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.

If you'd like to give this a try in your class, here is a link to my instruction and questions Google Slides document and a link to my recording sheet Google doc. Instead of printing off my own bills, I used money from an old board game (Monopoly Junior). One teacher created her own classroom cash to use for this activity and it was a really nice touch. |

I 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.

Towards the end of the class, the teacher sold pieces of candy for the money they had collected. I had originally thought about having an auction at the end of the activity or counting totals to declare a class winner. Selling candy was a better idea. It was much faster, so we could spend more time working on problems. It was also more discrete so teams didn't have to declare to the entire class what their group total was. I liked that everyone in the class got to use their cash and they all got something. |

I'm sure there are lots of variations of this activity. If you have some suggestions, I'd love to hear about them.

EL

]]>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 y-intercept). 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.

Instead 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.

Here 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)(x-1)(x-3)

C,G,H) y = (x+2)(x)(x-2)

A,B, F, I,J) y = -[(x+1)(x+1)(x-1)]/(x-1)

A,D,E,H) y = -(x+2)(x-1)(x-3)

C,G,H) y = (x+2)(x)(x-2)

A,B, F, I,J) y = -[(x+1)(x+1)(x-1)]/(x-1)

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

]]> Shut the Box is a traditional game that has a long history. The game is played with two dice and a sheet with the numbers 1 to 9 listed. These numbers are covered as dice are rolled. You might see special dice trays with wooden doors on hinges for the numbers but this equipment is not necessary. There are several variations to this game. I'll explain the rules to the version I play. |

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.

In 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.

In 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.

The 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).

There 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 grade-level 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

]]>We 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.

After 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 y-axis 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.

I thought that this activity was a nice way to incorporate both hands-on 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.

EL

]]> I was recently invited by a grade 6 class to work with them on decimal division. After discussing the learning outcome and reviewing the curriculum guide with the classroom teacher, we chatted about some possible activities. Based on a desired for a hands on activity we decided to try a design activity that we called Spinning Tops. I saw a similar activity posted on Twitter by Jen Carter (@jencarterbc). The activity was originally shared online by Mike Wiernicki (@mikewiernicki) on his blog, Under the Dome. |

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.

We asked students work in small groups (2 or 3) to design a spinning top, using linking cubes. The goal was to construct a top that will spin for as long as possible. Each group designed 3 different tops. They tested out lots of different ideas about what characteristics might lead to a top that spun for a long time. Most groups tried out a variety of shapes and sizes before settling on their three designs. |

Groups 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.

At 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).

EL

]]>- Vroom Vroom from Fawn Nguyen and Super Racers - Students use a toy pull back car to investigate how far it travels depending on how far it is pulled back.
- Knot Again! from Jon Orr and Ropes of Different Thickness from Alex Overwijk - Students investigate the relationship between the length that a rope can be stretched and how many knots are tied in the rope.
- Paper Clip Chain - Students investigate the relationship between the number of paper clips and the time it take to make a chain from them.
- Spaghetti Bridges from Mary Bourassa and Spaghetti Bridges from Andrew Busch - Students build a "bridge" using more and more strands of spaghetti and see how many coins/washers it can hold before breaking.
- Barbie Bungee from Fawn Nguyen and Barbie Bungee from Matt Vaudrey - Students investigate the number of rubber bands used in a bungee cord and how far the bungee stretches when attached to a toy doll.

- Down the Drain - Students measure how quickly water drains from a cylindrical tank.
- Speedy Squares from Mary Bourassa - Students time how long it takes them to make squares of different sizes using multi-link cubes.
- Pendulum Investigation - Ask students to measure the time it takes a pendulum to swing 10 times. Change the length of the pendulum and measure again. Take several more data points and a quadratic relationship will emerge.
- Penny Circle from Dan Meyer - Students count how many pennies can fit into circles of different diameters and then make predictions using this information.

- Bouncing a Ball from Andrew Busch - Students investigate how high a ball rebounds after each successive bounce. David Wees has a great suggestion of using audio to take more accurate measurements.
- Coin Flipping and Pennies and Dice from Andrew Busch - Students use coins, dice, candy (e.g. Skittles) or two coloured counters to investigate an exponential relationship based on the probability of landing on a specified side when thrown. I've seen variations of this activity in numerous textbooks.
- Too Hot to Handle from Andrew Busch - Students investigate how quickly a cup of how liquid cools using a thermometer.

I like the investigation above because they share several common features.

- They use inexpensive or easy to find materials
- They are hands-on and engaging.
- They don't require extensive preparations or difficult cleanup (Spaghetti Bridges probably requires the most cleanup as it will leave bits of broken dried spaghetti all over the floor, plan accordingly)
- The instructions are easy to explain. The majority of class time is spent carefully collecting and analyzing data.

Do you have a favourite activity or resource for activities? Please let me know what it is.

EL

]]>- Uses easy to find and inexpensive materials or materials that are commonly found in a mathematics classroom.
- Hands on activity for students to investigate in small groups
- The instructions are easy to explain and all students have an entry point into the activity.

There 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.

I 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.

Next we gathered some data to test our estimates. A practice round or two is a good idea as students' speed will increase as they figure out a good strategy for chaining the clips together. You might also ask students to do a few trials at each length of chain and take a mean (or perhaps a trimmed mean) to get more accurate data... or you could save this discussion until after students collect some data and then ask them if they feel their data is accurate. |

After 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.

EL

]]>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.

Hotel Snap from Fawn Nguyen (@fawnpnguyen)In this activity, students design a hotel using multi-link cubes. Students are given a number or restrictions on building as well as costs to minimize and profits to maximize. I blogged about this activity, that I called Hotel Cubed. NS Outcomes: Math 9 G01, Math 10 FM01 and Math at Work 10 N01 |

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 multi-link 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 multi-link cubes.

Each student creates a 3D object using multi-link 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.

Each group of students builds three terms of a linear pattern using multi-link 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/.

The Condo Challenge from Marilyn BurnsStudents are asked how many cubes are required to build a "Condo" that is 6-cubes high. Then they predict how many cubes would be required for a condo 12-cubes high. Later they create a mathematical model to predict the number of cubes for a condo of any height. Thanks to Halcyon Foster for suggesting this task! Although she doesn't mention linking cubes, Jo Boaler's Growing Shapes task (from her week of inspirational math) is a similar problem. NS Outcomes: Math 9 PR01, Math 10 RF04 |

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.

Students, in small groups, design a spinning top made of multi-link 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.

A few links to some documents that provide some additional suggestions for using linking cubes:

- Manipulatives Tip Sheet for Using Connecting Cubes from EduGains - This tip sheet describes a variety of uses for linking cubes.
- 101 Activities Using Linking Cubes - A couple of great activities described here by Jana Barnard and Cathy Talley. I really like the mean, median and mode stats activity.

The examples above are mostly from secondary math classes. Multi-link cubes are also incredibly useful in elementary math classes (counting, measuring with non-standard 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

]]>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 students 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 closer than I had expected. This activity could be easily extended for higher grade levels by incorporating linear relationships, linear equations and linear regression.

EL

]]>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 bi-monthly calculation, not bi-weekly.

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:

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:

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 bi-monthly calculation, not bi-weekly.

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!”

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:

- Talking Math with Your Kids
- Bedtime Math
- This blog post from denisegaskins.com
****

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.

]]>Do you know of any other great resources like these? Let me know!

K.