Why I Love This: This trick always gets kids excited and you can try it with students as young as 2nd grade. They are surprised that I am able to predict their sum even though that had “free” choice over their numbers. They start wondering if the sum is always the same and I invite them to try it on their own in small groups. Eventually the come to a common conclusion, but then the question becomes… WHY? Why does the sum not change? I challenge them to come up with a reason.
For late elementary and middle grades, you ask students to generalize — Is there a magic sum for a 5×5 or 6×6 grid? Can you predict the magic sum for a NxN grid? Exploring these questions can lead to computing sums of arithmetic series and much more!
Grade Band: 2nd – 7th
Math Content: addition, sequences, patterns
- 2.OA – Represent and solve problems involving addition and subtraction. Add and subtract within 20.
- 2.NBT – Use place value understanding and properties of operations to add and subtract.
- 3.OA – Solve problems involving the four operations, and identify and explain patterns in arithmetic.
- 3.NBT – Use place value understanding and properties of operations to perform multi-digit arithmetic
- 4.OA – Generate and analyze patterns.
- 5.OA – Write and interpret numerical expressions. Analyze patterns and relationships.
- 6.NS – Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Compute fluently with multi-digit numbers and find common factors and multiples
- 7.EE – Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Standards of Mathematical Practice:
- Make sense of problems & persevere in solving them
- Reason abstractly & quantitatively
- Construct viable arguments & critique the reasoning of others
- Model with mathematics
- Attend to precision
- Look for & make use of structure
- Look for and express regularity in repeated reasoning
Strategies to try:
- Try adding up all the numbers in the 4×4 grid. How does that sum relate to the magic sum?
- Try selecting four numbers, then change one of them. Will you have to move the other numbers so as not to break the rule of one number in each row and column? If so, what do you notice?
- Try the trick a few times and record your numbers on the grid. What do you notice when you compare your grids?
- What do you notice about the diagonal and the magic sum?
Questions to explore:
- Will this work for a 2×2 grid numbers 1, 2, 3, 4? If so, what is the sum?
- Will this work for a 3×3 grid of numbers 1-9? If so, what is the sum?
- What about a 5×5 grid?
- What about an n x n grid? What is the magic sum for any n?
- Will this work if we rearrange the rows or the columns?
- Will this work if we start from a number other than 1?
You can make a copy of this Jamboard to use with your students and let them explore their own strategies.
- One player chooses the paychecks and the other player collects the taxes.
- If possible, split students into breakout rooms with 2 or 3 students in each one. (If you place three students in a breakout, the third student can take notes and help the team develop better strategies)
- Direct each breakout room to a specific frame of the Jamboard so they can play without interfering with other students.