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High School — Mathematics
Measurement

Measurement allows us to quantify sizes that would otherwise be described with general phrases like "small," "large" or "very big." Units of measure provide such specifics as square feet, cubic centimeters, and meters per second.

In high school, students develop a better understanding of measurement. For example, they learn that length is measured using a unit, area is measured using a squared unit and volume is measured using a cubed unit. Students should understand the concepts and recall the formulas for area, surface area, and volume. They should be able to use these formulas both forward (finding the surface area of a prism given the dimensions of the figure) and backward (finding the length of one edge of a prism when given its volume and two other edge lengths.) Students should be able to convert between units and be able to use rates, such as velocity. Students should also be able to use the Pythagorean theorem and right triangle trigonometry. Each of these skills and concepts should be applied in real world contexts and students should be able to determine whether an answer is reasonable.

Even though students will not receive a score for the Mathematical Processes standard on the OGT, it is still an important part of the curriculum. Content and processes should be taught in tandem. To better understand Measurement, click on the dropdown menu above and select Mathematical Processes.

The content in this Teaching Tool is based largely on the Ohio Mathematics Content Standards and Benchmarks and includesreleased items from the Ohio Graduation Test. Additionally, these materials are aligned with the NCTM standards . While there are various suggestions and activities here to use when working with students, this Teaching Tool is designed to complement a rigorous, research-based curriculum, not to substitute for one.



Measurement


1. Measurement

Click on the following benchmarks for more information and for links to annotated OGT items.

a.

Benchmark A: Solve increasingly complex non-routine measurement problems and check for reasonableness of results.

In earlier grades students became familiar with basic area and volume formulas. The key to this benchmark is the application of skills that students should already have studied. They should be able to answer questions involving rates, velocity and density and decide if their answers are reasonable.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

b.

Benchmark B: Use formulas to find surface area and volume for specified three-dimensional objects accurate to a specified level of precision.

The focus of this benchmark is on using surface area and volume formulas. Students should be able to calculate measurements to an appropriate level of precision and to derive surface area and volume formulas.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

c.

Benchmark C: Apply indirect measurement techniques, tools and formulas, as appropriate, to find perimeter, circumference and area of circles, triangles, quadrilaterals and composite shapes, and to find volume of prisms, cylinders, and pyramids.

This benchmark focuses on using indirect measurement techniques. Students should be able to find perimeter and circumference of two-dimensional objects and surface area and volume of three-dimensional objects by using information about area and perimeter of two-dimensional figures.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

d.

Benchmark D: Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates.

In this benchmark, students focus on solving measurement and rate problems using proportional reasoning and indirect measurement techniques. Students should be able to work with U.S. customary units and metric units and convert within and between measurement systems. They should also be able to use properties of similar figures, central and inscribed angles and right triangles.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

e.

Benchmark E: Estimate and compute various attributes, including length, angle measure, area, surface area and volume, to a specified level of precision.

The focus of this benchmark is on estimating and computing various measurement associated with geometric objects. Students should be able to measure as precisely as needed for a given situation. They should also be able to find angle measurements with and without measuring (directly and indirectly) and to use surface area and volume formulas.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

Click here for an annotated item from the 2005 Ohio Graduation Test that addresses this benchmark.

f.

Benchmark F: Write and solve real-world, multistep problems involving money, elapsed time and temperature and verify reasonableness of solutions.

This benchmark focuses on the application of mathematical skills to solve problems. Students should be able to work with multistep problems involving money, elapsed time and temperature. Students should also be able to determine if their answer is reasonable within the context of a problem.

 

ABOUT THE MATH
  • Deriving Formulas

    You may find it useful to have students find the volumes of a variety of shapes and try to derive the formulas by noticing patterns. For example, the volume of a square prism is the area of the base times the height, while the volume of a square pyramid is of the area of the base times the height. Students should notice that the formulas are similar; the volume of a square pyramid is of the volume of a square prism.

    There are many specific formulas for calculating volume of specific objects, including the following most common formulas:

    • Right prism: where V = volume, B = the area of the base and h = height;
    • Right cylinder: where V = volume, B = the area of the base and h = height;
    • Right pyramid: where V = volume, B = the area of the base and h = height;
    • Right cone: where V = volume, B = the area of the base and h = height.

    The total surface area of any three-dimensional figure can be found by adding the areas of each of the individual surfaces of the figure. Rather than having students memorize the formulas for the surface areas of various three-dimensional figures, help them derive these formulas. For example, to find the surface area of a triangular prism, students must first understand that the bases are two congruent triangles and the lateral surfaces are three congruent rectangles. To find the surface area of the figure, they must add the areas of all the faces and bases.

  • Right triangle trigonometry

    Students should be able to find the side lengths of a right triangle given one acute angle measure and the length of one side. Or, they should be able to find acute angle measures given two side lengths. The equations for the trigonometric functions are the following:

    • Sine: , where the sine of an acute angle can be found by dividing the length of the side opposite the angle by the length of the hypotenuse. One way to remember the formula is the term SOH: Sine = Opposite divided by Hypotenuse.
    • Cosine: , where the cosine of an acute angle can be found by dividing the length of the side adjacent to (next to) the angle by the length of the hypotenuse. One way to remember the formula is the term CAH: Cosine = Adjacent divided by Hypotenuse.
    • Tangent: , where the tangent of an acute angle can be found by dividing the length of the side opposite the angle by the length of the side adjacent to the angle. One way to remember the formula is the term TOA: Tangent = Opposite divided by Adjacent.



Strategies

Help With Fundamentals

Listed here are a few of the difficulties students might have with this topic, with some suggestions to address these difficulties. Paying close attention to your students' work to identify other student difficulties. Once you have identified a particular difficulty, find a strategy that will help students overcome the problem.



Additional Instruction and Practice

If your students need additional instruction and practice, supplement your instruction with activities. Talk to your colleagues and use your curriculum to come up with activities for addressing these topics. Try these!

Activity 1

Help students determine which surface area or volume formula to use to solve a particular problem. Provide students with a list of problems that require different formulas. Ask students to decide which formula they should use for each problem. Later they can use the formulas to calculate the answers.

Activity 2

To help students understand surface area and volume, have students create a package for golf balls. Students can create prisms, cylinders, or pyramids. They should determine how much material they will need to create the package and how much room is inside. Challenge students by asking them to find the cost of their package given a specified cost per square centimeter for the material.



Advanced Work

Angle of inclination

Since students have learned about right triangle trigonometry, they may wonder whether, given the side lengths of a right triangle, they can find the measures of the angles that are non-right. This would be a good opportunity to teach them about the inverse trigonometric functions, whose output is an angle measure.

Problem 1

Helene is 5 feet tall and is standing 15 feet away from a tree that she knows is 45 feet high. What is the approximate angle of her line of sight to the horizon? Provide mathematical support for your solution.

Method

We need to find the measure of angle A. Since we can calculate the side opposite angle A and know the side adjacent to angle A, we can use the tangent function to set up the following equation: . Since we need to find the angle, we should use the inverse tangent function: 69°.


Answer

Answer Explanation

The angle of Helene's line of sight to the horizon is about 69°.

Extension

Have students create right triangles on paper. Have them measure the side lengths and use the inverse trigonometric functions to predict the angle measures. They can verify their answers using a protractor.