Lesson 1: Measuring the Volume of a Classroom

Narrative

Students use their yardsticks to measure the classroom and to compute the volume. They measure the length, width and height of the classroom. After they have the measurements, they compute the volume. They can use inches, feet or yards, each has advantages and disadvantages. The question is posed in terms of cubic ft, so that is the natural unit of measurement.

Inches: Easy to measure, they will multiply big numbers and get a big numeral as a result. Result is difficult to picture.

Feet or yards: Measurements will have fractional parts or decimals. Multiplication uses smaller numbers but includes fractions or decimals. The result is a smaller numeral.

Measuring the dimensions of the room will pose some logistic difficulties. It is easier if groups pool their resources and use all or at least two of their yardsticks end to end rather than iterate with just one.

Launch

Students work in their maker groups from the previous lesson and they use their yardstick. Have a box that is 1 ft on each side, i.e. 1 ft^3 cube to illustrate the question.

Task Statement

In the previous lesson we made yardsticks to help us figure out how much stuff would fit into our classroom. Today we’ll answer that question. How many 1 ft^3 cubes could we fit into the classroom?

Task Solution

The question asked about the number of cubic ft boxes that would fit into the classroom. So we need the length, width and height of the classroom up to whole feet to determine how many boxes could be stacked across the floor and up the walls of the classroom. One layer of boxes across the floor would be length x width and we can fit as many layers as the height of the classroom in ft, e.g. 1 layer is 30 rows of 20 boxes = 600 boxes. We can fit 9 layers into the classroom so the total number of boxes is 9 layers of 600 boxes = 5400 boxes. 

Synthesis

We multiplied 9 layers of 30 rows of 20 boxes = 9*30*20 = 5400 boxes. This gives us the volume computation 30 ft x 20 ft x 9 ft = 5400 ft^3. Notice if students multiplied the numbers in a different order and interpret the order, e.g. first stack up a layer against the wall and then repeat stacking layers until the room is full. Keep track of the different computations and their interpretations. Ask if there would be any gaps between the boxes and the walls. As an extension students could compute the volume of the gap using inches.

Task Narrative

Students generate dimensions of rectangular prisms with volume 60 ft^3. They make a model with linker cubes and then create the footprint of the actual prism on the floor of the classroom and hold up ropes to illustrate the height. The mathematical purpose of this task is to realize that different prisms can have the same volume and to deepen the understanding of volume as the number of unit cubes that fill a 3-d space. As students work, take note of groups that use different dimensions.

Launch

Have linker cubes or other cubes available for students to make a plan for how to stack the boxes in a rectangular prism. Other materials needed: painter’s tape to mark the outline on the floor and rope to illustrate the height.

Task Statement

The library is getting painted over the summer and they need to store 60 (50?) boxes that are 1 ft3 in our classroom. We need to figure out where to put them. The overall configuration has to be a rectangular prism. Use the linker cubes to make a plan for how to stack the boxes. Be prepared to share your reasoning with the class.

Task Solution

60 = 10*3*2=10*2*3=2*3*10=3*2*10=2*10*3=3*10*2 (length, width, height - some more practical than others, ceiling might not be 10 ft high)

60 = 5*3*4 (or any permutation)

60 = 6*2*5 (or any permutation)

60 = 60*1*1 (probably not feasible)

60 = 30*2*1 (not feasible?, not efficient)

60 = 15*2*2  (not feasible?, not efficient)

Synthesis

After students make their linker cube models, have groups that used different dimensions show their models and demonstrate that it gives the correct volume. Keep track of the different numerical equations on the board. 

Some configurations are mathematically possible but would not fit into the classroom. Discuss the pros and cons of different configurations for this particular purpose - too high is hard to stack, too low requires too much floor space. Have students agree on one or two and as a class mark the footprint of the prism on the floor of the classroom using painter’s tape. Have student hold up rope or stand in the corners and illustrate how high the prism would be. Make a point that the entire inside of this prism would be solidly filled with boxes.