Tuesday, November 24, 2015

Topologist's Map of the United States




To a topologist, two shapes are "the same" if one can be stretched, twisted, and distorted into the other shape without breaking or tearing it.  Shape, angle, size, distance, and position can be completely different, and yet the two shapes are topologically equivalent.

This is a topologist's map of the 48 contiguous United States.  While the states are not recognizable, you can work out which is which by how the states are connected to each other.  For example, which is the only state that borders exactly one state?  Can you find it on the topologist's map?  Can you identify all the states?  (One note:  Lake Michigan is included as one of the polygons, and the state of Michigan is represented as a single polygon, which was necessary to avoid some strange results.  So all together there are 49 polygons here.)

For a printable version click here.

For the solution, click here.

Thursday, October 15, 2015

Results of Mini Mental Math Survey

     Thank you to all who responded to my math survey.  People were asked to solve 100 divided by 7 mentally and to answer two questions about how they did it.  There was almost exactly a 50/50 split in how the problem was solved.  Of the 45 respondents, 22 reported using the long division algorithm in their heads, and 23 reported using some strategy that involved finding a multiple of 7 close to 100.

     Even more interesting to me was the mental images people reported.  Of the 45 respondents, 26 reported seeing digits and mathematical symbols in their heads when they performed the calculation, 8 reported seeing a spatial image such as a number line or bar (either horizontal or vertical), and 11 reported having no mental image at all.  Also interesting: there didn't seem to be a strong correlation between the strategy people used and what they saw in their heads when they did the calculation.


Thursday, October 1, 2015

Which Net?

The following problem is from an activity book on spatial thinking.  The booklet, along with the solutions, can be found at this website.  Look for more activity booklets to be available later this fall.



For the solution, click here.


Friday, April 10, 2015

Museum Guard Problem


For the solution, click here.

For more museum guard problems (and solutions), click here


Monday, March 9, 2015

Tuesday, February 10, 2015

Toothpick Perimeter Problem


It is possible to make an area as small as 2 square units (and perhaps smaller?) with 12 toothpicks!  For the solution, click here.



Monday, February 2, 2015

Stacked Dice


Solution: The sum of any two opposite faces of a die is 7.  So, excluding the top face, the sum of the numbers on the 3-dice tower is always equal to 7 x 2 x 3 = 42, and it doesn't matter how we arrange the dice.  The only thing that affects the total sum is the number showing on the top, which can be any number from 1 to 6.  So the minimum sum is 42 + 1 = 43 and the maximum sum is 42 + 6 = 48.  For 20 dice, the minimum sum is 7 x 2 x 20 + 1 = 281 and the the maximum sum is 7 x 2 x 20 + 6 = 286.  For n dice, the minimum sum is 7 x 2 x n + 1 and the maximum sum is 7 x 2 x n + 6.

Tuesday, January 20, 2015

Spilled Juice Problem

The Spilled Juice Problem

Henry had just finished his math homework when he spilled orange juice all over it.  Can you help Henry figure out what his homework problems (and solutions!) were?




For the solution, click here.





Wednesday, January 7, 2015

The Two Mangoes Problem


Solution:  

(1)  If one mango and one banana cost $4.70, then two bananas and two mangoes cost twice as much:  $9.40.
(2)  From there we can use the middle bag to find the cost of a single banana:  $13.00 (2 mangoes and 5 bananas) - $9.40 (2 mangoes and 2 bananas) = $3.60 (3 bananas).   If 3 bananas cost $3.60, then each banana costs $1.20.
(3)  From the first bag then we can see that a single mango costs $3.50 ($4.70-$1.20), and so the bag of 2 mangoes costs $7.00.