# 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.

# Linear Measurement

Sure middle schoolers know how to measure length! But if students need a little more experience with linear measurement, here are challenging resources that put linear measurement into a practical context, or can be used for review before introducing a new measurement topic.

Big Tree: Have You Ever Seen a Tree Big Enough to Drive a Car Through?
In this thought-provoking, discussion-generating challenge, students use common sense, given information, and their calculated tree diameters to answer the question. Teachers should investigate the links at the bottom of the web page. All activities on this site are printable.

Constant Dimensions
Students measure length and width of a rectangle using both standard and nonstandard units of measure. After creating a length vs. width graph, students observe an interesting and important fact — the ratio of length to width in a rectangle is constant.

Inclined Plane
Here is a terrific multiday lesson for teachers interested in having students apply their linear measurement skills and engage in mathematics discussions. Students use length and height data from hands-on experiments to draw conclusions. Useful teacher information is included.

Perimeter Explorer
Using this applet, students determine perimeters for irregular shapes on a grid. The applet is part of a complete lesson reinforcing students’ concept of perimeter and skills for finding perimeters. Shape area can be varied and a table comparing perimeter and area can be generated.

Reaching New Heights
Measuring, collecting and interpreting data, using variables—this complete lesson has it all! Students measure height and arm span, create a scatterplot, and draw conclusions about the correlation. This lesson is an excellent way to build the foundation for the study of functions. Teacher support and information about supporting research is included.

Rectangle: Area, Perimeter, Length, and Width
Using this simulation, students can instantaneously see the interplay among perimeter, area, and the rectangle dimensions. The rectangle size or shape is changed by dragging a point.