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 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:
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!
The basic game is easy to learn. Here are the basic rules:
Most 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. Skip-Bo, Zero Down, Krypto, etc). Some might need special cards to be printed off such as (Logarithm War).
Representing 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 (A-9) 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 fifty-three.”
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.
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.
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 Fountain
I 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 Tank
We 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. 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.
Pre-calculus 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, x-intercept and y-intercept, and to solve problems.
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:
It 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 self-checking 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, add-em up, tarsia puzzles, and question stacks.
Purposeful 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.
Just 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.
I'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.
I'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!
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 warm-up 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 Century
By 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?
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.
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.
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 Practice
I 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.
During a professional development session today with grade primary to grade 9 math coaches and support teachers, we spent some time working on solving some math puzzles. We used our work on these puzzles to reflect on what good group work looks and sounds like. We also discussed the characteristics of effective mathematical communication. It was great to see positive energy and teachers supporting and encouraging each other. Below are the five math puzzles and investigations that we worked on. We selected these puzzles because they are easy to explain, open to a wide range of students, and offered a fun challenge.
1-10 Card Investigation
This problem from Marilyn Burns asks you to find a special order for a stack of cards, Ace through 10. Place the stack of cards face down and turn over the first card. It should be the Ace. Place the next card on the bottom of the deck and then reveal the top card. It should be the number 2. Continue placing the next card on the bottom of the deck and then revealing the top card until all the cards are revealed. The face up cards should now be in order from Ace to 10. Marilyn recorded a video to demonstrate these rules that is much easier to understand than my written instructions. Marilyn's has a description of this problem on the MathSolutions website. This logic problem doesn't rely on any prerequisite mathematical knowledge and you can try it out quickly to see if you've come up with the correct solution. It gives students a chance to work together to try out strategies.
The Square-Sum Problem
Can you order all the numbers from 1 through 15 so that the sum of any two consecutive numbers are always a square number? For example, in the sequence: 4, 5, 11; 4+5=9 and 5+11=16.I really like this problem because there are some great extensions to take this problem farther and there is a very nice way to visualize the possible solutions. Numberphile has recorded a great video demonstrating both the problem and solution.
The Year Game - 2018
Use the digits in the year 2018 to write mathematical expressions for the counting numbers 1 through 100 (we only went to 20). Use any math operations (+, -, x, ÷, etc) and parentheses to write these expression. There is a more detailed description of this problem at the Math Forum website. For example, expressions for the number 1 might be: 10 ÷ (2+8) or 218^0. This problem is very similar to the classic Four Fours problem but with new digits each year. Which numbers are the hardest to find an expression for? Why do you think this is? I think this problem also leads nicely to a discussion about mathematical elegance and beauty. Look at a variety of expressions with the same value. Which expression do you think is the best? What makes for an "elegant" solution?
This problem challenged groups to make as many different shapes as possible with a perimeter of 12 units using a geoboard (or dot paper). Shapes were recorded on dot paper to make sure no shapes were repeated as reflections or rotations. I've seen variations of this problem in several places. One of them is Brian Bolt's book Mathematical Activities (1982). He suggests not only to find shapes with a perimeter of 12 but to also find the area of each shape. You can then find which shape has the maximum/minimum area. He also challenges students to find non-rectangular shapes (e.g. triangles) with a perimeter of 12 units. There were some good discussions about the lengths of diagonal line segments on the geoboard.
This problem is from NRICH. Finding a solution took some perseverance but most groups were eventually successful. The problem challenges you to create a square using 8 specific dominoes (0-1, 0-2, 0-3, 0-6, 1-2, 1-4, 2-2, and 3-5). In the 4 x 4 square, each of the columns and rows should sum to 8. The 3-5 domino seemed to be key as the rest of the squares in that row (or column) had to be blank.
What are your favourite math or logic puzzles?
This blog post could easily have been titled, "A Long Wait in a Really Long Line" but I like to be positive. I focus on the silver lining instead of the grey cloud. That's why I took a wait in a long long as an opportunity to practice estimation instead of a tedious and boring waste of time.
I spent a sunny Canada Day afternoon at the Halifax Commons. I went with my family to watch the SkyHawks, the Canadian Armed Forces Parachute Team. After the SkyHawks finished their jumps, we headed to the end of a a really long line so that my son could take a turn on a giant inflatable slide. Instead of dwelling on the length of the wait, we decided to focus on some fun estimation questions. How many people do you think are in this line? My son says he thinks it's more than 100 and I agree. How long will it take to get to the front of the line? I feel like we're in for a long wait. My initial estimate is at least 30 minutes. (Note: in order to answer this last question, I took a look at my watch to check the time... 12:06 pm).
Have we gotten any closer? It's 12:27 now (21 minutes in line) and it still feels like we've got a long way to go. There appears to be a strong correlation between the age of a child and the likelihood that they will have second thoughts at the top of the slide. This slows the line down dramatically as parents try to coax and cajole their child to make the leap. How many people in this line are kids waiting to slide and how many are parents/guardians? What is the average age of the kids going down the slide?
It's 12:40 now and we've been in line for over a half an hour. How many steps do you estimate their are to the top of the slide? I estimate that we're about half the distance to the slide from where we started. I realize that my initial estimate for how long we will be waiting was way off. At this point, I notice a group of upper elementary age students in front of us playing a hand clapping game called Concentration. After a bit, they shift to playing Chopsticks. This is a game I really like and I've used to introduce students to modular arithmetic so I watched their game to check out their strategy. It kept me entertained for a bit.
So close now I can taste it. It is 1:10 and we've been in line for over an hour. How tall do you estimate that slide is? I'd say it is about as tall as my house (a two story foursquare) which has a height of about 26 ft. (Note that the internet says this inflatable structure, called the 'Freestyle Combo', is 30 ft tall so my estimate seems pretty close).
We finally make it! It is 1:17 pm and we stood in line for 71 minutes. My son seems to think that this was a reasonable investment for such an awesome slide but I have my doubts. At least we got to do a lot of estimation while we waited in line. We definitely won't be heading to the back of the line for a second slide.
Swimming in the hotel pool I saw these depth markers. As a math teacher, they made me a bit uneasy. What do you notice in the photos below? What do you wonder?
Just look at those significant digits. They look so precise. I first thought... going from the shallow end to the deep end, it gets 1 foot / 0.2 metres deeper. That must mean that 1 ft = 0.2 m right? But then if 1 ft is 0.2 m then shouldn't 3 ft in the shallow end be 0.6 m instead of 1.0 m?
So I looked at it another way... 1 m is the same as 3 ft... So 1 ft must be about 0.33 m. Which would make 4 ft equal to about 1.33 m not the 1.2 m as shown. But I know that a meter stick is shorter than a yard stick so this is just an approximation. No problem, they just rounded off both values.
Then I had a moment of doubt... in the shallow end the values are in a ratio of 1/3 and in the deep end the values are in a ratio of 4/12 which is also 1/3 so shouldn't this work out? Then I realized the errors and misconceptions in this line of thinking.
Other Linear Conversions
Today, as I was driving around, I looked more closely at the clearance signs that I passed under. There doesn't seem to be much consistency in the units used or precision. Do people with tall cars know the height of their car? I just know that I'm about 6 ft tall and my car is shorter than I am. Of course you can always just wing it. If you clear the warning bar, you're good to go. Anyway, I know for sure that my car is less than 11 foot 8 if I ever end up in North Carolina.
I saw this relatively accurate sign at a parking garage today so I took a photo. 6'0" is approximately 1.8288 metres so these values are the closest I've seen.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 10 M02 - Students will be expected to apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.
Mathematics at Work 10 M01 - Students will be expected to demonstrate an understanding of the International System of Units (SI) by describing the relationships of the units for length, area, volume, capacity, mass, and temperature and applying strategies to convert SI units to imperial units.
Mathematics Essentials 10 D1 - Demonstrate a working knowledge of the metric system and imperial system.
Below is a simple probability game that I've played with students for many years in a number of different courses. I don't really remember where I got it from but if you've seen it before, I'd love to know where it originated. I've found that students enjoy it and they really develop some good ideas about probability. Its similar to the Two-Dice Sum Game from Marilyn Burns but with some additional mental math for older students.
This is a game for two or more players. Each player creates their own score card by drawing two concentric circles and then divide it into eight sectors. Choose eight different numbers between 2 and 12 and place them in the sectors in the outer ring. In the inner ring of the circle place numbers that add up to 100 (it is easiest to stick with multiples of 5). Roll two dice and add the numbers together, if the total is one of the numbers in the outer ring of your score card, you score that numbers value (the number in the inner ring). Players alternate rolling and scoring on their card until one of the players has 150 or more points (or to 100 for a bit shorter game). A google doc handout for this is available here.
Sample Score Card
Typically, the first time students play, they will randomly select numbers and values for each sector. After students have had the chance to play a few games, stop and ask them to look at different score cards to see if there any common characteristics of winning cards. What makes a good score card? Is it all just luck or are some cards better than others?
Developing a Strategy
What makes a good score card? Here are some common student observations:
You might ask students to roll two dice a bunch of times and record which numbers come up each time. Create a bit dot plot or a bar graph at the front of the classroom using all of this data. Ask students to predict what the shape of this graph will be. You could also use an online tool to model lots of dice rolls to see how it compares to the class data. You could then discuss the theoretical probability for each sum and compare it to the data you gathered.
I always find this a fun activity and a nice way to start a discussion about probability and strategy. A nice question to start the class on the following day would be a Would You Rather? math prompt, "Would you rather flip 2 coins and win if they match OR roll 2 dice and win if they don't match?" Another great follow up activity would be Don Steward's Dice Bingo.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics at Work 12 P01 - Students will be expected to analyze and interpret problems that involve probability.
Mathematics Essentials 10 G4 - Compare predicted and experimental results for familiar situations involving chance, using technology to extend the number of experimental trials.
Mathematics 8 SP02 - Students will be expected to solve problems involving the probability of independent events.
Mathematics 7 SP06 - Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events.