Problem Solving activity – Arithmagons are number puzzles. They require a mix of mathematical reasoning and basic fact knowledge to solve. These number puzzles are an excellent way to explore basic facts.
13 Printable Arithmagons
are number puzzles. They require a mix of mathematical reasoning and basic fact knowledge to solve. These number puzzles are an excellent way to explore basic facts.
What is an Arithmagon?
Arithmagons number puzzles with an empty circle sitting on each vertex of a regular polygon (in this case a triangle) and a number sitting on each of the adjoining lines. The number on each line is the sum of the two circles.
What to Do?
Use mathematical reasoning to work out what numbers are required in each circle so the sum of the two circles equals the number on the center of the line.
Skill Focus – This activity is wonderful for having students practice the problem-solving skill of, GUESS, CHECK, IMPROVE.
What You Need:
Arithmagon Puzzles (laminated or in plastic sleeves)
– I believe more teachable moments arise out of introducing these puzzles to a group of 6 or 8 students first. Then have those students pair off and introduce other small groups of students to the puzzles.
What to Do:
1. Explain what Arithmagons are and how they work.
2. Explain and demonstrate how the problem-solving skill of, ‘Guess, Check and Improve’ works.
3. Discuss why this strategy is often a good place to start when solving problems.
4. Demonstrate problem-solving skill using the online version of the puzzle. Be sure to speak your thinking / internal dialogue aloud as a model for students.
5. Have a group of students distribute the puzzles and counters. Students begin solving puzzles.
6. Note the behaviors you see that may need addressing e.g. ‘Whining in this class is not an effective problem-solving strategy’ hehehehe
7. About now is often a good time to discuss the ‘A-ha’ Feeling ie The feeling you get when you finally achieve success on something you have worked hard on. I often use the line when students are begging to be given the answer, ‘Who am I to rob you of the a-ha feeling?’ Ham it up! 🙂
8. Students complete and reflect on these questions in their Math Journals – a) What is an arithmagon? Describe what it looks like. b) Describe how arithmagons work. c) Outline how you solved an arithmagon puzzle. d) What strategies did you use to help you find solutions to these puzzles?
Session Two – Guided Discovery of Math Laws
1. Mathematicians throughout history have dedicated their lives to discovering, ‘Order in the Chaos’. What could this mean?
2. ‘Mathematicians look for patterns in order to explain the world.’ Discuss.
3. Examine a solved arithmagon. Add all the numbers in the circles. What is the total?
4. Add all the numbers in the squares. What is the total?
5. Repeat steps 3 and 4 for three or four. Notice anything?
6. Divide the sum of the circles on an arithmagon by two. Notice anything now?
7. Add the number in each circle with the number in the square diagonally opposite. Can you see a pattern emerging?
8. Based on this investigation and observations, write a math law for solving triangular arithmagons.
Before the Puzzle Possible Homework Task:
– Choose a famous mathematician from the past or present, try to find one the rest of the class has never heard of. Write a brief paragraph about their life’s work and how it affects our lives today.
e.g. Kepler and how planets move, Fibonacci and number pattern in nature, Newton and gravity, Mandelbrot & fractals, Descartes and co-ordinate geometry, Ada Lovelace’s first computer program, Al-Khwarizmi, Aryabhatta… See who you can find.
After the Puzzle Reflection:
– What have you learned about Math by investigating arithmagons? NB This question is a ‘big picture’ one. It is not asking you to describe arithmagons and how they work.
– Create an arithmagon that uses multiplication
– Create an arithmagon that uses decimals
– Traditionally arithmagons use only the numbers from 1 to 10. Is it possible to create arithmagons using the numbers 10 through 20? If so, create some for your friends.
– Develop a method of creating your own arithmagons using a given number of counters e.g 21, 26, 31 etc Is it possible? Publish these puzzles using computer software and distribute them for your friends to solve.