Students work with scale in a number of mathematics courses in Nova Scotia. In the Mathematics Essentials 12 course, they work with scale in conjunction with reading blueprints. This scale activity uses the blueprints for an iconic Canadian airplane, the DHC3 Otter. I started the lesson with some estimation. I didn't let students know the name of the airplane to start with so that they don't just Google the name of the airplane and find out the actual length. Instead they use clues from the photo to make an estimate. Some references might include the size of the dock or ramp, the height of the wing, or the size of the windows or door. We use a routine I learned from Estimation180 to ask for an estimate they know is too low, an estimate they know is too high and then a just right estimate. After everyone had made and estimate, I showed them an image from a blueprint and asked them if they'd like to revise their estimate using some additional information. On the blueprint, they can see a scale drawn above the plane. Students use this information to either confirm or adjust their estimate. We can then use the scale to measure the image to find out the actual length of the plane (about 41 ft). After this opening activity, I showed them a picture of a 1/48 scale plastic model and ask them to tell me what the size of the finished model is knowing the size of the actual plane and the scale factor. The opening estimation allowed students a chance to think about scale. Next I handed out the blueprints for the plane. Our task was to use the blue prints to help create a 1/10th scale drawing of the top view of the plane using painters tape on the floor.
Students worked in small groups to recorded their measurements and calculate the measurements for the scale drawing. Once they were finished each group was provided with some painters tape and measuring tapes in order to make their scale drawing on the floor. It could be nice to have one group do their drawing on the wall, then it could stay up as a reference to their work with scale. We also talked about writing their measurements on the painters tape as they put it down. Variations and ExtensionsStudents who finish quickly can continue to add additional details such as the pontoons. This activity could also be done with larger groups creating a life size drawing of this plan using sidewalk chalk outside (weather permitting). NS Outcomes:
Mathematics Essentials 12  3.1 Calculate the dimensions of actual objects using blueprints with various scales Mathematics 11  M02 Solve problems that involve scale diagrams, using proportional reasoning Mathematic at Work 11  G02 Students will be expected to solve problems that involve scale; and G04 Students will be expected to draw and describe exploded views, component parts and scale diagrams of simple 3‑D objects. Mathematics Essentials 11  E4 create 2D scale diagrams and 3D scale models Mathematics 9  G03 Students will be expected to draw and interpret scale diagrams of 2D shapes.
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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 GamesGames 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
Other StuffWhen 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. EL
Some colleagues recently told me about an activity they had used in class called "Math Market". I'm not sure who originally created it. The teacher who shared it with me learned it at a math conference several years ago. I decided to give it a try with a Calculus class that was just finishing up a unit on integration. Here is how the activity is run. Students work in small groups (we had groups of three). Each group starts with $5 and selects a captain who can buy questions of different levels of difficulty from the market. Easier questions cost less and have a smaller profit. More difficult questions cost more and have a higher profit. The captain takes the purchased question back to their group to solve. Once they all agree on a correct solution, the captain returns to the market to sell the solution for a profit. The card is added back to the bottom of the market pile and some other group will have an opportunity to buy it. If their solution is correct, they buy a new question and continue working. If the solution is incorrect, they have to buy the question again to attempt a revised solution (or they can purchase a new question at a different level of difficulty). We decided to purchase the solution at a reduced price ($1 less) if they forgot to include the "+C" at the end for the constant of integration. The easiest questions were free so that if groups went bankrupt with an incorrect solution, they would still be able to "buy" another a problem. I printed the questions on coloured card stock and cut them out. Each question was marked with its level of difficulty. I also added a letter to the card so that it would be easy to find its solution to check the answers. Resources
How it WentI like that students got immediate feedback on their work. If it was wrong, they had to work with their group to correct their mistake. This was a test review for the class so there were lots of different types of problems mixed together and students had to determine what strategy would be best to solve each problem. It is a nice way to introduce some interleaved practice. This activity could be done with nearly any topic but it worked really well for integration as the questions were challenging and took them some time to solve. This made the market area less crowded.
I'm sure there are lots of variations of this activity. If you have some suggestions, I'd love to hear about them. EL
I recently attended Nat Banting's presentation at the Ontario Association for Mathematics Education annual conference in Ottawa (#OAME2019). He talked about "Creating Mathematical Possibility by Constraining Mathematical Possibility" (you can check out the slides from this presentation on his website). Last week I saw several educators sharing variations of a task that Nat presented on his blog called a 'menu task'. In a menu task, students are given a list of specifications and are asked to create functions that satisfy these specifications. It would be a fairly straightforward task for students to create a different function for each single specification (e.g. create a function that has a positive yintercept). Students are challenged by asking them to use as few functions as possible to satisfy all the specifications in a list (in whatever combinations they desire). Amie Albrecht shared a Linear Relationships version of this menu task which inspired me to try creating one for high school calculus. Calculus Functions Menu TaskInstead of jumping right into the menu, I wanted to make sure that students were familiar with the expectation so I decided to build up to the menu. I started by asking students to come up with one function to satisfy each specification. Then I used the same specifications but asked students to satisfy them with only two functions. After getting used to the idea, I then continued with the Calculus functions menu task. The students have just started integration so this is mainly a review with just a bit of integration thrown in. If you'd like to give this task a try with your students, here is a link to my google slides. Example SolutionsHere are three example functions that I came up with to satisfy these ten specifications (some specifications are satisfied by more than one function): A,D,E,H) y = (x+2)(x1)(x3) C,G,H) y = (x+2)(x)(x2) A,B, F, I,J) y = [(x+1)(x+1)(x1)]/(x1) Students were able to come up with fewer functions for this task. Here is one student's two functions to satisfy the ten specifications. EL
Over the recent holiday weekend, I brought out my copy of Dice Games Properly Explained by Reiner Knizia (@ReinerKnizia). I enjoy playing dice games at home with my family as well as playing dice games in class with students. Dice games often have simple rules and typically don't require of lot of material other than dice. Dice are a great mathematics manipulative that can be used with wide range of ages. Even very young students can practice subitizing numbers from 1 to 6 by reading the dots on the face of the dice. Below are a few of my favourite mathy dice games. Shut the Box
The object of the game is to cover as many of the 9 boxes as possible. Any numbers left uncovered at the end of your turn are added together. "Shutting the box", or covering all the numbers, leads to a perfect score of 0. The player with the lowest score is the winner. On the players turn, they roll the dice and add them together. You can then cover any boxes that are not already shut that sum to the total you rolled. For example, if I rolled a 2 and 4 my total is 6. I could cover the 6 box, the 5 and 1, the 4 and 2 or the 1, 2 and 3 boxes if they are uncovered. If I can't partition my number to cover any boxes then my turn is over and I add up any uncovered boxes to determine my score. This is a fun game with some strategy to it. It really focuses on early addition skills as well as partitioning numbers in a variety of way. It is a great game for a lower elementary classroom. NS Outcomes: Primary  N04 Students will be expected to represent and describe numbers 2 to 10 in two parts, concretely and pictorially. Mathematics 1  N04 Students will be expected to represent and partition numbers to 20. Mathematics 2  N10 Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts. Even Minus OddIn this game, players take turns throwing six dice. You then total all the even dice together and all the odd dice together. Subtract the odd total from the even total to get your score. Take counters from the centre of the table equal to your score. If you have a negative total, pay that number of counters to the centre (don't worry, if you don't have any counters left, you're still in the game). When all the counters are gone from the centre, the game is over and the player with the most counters wins. Start with about 10 counters in the centre or more if you have a large group playing. NS Outcomes: Mathematics 2  N02 Students will be expected to demonstrate if a number (up to 100) is even or odd. Mathematics 2  N10 Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts. Game of SixIn this category game, you need only one die and a score sheet. Players take turns rolling the die over six rounds. One your turn, roll the die and decide which category to score. Multiply the number on your die by the category value (1, 2, 3, 4, 5 or 6). Each category can be used only once each game. For example, if you roll a six on your first turn, you could score it in category 6 and earn 6x6 = 36 points. At the end of the six rounds, each player adds up their total points. The player with the most points wins. NS Outcomes: Mathematics 4  N05 Students will be expected to describe and apply mental mathematics strategies, to recall basic multiplication facts to 9 × 9, and to determine related division facts. Mathematics 5  N03 Students will be expected to describe and apply mental mathematics strategies and number properties to recall, with fluency, answers for basic multiplication facts to 81 and related division facts. NinetyNineThe focus of this game is on the order of operations for whole numbers. Five dice are rolled in this game and the player who rolls the dice calls out any number they wish between 33 and 99. This is the target number for the round. Once the dice are rolled, players create an expression using all five numbers on the dice and any operations (+,  , x, ÷). The goal is to create an expression whose value is as close as possible to the target number without going over. Divisions must work out without a remainder. Players secretly write down their expression. Once everyone has an expression (or a reasonable amount of time has passed), players reveal their expression. The player closest to the target scores a zero. All other players score the difference between their expression's value and the values that was closest (to a maximum of 5). Play as many rounds as their are players so each person can have a round setting the target. NS Outcomes: Mathematics 5  N03 Students will be expected to describe and apply mental mathematics strategies and number properties to recall, with fluency, answers for basic multiplication facts to 81 and related division facts. Mathematics 6  N09 Students will be expected to explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers). A Focus on Mathematical ContentThere are lots of really fun dice games in Reiner's book although some are more suited to a mathematics classroom than others. I also really like the chapter on the theory of dice and probability (chapter 3). I think the games above are not only fun, but closely related to mathematics outcomes. A recent post I read from Hilary Kreisberg (@Dr_Kreisberg) discussed a protocol to assess good classroom tasks. One dimension of her protocol was to assess the mathematical content of a task and ask yourself if the task aligns well with specific gradelevel standards. I think this is an important aspect to remember and not just play games that are fun, but ones that also offer meaningful mathematical practice. EL
I recently had the opportunity to work with a calculus class on curve sketching and how derivatives affect the shape of a graph. The classroom teacher and I brainstormed some ideas about how we might infuse some hands on activity into the lesson. We decided to try an activity called Functions on the Floor. I originally saw an outline of this activity in a presentation from Liana Dawson called HandsOn Calculus Activities. Warm up  Functions and Their DerivativesWe started the class out with a Demos activity called Functions and Their Derivatives. We had students work in pairs on this activity. In the first part of the activity, students are presented with the graphs of three functions and they have to decide which is the original function, which is the derivative and which is the second derivative. In the second part of the activity, students create their own challenge and then get an opportunity to try out the challenges created by other groups. I really like the collaboration and discussions creating by working on these challenges. Functions on the Floor InstructionsAfter the warm up, we transitioned to the Functions on the Floor activity. We used masking tape to create several coordinate systems on the floor with the x and yaxis labeled from 3 to 3. At each of these stations was a list of information about a continuous function. Students used a small rope to create a function on the axis that satisfied all of the conditions listed. They then drew their function into a Desmos activity I had prepared. Using the drawings in the Desmos activity we could monitor students activity and plan for our review of the functions at the end of class. We heard some really constructive conversations taking place. The Google slides for the stations can be found here. Sharing our WorkI thought that this activity was a nice way to incorporate both handson physical problem solving while still leveraging some of the power of online tools like Desmos. I learned a few lessons doing this activity and the classroom teacher and I had a productive discussion after the lesson. We talked about how the lesson went and where we saw areas for improvement. We both agreed that eight stations was more than needed as it took some time for the discussions in the student groups to come up with a reasonable graph. We thought that some of the stations could have fewer constraints to consider as well. Also, I think next time I would use something besides rope. I had pretty inexpensive rope and it wasn't as pliable as I would have liked. I think a thick piece of yarn might have worked just a well. If you give this activity a try, let me know how it goes. Nova Scotia Mathematics Curriculum Outcomes Calculus 12 B15  Demonstrate an understanding of critical points and absolute extreme values of a function Calculus 12 B16  Find the intervals on which a function is increasing or decreasing Calculus 12 C5  Apply the First and Second Derivative Tests to determine the local extreme values of a function Calculus 12 C6  Determine the concavity of a function and locate the points of inflection by analyzing the second derivative EL
We started the lesson by reviewing strategies for decimal division. To create a need for division we had an activity where students found an average. We used the Estimating Time activity from NRICH. We challenged several student volunteers to estimate 5 seconds, giving each student three tries. We then asked the class to find the average of each student by dividing the total of their three attempts by three (just a note that students don't have a formal introduction to mean, median and mode until grade 7 SP01). After students finished, we highlighted several strategies that we saw student using and had the students explain their division strategies. We saw students making equal groups with base 10 blocks, partial quotients and the standard long division algorithm. Check out Graham Fletcher's progression of division video for some background on division strategies. Designing a Spinning Top
Testing DesignsGroups used a stopwatch (on their phone app or on https://stopwatch.onlineclock.net/ using a Chromebook) to see how long each top would spin. The tested each design three times and recorded their times. They then used division to find the average spin time for each design. They used this data to decide which of their designs was the best. Sharing DesignsAt then end of the class we led a class discussion about their design process. We asked students what characteristics made a top spin longest: size of the top (number of cubes used), shape of the top (symmetry). How it was spun seemed to be a big factor. One student spun the top from the sides instead of the top and was able to get a long spin (average of about 25 seconds). Further PracticeNova Scotia Mathematics Curriculum Outcomes Mathematics 6 N08  Students will be expected to demonstrate an understanding of multiplication and division of decimals (onedigit whole number multipliers and onedigit natural number divisors). 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 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. EL
Investigations where students discover the relationship between variables can help students build a deeper understanding of functions. Often these explorations are hands on and engaging lessons. They typically start with some sort of interesting video or question prompt such as "What makes for an exciting bungee jump?" or "Which cup will keep my coffee warm the longest?". I was prompted to think about my favourite investigations after seeing a post from Nat Banting on Twitter. Below are a collection of links and descriptions of my favourite secondary mathematics investigations. Linear Relationships
Quadratic Relationships
Exponential Relationships
I like the investigation above because they share several common features.
Do you have a favourite activity or resource for activities? Please let me know what it is. EL
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
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

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