# 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.

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.

# 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.

# Hands-On Measuring

Students need active learning experiences to understand measurement concepts and develop important skills. These resources provide opportunities for students to problem solve with hands-on and virtual measurements in real-world and online environments.

The Global Sun Temperature Project
Bigger than hands-on, this is an annual real-world, international and interdisciplinary research project for students. Classes gather local data, post data online, and use the aggregated data to see how average daily temperatures and hours of sunlight relate to distance from the equator.

It Takes Ten
Students use metric units to estimate and measure weight, length, and volume, and to determine area.

Open-Ended Math Problems: Get Ready, Get Set
Select a month and scroll down to find open-ended measurement problems at three levels of difficulty. Students build mathematics understanding and see how mathematics is used in everyday life.

Pentagon Puzzles
This measurement lesson is one of 37 hands-on projects focused on mathematics. See http://www.math.nmsu.edu/~breakingaway/lessons.html for more lessons.

Popcorn Math
Here is a volume estimating activity for students to do on their own or with others.

Surface Area and Volume
Examine prisms from multiple views, adjust dimensions, rotate prisms, and see how dimension changes impact volume and surface area. Students can also calculate volume and surface area.

# Scale and Powers of 10

Scale is a unifying concept in science. Whether one is considering the size and scope of the universe, an atom, or anything in-between, conceptual understanding of scale is a prerequisite to understanding. Other phenomena requiring knowledge of scale include geologic time, pH, and maps. Cognitively, most middle school students hold a concrete, incomplete conception of scale. Teachers can facilitate conceptual change to a more abstract conception with help from these resources.

Secret Worlds: The Universe Within
View the Milky Way at 10 million light years from the Earth. Then move through space toward the Earth in successive orders of magnitude until you reach leaf cells, the cell nucleus, chromatin, DNA, and finally the subatomic universe of electrons and protons. You can control the speed at which the images are flashed automatically or manually.

Powers of 10
This brief webcast approaches scale from the opposite direction of the tutorial above — moving from a picnic in a park to outer space and back to human cells — and indicates when the scale has changed by a magnitude of 10X.

Table of Images from Wordwizz.com
Bruce Bryson is the author of this award-winning site. It provides a table of contents in which teachers will find images of interest, such as the solar system, a bee’s eye, and the atomic nucleus. You may decide to have pairs of students investigate one item from the list and then share their findings in order of magnitude with the rest of the class.

Earthquake Scale and Magnitude
Not all scales are in powers of 10, as can be seen at this site.

Weather – Wind Chill
This interactive site explains how wind chill is estimated and allows students to calculate it. In doing so, students see a different manifestation of scale and magnitudes involving the interaction of variables, rather than a one-to-one corresponding change as in powers of 10.