Mathematics can be a nuanced subject. Subtle differences in the context or wording of a problem can lead to drastic differences in complexity. I find this especially true with the topic of combinatorics. I remember the first time I taught a course including counting with combinations and permutations. I created a worksheet for my students with what I thought were some fairly straight forward questions. It turns out that some of the problems I created were much more complicated to solve correctly than I had intended. I learned from this mistake and was much more careful from then on. My son and I were recently playing with a roll of tickets (a great math manipulative as it turns out). This led to some questions that appeared similar but were quite different in complexity. I took the pictures below and posted them to twitter. Sharing CandyThe answer to the question on the right can be calculated with a single combination. You can use the "stars and bars" approach to think about the calculation required. First you have to decide if everyone gets a candy or not. If everyone gets at least one candy, then you can think of the problem as putting the twelve candies in a row (the "stars") and inserting 3 dividers in between them (the "bars") to divide the row into four sections. As an example, {★★★★★★★★★★★★} would be one solution. Mathematically, there are 12 identical objects placed in 4 distinct bins, such that all bins contain at least one object. Given the 11 spaces between candies, how many ways are there to choose three of these spaces to place dividers. For the same situation, if you allow each person to receive zero candies, there are more possibilities. Using the same "stars and bars" approach, you can think of all twelve candies and the three dividers and being placed in a row. How many ways are there to do this? There are a total of 12+3 spaces and either a candy or divider is placed into each one. As an example, {★★★★★★★★★★★★} would be one solution. Ripping TicketsThe answer to the question on the left regarding tearing tickets is actually a much more complicated question than the one on the right despite appearing very similar. In this question, we are separating identical objects into identical bins. This means that {★★★★★★★★★★★★} is the same solution as {★★★★★★★★★★★★} since they are both three groups of 2 and one group of 6. This type of problem involves partition numbers and they have been studied by mathematicians such as Leonhard Euler, Srinivasa Ramanujan and more recently Ken Ono. Partition numbers are an open area of mathematics research. The solution for this problem is closely related to partition numbers. For every natural number n, its partition number, p(n), is defined as the number of ways we can write it as a sum of positive integers. For example, since the number 3 can be written as three different unique sums (1+1+1, 1+2 or 3), we say that p(3)=3. If we were looking for the total number of ways to partition the twelve tickets into any number of groups, our answer would be p(12) = 77 (from OEIS A000041). In our problem above however, we're looking for the number of ways to partition 12 into exactly 4 positive integers. We can do this either by counting with an organized list (brute force) or using recursion. For a description of the recursion method, see https://brilliant.org/wiki/identicalobjectsintoidenticalbins/.
Exploring Problem StructuresI recently tried out some problem sets from Craig Barton's SSDD problem website. SSDD stands for Same Surface, Different Deep Structure math problems. These are a set of problems (typically four) that have a very similar context but different solution strategies. The intent is for students to focus on determining the structure of each question and then to identify the corresponding strategy needed to solve it. I think this is an interesting routine for mathematics outcomes where there are a large variety of similar structures (like solving quadratics word problems or combinatorics problems). Michael Pershan wrote a blog post reflecting on the SSDD problem structure and how it might cause students to think in different ways. This type of reflection is why I write this blog and read other teachers blogs. Michael continued the conversation on Twitter and suggested that SSDD problems, “vary the deep differences while keeping the surface the same, and you draw attention to the way minor differences trigger different structure” I think that the type of Same and Different question prompt that I wrote about above also generates the same type of student thinking about the solution strategies required to solve a problem. I think that the SSDD structure could lead to a variety of similar question routines. For example, you could give students variety of questions but instead of answering them, they could be asked to group the questions together that share a similar solution strategy. Or perhaps, you could give students a general context and ask them to create several different questions from this context connected to a variety of mathematics topics (similar to a Notice and Wonder strategy). I think it is exciting to have so much collaboration and thoughtful conversation online between mathematics educators. Nova Scotia Mathematics Curriculum Outcomes Mathematics 12 P05  Solve problems that involve permutations. Mathematics 12 P06  Solve problems that involve combinations. EL
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An engaging puzzle with simple instructions is to ask students to create a rectangle using pentomino pieces. Let them figure out what rectangle dimensions are possible given the total number of squares in the pentominoes they're using (for a full set of 12 pentominoes, 3x20, 4x15, 5x12 and 6x10 are possible). You can also print off a variety of other shapes and challenge students to fill those shapes with pentominoes or have students create their own shapes and challenge their friends to solve them. Younger students can be challenged to fill an empty picture frame or tray with pentominoes. Here are five great games, puzzles and activities (in no particular order) that use pentominoes:
2. Pentomino Puzzles from Jon Orr (@MrOrr_geek)  Jon describes an activity where students are given a hundreds chart and a transparent pentomino. Ask students to place their pentomino on the hundreds chart so that it covers a sum of 135. Then ask students to share their strategies for finding the correct placement. Are there sums that are not possible to achieve? Continue the activity by selecting different tiles, giving different sums to find, creating equations and solving them. I also saw these questions posted by Amie Albrecht (@nomad_penguin) on twitter, "Can you find a pentomino that covers numbers that: sum to 150, sum to an even number, covers three multiples of 4, and more". Jon created an online Desmos Activity that allows you to do this activity virtually. 3. Pentomino Farms  I was introduced the the pentomino farms activity from Martin Gardener's book Knotted Doughnuts and Other Mathematical Entertainments. The task is to use the 12 pentominoes to build a fence around a field on your farm. The rule used to join the pentominoes to form a fence is that they must touch along the full edge of a square and not just at the corners. There are four types of farms to create. For each type, what's the largest field you can enclose?
4. How Convex is a Pentomino?  Which pentomino shape is the most convex? How do you measure "convexity"? This is an interesting question that generated lots of discussion on Twitter. Some interesting methods of measuring this were discussed by Alexandre Muñiz (@two_star) in his blog post, "Vexed by Convexity." I think it would make a great discussion for students to rank the pentominoes by convexity and then defend their choices.
If you have a favourite activity, puzzle or game featuring pentominoes, I would love to hear about it! EL
I prepared a lesson plan to work with a student. I carefully considered how I would introduce the topic, the path that the lesson might take and the questions that I would ask to prompt our discussion. I thought about the manipulatives that we might use to visualize and physically interact with the problem. I had a course carefully laid out. I started by drawing an irregular, kidney shaped area on the desk and asked the student how he would estimate the area of the shape. I was prepared for a number of different responses that I thought I might hear... but the student didn't follow my carefully plotted course for our lesson. Instead he replied, "I'd use Pick's Theorem." I grew up sailing on the Columbia River. When changing course on a sailboat, you can either turn the bow (the front of the boat) through the wind (i.e. tacking) or you can turn the stern (the back of the boat) through the wind (i.e. jibing). When tacking, the boom gently moves from from one side of the boat to the other. Jibing on the other hand can be dangerous as the boom suddenly jumps to the other side of the boat. When the student suggested Pick's Theorem, it felt like changing course by jibing instead of by tacking.
After our excursion through Pick's Theorem we found our way back to estimating the area with some manipulatives. First we covered the shape with square tiles and then we covered the shape with pennies. We found that we could cover the shape with 66 square tiles. I asked the student how the area we found with Pick's Theorem and the area we found with square tiles compared. Through our discussion we decided that we needed a common way to talk about these areas so we converted both to square centimeters. We found that the area from Pick's Theorem was 382.5 cm^2 and the area using square tiles was 412.5 cm^2. Next, we looked at our penny solution. We looked up the diameter of a penny online and found that 135 pennies at 2.85 cm^2 each gave us a total area of 384.75 cm^2. While discussing how this estimate compared to our others, the student started talking about Alex Thue and his theorem on circle packing (this student has a really good memory). The student remembered that the efficiency of hexagonal packed pennies was about 91%. So we used this efficiency to correct our penny estimate to make it even better. This led to another discussion that I hadn't planned on about tesselations and polygons that tile the plane. The student said he had read in a book that there were 14 irregular pentagons that tile the plane. His book was a few years old however so he didn't know that a 15th pentagon had been discovered in 2015 or other recent work in this area. While the lesson didn't go quite as I had planned, I was really happy to be able to take the student's contributions to the discussion and weave them into the overall narrative of our work. Being flexible, listening to students and incorporating their contributions into a discussion can sometimes throw you off course and you might end up someplace unexpected. The journey along these altered courses however can be incredible. EL
Over the Christmas holiday, the number of LEGO bricks in my house increased significantly. My son received LEGO sets as gifts from numerous grandparents, aunts and uncles. I was a LEGO fan when I was a child and now I have an excuse to play with them again as an adult. We've had lots of fun recently building sets and designing our own creations. At some point I became inspired to create a scale model of our home. Planning and BuildingI started this small project by building a test model to try out the proportions and to see what kinds of bricks I would need. The sizes of the door and window established the overall size. I continued revising the structure it until it looked right and then started collecting the bricks I needed. Building this model reminded me of working on an OpenMiddle.com math problem. In an "open middle" problem, there is a one starting point and one solution but many different paths to get to the solution. With LEGO, there are many different ways to create, revise and improve your model. There are lots of different building techniques that will all result in a well designed scale model. ScaleAfter I created my initial rough model I did some reading up on LEGO scale. It turns out that it is a fairly complex topic that lots of different people have investigated. I found the Brick Architect web site to be very helpful. For "classic minifigure" scale a ratio of 1:42 can be used. One major difficulty in discussing scale is that the proportions of a LEGO minifigure are not even close to the proportions of an actual person. A LEGO minifigure is about 4 cm tall and 1.6 cm wide. An average male human is about 175 cm tall and 40 cm wide... about half as wide as a minifigure would be at that height. Another challenge is converting units. The architectural drawings of my house are in feet, which I converted to metric (cm), then a scale factor is applied and finally the metric units are converted into LEGO bricks. I found an awesome tool that does this all for you, the LEGO Unit Converter. FinanceI used a lot of estimation to determine how many bricks of each type I would need. LEGO bricks are not cheap so you don't want to order more than you need (Check out Jon Orr's activity involving cost, Is LEGO Gender Biased?). I purchased the bricks I needed on BrickLink.com, a large online LEGO marketplace. BrickLink provides a detailed price guide for every brick available which makes it really easy to know if you're getting a good deal or not. I needed lots of 45 degree angle slope bricks for the roof of my house. These price stats let me know what a reasonable price is to pay for new or used bricks of this type. It is amazing to see how many bricks are sold on this site. I think that the stats from this site could make for an interesting grade 12 math research project. The Finished Project
EL
I recently stumbled across Richard Garfield's Balloon Balance puzzle from the 9th World Puzzle Championship in October, 2000. The puzzle is similar traditional balance puzzles such as SolveMe Mobiles but with a couple of twists. The first twist is the use of balloons to include negative numbers. The second twist is the inclusion of torque... the farther from the balance point (i.e. fulcrum), the more force is applied. I thought that a simplified version of this puzzle might work well for students who are practicing integer multiplication. First however I had to figure out the puzzle. I have to admit that this puzzle took me quite a while to solve and make sense of. I almost gave up a couple of times but I eventually figured it out. Solving this puzzle reminded me of how it feels to be a learner and to empathize with students struggling to understand a new concept. If you'd like to see the solution to the puzzle above, click this link. To use this with students, I simplified the puzzle a bit and then made some examples and easier challenges to start with. I modeled these from a version of this puzzle created by Joseph DeVincentis for the Boston Area Puzzle Hunt League (BAPHL) #5. Directions: On each set of balances, place the given balloons and weights in the open circles. Each beam should be balanced around its fulcrum (the small square) by the torque rule: sum of weight times distance from fulcrum for all weights on the left should equal that on the right. The beams themselves are weightless. The fulcrum with an x is the fixed anchor point for the system. The first example shows students how the basic puzzle works. The second example below add additional complexity to the system while retaining only 4 weights/balloons to place. I created two challenge puzzles for students to work on once they were confident with the examples. I haven't tried this out with students yet so I'm not sure what to expect. Feel free to check out my Google Slides with the examples, challenges and solutions. I'd appreciate feedback if you have suggestions to improve this. Nova Scotia Mathematics Curriculum Outcomes Grade 8 N07  Students will be expected to demonstrate an understanding of multiplication and division of integers, concretely, pictorially, and symbolically. EL
"Same and Different" is a math routine that I've been exploring recently. This is a routine that gives students a structure to compare and contrast two objects or ideas. I was introduced to this routine on twitter back in August 2017 when Brian Bushart launched the website https://samedifferentimages.wordpress.com/. Fueled by support from the #MTBoS, the site quickly gained momentum. In midDecember, I saw a tweet from Kristie Donavan. In it, she wrote "#samediffmath is one of my favorite structures for connecting new ideas with prior understanding!" She also attached a few examples that she has used. I really liked the idea of using this routine to help students construct mathematical connections. In the Nova Scotia mathematics curriculum, "Connections" is one of the seven Mathematical Processes that are intended to be infused throughout each course. Making connections between mathematical ideas helps students see mathematics as a consistent and integrated whole and not isolated units of study. Below are two images that I created to help students make connections between related representations and ideas. Below are two more #samediffmath images that I created to connect different representations. The first was inspired by a recent tweet from Kent Haines. He was exploring a variety of ways to model a linear equation. There were lots of replies and suggestions that I hadn't considered such as Howie Hua's reply suggesting using a "splat" model. Be sure to check out the hash tag #samediffmath on twitter or the Same or Different Images website for new examples of this routine. I also have a Google Slides document with my images on it if you'd like to use or revise my images above. EL
I'm a fan of selfchecking math activities. These activities give students immediate feedback and help them to find and correct errors. Many students will be able to correct their own computational errors, especially if students are working in pairs or small groups. When students are unable to fix their errors due to more serious misconceptions, the teacher can step in to help develop understanding. This helps the teacher use their time efficiently and focus on students facing challenges. I've recently seen one math activity used in a number of classrooms in a variety of forms. I'll call this activity a "question chain" although I've seen it referred to using lots of different names. This activity starts with a set of questions and associated answers. Students start by solving one of the questions. The answer to this first question leads the student to the next question. This process is repeated until the student arrives back at the starting question. The answers form a "solution bank." If the student can't find their answer, they know that they've made a mistake and need to find and correct their error. Below are three different ways that I've seen this activity implemented in classrooms. Questions on Cards
Links: Questions Posted on the Wall
Links: Questions on a Worksheet
Selecting a MethodDuring a recent professional development session with math teachers, we tried this activity using all three methods. Participants were split into three groups and each group was given a different method. All three versions of the activity included the same ten questions (see the files below).
After completing the activity we had a discussion to compare the three methods. All of them took about the same amount of preparation and could be quickly created using questions from a textbook or other problem bank. How would students record their work in each method (on paper, miniwhiteboard, etc.)? How would the teacher assess students work in each method? Would each method work better individually, in pairs or in small groups? How might this activity be used in a combined grade classroom? Which method might be most culturally relevant for your students and how does your knowledge of your students inform your selection of a method? Which method is the most engaging for your students? We had a very productive and rich discussion. Have you used this type of activity in your classroom? Another variation of this method is the "I have/ who has?" oral classroom activity. Have you used a different variation of any of these methods in your class? Do you have a favourite method? Why is it your favourite? EL
Practice is important. Whether it is playing the piano, shooting freethrows, 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. Open MiddleOne 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. Math GamesGames 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 19. Two cards are drawn to form a twodigit 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 3916 = 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 'goto' resource is a book titled Well Played, 68. Math PuzzlesThere 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.
Desmos ActivitiesFor 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. IncentivesThere are also activities that have a nonmath 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 nonmath 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). Please Share!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. EL
My son and I recently spent a lovely fall afternoon exploring the carnival games and amusement rides at a local fair. My son is quite adventurous when it comes to amusement park rides and is eager to try just about any ride that he meets the height requirement for. While we were walking through the midway, I spotted a carnival game called "Roll Down" that appeared to have a bit of mathematics involved. The object of this "game of skill" is to roll six balls down and inclined ramp to land in one of six numbered bins. If the sum of the six rolls is under 10 or over 31, you win. Is this game worth the $5 price to play? What are my chances of winning? Should I go for under 10 or over 31? The bins are just wide enough for a ball to fit so it is very difficult to aim a ball with accuracy. You also have to question if the balls roll straight and if the board is smooth and level. Lets just assume that the balls fall into a random bin (you could then play an equivalent game at home by rolling 6, sixsided dice). With six balls, the smallest sum possible is 6 (all 1's) and the largest is 36 (all 6's). How many ways are there to get each possible value? There are only 31 possible sums (636) that you can score. To roll a sum of under 10, you can score 6, 7, 8 or 9. To roll a sum of over 31, you can score 32, 33, 34, 35, or 36. At first glance, it looks like you have a 9/31 chance of winning but this is not correct.
This reminds me of a bet in the casino game craps that looks good, but on further inspection is really bad. The field bet is a bet on the sum of the next roll of two sixsided dice. If the sum of the two dice is 2, 3, 4, 9, 10, 11 or 12 you win. If the sum is 5, 6, 7 or 8 then you lose. It has the illusion that there are more ways to win than lose, but you are much more likely to roll one of the losing numbers. With practice at Roll Down, you might be able to achieve better than the random results that I detailed above. Instead of this practice, I decided to spend my $5 at the concession stand to buy a handbattered, deep fried corn dog. A midway concession stand can also be considered a bit of a gamble, but in this case it was a delicious win! EL
Activities that let students get immediate feedback on how their are doing are extremely beneficial. Activities that allow students to selfcheck their own work allow for this immediate feedback and correction. These types of activities can allow the teacher to focus their time with students who are having conceptual misunderstandings and not get bogged down helping students find and correct computational errors. While students are engaged in selfchecking activities, the teacher can also be working with small groups of students on minilessons or conversations/conferences. Below are a few of my favourite activities and routines that allow for students to check their own work: Team PostIts  I recently saw this activity described in a post by Julie Morgan. This activity is very easy to set up and does not require much front loaded time to create. The teacher posts a list of questions for small groups of students to work on. These might even be questions from the textbook. Each group solves the question, writes their answer on a sticky note and posts it on the whiteboard. As other groups complete the questions, they can compare their answers to those from other groups to see if they agree. If they don't agree, they double check their work. I would suggest that each group of students starts with a different question. Add ‘Em Up  In this activity, students do a set of problems, either on their own or as a group. These problems typically have numerical answers. The answers to the set of problems are added up and compared to the sum provided. If the sum is not the same, then the student knows that one or more of the problems in the set was done incorrectly and works to find the error. I first saw this activity described in a blog post from Kate Nowak. I later saw a blog post from Amy Gruen describing a simple and quick way to do this same activity that I used occasionally. There are many descriptions of how to organize this activity including one in a detailed blog post from Sara VanDerWerf. Row Game  I also first saw this activity described in a blog post from Kate Nowak. Typically, a row game is a worksheet of problems organized in two columns. The worksheet is completed by a pair of students, one doing the problems in column A and the other doing the problems in column B. The problems in each row have the same answer so if the students' answers don't match, they can work together to check their solutions to find the error. To make row games a bit easier to create, you can create an additional column with the sum of the solutions from column A and B (similar to the Add 'Em Up activity from above). This allows you to use any two problems and not have to create two problems with the same solution. Kate Nowak has a shared google folder with a large selection of crowdsourced row games. Added 03Oct017  I recently saw a great idea from Heidi Neufeld. She asked students who finished quickly to make a new row for the row game and create two different problems with the same answer.
Mathematical Circuit Training / Around the World / Star Chain / Question Stack  There are lots of different names for and variations of this activity. The essential part is that there is a series of problems and the answer for each problem leads you to the next question to ask. The answer to the final question leads you back to the starting question. This activity can be organized as a simple worksheet, a stack of cards to turn over, a set of cards to chain together or questions posted on signs around the classroom or hallway. If you make a mistake, you won't be able to find the next question and you know to try again. This can be done individually or in small groups.
Added 26Sep2017  Thanks Alicia! Invisible Ink  The description of this activity is from a blog post from David Petro. Students solve a set of problems on a card. When ready, they can check their solutions using an answer card. This card has the correct answers written with "invisible" ink that can only be seen by shining a small UV light on it. Once the student has solve the questions correctly, they move on to the next card containing more complex questions. David says, "Students really seem to like this style of activity as they feel empowered to move from card to card when they are ready and the added feature of checking the answers with the UV pen gives a sense of novelty." If you know of any other selfchecking activities that I've missed, please let me know and I'll add them here. EL

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