assessment Probes (NSTA Press, 2007). The probe unearths a subtle, yet fundamental concept of cell theory – that there are natural constraints that necessarily limit cell size. How can you help your students reach this conclusion?

An inquiry-based activity provides an opportunity for students to answer the question, “How does the size of a cell affect its ability to exchange materials with its environment?” Or, more simply stated, “Why are cells small?”

These two activities could be used as is, or they could be adapted for a guided inquiry experience. Both involve analyzing how the surface area to volume ratio affects the rate of diffusion in cubes of various sizes. Each uses different materials and a slightly different procedure.

Cell Size and Division
Students test”cells” made of agar and phenolphtalein with sodium hydroxide. They observe the rate of diffusion (evident by color change) in 1x1x1, 2x2x2, and 3x3x3 cubes.

Experiment on Cell Surface Area and Volume
In this lesson, students test “cells” made of potatoes with Lugol’s solution. They observe the rate of diffusion (evidenced by color change) in cubes with a length of 0.5 cm, 1 cm, 1.5 cm, 2 cm, 2.5 cm, and 3 cm. They also calculate the surface area, volume, and ratio of surface area to volume of all cubes.

How can you turn these activities into an inquiry-based lesson? We recommend starting with the probe to assess student thinking and spark interest. Next, you may ask students to generate a testable question based on the probe, or you may choose to provide the question for them. Students can then plan and conduct an investigation using the materials specified in either one of the lessons. Prompt students to generate claims and draw conclusions based on evidence from the investigation.

Teachers of younger students (grades 5 and 6) may wish to use the potatoes and Lugol’s solution, while an 8th grade teacher may have access to agar, phenolpthalein, and sodium hydroxide. Teachers may also modify this activity by removing the surface area and volume calcuations and instead relying on the qualitative observations made during the activity. Either way, students will gain new insight into why cells in any organism are small.

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# Close Encounters with Ratios

Understanding ratio and proportion, one of NCTM’s Focal Points for grade 7, presents a real challenge for all levels of middle school. Here are classroom-friendly ways to explore the topic from several angles. Each involves visuals or hands-on activities that bring students into close contact with the abstract concept of ratio. Let other teachers hear your ideas on teaching this topic! Post a comment below.

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.

Discovering the Value of Pi
Students measure the diameter and circumference of several circles, using a handy applet, record their data, and reach conclusions about the ratio of circumference to diameter. A genuine guided exploration!

Math-Kitecture
Math-Kitecture is about using architecture to do math (and vice versa). Activities engage students in doing real-life architecture while learning estimation, measuring skills, proportion, and ratios. In Floor Plan Your Classroom, for example, exact directions are set out and illustrated on how to make a copy to scale of your classroom.

What’s My Ratio?
What would happen to a picture in the pocket of someone who is shrunk or enlarged? This question hooks students into a study of similar figures. As they compare the measurements of corresponding parts of pictures that have been either decreased or increased in size, they can investigate concepts of similarity, constant ratio, and proportionality.

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 importantly, 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 experience!

Capture-Recapture: How Many Fish in the Pond?
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.

Size and Scale
This is a challenging and thorough activity on the physics of size and scale. The final 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 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.

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# History to Enrich Mathematics Learning!

Although the NCTM Standards do not have student expectations for learning mathematics history, exposure to this history can help students see real-world connections in mathematics.

Approximating Pi
Mathematics and science owe a great deal to Archimedes, including a way to approximate pi. Here is historical information along with an applet to approximate pi using the perimeter lengths of polygons inscribed within and circumscribed around a circle.

The Beginnings of Probability
Teachers can share some of this mathematics history as they work with students to compute probabilities for simple compound events, an NCTM expectation for students in grades 6-8.

The Golden Ratio
This rich site connects linear measurement, ratio and proportion, art, and mathematics history.

Measuring the Circumference of the Earth
This Internet project is hands-on, real-world, and historical. Students join with classes around the world to repeat the experiment of Eratosthenes — measuring the shadow of a meter stick and making calculations to approximate the circumference of Earth.

Pythagorean Puzzle
The Pythagorean theorem is at the intersection of algebra and geometry. At this site, learn about the life of Pythagoras and the development of the Pythagorean theorem. And use an applet to explore the meaning of the most famous equation in algebra.

Manipula Math with Java: Pythagorean Theorem
Here is another applet offering a more sophisticated approach to affirm in a visual way the validity of the Pythagorean theorem.

Tortoise and Hare Race
Uing an applet, students can vary parameters for the race. There is information about Zeno’s paradox along with exploration questions for students that can lead to a discussion about infinity.

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