Want to impress your students, get their rapt attention, make them want to ask “How did you do that?” in disbelief? Well, it may be time to break out “The Magic Problem”. I have used this problem over the years to get students hooked on math, to give students something they want to take home to their family members, and something they want to share with their friends. This problem is incredible, and perfect for 4th and 5th graders. I saw it many years ago and I can’t remember if it was from a website or a book, but when I saw it I knew I had to try it.

Before we get started with the nitty-gritty teacher talk, if you are working on transforming your math classroom into a space where students actively engage in their learning to practice fluency  and math concepts, grab my 10 Free Math Activities. No gimmicks, no sales, no nonsense with this freebie- just one way I can help with making math hour the favorite part of the day for every student! Looking for more practice on a math topic? Visit my TPT store. 

 

What is the Magic Problem?

The premise of this problem is that students will give you several 4-digit numbers, you will add these numbers along with a couple of your own 4-digit numbers to get a sum. The trick is, that you will predict the sum after they give you the first 4-digit number but before they give you any other numbers. How do you do it? Well, there is a method to the magic!

First things first. I always start out by asking students if it is possible to guess the answer to an addition problem before you know what all of the numbers, or addends are. After some careful thought they should answer no – not possible. Then comes the fun part – I lower my voice to a mock whisper and say…”What if I told you that I can? 

The Students Won’t Believe It

Of course they won’t believe you. So, you just have to show them and watch their amazement grow. This is great, because they are actively involved in trying to prove you wrong. Engagement is high when students are trying to prove something they are so sure about in math. First step is to ask them for a 4-digit number and write it down on a whiteboard, or large piece of paper. For this example, let’s pretend they give you 3,875. I write it at the top of the paper. 

Writing the Answer

The next step is “predicting” the answer. To do this we add 20,000 to the number and subtract 2. You can quickly rewrite the number with a 2 in front of the thousands and subtract 2 from the ones column.

 

 

Filling in the Middle

The next step is to ask for another 4-digit number. Try to steer them away from a 4-digit number with all of the same digits because it will be easy for them to see what you are doing. You want them to figure it out, but not right away! Usually the first few times they will give you totally random numbers. 

 

 

Your Turn

Now it is your turn to contribute a 4-digit number. In your number, each digit, when added to the digit directly above will add to 9. I think to myself, 8 + 1 = 9, 4 + 5 = 9 and so on. 

 

Next Steps

You are going to ask them for one more 4-digit number, which you will record. Then, you will follow up with another 4-digit number so that each set of vertical numbers add to 9. 

Wrap it Up

Okay, now comes the disbelief. I usually add it up in front of students and have them double check. When I come up with the same sum, they can’t believe it. They always want to repeat it because at this point, they think it was some kind of fluke. The more times you can repeat it the better. 

So How Does It Work?

Believe me, it is not magic. There is a logical, mathematical reason why this works every time. You may have already figured it out but here is what makes it possible to know the answer before you know all the numbers.

Remember at the beginning, when we added 20,000 to the number and subtracted two? 

 

 

When the students gave you a 4-digit number and you matched every number to sum to 9 it created a pair of numbers that equaled 9,999. Basically, you are adding 9,999 twice to your original number. Which is 20,000 minus 2!

Helping Students Figure It Out

Every time I have introduced this problem to students, they will eventually want to give you numbers like 4,444 or 1,111. Once they see your pattern matches their pattern, they begin to understand there is some kind of relationship between the two. Once they understand how it works, they will need to practice before sharing it with others. Engaging in math with a purpose is exactly what we want our students to be doing! 

For more ideas about using engaging activities to keep minds working and making the most of every minute check out this blog post by Minds in Bloom Make Every Minute Count