Inscribed Figures (w/ Circles)

Students enter silently according to the Daily Entrance Routine. As we continue working with circles I find that students are struggling most with the formulas and how to apply them to different situations. More practice with formulas and application to word problems is needed. One problem solving strategy we have been working to use is drawing a picture. I’ve been observing many of my students experience a deer in the headlights look and freeze when all they see is words on the page. Thus, I included the message at the bottom of the page.  The following questions are used to guide students through problem solving as I walk around during the first 6 minutes of class.

• What should you draw?
• How should you label the information given?
• What formula will you need for this circle? (Begin by writing it!)
• Where should you insert the information you are given?

Some students struggle solving equations which include the pi symbol and will often substitute the approximation 3.14, making the division step in solving the equation much for difficult than it needs to be.

I am looking out for this tendency while I am walking around the room without telling students NOT to do it. I am also looking for students who get stuck and cannot identify the next step in solving the equation because of the symbol for pi. As I walk around I am also encouraging students to persevere and put something down on paper (MP1). It is important that this encouragement be as positive as possible as students are already feeling negatively when they freeze. In my experience, scolding them has not yielded results.

After giving students 6 minutes to solve I guide students through the solution by spiraling through the questions included above and calling on students who successfully got to each of these steps. When we reach the point of solving the equation by dividing by pi (as the symbol), I call on different students to help. If they get stuck as a group I remind them about the steps to solving a different equation which does not include pi, such as ½ x = ½ (6).

Once we are finished reviewing or time has run out, we put away our Do Nows and move on to class notes.

I distribute class notes and class work. Students are asked to copy the aim off the board and read the definition of “inscribed” silently. I wait 1 minute and then ask a student to read the same definition out loud. Then I explain that the first picture included in the notes is a circle inscribed within a square. I ask two students to repeat what I have just said, “what is this a picture of?” – a circle inscribed within a square. Then I ask students to turn to their partners and preview the questions on their class worksheet. They need to tell each other what each problem describes as a picture. What shape is inscribed what other shape?

Students are given about 2 minutes to do this as I am walking around to ensure these statements are being made and helping students to pronounce or piece together the correct phrases.

After those two minutes I ask students to give me their attention one more time so that I can model how to solve the first example in their notes. The expectation is that students are copying the work exactly as it is being shown, and asking questions once I am finished.

I then ask a student to solve the way I did, and model his solution at the front of the black board.

After answering questions students are randomly assigned to groups of 4 using the random name generator on the smartboard. The first 2 – 3 groups are sent into booths and the rest are allowed to work in any area of the room. The expectations of classwork time are set and written on the blackboard:

• You must remain within your group or ask for permission to briefly speak with other groups
• You must complete at least one problem in the first 10 minutes.
• There will be a time counting down. You must be in your seat when the timer goes off.

Students are assigned to groups and the timer is set at the SMARTBoard. I am walking around to check work/answers, ensure students are on task and keeping track of time to complete at least one of the problems. The following questions will also guide them through some of the difficult steps to take in each problem.

Problem 1:

• How will you find the area where the grass is being planted only? (MP4 as students apply the math they know – areas of two different shapes – to solve problems)
• How do you find the area of a circle? (refer to your notes!)
• How can you use the fact about the square’s length to calculate the radius of the circle? (refer to your notes!)
• How do you find the area of a square?
• You found the area, but this is not the final answer. Re-read the question.  “…how much will he need to charge…”
• How can you calculate the total amount given the price per square foot and your calculated area?

Problem 2:

• How is this problem different from the rest? The circle is not inscribed within a polygon, a polygon is inscribed within a circle (I make sure to ask all groups this question)  
• Do you understand what part of the picture they want you to find?
• What two areas will you need to find?
• How will you find the area of the square? What information do you need?
• How will you find the area of the circle? What information do you need?

Problem 3:

Is already scaffolded on paper because it is most likely the problem most groups may not get to. Once they have worked in groups for ten minutes they must return to their seats to work independently for 10 minutes, most likely working to finish that last question. As I walk around during the last 10 minutes of this section I make sure students are accurately identifying lengths in the pictures and labeling them on paper. I also look out for students using and copying formulas to substitute unknown values. One strategy that can be used to make sense of the third problem is to ask students to draw in the diameter and the other 5 circles side to side along the side of the square. This will help them visualize the relationship between the fraction 1/6 and the operation 2*(1/6) to find the diameter of the smaller circle.  

Throughout the group work and independent work times of this section I plant seeds of information about presentations at the end of class. Student pairs are alerted during this time that they will be asked to present their solutions to the class at the end. I try to give different students a chance to present without only selecting the high achieving students. This is a good opportunity to pair up strategic pairs and ask students who may struggle to explain specific parts of a solution or concept. These pairs will also be given an opportunity to put work on the blackboard that they will need to present their solutions. At the end this section students will be asked to stop working and track the speaker at the front of the room.

I will be selecting three different pairs of students to explain their solutions to each problem based on the observations I made during group work and independent work time. These three pairs will be called up to the front one at a time to display their work on the board. I will be pushing them to answer questions that spiral through the essential understandings:

• Areas must be subtracted when figures are inscribed and we are asked to find parts only
• Answers can be left in terms of pi or approximated with a  calculator or other forms of pi
• The sides of a polygon could shed information about the radius and diameter of a circle

Once all solutions have been presented I distribute the homework and students are dismissed for their next class.