I have subscribed to The University of Waterloo CEMC Problem of the Week (follow this link to view their page and subscribe). One of the problems was called "Don't Get Vexed By this Hex!" and I presented the problem to my students during our Intervention time.
Suppose you had 11 green triangles, 5 rhombuses, and 5 trapezoids.
- How many hexagons congruent to the pattern block hexagon can you create?
- How many shapes are left over?
Once I posed the problem, I asked the students to make a prediction to the number of hexagons that would be created and the number of pattern blocks that would be left over. Next, I gave each student di-cut pattern blocks to manipulate to discover the answer (6 hexagons, no remainders). We glued or taped these to a piece of construction paper. Then, to explain why we did not have left over pieces mathematically, I used the "Something to Think About" portion of the answer given by CEMC.
With the remaining time, we explored what would happen if we changed the value of each shape. We began by making the value of the triangle = 1 whole, then we made the value of the rhombus = 1 whole, then we worked with the trapezoid = 1 whole, and finally the hexagon became our whole. With each change, we found the value of the other shapes (triangle, rhombus, trapezoid, and hexagon).
We were even able to create a figure with pattern blocks and used this as our whole and found the value of the other pattern blocks!