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Students will try to find a path across this city which crosses each bridge exactly once.
First, students will cross five bridges exactly once.
Next, they’ll try to cross seven bridges one time each. ⚠️ Note: This is impossible.
They come up with three more layouts using seven bridges. At least one should work.
Students count how many bridges come out of the cities. They’re looking for patterns to determine whether a map will work.
Finally, they try four other maps and learn about Euler and the Seven Bridges of Königsberg!
Students will find several solutions for magic triangles of various sizes.
First, students find a solution that adds up to 9 for an order-3 magic triangle.
Next, they find a solution that equals 17 for an order-4 magic triangle.
They revisit the order-3 triangle and find the remaining three solutions.
Finally, your students will return to the order-4 triangle to find as many solutions as possible. (There are 18 unique solutions).
Students will search for patterns of prime numbers within a triangle made famous by Laurence Klauber.
Students build their own Klauber triangle.
Then they highlight all of the prime numbers, looking for patterns.
I reveal a giant triangle and then challenge them to create their own shape to look for prime patterns.
Students will generate an Ulam Spiral, highlight the primes, and note what patterns they see.
Students arrange the first 100 (or so) integers into a spiral.
They will highlight (or circle) only the primes in their spiral, looking for patterns.
Finally, they will either extend their spiral or try to create a new shape or spiral and look for patterns.
Students will search for patterns, patterns, and more patterns within the fascinating Pascal’s Triangle.
Students will look for one pattern in this triangle and then use that pattern to add another row.
After I build out the triangle a bit more, students will search for even more patterns.
I show one set of interesting patterns, reveal the triangle’s name, and then point students towards more resources.
Goldbach’s Conjecture states that, “Any even number can be written as the sum of two primes.” Is it true?
Students will see that any positive integer is also the sum of four or fewer perfect squares.
Students find another solution for 10.
They see how many solutions they can find for 50 – and look for patterns along the way.
They work with 99 and any other number they’d like to explore.
Students arrange integers into squares so that each row, column, and diagonal will add up to the same sum.
I introduce the idea of a magic square and then start students off with a 3×3 square that has 5 in the middle.
As an extra hint, I reveal that the sums must all equal 15.
Finally, I reveal the solution and challenge your students to try a 4×4 magic square!
How few colors do you need to color in any map so that no two neighboring regions are the same color?
First, we introduce the idea of coloring in regions on a map with a very simple example that needs only three colors.
Then, we increase the challenge a bit with a second map that still only needs three colors.
Next, we present an even more challenging map.
We reveal the coloring problem’s true solution: no map needs more than four colors.
Students will try to explain why the first X odds add up to the same number as X2.
Students divide equilateral triangles over and over to create a Sierpinski Triangle.
Students learn to create their own Sierpinski Triangle by starting with an equilateral triangle.
Then, they create Sierpinski Carpets by starting with a square.
Finally, they experiment with three-dimensional versions, perhaps creating a Menger Sponge using Lego or in Minecraft.
Students will create a fractal known as The Koch Snowflake.
Students first create a Koch Curve – a simplified version of the Koch Snowflake.
They’ll take their curve from step 1 and extend it to become a snowflake.
Finally, students will create new versions of the Koch Snowflake by experimenting with different starting shapes.
Students will explore the unproven Waring’s Conjecture.
Students will determine if all perfect squares can be written as the sum of two primes.
Students will work with primes and perfect squares to investigate Legendre’s Conjecture.
Students will use the prime sieve to find primes up to 150.
Students will search for special primes and look for patterns along the way.
First, students search for twin primes: prime numbers with a difference of two.
Then, they look for cousin primes: prime numbers with a difference of four.
Finally, they look for prime quintuplets: five primes in which the difference between the largest and smallest is 12.
Turn any number into a palindrome by following these steps…
Start with any number and get to 1 using just two rules. It seems to always work…