Henry Ernest Dudeney was an English mathematician who was a prolific creator of logic puzzles and mathematical games. His "Perplexities" column was featured in the Strand Magazine from 1910 until his death in 1930. The Strand was a very popular publication during this time and was probably best know for The Sherlock Holmes short stories written by Arthur Conan Doyle. Below are four problems that were written by Dudeney and printed in the Strand. Each of them are problems that I think would be approachable by students and related to Nova Scotia mathematics outcomes. I've shown both Dudeney's solution as well as my attempts to solve them.
Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly. My Solution: I knew that the straight path up the side of the room and across the ceiling was unlikely to be the shortest path (of 42 ft). I created a net diagram, labeled points A and B and drew a right triangle to show the spiders path. I used Pythagorean Theorem to solve for the length of the path and got 40.7185 ft. I thought I was quite clever until I read Dudeney's solution and realized that there are several different ways to draw the net diagram and that I hadn't found the shortest path. I had stumbled upon a variation Dudeney's No. 3 solution. Students would probably benefit from modeling the net of the prism with Polydron pieces to see if different students come up with different net diagrams. Dudeney's Solution: Imagine the room to be a cardboard box. Then the box may be cut in various different ways, so that the cardboard may be laid flat on the table. I show four of these ways, and indicate in every case the relative positions of the spider and the fly, and the straightened course which the spider must take without going off the cardboard. These are the four most favourable cases, and it will be found that the shortest route is in No. 4, for it is only 40 feet in length (add the square of 32 to the square of 24 and extract the square root). It will be seen that the spider actually passes along five of the six sides of the room! Having marked the route, fold the box up (removing the side the spider does not use), and the appearance of the shortest course is rather surprising. If the spider had taken what most persons will consider obviously the shortest route (that shown in No. 1), he would have gone 42 feet! Route No. 2 is 43.174 feet in length, and Route No. 3 is 40.718 feet. Juggling with Digits  The Strand Magazine, Volume 77 (1929) page 312 Arrange the ten digits in three arithmetical sums, employing three of the four operations of addition, subtraction, multiplication, and division, and using no signs except the ordinary ones implying those operations. Here is an example to make it quite clear: 3 + 4 = 7; 9  8 = 1; 30 + 6 = 5. But this is not correct, because 2 is omitted, and 3 is repeated. My Solution: This puzzle reminded me of an Open Middle problem. Since you need three equations, only one of the numbers involved can be a two digit number. The rest must be single digits. Also, since you can't use any digit multiple times, the digit 1 can't be used to multiple or divide a number. Similarly, the digit 0 can't be used to add or subtract or multiple a number. This means that the 0 must be used as the ones digit of a twodigit number. For example, it could be used as 6 + 4 = 10 or as 2 x 5 = 10. After playing around with the numbers a bit I ended up with 4 x 5 = 20, 9  3 = 6 and 1 + 7 = 8. I was also came up with 3 + 7 = 10, 8  2 = 6 and 4 + 5 = 9 but then realized that I hadn't used three of the four operations. I haven't found any additional solutions, but i'm not positive that I've exhausted all the possibilities. Dudeney's Solution: 7 + 1 = 8; 9  6 = 3; 4 X 5 = 20 The Russian Motorcyclists  The Strand Magazine, Volume 53 (1917), pages 9597 Two Army motorcyclists, on the road at Adjbkmlprzll, wish to go to Brczrtwxy, which, for the sake of brevity, are marked in the accompanying map as A and B. Now, Pipipoff said: "I shall go to D, which is six miles, and then take the straight road to B, another fifteen miles." But Sliponsky thought he would try the upper road by way of C. Curiously enough, they found on reference to their cyclometers that the distance either way was exactly the same. This being so, they ought to have been able easily to answer the General's simple question, "How far is it from A to C?" It can be done in the head in a few moments, if you only know how. Can the reader state correctly the distance? My Solution: I labeled the distance from A to C as x and the distance from B to C as z. I then created a system of equations to solve algebraically. Since the distance both ways is the same, I knew that x + z = 21. Since this is a right triangle I also used Pythagorean Theorem to create the equation (x + 6)^2 + (15)^2 = z^2. Using the first equation I solved for z to find z = 21  x. I then substituted this equation into the second equation (x+6)^2 + 225 = (21x)^2 and then solved for x. The x^2 terms cancel out so you are not left with a quadratic to solve. Solving for x results in a distance of 3 1/3 miles. Dudeney's question implies that there is a simple way to solve this in your head so he is not talking about my method of solution! Dudeney's Solution: The two distances given were 15 miles and 6 miles. Now, all you need do is to divide 15 by 6 and add 2, which gives us 4 1/2. Now divide 15 by 4 1/2, and the result (3 1/3 miles) is the required distance between the two points. This pretty little rule applies to all such cases where the road forms a rightangled triangle. A simple solution by algebra will show why that constant 2 is added. We can prove the answer in this way. The three sides of the triangle are 15 miles, 9 1/3 miles (6 plus 3 1/3 miles) and 17 2/3 miles (to make it 21 miles each way). Multiply by 3 to get rid of the fractions, and we have 45, 28, and 53. Now, if the square of 45 (2,025) added to the square of 28 (784) equal the square of 53 (2,809) then it is correctand it will be found that they do so. I have not idea why Dudeney's "pretty little rule" works. If you have any idea what he is talking about, please let me know. I'm very curious and I'm hoping I haven't missed something obvious. I found a formula that relates the perimeter of a right triangle (P) to its two legs (a and b). The formula states that a = P * (P2b)/(2*(Pb)). Using this fomula a = 42 *(423)/(2*(4215)) = 21 * 12 / 27 = 28/3 = 9 1/3. This does not appear to be the relationship that Dudeney is using however. Does it have something to do with a generating function for Pythagorean Triples (such as m^2 – n^2 , 2mn , m^2 + n^2)?
My Solution: This question also reminds me of an Open Middle problem. To minimize the solution, we'd like to have the most single digit numbers possible, the rest will be two digit numbers. We will need at least 3 twodigit numbers since the digits 4, 6 and 8 must be in the tens place. If they were in the digits place they would be even and hence not prime. The digits 1 and 9 are not prime and therefore need to be the unit digit with 4, 6 or 8 in the tens place. The remaining four digits {2, 3, 5, 7} could be added as individual units but one of them must join the 1 and 9 as the unit digits in one of the 3 twodigit numbers. The digits 2 and 5 would make any of these two digit numbers composite, so they will be a prime on their own. Next, I looked at all the twodigit prime numbers that start with 4, 6 and 8 and found {41, 43, 47, 61, 67, 83, 89}. Since 89 is only number with a 9 in it, 89 must be one of the numbers in our answer. The remaining digits can be placed in three possible ways, each giving a total sum of 207 since we're just swapping the units digits in different numbers: 2+3+5+47+61+89, 2+3+5+41+67+89, 2+5+7+43+61+89. Dudeney's Solution: The 4, 6, and 8 must come in the tens place, as no prime number can end with one of these, and 2 and 5 can only appear in the units place if alone. When those facts are noted the rest is easy, as here shown: 47+61+89+2+3+5=207. I think that it is important, as mathematics educators, that we occasionally work on solving challenging problems. There was recently an article posted about how Math Teachers' Circles give educators a chance to remember why they love math and puts them in the learners' seat where they can better empathize with their students. When there are no organized Math Teachers' Circle in your area, sometimes trying out an occasional interesting logic puzzle or mathematics problem can have similar benefits. EL
0 Comments
Constructing Rectangular and Triangular PrismsDetermining the surface area of a prism can get a bit stale. Textbooks contain lots of pictures of various right rectangular and triangular prisms. These prisms are carefully labeled with the exact information that a student needs. Students are given the task of inserting these numbers into a formula and doing some basic calculations. These types of problems often don't require much thought. I've recently had the pleasure of working in some junior high classrooms. We were looking for a more handson and thought provoking activity for surface area. We were also looking for an activity in which students could be creative. This is what we came up with. Students, working in pairs, are given either a yellow or blue piece of coverstock. Students with a yellow piece are asked to design and draw the net of a right rectangular prism. Students with a blue piece are asked to design and draw the net of a right triangular prism. Students can draw whatever size or shape prism they wish as long as it covers the majority of the paper (at least half). Students use a ruler to carefully draw and measure the net. They measure and label the length and width of each face and calculate the area of each face on the net they have drawn. Once students have accurately drawn their nets and labeled the area of each side, a teacher will review their work. If it is an accurate net, the teacher will give the students a pair of scissors to cut it out. Make sure students do their calculations inside the net so that it is not lost when they cut it out. Once cut out, students can fold and tape their prism. Students found this activity to be more challenging than they expected. Several had to start over after realizing that the prism they started wouldn't fit on the page or their net wouldn't fold into a proper prism. You could extend this activity by having students tape their nets inside out (with the calculations on the inside) and then challenging them to order the prisms from least surface area to greatest surface area. Why I Like This Task
Double the Surface Area
Nova Scotia Mathematics Curriculum Outcomes Grade 8 M02  Students will be expected to draw and construct nets for 3D objects. Grade 8 M03  Students will be expected to determine the surface area of right rectangular prisms, right triangular prisms, and right cylinders to solve problems. Grade 9 G01  Students will be expected to determine the surface area of composite 3D objects to solve problems Math at Work 11 M01  Students will be expected to solve problems that involve SI and imperial units in surface area measurements and verify the solutions. EL
Text, Email and Phone PhishingIf I was playing some sort of financial scam bingo, I would surely be a winner. In the past three days, I've received phishing scam attempts through three different communication channels: text message, email and phone call. Two days ago I got a text message from the "Bank of Nova Scotia" alerting me to the fact that they had noticed unusual activity on my account. They suggested that I should go to a website to "verify my identity". In other words, they wanted me to give them my personal information (I don't have an account at Scotiabank). I recently learned that these types of text message scams are called "smishing" (short for "SMS phishing"). Yesterday I received an email from "RBC Royal Bank" letting me know that, "You have been locked out of your account!" Again, to unlock my account, I would have to visit a website and enter some personal information (I don't have an account at this bank either). Usually, these emails are blocked by my email providers spam blocker, but for some reason, this one sneaked through. This morning at 6:30 am, as I was getting my toddler dressed and ready for breakfast, I go a phone call. The caller, from an international phone number, greeted me by name and said his name was "Francis" and that he worked for the Visa Security Department. He wanted to alert me that someone had purchased a Western Union money transfer with my credit card and asked me to confirm the purchase. I told "Francis" that I would call Visa directly and hung up the phone. This early morning phone call credit card phishing scam has been around for a while, but this is the first time I got a call like this. Financial Literacy EducationThis recent surge in phishing scam attempts has highlighted the importance to me of teaching financial literacy to our students. High school students are at an age where they will soon be experiencing financial independence and need to make important decisions about budgeting, making purchases, investing and assuming debt such as student loans and credit cards. They will also have to learn how to protect themselves from fraud. The Nova Scotia high school mathematics curriculum contains a number of financial mathematics outcomes. These outcomes vary depending upon which mathematics courses a student takes. These outcomes deal with personal budgets, understanding pay and income, borrowing money, investing money and making purchases. Avoiding fraud and scams is not explicitly covered in any of these outcomes but could easily be included when teaching these topics. For example, when learning about banking services and credit cards, students could be taught how to recognize scams and what to do if they realize that they've been the victim of identity theft. They could also be taught how to keep their financial information secure and how they should and should not communicate with their bank and other financial institutions. The Canadian AntiFraud Centre maintains a list of fraud types and how to identify and report fraud. It is a very worthwhile website to check out: http://www.antifraudcentrecentreantifraude.ca/ Nova Scotia High School Mathematics Curriculum Outcomes relating to Finance Mathematics 10  FM01, FM02, FM03, FM04 Math at Work 10  N01, N02 Math Essentials 10  Earning and Purchasing, Banking Math at Work 11  N02, N03, N04, N05, A01 Math Essentials 11  Housing, Banking Mathematics 12  FM01, FM02, FM03 Math at Work 12  N02, N03 EL
Sherman K. Stein, in the preface of his textbook Mathematics, The Man Made Universe, says "Mathematics is completely the work of man. Each theorem, each proof, is the product of the human mind. In mathematics all the cards can be put on the table. In this sense, mathematics is concrete, whereas the world is abstract." The order of operations is a mathematical convention that has been developed and refined over centuries by many mathematicians. Just about every student, at some point in their mathematical education, will run across the mnemonic BEDMAS (or PEDMAS if you live in the US). The usefulness of this mnemonic can be debated (Tina Cardone, the author of Nix the Tricks, wrote an article regarding this for the NCTM). I recently chatted with a teacher who was looking for resources to help his grade 7 students practice using the order of operations with decimal numbers. I created a "row game" for his students to practice this skill. A "row game" is an activity for students to work with a partner to evaluate a series of expressions. The expressions are written in two columns. One student evaluates the expressions in the left column and the other student evaluates the expressions in the right column. The expressions in each row evaluate to the same value. If the students don't get the same answer, they can work together to determine where the error was made. There are two reasons that I like row games. The first is that students get instant feedback. They don't need to wait for a teacher to correct and return a sheet to know if they are being successful. The second reason is that it gets students to work together to find their mistakes. The teacher can then focus on providing assistance to students who are having real conceptual difficulties, not just making small computational errors. Kate Nowak has created a shared Google drive folder where a large collection of row games are stored. A link to my row game for order of operations with decimals is below.
Confusion regarding the order of operations has lead to many debates and arguments on Facebook about the correct way to evaluate expressions. You might see expressions such as "6 ÷ 2(1 + 2)" or "48 ÷ 2(9 + 3)" discussed. These might be used to start a lively debate in math class. I really like Vi Hart's take on these types of problems. Take a moment to watch the video below. Vi's contention is that expressions like these are ambiguous and it is incumbent upon the author of the expression to add brackets or other grouping symbols to make their mathematical expression clear. The placement of implicit multiplication (sometimes called multiplication by juxtaposition or simply putting symbols side by side) in the order of operations has not been settled by mathematicians such that is is part of the convention for the order of operations. Despite this, calculators and computers have to make a decision on how to interpret this. You will find calculators that give different answers for these types of expressions. The Math Forum writes, "I suppose I agree with you that it would be easier and perhaps more consistent to give multiplication precedence over division everywhere; but of course there is no authority to decree this, so the more prudent approach is probably just to recognize that there really isn't any universal rule. " One method of avoiding this confusion is to write expressions is using Reverse Polish (RPN) notation. I'm old enough to have owned an HP calculator that used this notation. In RPN, the operator always follows all of its operands. For example, instead of writing "(1+2)÷(3+4)", you would write "1 2 + 3 4 + ÷". RPN often requires fewer key strokes to enter on a calculator because parentheses are not required. It is however more difficult to learn. Nova Scotia Mathematics Curriculum Outcomes Grade 9 N04  Students will be expected to explain and apply the order of operations, including exponents, with and without technology. Grade 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). EL
I recently read Knowing and Teaching Elementary Mathematics by Liping Ma. I really enjoyed this book and I wanted share my thoughts on some of its key messages. But first, a brief description of the book. The book compares the responses to a series of interview questions given to both US and Chinese elementary teachers. These quesions are part of the Teacher Education and Learning to Teach (TELT) Study. The four questions from the book deal with subtraction with regrouping, multidigit multiplication, division with fractions and the relationship between area and perimeter. Liping Ma analyses teachers' responses to discern how their understanding of mathematics influence how they instruct and respond to their students. Richard Askey wrote ane extensive review of Liping Ma's book for the American Educator, a journal of the American Federation of Teachers. The key message that I took from this book is the importance for teachers to develop a profound understanding of fundamental mathematics. Liping Ma describes this as having, "an understanding of the terrain of fundamental mathematics that is deep, broad, and thorough." (p. 120) It's through this conceptual understanding of elementary mathematics that teachers are able to think flexibly about mathematics, appreciate and understand a variety of methods for solving problems and respond to their students' novel ideas and questions. As Tracy Johnston Zager noted in her TMC16 keynote address, K2 mathematics is not "the basics" rather it is, "student's introduction to what it means to do mathematics, what is math as a discipline" as well teaching kids "what it means to have a mathematical discussion, a mathematical idea, a mathematical question". Liping Ma says that "elementary mathematics is not a simple collection of disconnected number facts and calculational algorithms. Rather, it is an intellectually demanding, challenging, and exciting field  a foundation on which much can be built." (p. 116) Indeed, no matter what level of mathematics you teach, you should know it deeply and how it connects to related areas of mathematics. Liping Ma states, "Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations  the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics." (p. 112) One question that remains for me after reading this book is how teachers can best gain this profound understanding of mathematics. Liping Ma writes that, in China, teachers spend a significant amount of time studying teaching materials intensively. They closely exam both their curriculum guides and their textbooks. They study how the textbook has interpreted and presented the concepts in the curriculum and why the textbook's authors might have chosen this. Teachers review the examples and exercises in their textbook for each unit and reflect on the purpose of each exercise in relation to the curriculum outcomes. Teachers often spend significant time interacting with their teaching colleagues, "to share their ideas and reflections on teaching." (p. 136). The author notes that, "Time is an issue here. If teachers have to find out what to teach by themselves in their very limited time outside the classroom and decide how to teach it, then where is the time for them to study carefully what they are to teach? U.S. teachers have less working time outside the classroom than Chinese teachers, but they need to do much more in this limited time. What U.S. teachers are expected to accomplish, then is impossible. It is clear that they do not have enough time and appropriate support to think through thoroughly what they are to teach. And without a clear idea of what to teach, how can one determine how to teach it thoughtfully?" (p. 149). This is a current area of discussion in Nova Scotia, with working conditions being a significant issue in teachers' contract negotiations with the provincial government. The chapter in this book about division by fractions was especially timely. In the book, teachers were asked to how they determine the following quotient: 1 and 3/4 divided by 1/2. I recently had a conversation about the strategy of dividing fractions with a common denominator with a pair of junior high teachers (see Christopher Danielson's blog Overthinking my Teaching for a nice summary of this strategy). We discussed how the examples in the textbook were presented, what the curriculum guide had to say about this strategy, and how it could be used to enhance students' understanding of division. The discussion led me to investigate division of fractions more deeply (there is some great material on Ontario's EduGains website). This type of conversation is what Liping Ma suggests helps to promote a deeper understanding of mathematical content, and in my experience, has been a benefit of teachers' professional learning communities. If you are interesting in other teachers relfections on this book, below are two excellent blog posts to read: EL
Feliks Zemdegs (@Fazrulz) set a new Rubik's Cube World record at the POPS Open 2016 on December 11th. Even with this time, however, he did not win the competition. The winner is determined by a trimmed mean. Five times are recorded and the fastest and slowest are removed and the middle three are averaged. Trimmed means are used to score a number of sports and competitions such as diving and snowboard halfpipe. As a class activity, I would encourage students to guess what the World Record fastest time for solving a Rubik's Cube is. After some guessing, I would let them watch the video of Feliks solving the Rubik's Cube and have them estimate his time to the nearest hundredth of a second. As in many Estimation180 tasks, you might ask students to make a guess that they know is too fast and a guess that they know is too slow in order to establish a reasonable range for their estimates. (Note: You may want to download this video and shorten it. Since the title of the video includes the record time, you might give away the answer by showing the video from YouTube.) 4.73 seconds is really fast! How fast you say? As Anne Haensch stated, "That’s faster than you can say 'Hey Feliks, can you solve this Rubik’s Cube in under 5 seconds?'" You'll see in the video, when he turns the camera around, that the recorded time is actually 4.737 seconds. The official rules from the World Cube Association state that timed results are measured and truncated (not rounded) to the nearest hundredth of a second. These times are measured using a Stackmat timer and are accurate to 0.001 seconds. It was recently pointed out in a tweet from Matthew Oldridge (@MatthewOldridge) that there are few places in life where the thousandths digits are used. One of the places where people do see this is on gas pumps. The amount of fuel that you pump is shown to the thousandths of a litre (0.001 L is equal to 1 mL). This is a pretty small amount... just 1/5 of a teaspoon or about 20 drops (the size of a drop depends on the fluid and the dropper, but 20 drops in an mL is generally a good estimate). Using a photo like the one below, you might ask your students what they know about the price of gas for this sale.
To help students being introduced to decimal place value and operations, I created two variations of a game that can be played using either dice or a deck of number cards. Students either roll a die or draw a card to complete decimal numbers with the goal of creating the largest number. The links to the google docs of these handouts are below (https://goo.gl/vuNluM and https://goo.gl/oIUWSr) Nova Scotia Mathematics Curriculum Outcomes Math at Work 12 S01  Students will be expected to solve problems that involve measures of central tendency, including: mean, median and mode; weighted mean; trimmed mean. Math at Work 12 M01  Students will be expected to demonstrate an understanding of the limitations of measuring instruments, including precision, accuracy, uncertainty, and tolerance, and to solve problems. Grade 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). EL
When I got home, I did a search to find out why the machines were gone and ran across a story from Global News. It turns out that TD Bank had decided to retire all the coin counting machines in Canada in the wake of reports from the U.S. that the machines were shortchanging customers. In a segment on the Today Show called 'Rossen Reports', a team investigated the accuracy of a number of Coinstar machines as well as coincounting machines at various branches of TD Bank. The team tested the accuracy of the machines by carefully preparing bags filled with exactly $300 worth of pennies, nickles, dimes and quarters. They then used the machines to see how close their count was to $300. The Coinstar machines all checked out with the correct $300 total. The TD Bank machines did not fare so well. The totals on the machines tested at 5 different branches were: $299.95, $299.47, $299.30, $296.27 and $256.90. None of the machines returned an accurate $300 count. I don't think that machines can really be "100% accurate" all the time. What level of accuracy do you think is acceptable from a coincounting machine? How much time does it take to roll $300 worth of coins and how much is your time worth? I would probably accept $299.95 for the convenience of not having to roll that many coins. I would be a bit more hesitant to accept $296.27 and definitely would not accept $256.90. While the TD Bank machines were free for customers, in Canada, Coinstar machines apply a coin counting fee of 11.9 cents per dollar. For the $300 counted in this test, the fee would have been $35.70. That is a pretty hefty fee. Questions and EstimationsAccording to a class action lawsuit filed in New York in April 2016, TD’s coincounting machines processed 29 billion coins in 2012. Based on this figure and the data collected by the Rossen Report, how much money do you think customers lost? What factors did you consider when making this estimate? How would you design an experiment to test the accuracy of TD's coin counting machines? Would you test lots of different machines or a few machines multiple times? How many trials would you run to be confident in your results? What factors might contribute to the errors discovered in these machines? Nova Scotia Mathematics Curriculum Outcomes Mathematics 11 S02  Interpret statistical data, using: confidence intervals, confidence levels and margin of error. Mathematics 11 S03  Critically analyze society’s use of statistics. Grade 9 SP03  Students will be expected to develop and implement a project plan for the collection, display, and analysis of data by: formulating a question for investigation; choosing a data collection method that includes social considerations; selecting a population or a sample; collecting the data; displaying the collected data in an appropriate manner; drawing conclusions to answer the question. Grade 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. EL
Scale is a concept that is found at numerous grade levels in the Nova Scotia Mathematics curriculum. Scale drawings and models, similar polygons, and proportions are all found in mathematics outcomes. In math, scale is the ratio of the length in an image (or model) to the length of the actual object. Below is a question relating to scale factors. A scale factor is the ratio of any two corresponding lengths in two similar geometric figures. Take a look at the three different versions of Connect Four. Estimate the scale factor between each pair of game boards from the given pictures. Estimate the radius of each of the coloured chips. Is the scale factor of the radius of each coloured chip the same as the scale factor of their volume? You might ask students how scale is different from proportion. Try out this question: How big would a game board of Connect Four Hundred be (or even Connect Four Million) compared to Connect Four? In visual arts, scale refers to the size ratio between objects within an image. Using a consistent scale will make a drawing look more realistic. Objects do not appear too large or too small when compared to each other. Sometimes however, an artist might intentionally change the scale of certain objects in an image. One such technique is called 'Hieratic scale' or sometimes 'Hierarchical proportion'. This technique can be seen in paintings and sculpture from the middle ages where powerful or holy people were sometimes painted larger than ordinary or less important people to show their relative importance. The larger a person was, the greater their importance.
It would be fun to show students examples of how visual artists play with scale in order to make an impact on the viewer. Students might even be given an opportunity to create a piece of art that has an exaggerated or inconsistent scale or plays with forced perspective. Nova Scotia Mathematics Curriculum Outcomes Grade 6 N05  Students will be expected to demonstrate an understanding of ratio, concretely, pictorially, and symbolically. Grade 8 N04  Students will be expected to demonstrate an understanding of ratio and rate. Grade 8 N05  Students will be expected to solve problems that involve rates, ratios, and proportional reasoning. Grade 9 G03  Students will be expected to draw and interpret scale diagrams of 2D shapes. Math at Work 10 G03  Students will be expected to demonstrate an understanding of similarity of convex polygons, including regular and irregular polygons. Mathematics Essentials 11 D9  calculate scale factors in 2D scale diagrams and 3D scale models understand the relationship among the scale factor and the related change in area or volume.Math at Work 11 G02  Students will be expected to solve problems that involve scale. Mathematics 11 M03  Demonstrate an understanding of the relationships among scale factors, areas, surface areas and volumes of similar 2D shapes and 3D objects. EL
One of the things that I have enjoyed most about being a mathematics consultant at the Halifax Regional School Board is the outstanding group of educators that I get to collaborate with. My background is in senior high mathematics while most of the people I work with have a background in elementary and junior high mathematics education. I have learned tons about pedagogy, assessment and instruction as a result of professional conversations and collaboration with them. Over the summer I had the chance to watch Tracy Zager's Twitter Math Camp (TMC16) Keynote talk. There were a number of things that really resonated with me but I especially took home the idea that it is important to learn from a variety of people. She stressed that teachers at different grade levels have a lot to offer each other and that all teachers would benefit from 'vertical collaboration'. Unfortunately for most teachers, this type of forum in which people from all different grade levels share and learn, is limited to online communities (such as the MTBoS and Global Math Department) and not in the brick and mortar world. I encourage you to check out Tracy's talk. I was inspired by Tracy to share a few things that I've learned in the last year or so as a result of hanging out with elementary and junior high teachers. Many of them have to do with the importance of mathematical language. 1. Reading Threedigit NumeralsIt turns out that when you read a threedigit numeral, you're not supposed to say 'and' between the hundreds and the tens digit. For example, the number 237 is read, "two hundred thirtyseven" and not "two hundred and thirtyseven." This is explicitly taught in grade 3 mathematics (3N02.01 Read a given threedigit numeral without using the word 'and'). When reading numbers, the word 'and' is reserved for the decimal. Using the word 'and' for the decimal is covered in grade 4 mathematics. For example the number 3.6 is read, "three and six tenth." I've heard it suggested that reserving the 'and' for the decimal might be a difference between American English and British English. 2. Reading DecimalsSo to continue on with the 3.6 example, reading decimals is something that the elementary and junior high teachers that I work are very particular about. This is something that was also stressed during one of my math methods courses while working towards my B.Ed degree. The number 3.6 should be read, "three and six tenths" and not "3 point six" or "3 decimal six". I recently participated in a discussion on twitter with a group of teachers about this. Most teachers agreed that when introducing decimals to students, reading this as "three and six tenths" assists students with understanding place value. It also helps students as they move to write decimals as fractions. However, as students attain mastery of place value, there are situations where reading decimals with place value becomes cumbersome... how would you say 3.617829? (I've been told that if there were more than three decimal places you would say "point" or "decimal") Or how would you say $3.61 million dollars? I've always heard it read "three point six one million dollars" and not "three and sixtyone hundredths million dollars". 3. Reading PowersThe most recent thing I've learned about mathematics language is that the term 'power' is not to be used in the same sense as the term 'exponent'. In grade 9 mathematics, students are expected to identify the base, the exponent, and the power in an expression in exponential form (9N01). For example, the power 2^5 (where 2 is the base and 5 is the exponent), can be read “two to the exponent of five,” or “two to the fifth” and not “two to the power of five.” 4. 'Say More About That'Reading 'Opening Minds  Using Language to Change Lives' by Peter H. Johnston made a powerful impact on the way I observe and participate in classroom discussions. Working with HRSB math coaches and support teachers has taught me how important not only what you say is but also how you say it. A phrase I have heard a number of coaches use is 'Say more about that". It lets the person that you are talking to know that they are respected and you're interested in what they have to say. It also lets you know more about what they are thinking. 5. Simplify a Fraction (added Oct. 27th, 206)I learned today that, in the Nova Scotia curriculum, we simplify a fraction and use the terminology of simplestterm fractions. We don't 'reduce' a fraction to lowest terms because that makes it sounds like we are changing the fraction to make it smaller. The size of the fraction doesn't change, we are making an equivalent fraction that is in simplestterms. (from Grade 5 N07.02 and N07.05). I was also recently reminded about using precise language when reading fractions. For example, the fraction 5/4 should be read 'five fourths' instead of '5 over 4' ( which can confuse students by focusing on the physical arrangement of the digits) or 'five out of four' (which doesn’t support understanding of a fraction as a single number). I feel extremely fortunate to have a community of educators with a breadth and depth of different experiences with whom I can collaborate and learn at my workplace. What have you learned from the educators that you work with? Do you have the opportunity to have professional conversations with teachers from outside your school and/or grade level? EL
I recently visited Memory Lane Heritage Village in Lake Charlotte. It is a living history museum depicting coastal rural life in Nova Scotia during the 1940s. My five year old son had a great time. He especially enjoyed sitting in the 1928 Ford Model A car and pretending to drive. On the way back home I was thinking about other living history museums in Nova Scotia and realized that, based on my experience, it appears that the further away from my home in Halifax that I drive, the farther back in time the museums depict. When I got home, I dug up some data. Sherbrooke Village depicts a typical Nova Scotian village from the 1860s and the Fortress of Louisbourg allows you to experience life in Louisbourg during the 1740s. I used Google Maps to find the driving distance from my house to each of these locations and discovered a nearly perfect linear relationship. How perfect you ask? The correlation coefficient was 0.99906. I quickly created a scatter plot with a line of best fit to show my wife. Despite my exuberance, she appeared to remain unimpressed. It is also interesting to see that the points on the scatter plot are almost exactly where the sites are on a map of Nova Scotia as well. Mind Blown. A question that I still have is whether this apparent temporal relationship is based on distance or displacement. Perhaps I need to collect some additional data (or not intentionally disregard data that doesn't fit my hypothesis)? If I travel in the opposite direction, should a living history museum depict life in the past or in the future? I'd love to visit Yarmouth some day to experience what life will be like in rural Nova Scotia in the year 2213! EL

Archives
January 2017
Categories
All
