**Additional Resources**

**Materials Required**

**Estimated Time Needed**

*(Times are approximate and will depend on the needs of the students.)*

Lesson

- Assessment Task: Four Pentagons (15 minutes)
Have the students do this task in class or for homework a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. You should then be able to target your help more effectively in the follow-up lesson.

**Give out the assessment task,***Four Pentagons*.

**Introduce the task briefly and help students to understand the problem and its context.****Ask students to attempt the task on their own, without discussion.***Don't worry if you cannot understand everything because there will be lesson on this material [tomorrow] that will help.**By the end of the next lesson, you should expect to be more confident when answering questions like these.*

It is important that, as far as possible, students are allowed to answer the questions without assistance.

Students who sit together often produce similar answers, and then when they come to compare their work, they have little to discuss. For this reason, we suggest that when students do the task individually, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual seats. Experience has shown that this produces more profitable discussions.

- Assessing Students' Responses
Collect students' responses to the task and read through their papers. Make some notes on what their work reveals about their current levels of understanding, and their different problem solving approaches. The purpose of this is to forewarn you of issues that will arise during the lesson itself, so that you may prepare carefully.

We suggest that you do not score students' work. The research shows that this will be counterproductive as it will encourage students to compare scores, and distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this lesson unit.

We suggest you make a list of your own questions, based on your students' work. We recommend you either:

- Write one or two questions on each student's work, or
- Give each student a printed version of your list of questions, and highlight the appropriate questions for individual students.

If you do not have time to do this, you could select a few questions that will be of help to the majority of students, and write these on the board when you return the work to the students.

Common Issues: Suggested Questions and Prompts: **Student has difficulty in getting started.**- The student writes little in response to any of the questions.

*Write what you know about this diagram.**How might that information be useful?**What else can you calculate?*

**Student makes arithmetic errors.**- For example: The student writes, "Angle EJF = 180° − 144 = 46°."

*How can you be sure your answer is correct?*

**Student uses an incorrect formula.**- For example: The student does not identify the correct formula to use to find the interior angle of a pentagon (Q1).

*Find the correct formula for the interior angle of a regular pentagon.**What does n stand for in this formula?*

**Student produces a partially correct solution.**- For example: The student does not follow through the method s/he has written down.
- Or: The student calculates 540° but does not find interior angle.
- Or: The student calculates 108° or 216° but does not find angle AEJ.

*You have given an answer of [216°]. Which angle is this on the diagram? What do you need to do to complete your solution?*

**Student uses unjustified assumptions.**- For example: The student argues that supplementary angles sum to 180° without first establishing that the figure is a rhombus.

*The angles in a parallelogram are supplementary, but how do you know that this is a parallelogram?*

**Student provides poor reasoning.**- For example: The student calculates using a theorem but does not state what the theorem is.

*How do you know that this is the correct calculation to perform?**Would someone reading your solution understand why your answer is correct?*

**Student produces a full solution.**- The student provides a full and well-reasoned solution, and has justified all assumptions.

*Find another way of solving each part of the Four Pentagons problem.*