compare, estimate, and determine measurements in various contexts
describe the differences and similarities between volume and capacity, and apply the relationship between millilitres (mL) and cubic centimetres (cm3) to solve problems
Provide students with a variety of clean, empty milk cartons (250 mL, 500 mL, 1 L, or 2 L), and discuss what the capacity measurement means (capacity describes the amount of milk that the carton can hold, which does not include the gable). Have them use centimetre cubes (cm3) to find the volume of the space inside the carton by arranging the cubes inside the carton in rows and columns to form a layer and repeating that layer until the carton is filled. Discuss why there might be discrepancies in the measures (e.g., the container is not completely filled with milk; there may be gaps between the cubes), and use this to draw out the distinctions and relationships between these attributes. Have students use these relationships to identify benchmarks for volume and capacity (e.g., a bag of milk takes up approximately 1000 cm3 of space, which is the same as the base ten thousands block; 1 mL takes up the space of the base ten unit). Have students use their benchmarks to estimate the volume and capacity of a variety of things, such as everyday grocery items.
To reinforce the concept that 1 millilitre of liquid occupies 1 cm3 of space, have students use displacement to measure volume. For example, have them predict the displacement of water in a measuring cup when an object with a known volume (e.g., base ten rod or flat) is submerged in it (e.g., if a measuring cup had 50 mL of water and a rod was submerged in it, the measurement would now be 60 mL). Once the relationship and strategy are understood, have students use displacement to indirectly measure the volume of other objects.
Have students solve problems involving units of capacity and volume, such as the following:
solve problems involving perimeter, area, and volume that require converting from one metric unit of measurement to another
1 m= 100 cm
1 square metre= 1 m × 1 m
1 m2= 100 cm × 100 cm= 10 000 cm2
1 cubic metre= 1 m × 1 m × 1 m
1 m3= 100 cm × 100 cm × 100 cm= 1 000 000 cm3
Have students make a square that is 1 m × 1 m (1 m2) out of newspaper, newsprint, craft paper, or wrapping paper. Ask them to measure the perimeter of the square in metres, centimetres, and millimetres. Ask them to describe the relationship between:
Pose questions that require students to use these relationships to solve problems. For example, if the perimeter of a rectangular garden is 6 m, what are the dimensions of the garden in centimetres if its length is twice its width? Have students use the 1 m2 paper square from Sample Task 1 to visualize and answer the following questions:
Have students describe these area relationships and use diagrams and ratio tables to convert, for example, 8.25 m2 to square centimetres.
Have students make the faces of a cube that is 1 m × 1 m × 1 m by taking six 1 m2 paper squares from Sample Task 1 and taping them to a skeleton of a cube made with metre sticks. Have them determine how many centimetre cubes would be needed to fill this 1 m3 cube. Then, have students use this relationship to solve problems such as: How many 250 mL measuring cups are needed to fill a tank that is 1 m × 2 m × 3 m?
Have students solve problems involving units of capacity and volume (see E2.1), such as the following:
Support students in recognizing the relationship between cubic centimetres and millilitres and in using drawings to visualize the situation.
use the relationships between the radius, diameter, and circumference of a circle to explain the formula for finding the circumference and to solve related problems
To develop an understanding of the relationship between the radius, diameter, and circumference of a circle, have students use string or tape measures (paper or fabric versions) to measure several everyday circular objects (e.g., cups, cans, hula hoops). For each circle, have students measure the circumference and the diameter and determine the radius.
Have students organize their measurements in a table with columns to record radius, diameter, circumference (2πr or dπ), and the ratio of circumference to diameter. Ask them to write equivalent ratios in the form n:1. Have them compare their ratios, and then discuss this special ratio, known as pi (π), as a class. Have them find out more about pi and share their learning with the class.
Have students solve problems that involve circumference, diameter, or radius, such as:
Note: When bicycling, the distance a tire travels when it revolves once is equivalent to its circumference.
Have students solve these problems using approximations for pi, such as 3.14, then compare these results to those obtained using technology that uses pi (e.g., a calculator app that includes pi).
construct circles when given the radius, diameter, or circumference
Have students construct circles given the radius, given the diameter, and given the circumference.
Have students estimate the unknown measurements for the following circles, then construct them to verify:
Have students describe the calculations they used for the estimate and the construction. Have them compare different strategies for constructing circles (e.g., compass, technology).
show the relationships between the radius, diameter, and area of a circle, and use these relationships to explain the formula for measuring the area of a circle and to solve related problems
To support visualizing the relationship between the area of a circle and its radius, have students find the area of a circle by reshaping (decomposing and recomposing) it into a “parallelogram”. To do this, have them cut a circle into 8 or 16 sectors and rearrange the slices to form an approximate parallelogram (the more sectors, the closer the result is to a parallelogram):
Guide students to notice that the base of the parallelogram is half the circumference of the circle ( or = π × r) and that the height of the parallelogram is the radius of the circle (r).
Support students in recognizing that in this case, the area of the parallelogram (b × h) can be thought of as π × r × r = πr2.
Have students do research on the Internet to find other visual proofs or explanations of this formula and share them with the class.
For a more concrete experience, have students construct a circle with a given diameter or radius on grid paper, count the squares and partial squares to estimate the area, and compare their findings against the formula.
Have students use the formula for the area of a circle to solve real-life problems, such as:
Have students explain their thinking.
represent cylinders as nets and determine their surface area by adding the areas of their parts
Show students a closed cylinder. Tell them that you would like them to make a net for the cylinder. Ask them what they need to know. Now have them construct the net. Then, ask them to find the surface area of the cylinder using their net and the formulas for the area of a rectangle and the area of a circle.
Have students identify and compare the surface area of cylinders in real life that are closed on both ends, closed on one end, and open (no bases). Guide students to reformulate their formula to find the surface area of a cylinder to account for each of the three variations.
show that the volume of a prism or cylinder can be determined by multiplying the area of its base by its height, and apply this relationship to find the area of the base, volume, and height of prisms and cylinders when given two of the three measurements
Have students use pattern blocks to model the relationship between the volume of a prism and the area of its base. Have them stack pattern blocks to build a red trapezoid-based prism, a green triangle-based prism, a yellow hexagon-based prism, and a blue rhombus-based prism, all with the same height. As they do so, draw out the idea that every layer in the prism has the same area, and, by extension, every layer also has the same volume.
To support comparisons of the areas and volumes of the different pattern block prisms, ask questions such as the following:
Support students in generalizing a volume formula that would work with any pattern block prism (i.e., the area of the base times the height), and explain why this formula could be applied to any prism, including non-right prisms. (For different prisms, use examples generated from nets in E1.1, Sample Task 1.)
Guide students to consider whether this formula would also apply to cylinders. As students explain their reasoning, ensure that they recognize that, for both prisms and cylinders (and unlike pyramids), the area remains constant throughout their entire height. Have them inspect a variety of cylinders and prisms and identify the relevant dimensions needed to determine the volume of each, as follows:
After they have developed an understanding of the relationships between the volume, the area of the base, and the height of a prism or cylinder, have students apply these relationships to solve various problems, such as:
Guide students to recognize that if they have two of (a) the volume, (b) the area of the base, and (c) the height, they can multiply or divide to determine the unknown measurement. Support students in recognizing that the inverse relationship between multiplication and division enables them to use the formula to find more than just a missing volume. For example, they might consider the following:
Have students represent these situations with a diagram to show what is known and unknown, then use a chart to organize the various possibilities. Use these types of problems to help consolidate work with area, volume, and circles, and to reinforce the inverse relationship between multiplication and division.