Measuring the Volume of Solid Figures
Begin by telling the class that today we are going to learn how to measure volume. Tell students that the word volume is a homophone and it has more than one definition. Tell students that we all know that one definition of volume is how loud something is. See if there is anyone in the class who knows that mathematical definition of volume. Record all responses that are offered, without validating if any of these defintions are correct.
Tell the class that they will watch a short Brain Pop video titled "Volume of Prisms" found at http://www.brainpop.com/math/geometryandmeasurement/volumeofprisms/ that explains what volume is. Their job is to see if they can restate what volume is in their own words at the end of the video and then review the definitions given by the class to see if any of the definitions, or parts of definitions, given by the class is correct.
When the video is finished, ask students to review what they learned from the video and record their responses. What did you learn about what volume? What objects can you find the volume of? What are methods for measuring volume? Anything else that you think is important to remember?
Make a list of the learning goals for today’s lesson, making sure to highlight whichever of the learning goals the students already identified.
1. Volume means the space inside of a 3-dimensional object, which we will be measuring in cubic units. (Define cubic units, specifying that any unit of measurement can create a cubic unit, ex. cubic centimeters, cubic inches, etc.)
2. Volume can be measured in 2 ways - 1) by counting how many cubic units occupy the prism or 2) by using a formula
3. There are 2 formulas for volume, V=l*w*h or V=B*h
2 Direct Instruction
- Model for students how to find the volume of a few rectangular prisms with simple dimensions. Depending on your resources, you can draw simple prisms, labeling the length, width, and height, or you can use centimeter cubes to build a rectangular prism. A combination of the two is probably best, so that students who are tactile learners can actually see how the cubic units come together to form a rectangular prism.
- The first couple of examples should be simple examples with smaller dimensions, making it easier for students to simply visualize and count the cubic units inside the prism. Some examples are 2*2*2 or 1*2*3. With each example, you should first model how to measure volume using the formula l*w*h and then show b*h. This allows students to see you model both formulas.
- The following couple of examples should contain increasingly larger dimensions that would be too tedious for students to count in individual cubic units and even too tedious for you to build. This will force students to realize that using either of the formulas makes calculating volume more efficient. Some examples are 2*4*5, 3*5*5, etc. Continue to first model how to measure volume using the formula l*w*h and then show b*h. By this time, students have had ample models of how to find volume using both formulas, making it easier for them to do it on their own.
2. Draw or use the centimeter cubes to build a simple irregular solid by making some rows wider than other and some columns higher than others. Model that irregular solids require you to count indvidual cubic units, or use a combination of multiplication and addition (to find the volume more efficiently).
3 Guided Practice
- Provide 3 simple problems for students to solve. The first problem should be to find the volume of an irregular solid figure and the others should be finding the volume of rectangular prisms. Students can use centimeter cubes to create the prism themselves if necessary. Require the students to draw and label their own diagram of the figures you presented and to show which their answer using one of the formulas.
- Walk around as students solve problems to record difficulties that students are having and to offer quick assistance to students. Review with the class the general difficulties that you noticed as you walk around and allow a couple of students to share their solutions with the class. Address any questions before assigning their indepdendent work.
4 Independent Practice
Tell students that they will now have an opportunity to practice measuring volume on their own through Khan Academy. If your students are unfamiliar with Khan Academy, explain that Khan Academy is an amazing website that provides instructional videos on many different math topics, followed by short quizzes to assess how much understanding they have with that particular math skill. The quizzes are not part of their grade and are meant solely to give them an opportunity to practice, get immediate feedback about their answer, get hints for how to solve a problem, and provide them additional support through the tutorial video that they can review if they get confused about how to solve any of the problems. Tell them it’s like having their own teacher, right there with them if they need help solving the problems on their own.
First students should complete the “Volume with Unit Cubes 1” skill quiz, found at http://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-measurement-topic/cc-5th-volume/e/volume_with_unit_cubes, which follows the video “Measuring Volume with Unit Cubes”. Students do not have to watch the video to complete the quiz, although the video is linked to the quiz as a reference if students need it. The quiz contains 5 questions for students to complete, which show a combination of labeled rectangular prisms and irregular solids, which will require students to count individual cubic units or utilize multiplication and addition to solve. Tell students that they should note problems that are particularly challenging or confusing to them, even after the supports of the hints and the video.
Once finished with that, students should complete the quiz “Volume 1”, found at http://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-measurement-topic/cc-5th-volume/e/volume_1 ,which follows the video “Volume of a Rectangular Prism and Box examples”. Again, students do not have to watch the video to complete the quiz. These 5 questions focus solely on finding the area of rectangular prisms. These problems increase in difficulty, as some questions will give the volume and one or two dimensions and ask students to figure the remaining dimension.
Ask the students to share questions with the class that were challenging for them and ask their classmates who were more successful with those questions, to share their strategies for solving them.
Provide an exit slip for the students to assess their understanding of the 3 learning goals established at the beginning of the lesson. Students need to answer the following 3 questions:
- When we find the volume of a 3-dimensional object, what are we measuring? (cubic units)
- What are two strategies for finding the volume of an object?( Counting how many cubic units occupy the prism or by using the formula for volume.)
- What are the two formulas for volume? (V=l*w*h or V=B*h)
Key Standards Supported
Measurement And Data
|1.MD: Measure Lengths Indirectly And By Iterating Length Units.|
|1.MD.1||Order three objects by length; compare the lengths of two objects indirectly by using a third object.|
|1.MD.2||Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.|
|Represent And Interpret Data.|
|1.MD.4||Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.|
|Tell And Write Time.|
|1.MD.3||Tell and write time in hours and half-hours using analog and digital clocks.|
|2.MD: Measure And Estimate Lengths In Standard Units.|
|2.MD.1||Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.|
|2.MD.2||Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.|
|2.MD.3||Estimate lengths using units of inches, feet, centimeters, and meters.|
|2.MD.4||Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.|
|Relate Addition And Subtraction To Length.|
|2.MD.5||Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.|
|2.MD.6||Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.|
|Represent And Interpret Data.|
|2.MD.10||Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put- together, take-apart, and compare problems4 using information presented in a bar graph.|
|2.MD.9||Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.|
|Work With Time And Money.|
|2.MD.7||Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.|
|2.MD.8||Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?|
|3.MD: Geometric Measurement: Recognize Perimeter As An Attribute Of Plane Figures And Distinguish Between Linear And Area Measures.|
|3.MD.8||Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.|
|Geometric Measurement: Understand Concepts Of Area And Relate Area To Multiplication And To Addition.|
|3.MD.5||Recognize area as an attribute of plane figures and understand concepts of area measurement.|
|3.MD.5.a||A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.|
|3.MD.5.b||A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.|
|3.MD.6||Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).|
|3.MD.7||Relate area to the operations of multiplication and addition.|
|3.MD.7.a||Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.|
|3.MD.7.b||Multiply side lengths to find areas of rectangles with whole- number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.|
|3.MD.7.c||Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.|
|3.MD.7.d||Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.|
|Represent And Interpret Data.|
|3.MD.3||Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.|
|3.MD.4||Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.|
|Solve Problems Involving Measurement And Estimation Of Intervals Of Time, Liquid Volumes, And Masses Of Objects.|
|3.MD.1||Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.|
|3.MD.2||Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.7|
|4.MD: Geometric Measurement: Understand Concepts Of Angle And Measure Angles.|
|4.MD.5||Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:|
|4.MD.5.a||An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.|
|4.MD.5.b||An angle that turns through n one-degree angles is said to have an angle measure of n degrees.|
|4.MD.6||Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.|
|4.MD.7||Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.|
|Represent And Interpret Data.|
|4.MD.4||Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.|
|Solve Problems Involving Measurement And Conversion Of Measurements From A Larger Unit To A Smaller Unit.|
|4.MD.1||Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two- column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...|
|4.MD.2||Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.|
|4.MD.3||Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.|
|5.MD: Convert Like Measurement Units Within A Given Measurement System.|
|5.MD.1||Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.|
|Geometric Measurement: Understand Concepts Of Volume And Relate Volume To Multiplication And To Addition.|
|5.MD.3||Recognize volume as an attribute of solid figures and understand concepts of volume measurement.|
|5.MD.3.a||A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.|
|5.MD.3.b||A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.|
|5.MD.4||Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.|
|5.MD.5||Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.|
|5.MD.5.a||Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.|
|5.MD.5.b||Apply the formulas V=l×w×handV=b×h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems.|
|5.MD.5.c||Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.|
|Represent And Interpret Data.|
|5.MD.2||Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.|
|K.MD: Classify Objects And Count The Number Of Objects In Each Category.|
|K.MD.3||Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.3|
|Describe And Compare Measurable Attributes.|
|K.MD.1||Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.|
|K.MD.2||Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.|