This Factors and Multiples Puzzle from NRICH was a perfect way to start the school year with my sixth grade students. It is one of the best low threshold high ceiling tasks I've used and creates opportunities for students to work together and discuss numbers and sets of numbers while they review essential vocabulary in the context of an interesting challenge.

Here are some things that I did the day before we worked on the puzzle that I think helped students get started right away and work on the puzzle for the entire class period.

1) I did notice and wonder with square numbers and triangular numbers using these graphics which I recreated based on inspiration from this poster

2) I did some mini-challenges where students, using small whiteboards, wrote down as many numbers as they could think of that meet two conditions. Students loved this!

- Think of numbers that are
**Odd**and**Prime** - Think of numbers that are
**Even**and**Multiples of 5** - Think of numbers that are
**Less than 20**and**Factors of 60** - Think of numbers that are
**Square Numbers**and**Multiples of 3** - Think of numbers that are
**More than 20**and**Triangular Numbers** - Think of numbers that are
**Prime**and**Square Numbers**

3) I think this is a very good activity for partners. I copied half of the heading cards and number cards on a different color of copy paper so that two sets of partners working at the same table would not mix up their cards.

4) It was very easy for the students to accidentally disturb some or all of their pieces which caused some frustration. Next year I plan to enlarge the puzzle and copy on card stock and/or laminate. Several different colors of heading and number cards would also be helpful.

5) When partners believed that they had completed the puzzle correctly, I asked another partnership to look at their puzzle to see if they could find any errors. This on the fly extension helped push the ceiling even higher for early finishers.

5) When partners believed that they had completed the puzzle correctly, I asked another partnership to look at their puzzle to see if they could find any errors. This on the fly extension helped push the ceiling even higher for early finishers.

Here are some photographs of completed puzzles from my classes. There is at least one error on each of these puzzles and I am thinking I could use these photos next year to help introduce the activity and/or use as an extension.

This is a very cool chart that one of my students created on her own to help her logically and methodically determine where heading cards and number cards could or could not be placed in order to solve the puzzle.

And here is a link to some correct solutions to the puzzle posted on the NRICH website.