Math Focal Points
Table Of Contents
Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity.
Students extend their work with ratios to develop an understanding of proportionality that they apply to solve single and multistep problems in numerous contexts. They use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. They also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and identify the unit rate as the slope of the related line. They distinguish proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x).
The problems here deal with ratio, in the concrete as well as the abstract. Middle school learners will make actual scale models with paper or clay and find percentages in real-world situations. But they will also work hands-on with online images that make visual the abstractions of ratio and percentage.
Questions and hands-on work guide your class in these excellent investigative activities. In Math at the Mall, students calculate the ratio used in making a scale map of the mall, then figure areas and percentage of mall space for each category of shop. Math in the Park or City involves setting up proportions to find heights of buildings. In Gearing Up, students compare the ratio of the turn of the wheel to the turn of the pedal in various bikes.
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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.
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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!
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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.
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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.
(From Activities Exchange — MSP full record)
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.
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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.
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A problem straight from the mall! Here is a rack of clothing, originally on sale for 30 percent off the original price, but now discounted by an additional 50 percent. Is the new price actually 80 percent of the original price? Two complete solutions are set out, and several more problems in the shopping scenario are offered under "Try these."
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Math-Kitecture is about using architecture to do math (and vice versa). The author provides activities that engage students in doing real-life architecture while learning estimation, measuring skills, proportion, and ratios. In Floor Plan Your Classroom, for example, students make a rough sketch of the classroom, followed by a more exact scale drawing.
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Students 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. In other activities, they must find the scale heights for the tallest building in their neighborhood and create models for two other pyramids, given only their dimensions. A beautifully illustrated site!
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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!
(From Manipula math with Java — MSP full record)
This activity challenges students to determine which is worth more today: Babe Ruth's 1927 home-run record-breaking ball or Mark McGwire's 70th home-run record-breaking ball that sold in 1999 for three million dollars. The activity involves compound interest and rate of change.
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Using a 10 x 10 grid, students first represent simple percents, then move to percents less than 1 and greater than 100. Problems involving percent increase and decrease are illustrated on grids, offering visuals that reinforce the instruction in this one-period lesson.
(From Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics — MSP full record)
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.
(From Illuminations, National Council of Teachers of Mathematics Vision for School Mathematics — MSP full record)
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 surface area, and finally on the amount of sand that fits into each module (volume). The lesson is carefully described and includes handouts.
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Reference
NCTM. (2006).
Curriculum Focal Points for Kindergarten Through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: Author
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Copyright
May 2008 — The Ohio State University. This material is based upon work
supported by the National Science Foundation under Grant No. 0424671. 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.
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