My son and I were looking through an issue of Fun to Learn Friends magazine recently and we ran across a game called "Spring Bingo". He was quite interested and wanted to give it a try. He is in grade primary and can subitize the pips on a six sided die as well as confidently sum the values on two dice. Just the skills we need to play this game. We played several times. At the end of most of the games, my son got frustrated trying to roll either a 2 or a 12. He doesn't quite understand why it takes so long to roll one of these numbers. George's bingo card has both of these numbers out of the eight numbers on the card. This seems unnecessarily cruel. Ted's card has the 2 but not the 12. This made me wonder if this is a fair game or if one of the cards has a better probability of winning. Time for some math... It appears that Card B is slightly better than Card A. I'm not sure if this would make a significant difference in the outcome of the game (i.e. rolling all the numbers on your card before your opponent does). I wonder what the average number of rolls it takes to complete each bingo card is? How much of an advantage does going first in this game give? If the player with Bingo Card A goes first does this equalize the advantage of the better Bingo Card B? These would be great questions for the Mathematics 12 research project outcome (MRPO1). To try to answer some of these questions, I thought that writing some code would be helpful. I know a grade 8 student that completed the Introduction and Intermediate Programming with Python courses from Art of Problem Solving. I contacted him and he graciously created a very nice Python program to simulate this game for me. I modified the code a bit so that it just plays one bingo card and counts how many rolls it takes to complete the card. The average number of rolls in 100,000 games for "George's Card" was 58. The average number of rolls in 100,000 games for "Ted's Card" was 48. I was a bit surprised that there was this much difference. Playing with Ted's bingo card appears to be an advantage. Next I modified the code again so that it plays the game with George going first to count how often George wins. In 100,000 games, when George went first, he won 50703 of the games. I again modified the code so that Ted plays first. In 100,000 games, when Ted went first, he won 50806 of the games. It seems that going first is an even greater advantage than having the better bingo card. What I really like about this game is that there are mathematical outcomes that can be addressed with this activity across a wide range of grade levels. At younger ages, students are practicing subitizing and adding numbers. One variation of this game is to play it solo. This might be a nice option for a math station. I found several examples of "Roll and Cover" games where students have a sheet of paper filled with numbers (from 1 to 6 or from 2 to 12) and students roll the die or dice and cover the number (with a token or a bingo dauber) when they roll it. Just do a Google search for "roll and cover math game" and you'll find lots of examples posted online.
For students looking for an opportunity for enrichment, they can make variations of this game. They could also write computer code to simulate this game (using Scratch or Python or some other language). They could also do some statistical analysis of the game to see how fair it is. There are so many options with this simple game. Nova Scotia Mathematics Curriculum Outcomes Mathematics 1 N02  Students will be expected to recognize, at a glance, and name the quantity represented by familiar arrangements of 1 to 10 objects or dots. Mathematics 1 N09  Students will be expected to demonstrate an understanding of the addition of two singledigit numbers and the corresponding subtraction, concretely, pictorially, and symbolically in join, separate, equalize/compare, and partpartwhole situations. Mathematics 2 N10  Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts. Mathematics 5 SP04  Students will be expected to compare the likelihood of two possible outcomes occurring, using words such as less likely, equally likely, or more likely. Mathematics 6 SP04  Students will be expected to demonstrate an understanding of probability by: identifying all possible outcomes of a probability experiment; differentiating between experimental and theoretical probability; determining the theoretical probability of outcomes in a probability experiment; determining the experimental probability of outcomes in a probability experiment; comparing experimental results with the theoretical probability for an experiment. 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. Mathematics 8 SP02  Students will be expected to solve problems involving the probability of independent events. Mathematics 10 Essentials G1  Express probabilities of simple events as the number of favourable outcomes divided by the total number of outcomes Mathematics 12 P03  Solve problems that involve the probability of two events. Mathematics 12 MRP01  Research and give a presentation on topic that involves the application of mathematics. EL
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I know many teachers who say they wish they could go to a math conference but they are not sure of the logistics of applying for funding, applying for leave, getting sub coverage, etc… I have gone to many conferences over the past 10 years and always return feeling rejuvenated and excited to do my job!
I have attended the NCTM Annual Conference although I may try a regional conference in the future. The following is based on my own experience; please review your school board and Article 60 guidelines before making plans. Things to consider before booking your conference:
So now you have decided the conference, the city and travelling companions, here are some of the logistics of applying for funding:
Once you have gotten approval for leave and have confirmation on how much you will be reimbursed in expenses, it’s time to start planning your trip! I often scout out hotels and flights even before I have completed the steps above. My preference is to stay at one of the conference affiliated hotels. There are complimentary shuttle busses that run from the hotels to the conference centres all day. This is a big bonus in my eyes. Sometimes, I have used the shuttle busses just to get to another part of the city! You can find all the information about accommodations on the NCTM website. The best hotels (location or price) usually book up early. If I don’t have all the proper approval from my principal yet, I still reserve my hotel room. I can always cancel the reservation if plans fall through (unlike conference registration and plane tickets. Wait to book those!). Take note of the early bird registration dates for the conference. It will save you about $50 if you register before a certain date. If you know of a bunch of people who are attending the same conference, investigate group rates. Also, I usually scour the internet for NCTM conference registration coupon codes. I once saved myself fifty dollars by using a random code I found online. So now that you have gotten approval for leave and funding, registered for conference, reserved your room and booked your plane ticket, you need to plan your time at the conference. With thousands of sessions to choose from, selection can feel like a nightmare (similar to those many of us get in the days before a new school year begins). For this reason, you want to choose wisely. Like the first days of school, this planning you do in advance will certainly impact the days ahead. More on strategies to navigate the conference booklet and choosing your sessions coming soon! KZ . For anyone interested in data visualization, the r/dataisbeautiful subreddit is an amazing place to explore (you can also follow @DataIsBeautiful on Twitter). This subreddit has an active community of posters who create and discuss a wide variety of data visualizations. Shortly after the first successful landing by SpaceX of a reused Falcon 9 rocket, an incredible feat of engineering, the visualization below was created and shared by reddit user Brenden2016. Even more interesting than the visualization was the discussion that followed. Over 200 comments (as I write this) have been posted regarding this visualization and suggestions of how to improve its powerful story of the increased efficiency and success of SpaceX Falcon 9 landings. Several issues emerged:
Additionally, two users created and posted alternate versions of the graph. One user, nicocote, created a double bar graph using additional launch data that didn't appear in the original. Another user, azura26, created a step graph instead of a line graph. The full list of Falcon 9 landings can be found on the list of launches on Wikipedia. I took this data, summarized it and put it into a Google Sheets document to share (https://goo.gl/Ihr2Io). Please feel free to use it to generate your own data visualization or use it with students. In the ClassroomThis set of data appears to be a nice one to use in a classroom. It is a fairly small data set, is current and from the real world and (for me at least) is an engaging and interesting story. In a grade 8 classroom working on data presentation, I might start by showing the students some dramatic video of Falcon 9 boosters both crashing and landing successfully. Fiery explosions always make an impression! Then we could look at the data showing how SpaceX has done an impressive job of learning from their mistakes (and perhaps how this relates to a growth mindset in the mathematics classroom). We could then look at the data visualization from above and brainstorm, as a class, what features of the graph are positive and which could be improved. We could also answer the question, "what is the story this graph is trying to tell?" We could then break into groups, review that data and see if the groups could come up with a different way of telling that story using the data. Perhaps a different type of visualization or even an infographic could be created. What would you do with this data in your classroom? Please let me know. Nova Scotia Mathematics Curriculum Outcomes Grade 8 SP01  Students will be expected to critique ways in which data is presented. Mathematics Essentials 11 F2  Select an effective data display for a given set of data and explain the reasons for the choice. Mathematics at Work 11 S01  Students will be expected to solve problems that involve creating and interpreting graphs, including bar graphs, histograms, line graphs, and circle graphs. EL
A couple of weeks ago, I saw a tweet from Don Fraser (@DonFraser9) in which he noticed that every box of Raisin Bran says "2 scoops" no matter what it's size is. I was recently talking with a teacher about the Percent, Ratio, and Rate unit for grade 8 math and I was reminded of Don's question. This deserved some additional investigation so I headed to the grocery store to gather the facts. At my local store, Kellogg's Raisin Bran is available in 3 sizes as seen below. Questions to Consider1. Rate: Which size box of cereal has the best unit price (g/$)? 2. Ratio: If there are two scoops in the regular size box, to preserve the same proportion of raisins, how many scoops should there be in the Family Size and Jumbo Size box? Alternatively, if the ideal proportion of raisins is to be found in a different size box, how many scoops should there be in the other two sizes? 3. Ratio: A "scoop" is a nonstandard unit of measure. If the amount of raisins in each box stays the same proportion, then how should the size of the scoops change in order to maintain that proportion in each size box? 4. Volume: These boxes all have different volumes. Determine the volume of each and compare it to the weigh of cereal in that box. Does this rate stay the same? If it doesn't, what does it tell you about the amount of empty space in the box? How do the dimensions of each box compare? Are the different sized boxes similar shapes? Some additional Raisin Bran resources: Nova Scotia Mathematics Curriculum Outcomes 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 8 M04  Students will be expected to develop and apply formulas for determining the volume of right rectangular prisms, right triangular prisms, and right cylinders. 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 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. Mathematics 12 MRP01  Research and give a presentation on topic that involves the application of mathematics. EL
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
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

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