Volume of Square Pyramids - What is the relationship between and prism and pyramid?
Lesson 9 of 18
Objective: Students will be able to conduct a water lab to discover first-hand the relationship between the volumes of prisms and pyramids.
Opener: As students enter the room, they will immediately begin working on the opener. The opener is a mixture of previously learned questions, and students should work individually, and then as table groups to discuss the methods for solving the questions. After approximately 5 minutes, I will call on students to go to the board and solve the opener questions. As with all openers, I will take volunteers to go to the board – the volunteer is expected to explain their reasoning, and other students are expected to follow along with the work and ask questions/make suggestions as necessary. By having a student explain their reasoning while others listen and provide feedback, mathematical practice 3 – construct viable arguments and critique the reasoning of others – becomes a natural part of class. Instructional Strategy - Process for openers
Learning Target: After completion of the opener, I will address the day’s learning targets to the students. In today’s lesson, the intended targets are, “I can make connections between the volume of a prism and a pyramid with the same shape base and height” and “I can calculate the volume of a right square prism.” Students will jot the learning targets down in their agendas (our version of a student planner, there is a place to write the learning target for every day).
Activate Prior Knowledge: In order to activate their prior knowledge, I will ask how we calculated volume of a right prism in earlier lessons. What is volume of a prism? We will have a class discussion (using class rules of raising your hand to share your thoughts) on volume as we have studied it thus far. It is important to revisit the idea of volume of a prism being the area of the base times the height, as a pyramid is the same with one additional step.
Water Lab and Notes: The meat of this lesson will begin with a discovery lab. Students will work in their table groups to complete the discovery activity. My students are used to working in groups on a daily basis, and as they set off to begin the activity, I remind them of the most important rule of group work – be respectful of everyone’s opinion, though you are working as a group, you still have individuality and do not have to agree. During this time, I will walk around to listen to student conversation, and help promote appropriate talk where necessary. Students are forming their own hypothesis regarding the relationship between the volume of a pyramid and prism - when allowing students to hypothesize I try not to stifle their creativity and thoughts. Allowing students to hypothesize is a good use of mathematical practice 2 – reason abstractly and quantitatively. Normally, the look on my face reveals what I am thinking (I have a terrible poker face!), but during this activity I really try to keep my opinions and facial expressions out of the mix – I want the kids to think for themselves. The beauty of hypothesizing is that you can’t be wrong – it is just a guess! As groups get to part two of the lab, I will supply them with water and paper towels for the activity. Part two of the lab allows the students to actually test their hypothesis using water, and a prism and pyramid with the same base and height – which brings in mathematical practice number 5 – using appropriate tools strategically. It is most common for students to think that the prism is made up of only 2 pyramids; it is not often that I will have students correctly hypothesize the relationship, but after the experiment they have solid evidence that it really takes three pyramids to make a prism. What is this water experiment.wmv
Class Discussion: As groups finish the lab, I will call on a representative from each table to share their original hypothesis and final conclusions with the class. During this time, all other groups will quietly listen – they are not to interject their comments, as we are just reporting the facts. I will not comment until all tables have spoken. Most tables will conclude that it takes three pyramids to fill the prism with water, but groups may struggle to write that as the pyramid volume being 1/3 of the prism volume – which addresses mathematical practice number 6 – attending to precision. In mathematics, it is important that students are precise about the way they represent formulas and ideas; the common formula for a pyramid is 1/3 Bh – so it is important they understand that 1/3 is the same as dividing by 3. If I don’t have any groups that conclude the 1/3 relationship, I will guide them to it through questioning: If it takes 3 pyramids to fill the prism, then what could you do to the volume of the prism to get the volume of the pyramid? Is there another way we can write divide by 3? If I find the volume of this prism, what should I do next to turn it into a pyramid?
Instruction and Practice: After the class discussion, I will have students write the formula for finding the volume of a pyramid in their foldable, or on the notes sheet (back of water lab). I will use student responses, through raising their hand, to calculate the volume of one example pyramid, and then I will ask students to work in their table groups to complete 9 additional examples – which addresses mathematical practice number 4 – modeling with mathematics. We took time in class to derive our own formula, now the students are given the opportunity to complete additional problems using that formula – they are modeling what they learned. The routine for working in table groups in my classroom (we do this every day) is that students try the problem, wait for their table to finish, and then discuss the answer – which makes mathematical practices 1 and 3 – make sense of problems and persevere in solving them and construct viable arguments and critique the reasoning of others – a staple of every lesson. If there are discrepancies that they cannot solve on their own, they are to raise their hand and I will assist. As I notice groups finishing, I will ask that all groups stop working/talking, and we reconvene as a class to discuss the answers. Since this topic is new, I will ask for volunteers to go to the board and show their work/explain their reasoning (I allow them to bring a buddy if they don’t like to talk!)
Summarize + Homework
Exit Problems: To summarize the lesson, I will post three figures on the side board – one of each that we have studied: triangular prism, rectangular prism and square pyramid. I will have students calculate the volume of each figure, and write their responses on sticky notes from their baskets, and as I dismiss tables for the day, the students will post their answers next to each figure on their way out. The purpose of this activity is for me to gauge the learning that has taken place, and make adjustments to the next day’s lesson as needed. In cases where students do not show proficiency with any of the shapes, I will pull them in for tutoring during silent reading time the following morning.
Extra Thoughts From Me! I have done this particular lesson several times in my career – both in high school geometry and in 7th grade. I find that students really have a “aha!” moment when they pour the water from one figure to the other – and I would say about 90% of the students will remember that we always divide by 3, or multiply by 1/3 with a pyramid – the hands on activity really helps them to understand the “why” and when kids know why they are doing something, they are more likely to do it! Of course, now they will try to divide everything by 3…