Imagine this: your kingdom is in danger! You have a special herd of tiny dinosaur creatures that are very important to you, but to others, they’re just a tasty meal. Three neighboring kingdoms have joined forces to attack your fortress and capture the herd. Your walls are strong enough to hold them off for now, but tomorrow they will bring powerful siege weapons that could break through.
Fortunately, you have a wall-building machine that can work all night to strengthen your defenses. The catch is that it can only build wall segments of a specific length, and you need to figure out what that length should be. Each enemy kingdom has a special tool called a wall-buster that can destroy wall segments of certain sizes. The first kingdom’s wall-busters are 6 meters long, the second’s are 9 meters, and the third’s are 20 meters. They can also combine their wall-busters, like using two 6-meter ones to destroy a 12-meter wall, or a 6 and a 9 to break a 15-meter wall. But a 7-meter wall would be safe from any of them.
Your goal is to find the longest wall segment that can’t be destroyed by any combination of these wall-busters. This is where a clever idea from ancient Greece comes in handy: the sieve of Eratosthenes. Eratosthenes was a mathematician who wanted to find prime numbers, which are numbers that can only be divided by 1 and themselves. He created a method to filter out non-prime numbers by crossing out multiples of each number, starting with 2, then 3, and so on.
We can use a similar method to solve our wall problem. First, we list numbers and eliminate those that can be broken by the wall-busters. If a number can be destroyed, then all numbers that are a multiple of it can also be destroyed. For example, if 9 can be broken, then 15, 21, and 27 can too, because they are multiples of 9 plus 6.
By carefully eliminating numbers, we find that 43 is the largest wall segment that can’t be destroyed by any combination of the wall-busters. So, you set your wall-building machine to create 43-meter segments. After a tense night of construction, the sun rises, and your fortress stands strong, protecting your precious herd from becoming a meal.
And that’s how you save your kingdom with a bit of math and strategy!
Imagine you are defending your fortress. Create a simulation using blocks or paper strips to represent wall segments. Use different colored markers or stickers to represent the wall-busters of 6, 9, and 20 meters. Try to find the longest segment that cannot be destroyed by any combination of these wall-busters. Discuss your findings with your classmates.
Learn about the sieve of Eratosthenes by creating your own sieve. Use a grid of numbers from 1 to 50 and cross out multiples of each number starting from 2. Identify the prime numbers and discuss how this method helps in finding numbers that cannot be broken by the wall-busters.
Write a short story or comic strip about how you used math to save your kingdom. Include characters, the challenge of the wall-busters, and the solution you found. Share your story with the class and discuss how math can be used in real-life problem-solving.
Work in groups to come up with different wall lengths that could potentially withstand the wall-busters. Use trial and error to test different lengths and document which ones work and which do not. Present your findings and explain your reasoning to the class.
Design a creative defense strategy for your fortress using the concept of wall segments. Think about other materials or methods you could use to protect your kingdom. Create a poster or presentation to showcase your strategy and explain how it incorporates mathematical thinking.
Here’s a sanitized version of the transcript:
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**Transcript:**
Bad news: your worst enemies are at the gate. Your fledgling kingdom guards the world’s only herd of tiny dinosaur creatures. To you, they’re sacred. To everyone else, they’re food. The three closest nation-states have all teamed up in what they call an alliance to breach your walls and devour the herd. Your fortifications will hold off their armies for now. But when their siege weapons arrive tomorrow, you won’t stand a chance.
Luckily, you have a wall fabricator: if you run it all night, you may be able to reinforce your border before the weapons arrive. However, it can only create wall segments of a specific size that you must determine ahead of time. Your engineers have been in close consultation with your spymaster. Each rival kingdom has wall-busters that come in one specific size. The first group’s are all 6 meters, the second’s are 9, and the third’s are 20. Each wall-buster can level a wall segment of the matching size. They can also be combined; two 6s can take out a 12-meter wall, and a 6 and a 9 could break one that’s 15 meters. But a 7-meter wall would hold fast against any of them.
Meanwhile, large walls aren’t necessarily protected. Here’s how they could take down 70, 71, and 72 meters. Your fabricator takes the same amount of time to produce a wall segment no matter its length, and it’s not particularly fast. So to finish the wall in time, you need the longest segment that can’t be destroyed by any combination of the siege weapons, which your enemies have hundreds of. What wall length will save your kingdom? Pause here to figure it out yourself.
It’s possible to solve this problem by trial and error. But there’s also a remarkably quick and elegant solution inspired by an idea that’s thousands of years old: the sieve of Eratosthenes. Eratosthenes of Cyrene was a 3rd century BCE mathematician from ancient Greece interested in prime numbers, which are numbers only divisible by 1 and themselves. Presumably, he grew bored of manually checking whether a given number was prime, so he came up with the following technique: make a giant list of numbers. Cross out all of the multiples of 2, except 2 itself. Now do the same with multiples of 3. The even multiples have already been eliminated, and the odd multiples can all be found in this column. 4 was already accounted for when you did multiples of 2, so move on to 5. The multiples of 5 and 7 show up conveniently in diagonals. This method eliminates all possible composite numbers, leaving only primes behind.
We’ve already identified every prime less than 121, and it’s easy to go higher this way. We can use a similar technique with our wall problem to eliminate entire groups of numbers at once. A first, critical step is to be deliberate about the number of columns. If we use 6 again, the numbers in each column will be exactly 6 apart. What that means is that if we identify a number vulnerable to the wall-busters, then all the rest of the column below it would also fall. In other words, because your enemies can make 9, they can make 15, 21, 27, and so on by adding the 6-meter machines. So right away this eliminates 6, 9, and 20, and everything under them.
We’ve accounted for the 6s with the columns, so we can focus on combinations of 20s and 9s to eliminate more options. Your rivals can easily make 20 plus 9 and 20 plus 20 and everything below. Using this approach, we could have eliminated 70, 71, and 72, and infinitely many other options without having to do any calculations. In the remaining column, there are no multiples of 9 or 20, but 49 jumps out, as 2 times 20 plus 9. There’s no way to make 43, so that must be the largest wall segment that your enemies can’t destroy.
And there you have it. You plug 43 into the wall fabricator, and after a tense night, the sun rises on your now fortified fortress and a herd that won’t become unhappy meals.
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This version removes any inappropriate language and maintains the essence of the original content.
Kingdom – A realm or domain, often used metaphorically in mathematics to describe a set of numbers or a particular area of study. – In the kingdom of algebra, solving equations is a fundamental skill.
Wall – A barrier or limit, often used to describe boundaries in mathematical problems or graphs. – The graph of the function had a vertical wall at x = 3, where it was undefined.
Length – The measurement of something from end to end, often used to describe the size of a line segment in geometry. – The length of the rectangle’s side was calculated to be 8 units.
Numbers – Symbols or words used to represent quantities in mathematics. – In math class, we learned how to add and subtract positive and negative numbers.
Prime – A number greater than 1 that has no positive divisors other than 1 and itself. – The number 7 is a prime number because it can only be divided by 1 and 7.
Multiples – Numbers that can be divided by another number without leaving a remainder. – The multiples of 4 include 8, 12, and 16.
Destroy – To eliminate or simplify, often used in mathematics to describe reducing an equation or expression. – To solve the equation, we need to destroy the fractions by finding a common denominator.
Segments – Parts of a line that have two endpoints, used in geometry to describe portions of a line. – The line was divided into three equal segments, each measuring 5 units.
Solution – The value or values that satisfy an equation or inequality. – The solution to the equation 2x + 3 = 7 is x = 2.
Math – The study of numbers, quantities, shapes, and patterns, often involving calculations and problem-solving. – In math class, we explored different ways to solve quadratic equations.
