I was recently looking for an activity to explore a linear relationship, preferable one that included some practice with decimals. I found a few examples but none of them really seemed to satisfy what I was looking for. Here are a few of my criteria for good experiments to explore function relationships:
Notice and WonderThere were some great questions about volume and surface area, weight, and size of the paper clips (what is a #4 sized paper clip?). The questions the we went on to investigate was how long would it take to make a paper clip chain from all 100 paper clips. I was inspired by Dan Meyer's Guinness World Record for the longest paperclip chain in 24 hours. Dan blogged about breaking the record as well as asking student to see how many paperclips they could chain in one minute. EstimationI asked students to estimate how long they thought it would take to create a chain of 100 paper clips. I also asked them to think about an estimate that they know was too low (that creating a chain this fast was not possible) and too high (that they would have no problem creating a chain in this time even going slowly). Most students thought that a time between 5 and 6 minutes was a good "just right" estimate. Gathering Data
Revising EstimatesAfter collecting and analyzing some data, I ask students if they'd like to revise their estimate for 100 paper clips. Then we test their revised estimate using a plot of the values they collected and extrapolating. Below is one student's data plotted in Desmos. They estimated 300 seconds (5 minutes) to chain all 100 paperclips. This lesson could be modified to include outcomes from a number of different grade levels. I closed the lesson by showing students the record for the most paper clips linked together in one minute and asked students how they would compare. Nova Scotia Mathematics Curriculum Outcomes Mathematics 6 SP01  Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02  Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media. Mathematics 6 SP03  Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 7 PR02  Students will be expected to create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems. Mathematics 7 N02  Students will be expected to demonstrate an understanding of the addition, subtraction, multiplication and division of decimals to solve problems (for more than onedigit divisors or more than twodigit multipliers, the use of technology is expected). Mathematics 7 SP01  Students will be expected to demonstrate an understanding of central tendency and range by: determining the measures of central tendency (mean, median, mode) and range; determining the most appropriate measures of central tendency to report findings. Mathematics 7 SP02  Students will be expected to determine the effect on the mean, median, and mode when an outlier is included in a data set. EL
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Multilink cubes are an incredibly versatile manipulative for mathematics class. You may see these 2 cm interlocking cubes referred to by several different names including: linking cubes, multilink cubes, Snap cubes, CubeaLinks and HexaLink. It's a manipulative that I've seen used at nearly every grade level from Primary to 12. Recently, John Rowe (@MrJohnRowe) stated a conversation on Twitter about math manipulatives. The conversation prompted me to reflect on how I've used manipulatives, and especially multilink cubes, as part of instruction and inquiry. I thought it would be nice to list a few of my favourite examples here: Speedy Squares from Mary Bourassa (@MaryBourassa) Speedy Squares is an activity that asks students to predict how long it would take them to build a 26 x 26 square out of linking cubes. Students start by building smaller squares and recording their times. They can then use this time to extrapolate an answer. Students could use quadratic regression to make a more accurate prediction. Jon Orr (@MrOrr_geek) also blogged about this activity and how he introduced it to his class. NS Outcomes: Math Extended 11 RF02, PCAL 11 RF04 and Math 12 RF01
Skyscraper Puzzles from Brainbashers This is a logical reasoning puzzle that you can play with just pencil and paper. The game become more focused on spatial reasoning when you actually build the towers using multilink cubes. Lots of educators have written blog posts about how they use this puzzle in their math classrooms including Mary Bourassa, Sarah Carter, and Amie Albrecht. Mark Chubb (@MarkChubb3) has blogged about this puzzle and shared some great templates for using with multilink cubes. NS Outcomes: Math at Work 10 G01, Math 11 LR02 Orthographic Projections from Jocelle Skov (@mrs_skov) Each student creates a 3D object using multilink cubes. Next they draw the top, front and side views of their object. Once every student has finished the three views of their object, they trade drawings with another student. That student then tries to build the 3D object in the drawing. They check their work with the original object on the teachers desk. NS Outcomes: Math 8 G01, Math at Work 11 G03 3D Linear Relations inspired by Alicia Potvin (@AliciaPotvin1) Each group of students builds three terms of a linear pattern using multilink cubes. Ask students to use one colour for the part of the pattern that stays the same and another colour for the part of the pattern that changes. Groups then rotate through the room and for each pattern, record a table of values, a graph and the equation. You could also ask students to determine how many cubes would be in the 43rd term as suggested at http://www.visualpatterns.org/. NS Outcomes: Math 9 PR01, Math 10 RF04
Mean, Median, Mode and Range with Linking Cubes from Jana Barnard and Cathy Talley Ask each student to reach into a large box of linking cubes to grab as many as they can with one hand. Students then build a tower with their linking cubes. As a class, students organize their towers in order from shortest to tallest. To get the class range, subtract the height of the shortest tower from the tallest tower. Is there a height that occurs more often than any other? That is the mode. To get the median, find the tower in the middle of the row (if an even amount of towers, average the two middle towers). To get the mean, even out all the towers until they are the same height saving any "left over". Suppose you had 12 towers, each with a height of 10 and 5 remainder cubes. This would give a mean of 10 and 5/12 cubes. NS Outcomes: Grade 7 SP01 Spinners from Shaun Mitchell and Mike Wiernicki (h/t Jen Carter) Students, in small groups, design a spinning top made of multilink cubes. The goal is to design a top that spins the longest. Once the group settles on their design they collect some data. They spin the top and record the time it spins in seconds to the nearest hundredth (or tenth). They do this three or four times and then average the time (hence they have to add three or four decimal numbers and then divide that decimal by 3 or 4). They could also model their decimal numbers using decimal squares. NS Outcomes: Grade 6 N08, Grade 6 SP02 A few links to some documents that provide some additional suggestions for using linking cubes:
The examples above are mostly from secondary math classes. Multilink cubes are also incredibly useful in elementary math classes (counting, measuring with nonstandard units, composing and decomposing numbers, etc). What are your favourite linking cube activities? Let me know and I'll add them to this post. EL
I was recently invited by a class to work with them on collecting and analysing data. After brainstorming some ideas with the classroom teacher, we settled on collecting data from pull back cars. I check out Fawn Nguyen's Vroom Vroom lesson and Simon Job's Car Racing lesson to get some ideas on how to organize this lesson. We started the lesson by sharing the first half of Simon's video of cars racing across the floor. We had the students do some notice and wonder about the action taking place in the video and then introduced activity. We showed students the recording sheet that we would be using and how we would be taking measurements (A link to the record sheet Google Doc is here). Then we brainstormed some ways to make sure that we all collected good data and avoided errors: we would all use the same units (centimeters), all measure our distances the same way (from the front bumper), not use data if the car bumped into a wall or a desk, etc. We split up into racing teams of three students each. Each group got a measuring tape, a pull back car and a recording sheet on a clip board. The classroom teacher and I circulated the room (and a bit of the hallway) to help students and answer questions. After student finished collecting their data and plotting their values we came back together as a class. We asked several groups to plot their data on the whiteboard at the front of the room. We then had a discussion about general trends as well as why each car had a slightly different graph. Cars might have different wind up springs, different tire grip, dusty floors, aerodynamics, etc. We finished the class with a bit of excitement... the 150 Challenge. Each team had to use the data for their car to predict how much they would need to pull back to make the car travel as close to 150 cm as possible. Teams huddled to interpret their data and select a pull back distance. Each team brought their car to the front of the class to give it their best shot. There was lots of cheering and excitement as some teams got very close. The winning distance was only 2.5 cm. Much more close than I had expected. This activity could be easily extended for higher grade levels by incorporating linear relationships, linear equations and linear regression. Nova Scotia Mathematics Curriculum Outcomes Mathematics 6 SP01  Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02  Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media. Mathematics 6 SP03  Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 9 PR02  Students will be expected to graph a linear relation, analyze the graph, and interpolate or extrapolate to solve problems. Mathematics 10 RF07  Determine the equation of a linear relation, given: a graph, a point and the slope, two points and a point and the equation of a parallel or perpendicular line to solve problems. (including RF07.06 Determine the equation of the line of best fit from a scatterplot using technology and determine the correlation) Mathematics Extended 11 S01  Analyze, interpret, and draw conclusions from onevariable data using numerical and graphical summaries. EL
So a slight disaster struck our household recently. (and I mean slight; this is filtered through the lens of a child! My kids were recently at my husband's place of work (a large high school), waiting for him to finish up and drive them home. They kids are used to hanging around there and are often trying to find ways to amuse themselves while they wait for dad. My oldest son had his reaction ball with him and they decided to play catch with it in the open foyer of the school. In the usual brotherly fashion, they started to argue about who gets the ball. One of them (he asked to remain nameless!), threw the ball, hitting the trophy case in the foyer. The ball is deceptively heavy (about 272 g; a tennis ball is about 58 g); it ended up cracking the glass of the trophy case! The child who threw it says it hit the frame of the case three sections down from where the crack is; he’s floating a theory that the crack was already there. He’s also playing with the idea that the force of the hit on the frame sent vibrations throughout the whole case, causing the crack so far from the point of impact. #science #physics You can imagine how upset the kids were. They are really empathetic, kind, never any trouble at school and generally well behaved (A biased opinion, I know! But I’m their mom and #1 fan) When I arrived home and heard what went down, I encountered a very sheepish looking older brother and a very sad little brother who sent himself to his room. I went to have a chat with him and he was not in the mood to be cheered up. He told me that is was going to take more than 100 years to pay off the damage they had done. I tried to assure him it would not take that long and he said: "But Mom! I did the math" Here’s how the conversation went: Child: “My allowance is $10 every two weeks so that is only $120...wait $240 a year.” Me (in my head): Actually you are assuming that you get your allowance only 2 times a month, some months you get it 3 times. You are using a bimonthly calculation, not biweekly. Me (to my son): awww, honey :( Son (through tears): “and the glass is going to cost $30 000 dollars so that means like 100 years!!!! I’ll never be able to get that much money!!” Me (in my head): Well, you are assuming that your earning potential for the next 100 years is going to stay the same. As your mother, I am hoping that you will have a job at some point that pays more that $10 every two weeks. Also, how much do you think glass costs??? Hmmmm… 30 000 divided by 240 is 125. That’s a pretty good estimate using these assumptions. My child is a genius. Me (to my son): “How big was the glass? It can’t possibly cost $30 000!” (30 minutes of debating the cost of glass and listening to his various mental calculations) He went on to explain to me how he did his calculations and after a lengthy discussion, I finally convinced him that 1) the glass did not cost that much, 2) he was not going to be spending the next 100 years paying for this glass, and 3) he was most likely not the cause of the small crack in the glass. This conversation got me thinking about all of the little mathematical conversations parents have with their kids. I know my own kids are experts at negotiating timelines for bed, justifying how much screen time they should have and estimating how long it takes to get out the door for activities (factoring in travel time, and whether they are going to a game or a practice). As a math teacher, I notice and capitalize on these moments. My kids would argue I notice this too much. Sometimes when the kids ask a question that could be reasoned through mathematically, they preface the question with a “I just want the answer  don’t talk to me about the math!!!”. If you are looking to create these kinds of moments with your students or supporting their parents in having these kinds of conversations, check out and share the following websites:
As per my kid’s request, I’m working on not asking too many questions.
Do you know of any other great resources like these? Let me know! K.
The name of this card game can be traumatic for some students. For students who have lived in areas of armed conflict, "war" is not an appropriate name for a lighthearted card game. An alternative name you might consider is "Topit". The basic game is easy to learn. Here are the basic rules:
CardsMost of these versions can be played with a regular deck of playing cards. (You could remove the face cards, make all face cards equal 10 or use J=11, Q=12, K=13. Jokers could be removed or have a value of zero. Aces can be either 1 or 11.) If you want to use basic number cards, you can steal them from a UNO deck or any other numerical card game you might have laying about (e.g. SkipBo, Zero Down, Krypto, etc). Some might need special cards to be printed off such as (Logarithm War). Mathy VersionsRepresenting Numbers War  A special deck of cards is used showing a variety of ways to represent numbers from one to twenty. Representations might include numerals, tally marks, ten frames, dominoes, dot patterns, fingers, etc. Place Value War  Students flip two cards (A9) to make a two digit number (or three cards for a three digit number). They get to choose which represents the tens place and which is the units place. Ask students to read their number out loud, “five tens and three ones equals fiftythree.” Addition or Subtraction War  Each student flips two cards and either adds or subtracts them to get their value. Ask students to read aloud the number sentence created by their cards. For example, if the student draws a 4 and a 6, they should say, "four plus six equals ten" or "the sum of four and six is ten".
Decimal War  Similar to place value war, each student flips two or more cards to create a decimal number. You could create special deck to include fractions, decimals and percents as well as pictorial representations of rational numbers.
Evaluating Functions War  Students determine the value of a function at two different points to see which is greater. Students get practice evaluating functions from a table, equation, or graph. Sarah Carter has shared a set of cards she created for this activity. Radians and Degrees War  Practice mentally comparing angle measurements in radians and degrees using a special set of cards. Sarah Carter has a link to a free set and a description of this activity. Trig War  Play with cards with sine, cosine and tangent trig expressions and special angles on the unit circle (in both radians and degrees). Sam Shah has a link to a set of cards to download and a description of this activity.
Do you use any other versions of War in your classroom? Do you know a great deck of custom cards ready to download? Let me know. EL
I recently had the opportunity to work with a student to investigate parabolas and quadratic functions. We used one activity to investigate two different quadratic relationships. First we observed the shape of the stream of water coming out the side of a water bottle and then we observed the rate the water drains. The Water FountainI set up a cylindrical bottle of water on a crate. The bottle had a whole in it covered with a piece of tape. I asked students for some predictions. What will the shape of the water coming out of the side of the water bottle look like. What will happens to the stream of water as the water level goes down? I noticed that the student drew the water stream coming out of the bottle like it comes out of a water fountain (where we had just filled the bottle). We took the tape off the hole and then watched the water come out while making some observations and taking some photos. We selected a good photo (the black bulletin board in the background really helped) and loaded into Desmos. Then we used a table to record some points along the steam of water. After that we did a linear and then a quadratic regression on the point to see that the parabola was a much better fit than a line. We then had a chat about parabolas and projectile motion. Draining the TankWe set up the water bottle again but this time instead of looking at the shape of the stream of water, we focused on how fast the water level fell. I asked the student to predict what this might look like. You might ask students to predict what a graph of the water level might look like over time for the two situations below. How would the graph look when filling the tank compared to emptying the tank? The water flowing into a tank should rise at a linear rate. Students should expect that when the water drains from an open tank, the flow will be greatest at first and then gradually decrease as the water level decreases. (This is an application of Torricelli's Law). Next we taped a measuring tape to the side of the bottle and collected some data as the water flowed out of the bottle (A similar experiment is described in CanavanMcGrath, Foundations of Mathematics 12, 429). We used the stopwatch on my cell phone to record the time at each centimeter of height. This wasn't as accurate as I had hoped due to some distractions in the room. We set up the experiment again and the second time I recorded the water falling using a video (I used the CoachMyVideo app). We were able to get much more accurate values this way. We entered the data in a table on Desmos and then did a quadratic regression to fit a curve to our points. I was a bit surprised at how well the data from the video analysis on our second attempt fit to a quadratic curve (R^2 = 0.9999. I really liked how we could use the exact same setup to investigate two different quadratic relationships. Nova Scotia Mathematics Curriculum Outcomes Mathematics 11 RF02  Demonstrate an understanding of the characteristics of quadratic functions, including: vertex, intercepts, domain and range and the axis of symmetry. Precalculus 11 RF04  Students will be expected to analyze quadratic functions of the form y = ax^2 + bx + c to identify characteristics of the corresponding graph, including vertex, domain and range, direction of opening, axis of symmetry, xintercept and yintercept, and to solve problems. EL
I've been thinking a lot lately about the role of practice in the mathematics classroom. Reading Mark Chubb's blog post made me reflect on my teaching. Practice provides students an opportunity to enhance and refine newly acquired mathematical concepts and skills. I have lots of questions about practice:
I think that it is important to be reflective about what I devote time to in class. You don't want to invest time in something that is not going to pay dividends in student understanding. Jon Orr has an insightful Ignite talk where he talks about "being picky" with the technology tools he uses with his students. I think that teachers should think critically about practice routines as well. Below are a few criteria that I consider when making decisions about student practice. Characteristics of Effective Practice:
Immediate FeedbackIt is not desirable for students to spend time practicing and have no idea if they are producing accurate and correct mathematics. Self checking activities allow students to know immediately if they are on the right track or if they need to ask for clarification. One of my favourite activities that are selfchecking are row games. I first learned about row games from Kate Nowak's blog. In a row game, students work with a partner. Each of them completes a different question, but the answer to both questions are the same. If they don't get the same answer, they collaborate to find out where the mistake is. About a year ago I wrote a blog post describing a number of other self checking activities including row games, addem up, tarsia puzzles, and question stacks. PurposefulPurposeful practice is practice with a goal to achieve. An example of a question with purpose is an Open Middle question. The Two Fractions Challenge from Michael Fenton is a great example of this type of question. In this problem students create an expression using 4 digits and one operation. The goals is to make the value of this expression the largest, smallest, or closets to zero. Students will evaluate a great many fraction expressions as they hunt for an optimum solution. Another practice activity with a clear goal is a Tarsia puzzle. I explored a few math practice routines with purpose in a previous blog post. EngagingJust about every student loves a good game or puzzle. The challenge is to find a game that is easy to learn and targeted to the math skill you want to practice. An example is playing the card game war to practice adding integers. Remove the face cards and jokers from a deck of cards. Shuffle and deal the cards to two students. Red cards are negative numbers and black cards are positive numbers. Each student lays down two cards and adds them together. The with the largest value wins the cards. A couple of puzzles that I've seen used a number of times in class are KenKen and Shikaku (aka Rectangles) puzzles. Both of these puzzles come in a variety of difficulty levels and require lots of number sense and logical reasoning. You might also consider adding to movement to an activity in order to boost engagement. For example, have questions posted around the room (i.e. a math scavenger hunt) instead of printed on a handout. Another way to add a bit of movement is with stations set up around the classroom that students move between. When practice includes one or more of the criteria above, I believe it will be more effective. Once you've though about how you're going to practice math, the next step is to thing about what you're going to practice. Often it is the topic you're exploring in class but sometimes you might include some cumulative review as well. Retrieval PracticeI've been exploring retrieval practice lately and looking for strategies to incorporate it into classroom practice. The goal of retrieval practice is to cement understanding in long term memory. "Retrieval practice is a strategy in which bringing information to mind enhances and boosts learning. Deliberately recalling information forces us to pull our knowledge “out” and examine what we know."  https://www.retrievalpractice.org/ One strategy is to start the class with four quick questions for students to do in 5 minutes. These questions relate to a mixture of outcomes from previous units of study. Students have to reach back into memory in order to determine the methods and strategies to solve the questions. After students have had a chance to work on them spend the next 5 minutes reviewing solutions. My PracticeI've been taking Mandarin Chinese lessons with my son for the past year or so. I know that if we don't practice in between weekly lessons then we quickly forget what we've learned. Instead of working through workbooks and study sheets, I try to include some conversational practice throughout our day, while eating breakfast or in the car on the way home from school. Our favourite ways to practice are playing games and singing songs. We've made up a couple of our own games to practice together. The practice helps keep our skills fresh and helps solidify our learning. My next challenge is to learn to knit. My son is learning at school and is trying to teach me. He makes it look easy... I've got a lot of practice to do! EL
Brian Bolt has written numerous resource books for teachers containing collections of rich mathematical problems, puzzles, investigations and games. Some are descriptions of classic problems and puzzles while others are new creations. I think these books are a great resource and I wanted to share three of my favourite problems from them. Number the Sectors
This is problem #53 from Even More Mathematical Activities (1987) and problem #72 from The Mathematical Funfair (1989). Instead of starting by asking students to find a set of numbers that gives all the integers from 1 to 25, I like to create an example as a class and then challenge them to do better (get to a larger number). You can also ask them to prove what the maximum value is. Bolt has an alternate version of this puzzle in A Mathematical Pandora's Box (1993) (#12 Can you Do Better), which has 5 sectors around a central circle. This version can be found is online at NRICH Maths as the Number Daisy. How Large a Number Can You Make?Make the largest number with just the digits 1, 2, and 3 once only and any mathematical symbols you are aware of, but no symbol is to be used more than once. The challenge is to see who can make the largest number. Here are some numbers to get the ball rolling: This is problem #83 from Even More Mathematical Activities, (1987). I've given this as a warmup problem for high school students and this often leads to a discussion of how to know which is bigger, 2^31 or 3^21? Make a CenturyBy putting arithmetical signs in suitable places between the digits make the following sum correct: 1 2 3 4 5 6 7 8 9 = 100 There is more than one solution. See how many you can find. This is problem #127 from Mathematical Activities (1982). I would start this challenge with students by asking them to make an expression using the numbers from 1 to 9 to make a value as close as possible to 100. I would then add on the challenge to try to find an expression exactly equal to 100. There is a very similar problem called Make 100 on NRICH Maths. I saw an earlier version of this as problem #94 in Amusements in Mathematics (1917) by Henry Ernest Dudeney. In Dudeney's version, he includes an additional challenge to try to find a solution which "employs (1) the fewest possible signs, and (2) the fewest possible separate strokes or dots of the pen. That is, it is necessary to use as few signs as possible, and those signs should be of the simplest form. The signs of addition and multiplication (+ and ×) will thus count as two strokes, the sign of subtraction () as one stroke, the sign of division (÷) as three, and so on." What are Your Favourite Problems?Do you have a favourite problem or puzzle from one of Brian Bolt's puzzle books? Do you have other favourite collections of puzzles? EL
Being on Twitter and following hashtags like #MTBoS and #ITeachMath allows me to see classroom mathematics well beyond my physical horizons. I get to glimpse creative and engaging mathematics education around the globe. Recently I saw a couple of different ideas that I've tried to adapt and apply for myself. Mysteries
Since the Nova Scotia grade 8 classes are working on integer multiplication and division, I decided to create a math mystery of my own. Another nice source of math mysteries is the book Mathematical Team Games: Enjoyable Activities to Enhance the Curriculum by Vivien Lucas. Treasure Hunt
I liked this idea because it is relatively easy to create; just a find a series of questions with unique answers. Also, students get instant feedback. If their answer isn't on the map, they know they've made a mistake. I would call this purposeful practice as there is a goal to achieve at the end of the activity. There is a reason to persevere. Once students are familiar with the activity, you could give them a blank template (or they could hand draw their own version) and they could work in small groups to make their own treasure hunt activity (and answer key) and share it with each other. The Role of PracticeI recently read Mark Chubb's (@MarkChubb3) blog post on the role of practice in math class. He discussed the differences between "rote practice" and "dynamic practice". Rote practice involves following procedures, drill and repetition while dynamic practice involves active student thinking, playful experiences and puzzles. I think that the Mystery activity is a more "dynamic" activity than doing the Treasure Hunt activity. However, I think that creating your own Treasure Hunt activity does involve additional characteristics of dynamic practice. EL
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. 110 Card InvestigationThis 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 SquareSum ProblemCan 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  2018Use 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? Perimeter 12This 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 nonrectangular 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. Eight DominoesThis 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 (01, 02, 03, 06, 12, 14, 22, and 35). In the 4 x 4 square, each of the columns and rows should sum to 8. The 35 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? EL

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