I created the above WODB question for a group of elementary teachers. I spent quite a bit of time creating and revising it (you can see my drafts below). Making a good WODB problem takes a bit of practice I've found. I'm a bit embarrassed about how poorly designed (and sometimes goofy) some of my earlier attempts were. Here are some suggestions why each image might not belong: Top Left: It is the only shape with more than one interior shape (a triangle and two trapezoids). It is the only shape that is not composed of congruent shapes. The only shape without rotational symmetry. Top Right: It is the only shape with 4 interior shapes. It is the only shape whose interior shapes are all similar to its exterior shape. It is the only shape with a single angle measure (60 degrees). Bottom Left: It is the only shape with right angles in it. It is the only triangle that is point down (it tips over). Bottom Right: It is the only shape which includes concave shapes. It is the only shape with interior reflex angles in it. The only shape without reflective symmetry. Updated 9/2/2016  Added some additional suggestions regarding symmetry from Jennifer Bruce. I've been working with Van Hiele's levels of geometric thinking lately and how it relates to the geometry outcomes in the NS Mathematics curriculum. This got me thinking about how a student's level of geometric thinking relates to WODB problems. In designing the problem above, I was intentionally trying to create something aimed at students at Van Hiele's descriptive level vice the visual level. Given a student's level of geometric thinking, how might they respond to this problem? Are some WODB problems better suited for students at a visual or descriptive level or are WODB problems inherently differentiated and students at all levels can relate meaningfully to these style of problems. Another question that I've been thinking about recently is how to use WODB questions. Should I use them to introduce new terminology or to review terminology that students have already been exposed to. For example, in the problem above, I made sure to include a concave shape. Would it be more effective to wait until students learn concave and convex polygons and then reinforce it with this problem or should I introduce the vocabulary here and then talk about its definition? These are questions that I'm still exploring. Nova Scotia Mathematics Curriculum Outcomes Grade 2 G03  Students will be expected to recognize, name, describe, compare and build 2D shapes, including triangles, squares, rectangles, and circles. Grade 3 G02  Students will be expected to name, describe, compare, create, and sort regular and irregular polygons, including triangles, quadrilaterals, pentagons, hexagons, and octagons according to the number of sides. Grade 4 G02  Students will be expected to demonstrate an understanding of congruency, concretely and pictorially. Grade 5 G02  Students will be expected to name, identify, and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms, and rhombi, according to their attributes Grade 6 G01  Students will be expected to construct and compare triangles, including scalene, isosceles, equilateral, right, obtuse, or acute in different orientations. Grade 6 G02  Students will be expected to describe and compare the sides and angles of regular and irregular polygons. Grade 9 G02  Students will be expected to demonstrate an understanding of similarity of polygons. EL
12 Comments
The final chapter of the Math at Work 12 textbook deals with Trigonometry and the Law of Sines and Law of Cosines. Towards the end of the chapter there is a puzzle (p351) that asks students to create a triangle using 9 of the numbers from 1 to 10. Each side of the triangle is the sum of 4 of these numbers. I liked the construct of this puzzle but I wasn't a big fan of the questions that it asked students so I decided to give it an overhaul. An image from the textbook is below.
We followed up this warmup with an open middle style problem using the same situation that would require students to apply the law of cosines. A challenge like the one below gives the students a reason to practice the law of cosines without feeling tedious or repetitive. Directions: Use the numbers 19 (using each number no more than once) to fill in the circles. The sum of the numbers on each side of the triangle is equal to the length of that side. What is the triangle with the largest (or smallest) angle that you can make? Hints:
A triangle with the largest angle (there are several variations with the same angle): Side A: 6+8+9+7=30 Side B: 7+4+1+3=15 Side C: 3+2+5+6=16 Angle A: 150.799 Angle B: 14.119 Angle C = 15.082 A triangle with the smallest angle (there are several variations with the same angle): Side A: 1+2+3+4=10 Side B: 4+8+9+7=28 Side C: 1+5+6+7=19 Angle A: 10.844 Angle B: 148.212 Angle C = 20.944
Another challenging question that could be asked is how many different arrangements of the numbers 1 to 9 in the triangle diagram could you make? You have to consider that rotations of the triangle are the same. This would be a challenging combinatorics question even for Precalculus 12 students. Nova Scotia Mathematics Curriculum Outcomes Mathematics 11  G03 Solve problems that involve the cosine law and the sine law, including the ambiguous case. Math at Work 12  G01 Students will be expected to solve problems by using the sine law and cosine law, excluding the ambiguous case. Math at Work 12  N01 Students will be expected to analyze puzzles and games that involve logical reasoning, using problemsolving strategies. Mathematics 12  LR01 Analyze puzzles and games that involve numerical and logical reasoning, using problemsolving strategies Precalculus 12  PC03 Determine the number of combinations of n different elements taken r at a time to solve problems. EL
Coins in a Row is a simple game that I have used in lots of classrooms at different grade levels. It is a great way to tackle some mental math and logical reasoning. I first learned about this game in an article from Ivar Peterson's Math Trek. The game originally appeared in Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection (2004). The Rules: Arrange a bunch of coins of various denomination in a row (any even number of coins will do). The first player chooses a coin from one of the ends and slide it over to her pile. Your opponent then chooses a coin from one of the ends of the row of remaining coins. You and your opponent take turns removing a coin in this manner until they are gone. The player with the highest total value of coins wins. The fun part of this game is that it is unfair. Unfair games are a great way to get out of doing some chores while you're visiting inlaws over the holiday break (e.g. "Lets play a game, the loser has to wash the dishes"). The player who goes first can always win if they know the correct strategy. After playing this game a few times, I let students know that it is an unfair game and ask them if they can find a winning strategy. Here is how the winning strategy works. Label the coins from 1 to n going from left to right. Add up the value of all the odd labeled coins then add up the value of all the even labeled coins. The first player can choose either an odd or even labeled coin. The second player only has two evens to choose from if the first player took the first coin from the row. The second player only has two odd coins to choose from if the first player to the coin from the end of the row. The first player then just takes a coin from whichever end the second player chose from and the first player can guarantee that they get either all the odd or even labeled coins. This strategy guarantees at least a tie for the first player. It is not necessarily the "best" strategy however. According to Ivars Peterson, no one has yet worked out an optimal strategy that works for any number of coins.
Alternate versions are quite easy to create for this game. For example, Coins in a Circle. In this version, the first player can pick any coin, then play continues as in the previous version. Another way to play would be to use a deck of cards (remove the face cards) and deal out 10 cards in a row. EL

Archives
March 2020
Categories
All
