Working from home and looking after an 8 year old at the same time throughout the past three months has proven to be complicated. Providing an enriching, engaging and educational home environment for my son while still attending to work deadlines and schedules has been a difficult balancing act.
A provincial state of emergency required us to adapt our daily routines. One positive change has been playing board games. My son and I have played daily board games and card games. We have gone from simple children's games such as Snakes and Ladders to more complex European-style board games like Alhambra. Our favourites (and most played) games have been: Splendor, Tiny Towns, and Kingdomino. These games give players lots of opportunities to use spatial and numeral reasoning to make informed choices. These games all work well as two player games or with larger groups.
A game that I have come to appreciate is Kingdomino. It is easy to learn, quick to play and has complex strategy. In this game, players take turns drafting domino shaped tiles to add to their kingdom. Players have to build within a size constraint. Players start with a 1x1 home castle tile and your kingdom, including this home tile, can not exceed 5x5 (meaning you can add a maximum of 12 tiles to your kingdom). This size constraint means that if your kingdom is not well planned, you might end up with empty spots that can't be filled. The goal is the maximize the value of your kingdom at the end of the game. To calculate the final value, you multiply the area of each contiguous region of a single terrain type by the number of crowns contained in the region (there are six different terrains types: pastures, wheat fields, lakes, mines, forests, and swamps). If a region doesn't have any crowns in it, it scores zero points (e.g. there is a region of 4 lakes in the kingdom below but because it has zero crowns, it scores 4x0=0 points). There are four different regions with crowns in the kingdom below. The wheat field (outlined in orange) scores 4x1=4 points. The pasture (outlined in blue) scores 6x6=36 points. The forest (outlined in red) scores 5x2=10 points. And finally, the mine (outlined in green) scores 1x1=1 point. This gives a total values of 4+36+10+1=51 points... a decent total score.
Bringing Math to the Surface
When I first looked at the mathematics applied in this game, I only considered adding up the scores to determine a winner at the end. Playing this game with my son however led me to see how mathematical decision are used throughout the game. Each turn, you must decide which domino (or dominoes if you're playing with two people) you will select for your kingdom. In order to bring mathematical reasoning to the surface, I began reasoning out loud. This let my son hear all the considerations that go into my selection. Given the growing kingdom below, which of the four dominoes would you select and why?
The first domino doesn't have any crowns, but it could be added to the bottom of the kingdom to make a region of 4 wheat fields where crowns could be added on a future turn. The first domino also gives me the benefit of going first next turn (dominoes are placed in order from top to bottom each turn). The second domino has a crown in a forest. This could be added to existing region of 3 forest that has 2 crowns already. This would turn your forest region from scoring 3x2=6 to 4x3=12... a gain of 6 points. The third domino would be even better. Playing it in the bottom right would still gain you the 6 points in the forest region but also turn your 2 point lake region in to a 3 point lake region. The last domino could add a crown to your lake region and a space to your forest region. This would make the forest region worth 8 (a gain of 2) and your lake region worth 6 (a gain of 4). Looks like the third tile would give us the most immediate gain in score (a gain of 7).
Older players will also need to consider the unequal distribution of terrain types and crowns. There are lots of wheat fields (26) but only 5 have a single crown. There are 14 pastures but two of these have one crown and two have double crowns. Mines are rare with only 6 of them in the game but they can generate big scores. Three of the mines have double crowns and one mine has the only triple crown in the game. The trick is to grab them when you can and place them next to each other.
Strategy vs. Luck
There is a fair amount of luck in this game. The dominoes are shuffled and show up in random order. If you have the first pick in the round you can get a great tile but if you go last, you may get stuck with a tile that you don't really want. Also, if you're not playing with four players, you won't be playing with the full set of dominoes. A domino you were hoping to get might not even be in the game. With that being said, there is enough strategy to keep both kids and adults thinking critically about their decisions throughout the game.
I think this is a great example of a board game that demonstrates a playful and entertaining way to practice mathematical skills and reasoning. It's easy to learn rules and quick play time make it a great choice for playing with younger family members. It has been in regular rotation with my family during our at home learning.
If you want to quickly learn how to play Kingdomino, you can watch Rodney Smith's Watch It Played video on YouTube. I have come to appreciate how much skill there is to teach an 8 year old, with limited patience, the rules to a new board game. If you want some tips on teaching your kids (or friends) how to play a new game, I highly suggest Shut Up & Sit Down's video How to Teach Board Games Like a Pro. A lot of these tips apply equally to the classroom and teachers will be able to see how good pedagogy is universal.
Summer is here and I've recently returned from a family vacation that included several lengthy flights. My son just turned eight years old and enjoys flying but can get a bit restless after a few hours on an airplane. Below are a few of the games and activities that we packed to make the time pass enjoyably. These games are best when they are small, lightweight, easy to pack and can be played on the limited space of an airplane tray table (often around 16.5" x 10.5" but there is no standard size). Games are ideally two player but if you're on a larger plane, you might be able to play a three player game.
Pencil and Paper Games
Games that can be played with just pencil and paper are ideal for an airplane. We bring a few pencils and a tablet of paper with us for games, sketching or making notes. Here are a few of our favourite pencil and paper games:
Games with Dice, Cards and Counters
Commercial Board Games
When on a long play ride I also bring snacks, a small LEGO set, and some art supplies to draw or colour with.
Do you have suggestions for travel games to play with with your children? I'd love to hear your suggestions.
During a professional development session today with grade primary to grade 9 math coaches and support teachers, we spent some time working on solving some math puzzles. We used our work on these puzzles to reflect on what good group work looks and sounds like. We also discussed the characteristics of effective mathematical communication. It was great to see positive energy and teachers supporting and encouraging each other. Below are the five math puzzles and investigations that we worked on. We selected these puzzles because they are easy to explain, open to a wide range of students, and offered a fun challenge.
1-10 Card Investigation
This problem from Marilyn Burns asks you to find a special order for a stack of cards, Ace through 10. Place the stack of cards face down and turn over the first card. It should be the Ace. Place the next card on the bottom of the deck and then reveal the top card. It should be the number 2. Continue placing the next card on the bottom of the deck and then revealing the top card until all the cards are revealed. The face up cards should now be in order from Ace to 10. Marilyn recorded a video to demonstrate these rules that is much easier to understand than my written instructions. Marilyn's has a description of this problem on the MathSolutions website. This logic problem doesn't rely on any prerequisite mathematical knowledge and you can try it out quickly to see if you've come up with the correct solution. It gives students a chance to work together to try out strategies.
The Square-Sum Problem
Can you order all the numbers from 1 through 15 so that the sum of any two consecutive numbers are always a square number? For example, in the sequence: 4, 5, 11; 4+5=9 and 5+11=16.I really like this problem because there are some great extensions to take this problem farther and there is a very nice way to visualize the possible solutions. Numberphile has recorded a great video demonstrating both the problem and solution.
The Year Game - 2018
Use the digits in the year 2018 to write mathematical expressions for the counting numbers 1 through 100 (we only went to 20). Use any math operations (+, -, x, ÷, etc) and parentheses to write these expression. There is a more detailed description of this problem at the Math Forum website. For example, expressions for the number 1 might be: 10 ÷ (2+8) or 218^0. This problem is very similar to the classic Four Fours problem but with new digits each year. Which numbers are the hardest to find an expression for? Why do you think this is? I think this problem also leads nicely to a discussion about mathematical elegance and beauty. Look at a variety of expressions with the same value. Which expression do you think is the best? What makes for an "elegant" solution?
This problem challenged groups to make as many different shapes as possible with a perimeter of 12 units using a geoboard (or dot paper). Shapes were recorded on dot paper to make sure no shapes were repeated as reflections or rotations. I've seen variations of this problem in several places. One of them is Brian Bolt's book Mathematical Activities (1982). He suggests not only to find shapes with a perimeter of 12 but to also find the area of each shape. You can then find which shape has the maximum/minimum area. He also challenges students to find non-rectangular shapes (e.g. triangles) with a perimeter of 12 units. There were some good discussions about the lengths of diagonal line segments on the geoboard.
This problem is from NRICH. Finding a solution took some perseverance but most groups were eventually successful. The problem challenges you to create a square using 8 specific dominoes (0-1, 0-2, 0-3, 0-6, 1-2, 1-4, 2-2, and 3-5). In the 4 x 4 square, each of the columns and rows should sum to 8. The 3-5 domino seemed to be key as the rest of the squares in that row (or column) had to be blank.
What are your favourite math or logic puzzles?
Practice is important. Whether it is playing the piano, shooting free-throws, popping an ollie or solving a quadratic equation you need to practice to improve. Some practice routines are more effective than others at helping students solidify their understanding. Practice can often seem tedious and it can be difficult to maintain the motivation to practice.
In mathematics, students practice skills in a variety of ways. One style of practice that can help students stay motivated and engaged is purposeful practice. Instead of a page full of repetitive problems, students focus on an activity that has a mathematical goal to achieve. Dan Meyer wrote a blog post a few years ago titled "Purposeful Practice & Dandy Candies" that started me thinking about how to make activities in my classroom more purposeful.
One of my favourite sources of problems with purposeful practice is Open Middle. There is a large selection of questions organized by topic and grade level. Each question has an "open middle" meaning there are many ways to explore and solve the problem. Below is a question submitted to the Open Middle site by Robert Kaplinsky.
In this question, students try to find the arrangement of digits that yields the product closest to 50. Students will try numerous different arrangements of digits and get lots of practice multiplying decimal numbers without it seeming tedious. The question can also be quickly modified to give additional practice. For example, just add a hundredths place onto one of the factors and use 5 different digits.
Games can be a great way to encourage students to practice. There are lots of examples of but I'm going to mention just two. The first goes by several names. Joe Schwartz wrote a great post about Factor Captor. A similar game is described on the NCTM Illuminations site called the Factor Game. Students alternate turns playing on board filled with numbers. The first player selects a number to cover and adds that number to their score. The second player finds all the factors of that number, covers them and totals those number to add to their score. The roles are then reversed and play continues until there are no uncovered numbers remaining. There is a lot of math in this game and it is fun to play.
The second game is Horseshoes from Math4Love. This game is played with a deck of cards numbered 1-9. Two cards are drawn to form a two-digit target number. Then four more cards are drawn. Players use these four digits to create an equation using addition and subtraction that is as close as possible to the target number. For example, let's say that the target number is 25 and the four digits given are 1, 3, 6 and 9. A student might create the equation 39-16 = 23. Another student might make the equation 13+6+9 = 28. There are lots of way to tweak this game for different levels of complexity.
Both of these games allow for lots of numerical practice in a format that engages students. There are many excellent sources of ideas for mathematical games. My current 'go-to' resource is a book titled Well Played, 6-8.
There are several mathematical puzzles that include lots of practice with numerical computations in pursuit of a solution. KenKen puzzles and Maze 100 from NRICH are two such puzzles that I've used. I also think that Yohaku puzzles are great. They are numerical puzzles where you need to determine the number that is in each square in order to make the column and row sum/products.
For classrooms with the available technology, activities from Desmos.com are another way to practice with purpose. An activity that incorporates a lot of meaningful practice is Marbleslides. In this activity, student try to capture stars on a Cartesian grid by creating a path using functions that marbles roll down. Students work to refine their functions to capture as many marbles as possible. Another activity that generates lots of practice is Transformation Golf. Students use a series of rigid transformations to move a shape to specified location. They have to find an efficient path around several obstacles. Each successive challenge increases in complexity.
There are also activities that have a non-math goal for students to achieve. Lots of online math games have incentives for students such as badges to earn, experience points to accumulate or virtual prizes to win. There are also worksheets and activities with non-math goals. Worksheets such as "Algebra with Pizzazz" and "Punchline Algebra" have a riddle to be solved once all of the questions are completed.
In my classroom practice, I used a number of activities with these types of incentives and I think that many students find them exciting and fun. If students are excited to do math, I consider that a win. However, I think that these types of activities should be used with caution. We don't want to inadvertently send a message to our students that math isn't fun by itself so we have to disguise it (like sneaking vegetables into their favourite foods so kids will eat them).
If you have a favourite math activity, game or puzzle that gets students practicing math with a purpose, I'd love to hear about it. Please leave me a comment.
The Riddle of the Tiled Hearth is one of many mathematical puzzles from Henry Ernest Dudeney's 1907 book titled The Canterbury Puzzles And Other Curious Problems. The first group of puzzles in this book are based on the characters from Geoffrey Chaucer's Canterbury Tales. Puzzles from this book could be used as part of a cross curricular unit on history, literature and mathematics. There are a number of very interesting puzzles and games including the first pentomino puzzle called The Broken Chessboard and a clever variation of the game Nim called The Thirty One Game. Since this book was first published in 1907, the copyright has expired and it is freely available on Project Gutenberg.
It seems that it was Friar Andrew who first managed to "rede the riddle of the Tiled Hearth." Yet it was a simple enough little puzzle. The square hearth, where they burnt their Yule logs and round which they had such merry carousings, was floored with sixteen large ornamental tiles. When these became cracked and burnt with the heat of the great fire, it was decided to put down new tiles, which had to be selected from four different patterns (the Cross, the Fleur-de-lys, the Lion, and the Star); but plain tiles were also available. The Abbot proposed that they should be laid as shown in our sketch, without any plain tiles at all; but Brother Richard broke in,--
To use this activity with students, I would start by introducing using the tiled hearth story as written above. Then I would introduce some manipulatives that would let them physically explore and work with the puzzle. I would give each group of students a large 4x4 grid on a sheet of paper and some multi-link cubes of 4 different colours.
One aspect of this puzzle that I like is that students can play around with it and have some intermediate success. They might just place a few cubes on the grid. With time, they can refine their solutions to get better and better. Below shows how a student might explore to place more and more cubes.
The Solution from Canterbury Puzzles shows that the best solution leaves 3 blank spaces. Dudeney states, "The correct answer is shown in the illustration on page 196. No tile is in line (either horizontally, vertically, or diagonally) with another tile of the same design, and only three plain tiles are used. If after placing the four lions you fall into the error of placing four other tiles of another pattern, instead of only three, you will be left with four places that must be occupied by plain tiles. The secret consists in placing four of one kind and only three of each of the others." Below are both my solution using cubes and Dudeney's equivalent solution.
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 warm-up 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 1-9 (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?
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 Pre-calculus 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 problem-solving strategies.
Mathematics 12 - LR01 Analyze puzzles and games that involve numerical and logical reasoning, using problem-solving strategies
Pre-calculus 12 - PC03 Determine the number of combinations of n different elements taken r at a time to solve problems.