There are only a few slides in this puzzle so before heading on to the next one, your students and you should study the image and come up with conjectures about how the images are created and what to expect on the next slide. This first slide has too little information to speculate wisely… so start speculating with the next slide.

This is an example of a mini mathematical universe. It is to be used to teach the scientific method in the math classroom.

What is going on here?

Your students and you should make conjectures about what is going on and what will appear on the next slide.

List observations from your class. Here are examples:

  1. All twenty-one of the images are composed of four squares.
  2. All squares in an image lie on a vertical axis.
  3. If squares intersect – they intersect at two points – never an edge.
  4. These are Venn diagrams.

The last student idea is wrong, but you would put it up on the board for discussion without passing judgment. What do your students expect on the last slide? Perhaps students will manage to predict the bottom left figure? Maybe the whole left column?

Andy Juell created a numerical version of this same pattern:

1 (7, 5, 3, 1) 2 (7, 5, 1, 1) 3 (7, 4, 2, 3) 4 (7, 4, 1, 2) 5 (7, 3, 1, 1) 6 (7, 2, 3, 2) 7 (7, 1, 1, 1) 8 (6, 4, 5, 1) 9 (6, 4, 1, 3) 10 (6, 3, 5, 2) 11 (6, 3, 1, 2) 12 (6, 2, 5, 1) 13 (6, 1, 4, 1) 14 (6, 1, 2, 3) 15 (6, 1, 1, 2) 16 (5, 2, 4, 3) 17 (5, 1, 4, 2) 18 (4, 5, 1, 2) 19 (4, 1, 3, 2) 20 (3, 5, 2, 2) 21 (2, 5, 1, 2)

See if you can figure out which of the 21 square patterns are connected to each of Andy’s numerical patterns.

Look at the vertical axis going through each clump of squares. It is intersected at regular intervals by exactly one square. Did your students find that? More difficult is to figure out how the squares are organized.

This image is by Andy Juell who discovered that I had missed some square patterns. Thank you Andy!

Andy didn’t stop here – his computer program actually calculated the number of square dance patterns for six squares. Get your class to guess how many. The answer is 1493 as pictured! 

Thank you so much Andy – for finding the errors and extending the beauty!

Square Dance

(MathPickle, 2016)

This mini mathematical universe is used to get your students generating and discussing true and false conjectures. Make sure to reward true AND false conjectures. If students are hesitant to come up with false conjectures you should suggest your own or prod them into making one.

After you look at the slide show above you might be interested in seeing Andy Juell’s numerical pattern that describes the 160 patterns that appeared in the second to last slide:

1 (9, 7, 5, 3, 1) 2 (9, 7, 5, 1, 1) 3 (9, 7, 4, 2, 3) 4 (9, 7, 4, 1, 2) 5 (9, 7, 3, 1, 1) 6 (9, 7, 2, 3, 2) 7 (9, 7, 1, 1, 1) 8 (9, 6, 4, 5, 1) 9 (9, 6, 4, 1, 3) 10 (9, 6, 3, 5, 2) 11 (9, 6, 3, 1, 2) 12 (9, 6, 2, 5, 1) 13 (9, 6, 1, 4, 1) 14 (9, 6, 1, 2, 3) 15 (9, 6, 1, 1, 2) 16 (9, 5, 3, 1, 1) 17 (9, 5, 2, 4, 3) 18 (9, 5, 1, 4, 2) 19 (9, 5, 1, 1, 1) 20 (9, 4, 5, 1, 2) 21 (9, 4, 2, 3, 1) 22 (9, 4, 1, 3, 2) 23 (9, 4, 1, 2, 1) 24 (9, 3, 5, 2, 2) 25 (9, 3, 4, 2, 1) 26 (9, 3, 1, 3, 1) 27 (9, 3, 1, 1, 1) 28 (9, 2, 5, 1, 2) 29 (9, 2, 4, 1, 1) 30 (9, 2, 3, 2, 1) 31 (9, 1, 3, 1, 1) 32 (9, 1, 1, 1, 1) 33 (8, 6, 7, 3, 1) 34 (8, 6, 7, 1, 1) 35 (8, 6, 4, 2, 5) 36 (8, 6, 3, 1, 3) 37 (8, 6, 2, 3, 4) 38 (8, 6, 1, 5, 1) 39 (8, 6, 1, 2, 4) 40 (8, 6, 1, 1, 3) 41 (8, 5, 7, 4, 1) 42 (8, 5, 7, 2, 3) 43 (8, 5, 7, 1, 2) 44 (8, 5, 3, 1, 2) 45 (8, 5, 2, 6, 2) 46 (8, 5, 1, 3, 4) 47 (8, 5, 1, 1, 2) 48 (8, 4, 7, 1, 1) 49 (8, 4, 5, 6, 2) 50 (8, 4, 5, 1, 3) 51 (8, 4, 2, 6, 1) 52 (8, 4, 2, 3, 2) 53 (8, 4, 1, 5, 1) 54 (8, 4, 1, 2, 2) 55 (8, 3, 7, 4, 1) 56 (8, 3, 7, 2, 1) 57 (8, 3, 5, 6, 1) 58 (8, 3, 5, 2, 3) 59 (8, 3, 4, 6, 2) 60 (8, 3, 4, 2, 2) 61 (8, 3, 1, 4, 1) 62 (8, 3, 1, 2, 3) 63 (8, 3, 1, 1, 2) 64 (8, 2, 7, 3, 1) 65 (8, 2, 7, 1, 1) 66 (8, 2, 5, 2, 4) 67 (8, 2, 5, 1, 3) 68 (8, 2, 4, 5, 2) 69 (8, 2, 4, 1, 2) 70 (8, 2, 3, 5, 1) 71 (8, 2, 3, 2, 2) 72 (8, 1, 6, 3, 1) 73 (8, 1, 6, 1, 1) 74 (8, 1, 4, 5, 1) 75 (8, 1, 4, 1, 3) 76 (8, 1, 3, 5, 2) 77 (8, 1, 3, 1, 2) 78 (8, 1, 2, 5, 1) 79 (8, 1, 1, 4, 1) 80 (8, 1, 1, 2, 3) 81 (8, 1, 1, 1, 2) 82 (7, 5, 6, 1, 4) 83 (7, 5, 3, 6, 4) 84 (7, 5, 2, 6, 3) 85 (7, 4, 6, 3, 5) 86 (7, 4, 6, 1, 3) 87 (7, 4, 2, 6, 2) 88 (7, 4, 2, 5, 3) 89 (7, 4, 1, 5, 2) 90 (7, 3, 6, 2, 3) 91 (7, 3, 4, 6, 3) 92 (7, 3, 1, 4, 2) 93 (7, 2, 6, 5, 1) 94 (7, 2, 6, 2, 4) 95 (7, 2, 6, 1, 3) 96 (7, 2, 4, 5, 3) 97 (7, 2, 3, 5, 2) 98 (7, 2, 3, 4, 3) 99 (7, 1, 6, 4, 1) 100 (7, 1, 6, 2, 3) 101 (7, 1, 6, 1, 2) 102 (7, 1, 5, 2, 4) 103 (7, 1, 5, 1, 3) 104 (7, 1, 2, 5, 2) 105 (7, 1, 2, 4, 3) 106 (7, 1, 1, 4, 2) 107 (6, 7, 5, 1, 4) 108 (6, 7, 3, 4, 5) 109 (6, 7, 3, 1, 2) 110 (6, 7, 1, 5, 2) 111 (6, 7, 1, 3, 4) 112 (6, 7, 1, 1, 2) 113 (6, 4, 2, 5, 2) 114 (6, 4, 1, 5, 1) 115 (6, 3, 1, 4, 1) 116 (6, 3, 1, 3, 2) 117 (6, 2, 3, 5, 1) 118 (6, 2, 3, 4, 2) 119 (6, 1, 5, 3, 4) 120 (6, 1, 5, 1, 2) 121 (6, 1, 4, 5, 3) 122 (6, 1, 2, 5, 1) 123 (6, 1, 2, 4, 2) 124 (6, 1, 1, 4, 1) 125 (6, 1, 1, 3, 2) 126 (5, 7, 4, 1, 2) 127 (5, 7, 2, 4, 3) 128 (5, 7, 2, 3, 2) 129 (5, 7, 1, 2, 2) 130 (5, 6, 1, 4, 3) 131 (5, 3, 6, 4, 3) 132 (5, 2, 6, 2, 2) 133 (5, 1, 3, 4, 2) 134 (4, 7, 5, 3, 4) 135 (4, 7, 5, 2, 3) 136 (4, 7, 1, 4, 1) 137 (4, 7, 1, 2, 3) 138 (4, 7, 1, 1, 2) 139 (4, 6, 3, 5, 3) 140 (4, 6, 1, 4, 2) 141 (4, 5, 6, 2, 2) 142 (4, 5, 1, 4, 1) 143 (4, 5, 1, 3, 2) 144 (4, 2, 6, 2, 3) 145 (4, 2, 6, 1, 2) 146 (4, 2, 5, 4, 2) 147 (4, 1, 5, 2, 3) 148 (4, 1, 5, 1, 2) 149 (4, 1, 3, 4, 1) 150 (3, 7, 5, 1, 3) 151 (3, 7, 4, 1, 2) 152 (3, 7, 2, 2, 3) 153 (3, 7, 2, 1, 2) 154 (3, 5, 6, 1, 2) 155 (3, 5, 2, 3, 2) 156 (3, 4, 6, 2, 2) 157 (2, 7, 3, 1, 2) 158 (2, 7, 1, 1, 2) 159 (2, 5, 2, 4, 2) 160 (2, 5, 1, 3, 2)

 

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

 
MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

 
MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.

(http://www.corestandards.org/Math/Practice/)

Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)

 

Lora Saarnio

(CEO)