I begin part two of this lesson with two carefully selected questions.
Question 1: Find the relationship between the volume of a sphere and a cubical box of same height and diameter.
Question 2: What is the radius of the circle whose circumference is 30 cm? (round to nearest 1/100th)
My objective is to get students thinking in ways that will help them carry out the application task that follows. These are my Cue Questions. I either write these on the board before students settle in, or hand them out as an Entrance Ticket. Either way, students in my class are so used to me allowing them to discuss things with their partner, that unless I indicate otherwise, they will begin to do just that.
As my students work on these questions, I walk through listening and informally observing. Eventually I will select two students to go to the board and share their work.
For Question 1 I tell my students that they should try and work out more than one case. I encourage them to compare volumes to see if the relationship they've derived holds true.
For Question 2, students may ask for the circumference formula which I politely ask that they find it themselves. I do make sure they don't use the circle area formula instead.
Before beginning I gather the following materials for the modeling application that follows:
For this exploration I form groups of 3 students, giving each student a copy of Group Data Slip. Each group will share the following responsibilities (self-assigned):
Each group will receive one sports ball to analyze. I address the class...
Each group is to decide on a method of finding the volume of their sports ball. Then, measure the ball and determine its volume. You have 15 minutes to complete this task.
I ask that they put on their "creativity caps" for this exercise refering back to the Launch Tasks if necessary. Pretty clever ideas are raised sometimes so I may remind students of some of the interesting thoughts shared today.
As students work I walk through asking questions about their plans to find the volume of their sports ball. I expect most groups will measure the circumference, calculate the radius, then calculate the volume. I am on the lookout for promising ideas and approaches that are different from the other groups (See Timing Myself reflection). I encourage each student to write the steps taken on their data slips.
Once each group is finished, students are to form new groups, this time by number....all #1s, #2's, ect. These groups will now contain students who calculated the volume of different sports balls. See my class example: Grouping scheme.docx In my case we went from six groups of three students to three groups of six students. I ask the groups to discuss the different methods used by their original group to find the volume of their sports ball. I encourage students to use the white board to make the explanation visible to the rest of their group and not allow repetition of methods used to save time. After about 15 minutes, students will return to their original groups.
Here are some examples of methods that my students used to find the volume of their sports ball:
A third method that may come up is to find the volume of a sphere with the same diameter and height as the cylinder it's in, then multiplied thie cylinder volume by 2/3.
Once the original groups re-form, I ask that they take a few of minutes to discuss and answer these questions on the back of their Group Data Slip to be collected at end of class: