I recently discovered that the way that prices are shown on rate signs for motor vehicle fuel varies quite a bit. Some display the current price in dollars, some in cents and some, confusingly, show both dollars and cents. None of the signs that I found actually showed the units being displayed ($, ¢, L, gal., etc). I decided that it would be a good topic for a Which One Doesn't Belong discussion that would focus in on decimals, place value and fractions as well as currency and volume measurement. Here are some suggestions why each image might not belong as well as some additional information for discussion: Top Left: This is the only fuel price sign with both a decimal and a fraction. It is from a Costco in Oregon. The price is $2.29 and 9/10¢ per gallon. Both dollars and cents are used in this price. If you assumed that the entire price was listed in dollars, then $2.29 and $9/10 would actually give a price of $2.29 + $0.9 = $3.19. Top Right: This is the only fuel price sign with just one decimal. It is from a Shell station in Halifax, NS and is in Canadian cents per litre. In Nova Scotia, the price often doesn't end with the standard 9/10¢ since the price is regulated by the Nova Scotia Utility and Review Board and only changes once per week. Bottom Left: This is the only fuel price sign with two decimals. It is from an Atlantic Superstore gas station in Dartmouth, NS. Similar to the sign above it (top left), it displays a price of $1.00 and 0.5¢ per litre. While probably not confusing to the average motorist, I find the use of two decimals to be mathematically troubling. Why not just list the price as $1.005? Bottom Right: This is the only fuel price sign with no decimals. It is from a 76 gas station in Oregon and is in US cents per gallon. It is interesting to note that Oregon and New Jersey are the only two US states with laws prohibiting self serve at gas stations. This sign mentions "Mini Serve" to note that an attendant operates the pumps but does not provide any of the other services you'd expect with full service.
Nova Scotia Mathematics Curriculum Outcomes Grade 5 N09  Students will be expected to relate decimals to fractions and fractions to decimals (to thousandths). Grade 5 N10  Students will be expected to compare and order decimals (to thousandths) by using benchmarks, place value, and equivalent decimals. Grade 6 N01  Students will be expected to demonstrate an understanding of place value for numbers greater than one million and less than onethousandth. Grade 7 N07  Students will be expected to compare, order, and position positive fractions, positive decimals (to thousandths), and whole numbers by using benchmarks, place value, and equivalent fractions and/or decimals. EL
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I am trying to be purposeful and organized in taking the "Calls to Action" from the recent NCTM conference in San Francisco. I am going to play with "Deleting the Texbook" over my next few blog posts (watch Meyer's most recent NCTM presentation here for elaboration). Since I attended my first conference about 7 years ago, I have been heavily influenced by the work of Dan Meyer. His Ted Talk "Math Class Needs a Makeover" really changed the way I thought about teaching math. I want to share something I did with first as a math coach with a class of grade 3 students and then as a presentation for teachers at the Nova Scotia MTA conference a few year back. I feel like I have been taking this particular "Call to Action" for many years now; I am excited about the evolution of this thinking and I wanted to show how I started with this work. The students in this class were a few weeks out from writing a provincial math assessment. The classroom teacher wanted to do some questions from the practice booklet. This was the original question: Instead of posing the question as is, I took out as much information as possible while keeping the math I wanted the students to learn in mind. The students needed to get curious before I hit them with the math. All staff at schools need to wear an ID badge and laynard (wait for it...it will make sense in a moment). Lots of people keep keys, change, notes, etc in the plastic badge holder. Instead of asking the above question, I announced to the class that I had some change in my ID holder (I put the correct amount in and shook it for dramatic affect!). Their job was to figure out how much money I had. They were a little confused at first, so I prompted them; I told them they could ask me questions about the coins. I gave them a few minutes to think, then to confer with a partner before I took questions from the class. I recorded and answered their questions. Here are the questions they came up with along with my answers in brackets:
Each time I took a question, I asked students "Why do you want to know that?" or "How will that information help you solve the problem?" Once they had all the information they felt they needed, they set off to solve the problem. They basically constructed the original question on their own. And because they had a part in building it, they really understood the context of the problem and had ideas of how they might solve it. When they were confident that they had the solution, I let them empty my ID badge and count the coins. That is much more satisfying than just giving the correct answer. Students got to prove that they were right, rather than the teacher being in charge of all the answers. I often hear from teachers that their students don't understand word problems and that it's not the math, it's reading comprehension. I see this strategy of building a question with the class as a way to help them get inside the problem before they start. Another reason I like this strategy so much is that I was able to create it by modifying a resource (practice booklet) that I already had. We can get bogged down trying to find appropriate resources for students at appropriate levels. By using a resource I already had, taking out parts, and layering them back in as needed I was able to differentiate on the fly. No tiered lesson plan, no separate handouts. It was one problem that we worked together as a class to first build, then solve. The Most Interesting Question I Almost Overlooked"What kind of coffee do you drink?
I didn't understand where the student was going with the question but I asked the usual questions of him: "How will that help you solve the problem?" and "Why do you want to know?" The student thought I probably kept my coffee money in my ID badge. If he knew the coffee I drank, he would have the problem solved.** Based on his question, we created a series of extension questions to keep the problem solving going. **All Canadians know the price of a medium double double. :P KZ I've mentioned in the past that I think WODB questions are a great way to begin a lesson and engage students in a mathematical discussions. Below is one such question I created for grade 7 students. Most are working in or have recently finished Unit 3 on Fractions, Decimals and Percents. The question below might be a good review now that students have the language and vocabulary to debate which one doesn't belong. Here are some suggestions and why each number might not belong: Top Left: It is the only percentage and it is the only number with a tens place. It is also the only number without a vinculum. This is also the only number that is not an exact multiple of 1/3. 33% would be 33/100 and so is just a bit less than 1/3. If all the numbers were written as decimals, this would be the only terminating decimal. Top Right: It is the only number that would be a unit fraction when written as a fraction Bottom Left: It is the only value "greater than one". 33 is great than one but since it is a percentage, it represents 33/100. Bottom Right: This is the only even number and also the only term without the digit 3 in it. It is also the only number written as a decimal and the only box with just one digit in it. If your students are not used to participating in this type of discussion, you might scaffold it by first having students brainstorm and list the vocabulary and math terminology that might be used in the coming conversation. This might be especially helpful in a late French Immersion classroom where all students might not be familiar with the correct vocabulary. Having it listed on the board might help students feel more confidence in joining in the conversation. For the fractions, decimals and percent WODB above here is a list of vocabulary students might come up with: unit fraction, improper fraction, mixed number, percent, terminating decimal, nonterminating decimal, even and odd. A special thanks to Matt Murphy (@Lemurph42) for the following translation of the above post! Lequel n’appartient pas? Les Fractions, Décimaux et Pourcentages.J’ai déjà dit que je trouve les questions de WODB font un bon départ pour engager les élèves en discussions mathématiques. Cidessous il y a une telle question que j’ai créé pour les élèves de 7e année. La plupart de nos classes ont déjà complété chapitre 3 (fractions, décimaux et pourcentages), ou le finiront bientôt. L’exemple suivant pourrait être une bonne révision maintenant que les élèves ont le vocabulaire nécessaire pour s’engager en débat. Voici les arguments pour chaque choix : Gauche supérieure: c’est la seule pourcentage et c’est le seul nombre avec un chiffre à la position des dixaines. C’est aussi le seul nombre qui n’a pas une surlignéation. Ce n’est pas un multiple de 1/3 (33 % sera 33/100, un peu plus petit que 1/3). Si tous les nombre seront écrit comme décimal, c’est le seul qui termine. Droite supérieur: c’est le seul qui est une fraction unitaire. Gauche inférieur: le seul nombre qui est « plus grand qu’une ». C’est vrai que 33 est plus grand qu’une, mais puisque c’est une pourcentage, la valeur est 33/100 en réalité. Droite inférieure: c’est le seul nombre paire et il ne contient pas le chiffre 3. C’est le seule nombre en forme décimal et le seul qui ne contient qu’un chiffre. Si vos élèves ne sont pas habitués à tel sorte de discussions, vous pouvez utiliser l’échafaudage. D’abord, demandez aux élèves de faire un remueménginges du vocabulaire mathématique qui pourrait être utile dans la conversation. Ceci pourrait être utile dans une classe d’immersion tardive dont les élèves ne seront pas familiers avec tous les mots de vocabulaire. La confiance des élèves augmentera avec une liste mots disponible au tableau, faisant qu’ils seront plus capables à participer. Pour la WODB des fractions, décimaux et pourcentages en haut, voici quelques mots de vocabulaire que, peutêtre, vos élèves fourniront: fraction impropre, nombre mixte, pourcentage, décimal terminé, décimal illimité, paire, impaire Nova Scotia Mathematics Curriculum Outcomes Grade 7 N07  Students will be expected to compare, order, and position positive fractions, positive decimals (to thousandths), and whole numbers by using benchmarks, place value, and equivalent fractions and/or decimals. Grade 7 N03  Students will be expected to solve problems involving percents from 1% to 100% (limited to whole numbers). EL

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