Exploring Dilations 2
Objective
Big Idea
Launch
On the second day of the Dilations lesson I begin by handing each student an Entrance Slip in the intent of applying some prior knowledge and preparing for the lesson's tasks. I ask that they quietly and independently complete the slip.
Questions 1 to 3 are multiplication questions involving fractions, which learners will be doing when muliplying using conversion factors.
Just like the dilations, which students will be analyzing and performing in this lesson, questions 4 and 5 demonstrate change of size without change of proportions.
Once all students are done, I ask volunteer students to explain their answers and I discuss any doubts or misconceptions students may have before going on to the next section of the lesson.
Partner Activity
To begin this section of our lesson I randomly pair students up and give each pair a Cat Dilations 2 Worksheet. I assign each pair of students one and only one of the four tasks on the sheet. The goal of the task is to determine how the image was drawn from the pre-image. My plan is that towards the end, student pairs that have answered the same problem will group up to discuss their work. Calculators can be handy for this task.
Each partnership will determine how the image was produced from the pre-image based on the three points that are labeled on the cat. I give the class the following instructions:
- Analyze the figures in your assigned coordinate plane. Make sure you know which figure is the image and which is the pre-image.
- With your partner, discuss and determine how the image was created from the pre-image given the scale factor and center of dilation.
- Write the procedures on your sheet such that another person can perform the dilation by following your procedures.
At this point I walk around aiding students but allowing them to tussle with the task pretty much on their own. With students that verbally or non-verbally show they are struggling, I help by hinting things out like, “start by finding distances; see what you can do with those” or I’ll say “refer back to yesterday’s lesson….see if that can help you.” I also advise all students to make a table to organize their data, and write the procedures as they go along.
I expect to see students undertaking the following procedures as they work:
- Find the horizontal and vertical distances from the center of dilation to each of the 3 pre-image points
- Multiply the distances found by the given scale factor. These will be the “new” distances.
- Starting from the center of dilation point, use the “new” horizontal and vertical distances to plot each image point.
To avoid difficulty and confusion, I advise students to take care of one pre-image-image point at a time and literally write down what they do and what they calculate. Later, they can organize and “polish” their procedures.
Group Pairings
After considerable time has elapsed and students are done, I ask all pairs working on the same coordinate plane to group together and discuss their procedures. (Standing up and moving about to find each other is a good break for the brain)
Once they are grouped, each large group must reach a consensus and agree on a set of procedures that will work. I walk around to each of these groups and randomly point to any student to be the secretary of the group. This student will record the final set of procedures on producing image points from the pre-image points.
This part of the lesson is a test of teamwork. Students should be able to read their work to each other and decide on a set of procedures to follow. Students may argue, sometimes in vain because usually, there work coincides with what to do, but the instructions are just worded differently. It is a challenge for students to be able to recognize other student work, sometimes even when the same idea is being expressed.I try and stay alert about this in case I need to immediately “put out the fire” and help group members harmonize their ideas.
Once students agree on the procedures to follow and the secretary of each large group is ready, I ask that he/she share these with the whole class.
To end this section of the lesson , I address the whole class and ask if their set of procedures will work with any scale factor as well as the given ones. They may not answer with assurance. The closure questions should help with this.
Closure
To close the lesson, I ask each student to use the set of procedures his/her group concluded with to dilate the pre-image on this Exit Slip. I randomly assign each student one of the four tasks on the Exit Slip. I allow the large group to remain together but each student actually performs his/her own dilation. Students will always find someone nearby that is doing the same dilation and compare their result. This is fine actually because they can find possible mistakes, which usually is a minor math error or a wrongly plotted poing.
I ask that they put their names on the slips and I collect them. I like to hand them back the following day and I plan to ask that they correct any errors, if any.
