The objective of the launch section in today's lesson is to help my students access their prior knowledge. I want them to connect their extensive prior knowledge of area to volume. We'll use this knowledge to move forward with both prisms and cylinders in the New Info section of the lesson.
I open the Area Volume Slides 2 and proceed to tell students that when they studied area they were provided with a definition for the meaning of area which rested upon the unit square. The sample figure in Slide 1 shows that the area of a figure was nothing more than the count of the unit squares in the figure. On Slide 2 we move to the volume of a solid with a similar visual definition employing the unit cube, which can be 1 inch on all sides, or 1cm, or 1ft, or any other unit. I'll then ask:
Looking at this figure in slide 2, how would you define volume?
Most students have worked with unifix cubes or wooden cubes in prior years, so this activity generally builds confidence and momentum. The light bulbs begin to light up as students make connections that they might have missed yesterday.
I created this Volume Cylinder Video to introduce my students to the formula for the volume of a cylinder. With Cavalieri's Principle in mind I try to make students understand how the formula is derived. The rewards of understanding the concept here are substantial because it lays groundwork for understanding the concept of the volume of any figure in both concrete and abstract terms. In upcoming lessons, students will be deriving and applying formulas for cones and spheres.
Partners for this activity should be of similar ability level. Each student will work on their own Applying your knowledge activity sheet. The problems require ample thinking and writing time and I make sure not to rush them.
As I handout the sheets I ask my students to show all of their math work. In Problem 1 students will be surprised to see that the salinated water occupying the outer space of the cylinder is more abundant than the fresh water in the inner cylinder. As students work I look out for students who refer to the larger surface area in their explanation. If students are confused, I ask probing questions to push them in this direction.
Problem 2 is often challenging for my students. The thinking required to solve the problem is challenging. But, I have found that this is an important opportunity for students to persevere and come up with a solution (MP1).
Many students choose random values for their first two types of cylinders and then use the volume formula and the remaining wax for the 3rd type of candle. Here is an example of student work:
This student was guided into choosing random values for the first candles and subtracting from the "stock" of wax. In his work he chose the same radius for the first two candles, only changing the height. Then he used the leftover wax for the third candle. I must hightlight that he skillfully varied the height of this 3rd candle (ended with 7.5in), in his equation until, dividing by 190.32, he obtained a value close to a perfect square (25.375), just so he could obtain a radius close to a whole number.
See my Students may need help structuring reflection for more on this exploration.
At the end of their work, I ask that each student circle the hand gesture (below the diagrams), that indicates how they feel about their overall understanding of each problem. Then, they turn in their papers.
The Homework assignment is short and to the point. The first two questions are straight forward substitution questions for practice using the volume formula. The 3rd question is a vocabulary questions, refreshing students' ideas of difference between volume and surface area. The 4th question is a mental math questions where students, if they know the volume formula well, should easily see that both cylinders have the same volume.