A gear is a simple machine that is very useful to increase the speed or torque of a wheel. In this activity, students learn about the trade-off between speed and torque when designing gear ratios. The activity setup includes a LEGO(TM) MINDSTORMS(TM) NXT pulley system with two independent gear sets and motors that spin two pulleys. Each pulley has weights attached by string. In a teacher demonstration, the effect of adding increasing amounts of weight to the pulley systems with different gear ratios is observed as the system's ability to lift the weights is tested. Then student teams are challenged to design a gear set that will lift a given load as quickly as possible. They test and refine their designs to find the ideal gear ratio, one that provides enough torque to lift the weight while still achieving the fastest speed possible.

Students explore the concept of similar right triangles and how they apply to trigonometric ratios. Use this lesson as a refresher of what trig ratios are and how they work. In addition to trigonometry, students explore a clinometer app on an Android® or iOS® device and how it can be used to test the mathematics underpinning trigonometry. This prepares student for the associated activity, during which groups each put a clinometer through its paces to better understand trigonometry.

The goal of this task is to use geometry study the structure of beehives. Beehives have a tremendous simplicity as they are constructed entirely of small, equally sized walls. In order to as useful as possible for the hive, the goal should be to create the largest possible volume using the least amount of materials. In other words, the ratio of the volume of each cell to its surface area needs to be maximized. This then reduces to maximizing the ratio of the surface area of the cell shape to its perimeter.

This lab exercise exposes students to a potentially new alternative energy source hydrogen gas. Student teams are given a hydrogen generator and an oxygen generator. They balance the chemical equation for the combustion of hydrogen gas in the presence of oxygen. Then they analyze what the equation really means. Two hypotheses are given, based on what one might predict upon analyzing the chemical equation. Once students have thought about the process, they are walked through the experiment and shown how to collect the gas in different ratios. By trial and error, students determine the ideal combustion ratio. For both volume of explosion and kick generated by explosion, they qualitatively record results on a 0-4 scale. Then, students evaluate their collected results to see if the hypotheses were correct and how their results match the theoretical equation. Students learn that while hydrogen will most commonly be used for fuel cells (no combustion situation), it has been used in rocket engines (for which a tremendous combustion occurs).

Student teams design, build and test small-sized gliders to maximize flight distance and an aerodynamic ratio, applying their knowledge of fluid dynamics to its role in flight. Students experience the entire engineering design process, from brainstorming to CAD (or by hand) drafting, including researching (physics of aerodynamics and glider components that take advantage of that science), creating materials lists, constructing, testing and evaluating—all within constraints (works with a launcher, budget limitation, maximizing flight distance to mass ratio), and concluding with a summary final report. Numerous handouts and rubrics are provided.

Students learn about radar imaging and its various military and civilian applications that include recognition and detection of human-made targets, and the monitoring of space, deforestation and oil spills. They learn how the concepts of similarity and scaling are used in radar imaging to create three-dimensional models of various targets. Students apply the critical attributes of similar figures to create scale models of a radar imaging scenario using infrared range sensors (to emulate radar functions) and toy airplanes (to emulate targets). They use technology tools to measure angles and distances, and relate the concept of similar figures to real-world applications.

Working as a team, students discover that the value of pi (3.1415926...) is a constant and applies to all different sized circles. The team builds a basic robot and programs it to travel in a circular motion. A marker attached to the robot chassis draws a circle on the ground as the robot travels the programmed circular path. Students measure the circle's circumference and diameter and calculate pi by dividing the circumference by the diameter. They discover the pi and circumference relationship; the circumference of a circle divided by the diameter is the value of pi.

Rock out to a musical demonstration of math involved in constructing rhythms from groups of beats in this video from KQED. In the accompanying classroom activity, students consider how the note durations comprising a measure in 3/4 time relate to fractions that sum to 1. Then, they create and perform their own 3/4 measures comprised of quarter, eighth, and sixteenth notes. To get the most from the activity, students should have experience adding fractions with unlike denominator.

In this KQED infographic, find out how much an adult in different-sized households needs to make to pay for basic monthly living expenses. In the accompanying classroom activity, students solve real-life problems involving rate and ratio that involve calculating whether minimum wage in their state is a living wage. They learn whether minimum-wage pay can cover expenses such as food, housing, transportation, and medical care. They then compare their findings with data shown on bar graphs representing living wage and minimum wage in California. To get the most out of this activity, students should be comfortable with arithmetic with dollars and cents into the ten thousands.

As part of a design challenge, students learn how to use a rotation sensor (located inside the casing of a LEGO® MINDSTORMS ® NXT motor) to measure how far a robot moves with each rotation. Through experimentation and measurement with the sensor, student pairs determine the relationship between the number of rotations of the robot's wheels and the distance traveled by the robot. Then they use this ratio to program LEGO robots to move precise distances in a contest of accuracy. The robot that gets closest to the goal without touching the toy figures at the finish line is the winning programming design. Students learn how rotational sensors measure distance, how mathematics can be used for real-world purposes, and about potential sources of error due to gearing when using rotation sensor readings for distance calculations. They also become familiar with the engineering design process as they engage in its steps, from understanding the problem to multiple test/improve iterations to successful design.

In this lesson, students create a class mural by using scale, ratio, and proportion. Media resources and teacher materials are included.

This QuickTime movie gives examples of writing ratios. As you watch and listen to the teacher and students interact it helps clarify the thinking behind applying this concept.

This QuickTime movie explains what ratios are and models how to solve a ratio problem. As you watch and listen to the teacher and students interact it helps clarify the thinking behind applying this concept. Click the start button to access the video. [4:36]

Zooming In On Figures

Unit Overview

Type of Unit: Concept; Project

Length of Unit: 18 days and 5 days for project

Prior Knowledge

Students should be able to:

Find the area of triangles and special quadrilaterals.
Use nets composed of triangles and rectangles in order to find the surface area of solids.
Find the volume of right rectangular prisms.
Solve proportions.

Lesson Flow

After an initial exploratory lesson that gets students thinking in general about geometry and its application in real-world contexts, the unit is divided into two concept development sections: the first focuses on two-dimensional (2-D) figures and measures, and the second looks at three-dimensional (3-D) figures and measures.
The first set of conceptual lessons looks at 2-D figures and area and length calculations. Students explore finding the area of polygons by deconstructing them into known figures. This exploration will lead to looking at regular polygons and deriving a general formula. The general formula for polygons leads to the formula for the area of a circle. Students will also investigate the ratio of circumference to diameter ( pi ). All of this will be applied toward looking at scale and the way that length and area are affected. All the lessons noted above will feature examples of real-world contexts.
The second set of conceptual development lessons focuses on 3-D figures and surface area and volume calculations. Students will revisit nets to arrive at a general formula for finding the surface area of any right prism. Students will extend their knowledge of area of polygons to surface area calculations as well as a general formula for the volume of any right prism. Students will explore the 3-D surface that results from a plane slicing through a rectangular prism or pyramid. Students will also explore 3-D figures composed of cubes, finding the surface area and volume by looking at 3-D views.
The unit ends with a unit examination and project presentations.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit’s concepts or to assist students who may have fallen behind on work.Problem DescriptionsSprinklersExplore different sprinkler layouts, looking at circular areas (and partial circles) to decide which will be best to water a lawn.Leaning TowerChoose a scale and use a ruler and protractor to make a simple scale drawing.Pizza DoublerIf you could choose between doubling the fraction of the pizza that a slice is, or doubling the radius, which option would give you more pizza? In this problem you will investigate which choice gives a bigger slice.Area and ScaleWhen a figure is redrawn at a larger scale the side lengths increase by the factor of the scale (if the scale doubles the size, the side lengths double also). But, does the area increase the same way? Explore a dynamic sketch and see how area changes when the scale changes.Tree House 1Given plans for a tree house, redraw the plans at a different scale.

Students will explore scale and use it to find measurements in scale drawings.Key ConceptsScale drawings are drawn proportionally so that there is a ratio between a given length on the drawing and the actual length. This ratio is used to set up a proportion to find other measurements.GoalsUnderstand that scale drawings are proportional.Use scale to find actual measurements.ELL: Define these terms in the context of the discussion:scalescale drawingscaled to fitproportionalAllow ELLs to use the dictionary if they wish.

Learn Alberta offers a real-world scenario of skateboarding to introduce the concept of square roots. This multimedia site offers two videos, printable exercises and interactive exercises of finding the square root through visual prompts of squares.

The use of rate, ratio, and proportion is illustrated in this video about action photography. An additional interactive element allows students to explore equivalent ratios equivalencies through enlarging and reducing images to compare an original ratio and a target ratio. Finally, a print activity is also provided.

Learn about ratios with this handy reference site. Detailed explanations and examples are used to explain ratios. Also highlights common mistakes that can be made when learning about ratios.