Why I love this problem:
First, you can easily make this hands-on by asking students to stand in a circle, choosing a “skip number”, and then passing a ball around to every nth person. Students can see for themselves exactly who touches the ball and who gets left out.
Second, once you start drawing stars, you start getting beautiful patterns!
Third, students will discover patterns involving primes, composites as well as the notion of “relative prime”. Two integers are relatively prime (or coprime) if there is no integer greater than one that divides them both
Tip: Start with the physical versions of this activity if possible.
Grade Band: 4th – 6th
Math Content: Prime and composite numbers. Factors and multiples. Angles.
- 4.OA – Gain familiarity with factors and multiples
- 4.OA – Generate and analyze patterns.
- 5.OA – Analyze patterns and relationships.
- 6.NS – Compute fluently with multi-digit numbers and find common factors and multiples.
Standards of Mathematical Practice:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
Strategies to try:
- Start with a small number of points and explore all the different skip numbers.
- Then add one more dot and repeat
- Pause to look for patterns.
- What do you notice about the times when not every point is included in the picture and when every point is included?
- What happens if you double the number of points?
Questions to explore:
- When do you get “pointy stars”?
- When do you some of the dots get skipped?
- Which stars are the prettiest to you?
- What are the angles are formed?
- If you had 24 points, would every skip number create a star? What about 23 points?
Here are two Desmos activity builders you can use. Students can use these to quickly draw and explore many examples of stars.