Understanding Perfect Cubes and Cube Roots Continued
Lesson 4 of 19
Objective: Students will be able to use a visual understanding to make connections to area and volume as applications of squares, square roots, and cubes, cube roots
Beginning the Activity
Clarifying your learning goals before beginning a lesson is very important. To watch a short video about clarifying learning intentions and criteria for success see the resource section.
Clarify for a student that today is focused around extending their current understanding of Perfect Squares to include an understanding of Perfect Cubes. The goal is to complete the rest of the Perfect Cubes activity including any parts of the extension activity you choose to use. Lead a short two minute review of the work completed on day one using any pictures taken of student models and a quick review of the answers to the table using a student paper and the document camera
After the quick review, pull a few of the perfect square and cube pictures up on your projector board and tell students these images are inspiration for good discussion as students work within cooperative groups to complete all parts of question one, a-e. These are some thinking questions that should generate some good discussion within cooperative groups and I like to see full sentence explanations so that I understand student’s thinking and they begin to practice explaining themselves in complete thoughts.
As students work within groups and provide feedback to each other, walk about the room providing feedback that also moves learning forward. I usually begin with the groups that I know have less prior experience with exponents to ensure they are beginning productively. After checking on all the groups who may struggle, I check with the groups who showed more prior knowledge to ensure they are also working productively. As you provide feedback to groups make note of different approaches to the answering the questions and begin to ask groups to present ideas to specific questions during the mini-wrap up session. I like to tell groups ahead of time which questions they will present and what thinking I really like from their work so that they are ready to speak and they know what it is I really want them to mention.
After about 15 minutes when most or all groups have had time to complete the questions thoughtfully, pull the class together for a mini-wrap up. Your goal with this discussion is to ensure students understand conceptually the meaning of cubing a value and the inverse cube rooting a value. Students need to see the math concept visually and then move towards the algebraic understanding and estimation of perfect and non-perfect cubes.
Several teaching strategies are employed throughout this cooperative group time. Click on any of the following resources to watch a short video explaining each strategy:
The extension lessons can be used as homework if not completed during class. Group discussion is desirable but not essential to completing each of the extensions.
The goal of the extension lessons is to connect the visual models of perfect squares and cubes to the geometry they represent with area and volume. The perfect square models are useful for making connections to perimeter and area. Application questions such as using area to find the perimeter of a square are a natural extension of this lesson and useful applications of squares and square roots. Similarly, if students can connect their 3-dimensional cubes to volume and apply cube roots to finding length of sides then again this is a direct application of cube roots. I like to give math concepts a context when possible so students begin to see all the many connections that exist between math skills. This is also a good time to discuss a conceptual understanding of volume – a 2-D base is repeated and layered on top of itself to create height (volume is found by area of the base times height). You can bring so many additional concepts into the mini-wrap up discussions.
The extensions activities are divided into two parts – part 1 is the straight application of area and volume. Part 2 is graphing perimeter, area, and volume to begin strengthening student graphing skills and ability to discern patterns of change from a graph. Once graphs are complete, it would be a good time to introduce linear patterns visually (perimeter) and non linear patters (area and volume).