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, non-terminating 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. Ci-dessous 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 remue-ménginges 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, peut-être, 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).
I'm a big fan of using open-ended problems in math class. An open-ended problem is one that has a number of different correct answers or several ways of getting to a correct answer. Below is a question from page 140 of the MMS9 textbook that is not an open problem but has lots of potential.
This problem asks students to evaluate four different versions of a nearly identical expression. It tells students exactly what to evaluate... snore. A bit tedious and not very engaging. Lets open this question up by putting students into pairs and giving them the equation with no (parentheses) or [brackets].
Ask students to find out how many different values they can get by putting in one set of brackets. What if they could put in two sets of brackets? What if they could put in as many brackets as they'd like? What is the biggest value they could create? How about the smallest value that they can find? How about the value closest to 0? Once they've created and evaluated a bunch of different expressions, each group could take turns placing their numbers on a clothesline number line. This will get students practicing order operations with a goal and a challenge.
There is a very similar version of this problem in the Grade 9 Mathematics Curriculum Guide for outcome N04. The following problem is in the Assessment Tasks section:
Without the use of a calculator, simplify the expression 1/2 - 1/3 x 1/2 - 1/3 and express your answer as a fraction.
− By inserting one pair of brackets, how many different answers are possible?
− By inserting two pairs of brackets, is it possible to receive a different answer?
Want an even more open version of this problem? Just give students the four numbers and ask them to use the operations of addition, subtraction, multiplication and division and do the same activity. There is an activity from Illustrative Mathematics that describes this problem using the numbers 1, 2, 3 and 4. Perhaps that could be a warm-up for the more complicated question with fractions.
Nova Scotia Mathematics Curriculum Outcomes
Grade 9 N03 - Students will be expected to demonstrate an understanding of rational numbers by comparing and ordering rational numbers and solving problems that involve arithmetic operations on rational numbers.
Grade 9 N04 - Students will be expected to explain and apply the order of operations, including exponents, with and without technology.
One of my favourite activities recently has been Fawn Nguyen's Snap Hotel. This activity can also be found on the NCTM Illuminations website. You might see this activity online under a few different names... my favourite is Hotel^3. It is an engaging problem solving task. Students are given 50 multi-link cubes and instructed to build a model hotel. Each cube represents one hotel room. Some rooms are more desirable than others and can be rented for higher prices. The number of windows and whether or not the room has a roof determine the rental price. Students also have to consider costs associated with property and height. The hotels that students create in this activity remind me of Montreal's Habitat 67. Students might talk about the costs and benefits of this type of architecture. This activity can be used to assess a number of Nova Scotia mathematics curriculum outcomes.
The Rules - As a team, build a hotel that yields the highest profit.
Below are some hotel's created by teachers during a session at the NS MTA conference in Oct. 2015.
Suggestions for Improvement
Below are my Powerpoint introduction and a student handout. Also included is an analysis of several different hotels to see how their profits compare.
Why I Like This Task
Nova Scotia Curriculum Outcomes