Around and Around: Investigating Multiples

In my prior posts (When Factors Put on Their Dancing ShoesWhen Factoring Gets PersonalOpen the Safe, and Reasoning with Multiples to Find the Mystery Number), I’ve given examples of how learning about multiples and factors can be made more engaging through the use of puzzles. Now, to add to this collection, is the lock puzzle below.

The first page of the lock puzzle shows a lock consisting of a single dial with 4 tick marks. The goal is to open the lock by having its pointer move clockwise around the dial and end pointing straight up, back where it began. In the lock’s initial state, notice that the pointer is set to move 3 times. Press Go Slowly or Go Quickly — you’ll discover that the pointer doesn’t quite make it once around the dial. If you press Reset, change the “Pointer Movement” value to 4,  and set it in motion again, you’ll see that the pointer makes one complete revolution and opens the lock.

For students just learning about multiples, this lock makes for an engaging model. They soon discover that while 4 is the most efficient value for opening the lock, they can also enter any multiple of 4 to get the job done. Students are eager to make new locks with more ticks by entering a new value for the parameter “ticks.” And as you can well imagine, they enjoy experimenting with locks with considerably more than just 4 ticks on the dial!

To view the second lock, press the right arrow in the bottom-right corner of the sketch. This new lock ups the ante by adding a second dial. Now the goal is to determine how many times the pointers should move so that when they come to a rest, both are simultaneously pointing straight up, back in their starting positions. As with the one-dial lock, students can change the number of ticks on each dial to create new locks of their own.

And finally the third page of this model presents one further evolution of the lock—a lock with three dials.

Below are just some of the questions that you or your students might ask about the three locks. Let me know what other investigations your students suggest!

  • Will more than one number open a lock? What do these numbers share in common?
  • How can you use the one-dial lock to identify the number closest to 1,000 that is a multiple of 17?
  • When does the minimum number of moves needed to open a lock equal the product of the ticks on its dials?
  • How can we use the two-dial lock to check if two numbers have any factors in common other than one?
  • How many two-dial locks can you make that take a minimum of 30 moves to open? How about 16 moves?
  • Suppose you commission me to build a two-dial lock. The two dials must be different and each must have more than one tick mark. You specify how many moves for the lock to open, but I reply, “That’s impossible!” What are some possible numbers you might have proposed?
  • How many three-dial locks can you make that take a minimum of 210 moves to open?
  •  Suppose we decide that each dial on the two-dial lock must have more than one but fewer than 10 ticks.  Which such lock will take the most moves to open?
  • Describe a two-dial lock that would take a really long time to open.
  • Create a two-dial lock that takes approximately one minute to open once its pointers are set in motion (You decide whether the pointers move slowly or quickly.) Create a lock that will open in approximately five minutes. How about one class period or one school day? Be sure to check!

An annotated list of all our elementary-themed blog posts is here.

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