The Riddle of the Tiled Hearth is one of many mathematical puzzles from Henry Ernest Dudeney's 1907 book titled The Canterbury Puzzles And Other Curious Problems. The first group of puzzles in this book are based on the characters from Geoffrey Chaucer's Canterbury Tales. Puzzles from this book could be used as part of a cross curricular unit on history, literature and mathematics. There are a number of very interesting puzzles and games including the first pentomino puzzle called The Broken Chessboard and a clever variation of the game Nim called The Thirty One Game. Since this book was first published in 1907, the copyright has expired and it is freely available on Project Gutenberg.
It seems that it was Friar Andrew who first managed to "rede the riddle of the Tiled Hearth." Yet it was a simple enough little puzzle. The square hearth, where they burnt their Yule logs and round which they had such merry carousings, was floored with sixteen large ornamental tiles. When these became cracked and burnt with the heat of the great fire, it was decided to put down new tiles, which had to be selected from four different patterns (the Cross, the Fleur-de-lys, the Lion, and the Star); but plain tiles were also available. The Abbot proposed that they should be laid as shown in our sketch, without any plain tiles at all; but Brother Richard broke in,--
To use this activity with students, I would start by introducing using the tiled hearth story as written above. Then I would introduce some manipulatives that would let them physically explore and work with the puzzle. I would give each group of students a large 4x4 grid on a sheet of paper and some multi-link cubes of 4 different colours.
One aspect of this puzzle that I like is that students can play around with it and have some intermediate success. They might just place a few cubes on the grid. With time, they can refine their solutions to get better and better. Below shows how a student might explore to place more and more cubes.
The Solution from Canterbury Puzzles shows that the best solution leaves 3 blank spaces. Dudeney states, "The correct answer is shown in the illustration on page 196. No tile is in line (either horizontally, vertically, or diagonally) with another tile of the same design, and only three plain tiles are used. If after placing the four lions you fall into the error of placing four other tiles of another pattern, instead of only three, you will be left with four places that must be occupied by plain tiles. The secret consists in placing four of one kind and only three of each of the others." Below are both my solution using cubes and Dudeney's equivalent solution.
Revisiting the Classic Ferris Wheel Problem
This type of pseudo-context word problem robs students of the opportunity to explore and analyze real-world problems in much depth. Dan Meyer has written quite quite a lot about pseudocontext. My concern with the Ferris wheel problem is not that you can't model the height of a seat on a Ferris wheel with a sine function, it is why would you do it?
Instead of doing a textbook problem with a fictional Ferris wheel, I decided to use a real Ferris wheel from a nearby amusement park that some of my students would be familiar with. I visited the park to take a video of the Ferris wheel in action. Below is a 30 second clip of the "Big Ellie" Ferris Wheel at Atlantic Playland.
Notice and Wonder
I started by asking students what they noticed in the video. After brainstorming and recording the students observations I asked students what they wondered about in the video. They asked questions like "how fast is the ride going?", "how tall is this Ferris wheel?", "how far can you see from the top of the ride?", "how long does the ride last?". In order to investigate these questions further we needed to estimate some values such as the radius of the wheel, how long it takes to make one revolution, and the height of the central axis about the ground. I asked students to estimate these values using the clues in the video we watched. We watched it several times in order to get some good estimates.
I also talked about some of the mental math required to operate a ride like this. Because it is belt driven, you have to load the Ferris wheel so that it is equally balanced around the wheel. Otherwise, one side of the wheel would become too heavy and the drive cable would slip in the rim and the wheel wouldn't be able to turn! This requires a lot of on the fly estimates of weights of the riders as it is being loaded.
In order to get a see how good we did with our estimations we turned to the internet in order to try to hunt down some of these values with a Google search. This lead to a discussion about what keywords we could use to hunt down this information. A search of "height of the central axis of the Ferris wheel at Atlantic Playland" was not very fruitful... an essential skill to solve a problem like this is to translate mathematical language into common terms that you can use for a Google search. Ve Anusic has a great blog post where he discusses a similar problem and the discussion with his students about the information you need and the information you might find online. First we did a search to find Atlantic Playland's website and found that they called their ride "Big Ellie". A search for this name lead us to believe that this Ferris wheel is a No. 5 Big Eli wheel made by Eli Bridge (I later emailed the park and confirmed that this is indeed the model of their Ferris wheel). Eli Bridge's website gave us some interesting information but not exactly what we were looking for. A bit more searching and we were able to find a pdf of the owner's manual for this ride that included a helpful diagram.
It is only after we were able to answer some of the students' questions regarding the video of the Ferris wheel did we start to talk how we might mathematically modeling the height of a person riding the wheel over time and the periodic nature of this function. Students were much better able to make sense of this visual model once they had a good grasp of the context of the problem.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 12 - RF03 Represent data, using sinusoidal functions, to solve problems.
Pre-calculus 12 - T04 Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems.
I drive past this building every day on my way to work. It is Young Tower at 6080 Young Street in Halifax. I think it is pretty interesting... I used this picture as a problem solving warm up activity for a group of grade 10 math teachers recently. I gave each group of teachers a large piece of chart paper and asked them to divide the paper in half with a line. I asked teachers to brainstorm what they notice about this picture and record it on one half of their chart paper. I asked them to look at the picture using a number of lenses. What would an architect notice about this image? What would a person who worked at this building notice about this picture? What would a mathematician notice about this picture? After about 5 minutes of brainstorming, I asked each group to tell me one thing they noticed and I recorded it at the front of the room. Groups noticed things like the number and size of windows on the building ("about half the lateral surface is glass"), the shape of the building ("almost a cube"), the picture must have been taken on a weekend because there are very few cars in the parking lot, and the weather was really nice that day. Next I asked them to brainstorm what they wonder about this picture and record in on the other half of their chart paper. If this picture was the start of a math problem, what could that math problem be? What things that they noticed sparked their curiosity? After another 5 minutes, I asked each group once again to tell me one thing that they wondered. After looking at all the questions that the groups posed, we selected one and asked everyone to estimate an answer to that question. I also asked them what information would they need to make a more accurate estimate. Once they had an initial estimate, I gave them some additional information about the building and let them revise their estimate. We had several really interesting questions posed by groups. Some questions concerned the shape of the building, like "How close to a perfect cube is this building?" Other questions focused on finance such as, "How much revenue is generated by leasing all of the office space in this building?" One of my favourite 'wonderings' was, "How much wrapping paper would it take to wrap this building up like a Christmas present?"
This "I Notice/I Wonder" problem solving strategy is one that I saw shared by Max Ray-Riek from the Math Forum. He has a blog where he talks about Noticing and Wondering in High School. This strategy starts off with brainstorming to let students get familiar and engaged with a problem situation before jumping into a specific question to solve. By having students come up with questions, you'll often get more engagement and interest. It also allows you to respond to interesting suggestions from students that you might not have considered. It allows everyone in the class meaningful participation in the conversation because everyone has something that they can notice. This strategy might also create additional opportunities for differentiation by using several different questions that students suggested.
I really like some of the questions found on the Openmiddle.com website. It is a great resource for questions that really get students thinking. They are often formatted so that there is a very low threshold for entry to the problem but they allow for enrichment and extensions. I created the problem below for a professional development session for Math 10 teachers. Students in the Math 10 course are near the following outcome in the yearly plan: AN03 Students will be expected to demonstrate an understanding of powers with integral and rational exponents.
Fill in the boxes with whole numbers 1 through 6, using each number at most once, so that the value of the expression is as large (or as small) as possible.
In order to verify that these were correct values, I wrote a short program in Python (see below) to check all of the 720 possible (6!) values. It was my personal Hour of Code activity. I`m just learning to use Python so my code could definitely be more efficient. If you can write some better code, let me know and I`ll post it here and give you credit.
largest = 0
smallest = 10000
for i in ValuesList:
for j in ValuesList:
for k in ValuesList:
for l in ValuesList:
for m in ValuesList:
for n in ValuesList:
if i!=j and i!=k and i!=l and i!=m and i!=n and j!=k and j!=l and j!=m and j!=n and k!=l and k!=m and k!=n and l!=m and l!=n and m!=n:
if z > largest:
largest = z
if z < smallest:
smallest = z
Update: I submitted this problem to the OpenMiddle.com website and it has been posted there.
I'm a big fan of using open-ended problems in math class. An open-ended problem is one that has a number of different correct answers or several ways of getting to a correct answer. Below is a question from page 140 of the MMS9 textbook that is not an open problem but has lots of potential.
This problem asks students to evaluate four different versions of a nearly identical expression. It tells students exactly what to evaluate... snore. A bit tedious and not very engaging. Lets open this question up by putting students into pairs and giving them the equation with no (parentheses) or [brackets].
Ask students to find out how many different values they can get by putting in one set of brackets. What if they could put in two sets of brackets? What if they could put in as many brackets as they'd like? What is the biggest value they could create? How about the smallest value that they can find? How about the value closest to 0? Once they've created and evaluated a bunch of different expressions, each group could take turns placing their numbers on a clothesline number line. This will get students practicing order operations with a goal and a challenge.
There is a very similar version of this problem in the Grade 9 Mathematics Curriculum Guide for outcome N04. The following problem is in the Assessment Tasks section:
Without the use of a calculator, simplify the expression 1/2 - 1/3 x 1/2 - 1/3 and express your answer as a fraction.
− By inserting one pair of brackets, how many different answers are possible?
− By inserting two pairs of brackets, is it possible to receive a different answer?
Want an even more open version of this problem? Just give students the four numbers and ask them to use the operations of addition, subtraction, multiplication and division and do the same activity. There is an activity from Illustrative Mathematics that describes this problem using the numbers 1, 2, 3 and 4. Perhaps that could be a warm-up for the more complicated question with fractions.
Nova Scotia Mathematics Curriculum Outcomes
Grade 9 N03 - Students will be expected to demonstrate an understanding of rational numbers by comparing and ordering rational numbers and solving problems that involve arithmetic operations on rational numbers.
Grade 9 N04 - Students will be expected to explain and apply the order of operations, including exponents, with and without technology.
One of my favourite activities recently has been Fawn Nguyen's Snap Hotel. This activity can also be found on the NCTM Illuminations website. You might see this activity online under a few different names... my favourite is Hotel^3. It is an engaging problem solving task. Students are given 50 multi-link cubes and instructed to build a model hotel. Each cube represents one hotel room. Some rooms are more desirable than others and can be rented for higher prices. The number of windows and whether or not the room has a roof determine the rental price. Students also have to consider costs associated with property and height. The hotels that students create in this activity remind me of Montreal's Habitat 67. Students might talk about the costs and benefits of this type of architecture. This activity can be used to assess a number of Nova Scotia mathematics curriculum outcomes.
The Rules - As a team, build a hotel that yields the highest profit.
Below are some hotel's created by teachers during a session at the NS MTA conference in Oct. 2015.
Suggestions for Improvement
Below are my Powerpoint introduction and a student handout. Also included is an analysis of several different hotels to see how their profits compare.
Why I Like This Task
Nova Scotia Curriculum Outcomes
I was inspired recently by Jocelyn Procopio's new site Storied Math. She posts a video or a picture and asks students to submit a question and solution that goes with the image. I really like the invitation to students to get in on the action of creating real world problems and contexts that are interesting to them. Thinking about a question that I could suggest made me think of a question about tree planting from the Grade 10 Math Finance textbook. When I did this question in class, my students questioned the numbers given in the problem. We decided to research at bit more using the internet to find a tree planting company and find out how they pay their tree planters. The process reminded me of Dan Meyer's Makeover Monday blog posts from a couple of summers ago where he invited teachers to reconstruct a problem from a textbook. Below is a problem ripe for a makeover.
How could we make this question better... lets start by removing all of the information and instead give the students a video of a young person planting trees and ask students what they notice and wonder.
Some questions that I had while watching this video were... How many trees can he plant in a hour/day? How much does that bag weigh when it is full of tree saplings? How much money does he get paid for each tree he plants? How many hours a day does he work? How many of those tree saplings survive and grow? Lots of interesting questions could be asked here. Like "How much of my daily income will be spent on massage therapy for my aching back?" This is a physically demanding job!
Selecting a Question to Explore
The next step is settling on an interesting question to explore and pose a solution. Once students have formulated a question, we can start to figure out what information we will need in order to answer the question. Not only does the textbook problem tell you what question to answer, it gives you exactly the information you need to solve that particular problem. There is no place for the students to think or be curious. They just take the numbers from the text and do some mathematical operations on them and hope they get the correct answer.
Let's decide to answer the same question that the textbook asked... "What would the tree planter's gross pay be for 6 weeks of work?" Now we have to figure out what information we need in order to answer this question.
Digging up the Facts
Solving the Problem
So what will the tree planter make in six weeks of work? The answer to the textbook question would be 3500 trees/wk * 6 wks * $0.17/tree = $3570 gross pay. This is not a whole lot better than working at a fast food restaurant earning minimum wage. 40 hrs/wk * 6 wks * $10.60 hr = $2544 (minimum wage in Nova Scotia as of April 1, 2015 is $10.60 per hour). Consider the cost of travel required to get to BC as well as the special equipment you might have to invest in (shovel, boots, tree bag, camping gear) and the tree planting job listed in the textbook doesn't sound so great. Not to mention days of backbreaking labour in hot and difficult conditions.
Lets say that our planter can plant 2000 trees per day for 5 days per week and earn $0.14 per tree. This would give us a gross pay of $8400 for 6 weeks. This seems a bit more enticing for a student to go out west to plant trees instead of working for minimum wage.
Instead of gross pay, you could have students figure out net pay. What will his deduction be for CPP, income tax, EI, etc? How much will have actually have in his bank account at the end of the six weeks? What are some high-paying summer jobs that are available to young people in Nova Scotia?
There is an article in the October 2015 NCTM publication Mathematics Teaching in Middle School titled Social Justice and Proportional Reasoning. The author, Ksenija Simi-Muller, has a great table at the end of the article listing strategies and advice for modifying textbook tasks to become real-world problems. One of her strategies is, "Require students to create a written argument based on the information given in the textbook problem. This is one of the most effective ways to engage students with real-world problems." I really like this suggestion to get student to critically think about the questions being posed and not just plugging numbers into equations.
Would you rather install and maintain an electric hand dryer or a paper towel dispenser in a public washroom?
Public school students are familiar with public washrooms since every school has them. Does your school have paper towels or an electric hand dryer? Most public schools that I have been in have paper towel holders... why do you think that is?
Cost of Installation
A hand dryer cost considerably more to purchase and install than a paper towel dispenser. A typical hand dryer costs in the range of $600 plus installing electric hookup. A good paper towel dispenser costs around $100 and only requires a few screws to install.
Cost of Operation
The cost of operation can vary widely depending upon where the washroom is and how often it is used. A washroom in a big sports stadium, an airport or a large school will see a lot more visits than a small business.
Electricity to run a hand dryer is quite a bit cheaper than purchasing paper towels. In Nova Scotia in 2015, the electricity rate is 14.947 cents per kilowatt-hour. To find the annual cost of operation of a hand dryer, multiply the dry time of the hand dryer (in hours) by the rated wattage of the hand dryer (kW) by the average number of uses per day by the number of days in a year by the electricity price per kWh.
To find the cost of operating a paper towel dispenser, multiply the number of visits per day by the number of paper towels used per visit (about 2 probably) by the cost of each paper towel (about $0.01 on average).
Dyson has a really great website on how they calculate costs for their Dyson Airblade electric hand dryer and compare the costs to traditional hand dryers and paper towels.
Update: This question is now on the Would You Rather website!
Would You Rather have the revenue from an amusement park Ferris wheel or carousel ride?
I'm a big fan of questions based on the "Would You Rather?" prompt. This weekend, my son and I visited Atlantic Playland for the first time. We bought a book of tickets and started exploring some of the rides. Our favorites were the Ferris wheel and the Nostalgic Carousel. The Ferris wheel costs us 4 tickets per ride and the carousel cost us 3 tickets per ride. If you were an amusement park operator, which ride would you rather have? Why do you think these rides cost a different number of tickets? Both rides lasted about the same amount of time. The carousel holds more people and is much faster to load and unload. Operating a carousel does not take as much training as operating a Ferris wheel. There is a surprising amount of mental math and estimation require to operate a Ferris wheel. Loading and unloading a Ferris wheel takes a bit of time as the weight of the riders has to be balanced. The Ferris wheel appears to be a more popular ride however and often lots of riders while the carousel was never very full (the Ferris wheel can hold up to 24 people in 12, 2-person seats while the Ferris wheel can hold over 30 at a time).
What do you think? If you operated an amusement park, which ride would you want to have?
Update: This question is now on the Would You Rather website!