The Magic of Math: Fun with Final Activities
By Kathleen Palmieri
The school year is winding down and the prescribed math curriculum is in its final pages.
This is the time when I like to offer class activities that not only help to review concepts and skills, but allow students to engage in math conversations and have fun with the magic of math.
My students are fifth graders and our activities are grade-appropriate but teachers in higher graders might find some fun things here to try or adapt as well. Dr. Shah (below) has plenty of ideas!
Six Digits Become Three
The first activity is a student favorite! It involves using place value, division, multiplication, a lot of student discussion and wonder!
►Take any three digit number – 671
►Write it twice to make a six digit number: 671,671
►Next, divide that number by 7, 11, and 13 (the order you divide in does not matter).
Example: 671,671 ÷ 7 = 95,953; 95,953 ÷ 11 = 8,723; 8,723 ÷ 3= 671
The final quotient will be the number that you originally started with.
How did this work? Let’s break it down:
►Any three digit number has a ones place, tens place, and hundreds place.
So that means 671= 6 x 100 + 7 x 10 + 1
►Now, what happens when you write the three digit number twice, in this case 671,671?
Well, now we have 6 x 100,000 + 7 x 10,000 + 1 x 1,000 + 6 x 100 + 7 x 10 + 1
What do you notice? Each digit is used twice.
►Look all the way to the right and notice the 1 has a value of 1, but the next time the 1 is used it is being multiplied by 1,000.
►What about the 7? First it was multiplied by 10 then by 10,000.
►Finally. the 6 was first multiplied by 100 then 100,000.
►The 671 at the start is really the same as the original 671 but was multiplied by 1,000.
Factor out the 1,000 and you have:
= 1,000 (6 x 100 + 7 x 10 + 1) + 6 x 100 + 7 x 10 + 1
►Now, collect like terms (6 x 100) (7 x 10) + 1. There are 1,000 of them on the left and one of them on the right. So, if you cancel out the like terms you get 1,001 which will bring you back to the original 671.
►The trick is actually the 1,001 because if you factor it, it is 7 x 11 x 13, the numbers that were divided in the beginning to make this trick work. When we divide by 7, 11, and 13, it will cancel out the 1,001 and the number left is 671.
I should probably warn you that your students will continue to use this “trick” with different numbers for many days after it’s been done in class.
A few months ago I attended the Midschoolmath virtual conference where I was introduced to the work of Dr. Raj Shah. His presentation was amazing, offering several math problems that I then shared with my students.
The “Handshake Scenario” is the problem that was hands-down my students’ favorite – pun intended! Here’s how it works:
►Imagine a party where every guest shakes hands with every other guest exactly once.
►Question: If there are 15 people at the party, how many handshakes are there?
Dr. Shah offers these strategies:
Solve a simpler problem – Students will start with a small number of people like 3 and try to count the handshakes. The first person shakes hands with two others, then the second person shakes with the third person. That’s three total handshakes. Students can continue this logic for larger numbers of people to solve the problem.
Draw a picture – Students can draw a diagram with dots to represent people and lines connecting people that represent the handshakes. Then they can count the lines. This strategy goes well when matched with the one above.
Here is the catch with the problems Dr.Shah generously shared: he does not reveal the answers! While at first that is a bit alarming, once logic sets in it makes perfect sense. What do we want our students to do? Work through problems, break them down, and truly make sense of the information provided to get to the solution. As educators, we should also dig in and solve the problems too!
As Dr. Shah writes, “I (almost) never tell the answers because the moment a person finds out the answer, all thinking stops!” However, he writes, “If you try these for yourself and you are stuck, talk to a friend. If you are desperate, send me an image of your work so far and I’ll write back with some questions that might give you some hope.”
I’ve included links below if you’d like to learn more about incorporating “magical math” problems into your lessons.
Dr. Raj Shah – I Help Teachers Make Math Irresistible
Kathleen Palmieri is a National Board Certified Teacher and NBCT Professional Learning facilitator. She is a fifth grade educator in upstate New York who reviews and writes regularly for MiddleWeb. With a passion for literacy and learning in the classroom, she participates in various writing workshops, curriculum writing endeavors, and math presentations and loves to blog about ELA and math topics.
As a lifelong learner, Kathie is an avid reader and researcher of educational practices and techniques. Collaborating with colleagues and globally on Twitter @Kathie042500 and expanding her education adventures at www.kathleenpalmieri.com are ongoing practices.
Using magic with numbers is a great way to interest students in algebra. You can catch their interest and then use the algebra a a way to show why the magic works. I have a collection of “Magic with Numbers” which can be found at http://bit.ly/lennyvalgebra
Kathleen Palmieri and her readers may be interested in another article about the classic “handshake” problem. See my post.
It’s a wonderful problem to use (in non-pandemic times) as either an introduction to or a practice for Polya’s problem solving process (especially “act it out”, “make a table”, and “find a pattern”). Enjoy!