This problem is available as a printable worksheet: Polygon Pictures.
This problem offers an engaging context to practise applying knowledge about angles in polygons. There is an opportunity to use Dynamic Geometry programs such as GeoGebra for students to recreate the patterns in the problem and create patterns of their own.
You may wish to show these slides of the two pictures from the problem.
"Here is a pattern made by rotating a regular polygon by a fixed angle around one of its vertices. Can you see what polygon was used? How could you work out what angle it was rotated by?"
Give students a short while to think, then discuss with their partner.
Then share answers with the rest of the class.
"Let's list all the properties about angles that might be useful in finding all the missing angles in the picture."
List students' ideas on the board, then hand out the worksheet.
Invite them to work in pairs to write in as many of the missing angles as they can.
They could colour-code the angles and then use the space under the picture to explain how they worked out each angle.
Finally, bring the class together to share how they calculated different angles. Draw attention to the fact that some angles can be calculated in more than one way, allowing students to check their solutions make sense.
Which polygon was used?
What do you know about the angles in regular polygons?
What shapes are formed by the overlaps?
Angles and Polygons Short Problems 1 star Sheet 3 would make a good pre-lesson task to give students some angle calculation practice before tackling this problem.
Students could use GeoGebra to recreate the images and create similar images of their own.
The thinking in this task would be good preparation for the problem Semi-regular Tessellations.
The two star Angles and Polygons Short Problems worksheets may also make a good extension.