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
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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
There are some really big doors around Halifax. The door on Irving Shipbuilding's Halifax Shipyard Assembly and Ultra Hall facility is big enough for large "megablocks" of ships under construction to pass through. The doors at IMP Aerospace's Hangar #9 at the Stanfield International airport is big enough for large aircraft to pass through. Which door do you think is the largest? What Do You Mean by Largest?The first thing you might want to do is settle on what you mean by "largest". Do you mean width, height, area, mass or some other measurement? Each of these doors might be the largest for a specific measurement. For example, the aircraft hangar door is made of metal and quite probably has more mass than the shipyard door which is constructed of a polyester fabric. Estimating DimensionsThe shipyard door is really tall but the aircraft hangar door is really wide. Below are pictures of the two facilities from Google earth with the same scale so that you can compare the buildings that these doors are on. https://www.google.ca/maps/@44.6674497,63.5972041,542m/data=!3m1!1e3 https://www.google.ca/maps/@44.8701869,63.5322158,573m/data=!3m1!1e3 Door Dimensions and Surface Area
So the shipyard door has the largest height and the largest area but the hangar door has the largest width and the largest mass. Would you call this a tie? How would you determine the winner?
More Big DoorsDo you know of other big doors around Halifax? Have you seen bigger doors in other parts of Nova Scotia or the rest of Canada? What is your definition of door? Note of Thanks: I want to say thank you to the people at both Irving Shipyard and IMP Aerospace who were very helpful providing information for this post. EL
This is the second year for the Math Photo Challenge on twitter. Each week a new challenge topic is presented. During the challenge, you can follow @mathphoto16 on twitter to see the weekly prompt. You can also visit https://mathphoto16.wordpress.com/challenges/ to see the weekly prompt. Participants post photos on twitter and use the hashtag #mathphoto16 and the hashtag for the weekly prompt. This year, the challenge is being hosted and organized by Amie Albrecht (@nomad_penguin) and John Rowe (@MrJohnRowe) from Australia. Last year, I sat back and watched as people tweeted photos but this year I decided to participate and encouraging other teachers and students to do so as well. With this post, I'm going to recap the challenges and the photos I tweeted throughout the summer.
The Week #2 (June 19  25) challenge was #scale. Scale is a ratio that measures the relative size of two objects. We encounter scale in our daily lives such as longer/shorter, faster/slower and bigger/smaller. I posted two photos from the Halifax Stanfield International Airport baggage carousels. Each of the three baggage carousels in the domestic arrivals area has a scale model built by Anchor Models. The MacKay and Macdonald Bridge models are 47 ft long and have a scale of 1/60. The model of the Halifax Town Clock has a scale of 1/12. The Week #3 (June 26  July 2) challenge was #lines, curves and spirals. I posted a photo of a pediment portico and eyebrow dormer from a house in the South End of Halifax. I really like this style of architecture and it is pretty common on the peninsula in Halifax. A few weeks later, I was at the Halifax Central Library and took a photo of the lines and shadows in the buildings central atrium. The Week #4 (July 3  9) challenge was #multiples. Many objects can be arranged in multiples and groups. This helps us count them more easily and share them out equally. I posted two photos for week 4. The first is an arrangement of tea cups from a tea house in Beijing. The second is a picture of the 3 chimneys from the Tufts Cove Generating Station on the Halifax Harbour. I like how each of the 500 ft tall smokestacks are partitioned into alternating white and red bands. The Week #5 (July 10  16) challenge was #zero. Without zero, mathematics would be nothing. Search out the concept of zero. Be creative in your interpretation. I posted a photo of the sign for Exit 0 on Nova Scotia Highway 102. While most Canadian provinces use kilometre based exit numbers, Nova Scotia uses sequential exit numbers. The only other Canadian province which uses sequential exit numbers is Newfoundland and Labrador. The Week #6 (July 1723) challenge was #infinity. Infinity characterizes things that never end. Look for interpretations of infinity. This was a difficult concept to capture with a photo and there were relatively few photos submitted on Twitter. I posted two photos. The first depicted how the imagination is infinite. A toy such as a set of blocks can be used for an limitless number of different games and activities. The second photo I posted was from the television show The Big Bang Theory. In the episode titled "The Friendship Algorithm" (S2, E13), Sheldon gets stuck in an infinite loop in his algorithm. The Week #7 (July 2430) challenge was #shapes. Shapes enclose the space around us into areas, in both regular and irregular ways. Along with single shapes, look for congruence, similarity, tessellations and more. I posted a photo that I took of the The Beijing National Aquatics Center. The bubble like geometric shapes covering the exterior of the building are based on the Weaire–Phelan structure. I really enjoyed participating in this photo project and seeing all of the interesting and creative photos that were posted by people from all around the world. It gave me a purpose to view my community through a mathematical lens and connect mathematical ideas to the world around me. EL
The Sno Cap Drive In in Sisters, Oregon is a diner that serves homemade ice cream and old fashioned burgers. The food here is great and the homemade ice cream is delicious. I stopped in this summer for a scoop of Cookie Monster ice cream. While waiting in line, I had some time to notice and wonder about the prices listed on the menu. What is going on here? NoticingI noticed the differences in price between having your burger with potato chips or with french fries. Depending on the burger you're having, it costs anywhere from 45¢ to 95¢ more to have fries instead of chips. I also noticed the difference between the burgers with cheese and those without cheese. To add cheese to your deluxe burger is an additional 80¢ but to add cheese to your bacon burger is only an additional 30¢. WonderingIs there a pattern or rule to the price differences between having a burger with chips or fries? The list of differences between chips and fries is: 70¢, 70¢, 65¢, 60¢, 45¢, 80¢, 75¢, 75¢, 95¢, 70¢, 65¢, 70¢, 70¢, 65¢, 80¢, 60¢. The values range from a minimum of 45¢ to a maximum of 95¢. The mean difference is 70¢ and the standard deviation (a measure of variation) is 11¢. I tried sorting the burgers in a variety of ways but there appears to be no pattern to the difference in price between chips and fries. This restaurant is in Oregon, which has no sales tax (one of only five such states), so the prices are not set so that when tax is applied, the total is a round number. I'm assuming that every burger gets the same amount of fries, but perhaps this isn't the case. I wonder if they get many questions about this? If you wanted to give the prices on this menu an overhaul, how would you price these burgers, chips and fries? Could you come up with a more logical system of pricing? What factors would go into making these price decisions? Which burger do you think is currently the most profitable based on the current prices? Nova Scotia Mathematics Curriculum Outcomes Grade 7 SP01  Students will be expected to demonstrate an understanding of central tendency and range by: determining the measures of central tendency (mean, median, mode) and range; determining the most appropriate measures of central tendency to report findings. 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). Mathematics 11 S01  Demonstrate an understanding of normal distribution, including: standard deviation and zscores. EL

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