Thought for the day: Follow up on the unexpected ideas that your students come up with. Don’t feel limited to the ideas I have written about here!
Concepts: place value patterns in tens and ones; the 100-board representation for whole numbers, recognizing and extending complex visual patterns
I notice eight “blocks” made of squares with symbols in them.
I notice that each block has a different shape.
I notice total of 25 squares in the blocks.
I notice a total of 7 different symbols
I notice that the first symbol in each square stays the same in every row.
I notice that the second symbol in each square stays the same in every column.
I notice that the blocks look sort of like puzzle pieces.
I wonder If I can put the pieces together to make a pattern.
I notice that the symbols seems to go in a certain order.
I wonder if the symbols could represent numbers (digits).
I wonder if I there is a way to figure out which symbol represents which number.
As they notice and wonder, students may try putting blocks together in different ways. The patterns and ideas that they create may come from what they have noticed and wondered. For example, they may:
Create designs by putting the blocks together in different ways. (Consider cutting them out to make a “puzzle.”}
Create patterns that they can analyze.
Create “number codes” from the symbols.
Create connections to other things they have seen in math class (such as 100-boards).
Create questions and problems that use their codes.
Create puzzles of their own like this one.
Reflecting and Extending
I notice that the pieces can be arranged into a 5-by-5 square.
I notice that I can make the first symbol in each square the same in every row.
I notice that I can make the second symbol in each square the same in every column.
I notice that the patterns in the symbols remind me of the patterns in a 100 board.
I wonder how many different codes I can make by comparing my 5-by-5 square to a number board.
I wonder if I can extend my 5-by-5 square to make a complete 100-board in symbols.
I wonder if I can make up math problems using my code.
I wonder if I can create my own puzzle like this.
This prompt was inspired by an activity from the NRICH Maths website. Their activity uses an entire 100-board while the prompt uses just a portion of it, which creates multiple possibilities for codes that connect the symbols to digits. Also, you will not necessarily tell your students about the connection to a 100-board. They can try to make this connection themselves!
The completed puzzle looks like this:
The symbols may take on different values, but once students choose a value for one of the symbols, the rest are determined automatically, because they occur in a definite order:
The wedge symbol on the left may be either 1, 2, or 3—or maybe 0 if students agree that it is okay for the tens digit to be 0! As an example: If the wedge equals 1, then the seven symbols are 1, 2, 3, 4, 5, 6, and 7 respectively.
Ask students to think about what goes wrong if the wedge represents a number greater than or equal to 4!
Once students have figured out the puzzle and imagined possible values for the symbols, they may find the place inside a full 100-board where their completed puzzle fits, try to use symbols to create an entire 100-board (they will have to invent three new symbols!), create arithmetic problems to solve using their codes, or create their own puzzles like this one!
Older or more advanced students may be able to think of the completed puzzle as a representation of patterns in a different number base!