Thursday, 20 April 2017

Fraction Operation Continuum

As math teachers we definitely want our students to practice to become proficient but piles of problems or worksheets aren't going to be very engaging to students. We think this tweak to the standard worksheet is a way to turn those boring questions into something more engaging.

In Ontario, grade 7s are introduced to operations with fractions. Addition & subtraction and multiplication & division with whole numbers. The premiss here is fairly simple. Students are presented with multiple cards of questions (in this case of adding and subtraction of fractions (with a little of division and multiplication with whole numbers). The cards represent problems that increase in difficulty as you go from one to the next. Students can all start at the first envelope or you could give them an exit card the day before to help place them in a particular card to start. Students check their own answers using answer cards with the answers written with "invisible" ink that can be revealed by shining a UV light on it.

Students really seem to like this style of activity as they feel empowered to move from card to card when they are ready and the added feature of checking the answers with the UV pen gives a sense of novelty. This could be used as practice or review.

  • Gr7 - divide whole numbers by simple fractions
  • Gr7 - use a variety of mental strategies to solve problems involving the addition and subtraction of fractions
  • Gr7 - add and subtract fractions with simple like and unlike denominators, using a variety of tools (e.g., fraction circles, Cuisenaire rods, drawings, calculators) and algorithms;
  • Gr7 -  demonstrate, using concrete materials, the relationship between the repeated addition of fractions and the multiplication of that fraction by a whole number
  • Gr8 - As review (we plan an extension so that this could be used for grade 8 with multiplying and dividing fractions)
  • Enough copies of each of the question cards for your class (there are four cards per page at each level) in different colour card stock for each level,  laminated (use colours that allow seeing the magic pen writing - you may want to test this). You will likely not need as many cards in the last few envelopes as students work at different paces. You will need as many as you have in your class if you decide to start everyone at the first level. Fewer if you let students start at different levels (see below)
  • 3 sets of the answer cards (use magic pen to write the answers anywhere along each equation, they could be sideways, upside-down, (the answers are on the last page of the Google Doc). The answer cards are the same as the question cards but you write the answers in invisible ink on them. To help distinguish the answer cards to the question cards you should put a stamp or sticker on the back. Write on the cards first then laminate them. If you write on the card after lamination then the ink tends to wear off. There is a separate answer card on the last page of the download. That is for you to carry around (or not) but not for showing students but more for your reference.
  • 3 "magic" pens can be purchased at Chapters/Indigo or we found these at a Scholastic's book fair. We have since purchased some on eBay or Amazon.

  1. Place the questions in piles (or in envelopes taped to the wall) in order of difficulty and set up three stations for the answer cards. Students will get a card and answer the first 5 questions. 
  2. You could have all students start at level 1 but for this activity to be most successful, students should start at the appropriate envelope. If they start in one that is too hard they will be frustrated and if they start in one that is too easy they will be bored. Use an exit card (the day before) to help you decide which envelope each student should start in. When given back the exit card write down the level they will start in. 
  3. To check their answers, they will go to a station and use the magic pens. Students may decide to do one question at a time and then go check their answer or they may do all 5 and then check. Students are monitoring themselves so they decide. If they get the first 5 right, they have a level of mastery to move themselves to the next card. If not there are more questions on the card until they master that type. You can decide whether you want them to do the other 5 or just do enough to get a total of five correct. 
  4. As they move through the continuum, the hope is that they reach level 6 which matches the grade 7 curriculum. Since our goal is to get them to level 6, students should solve ALL equations on that card instead of just five. 
  5. The seventh level is set up to challenge students who are moving forward quickly. They should solve all questions on this card.  


  • Fraction Operation Continuum (with Answers) - (PDF, Google Doc)
Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Friday, 24 February 2017

Investigating Graphs of Rational Functions

In Ontario in our grade
12 advanced functions course we are to graph rational functions that are reciprocals of linear and quadratic functions and ones where both the numerator and denominator are linear functions. In this post there are several activities (both hands on and electronic) that could be used throughout a unit on graphing Rational Funcitons. Here, we start with an investigation that is done with students in groups graphing reciprocal linear and quadratic functions by hand and then ends with a Desmos Card sort. There is also a Desmos investigation on functions in the form   and a consolidation activity using Desmos Marbleslides. Lastly, there is an assignment challenging students to create their own Which One Doesn't Belong. Note that you may not want to do all of these activities while doing this unit. Just pick and choose.

  • MHF4U 2.1 - determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions.
  • MHF4U 2.2 - determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that have linear expressions in the numerator and denominator, and make connections between the algebraic and graphical representations of these rational functions
  • MHF4U 2.3 - sketch the graph of a simple rational function using its key features, given the algebraic representation of the function
  • Chart paper or vertical surfaces for the investigation (markers etc)
  • If you choose to do the hands on card sort then you need to make copies of this set of cards (one per group). We usually make these on card stock and laminate them for durability.
  • If you choose to do the Desmos card sort then you need to have technology for your students. 

Activity 1: Reciprocal linear and quadratic functions

  1. Put students in groups of no more than three. They can work on vertical surfaces or at a table.
  2. Each group is given one set of a linear equation and its corresponding reciprocal (or rational) function. There is enough sets for 11 groups. They are to graph each on the same axis. Afterwards they should determine any intercepts and/or equations of asymptotes (see answers to the right).
  3. Once complete they should walk around the class to see other sets and come to some conclusions as to properties of linear functions and their corresponding rational functions.
  4. Once complete there is a couple of follow up questions to check their thinking. 
  5. Next each group gets a set of quadratic functions and corresponding rational function. They are to graph each on the same axis. Afterwards they should determine any intercepts and/or equations of asymptotes. 
  6. Again, once complete, they should walk around the class to see other sets and come to some conclusions. 
  7. There is another set of follow up questions to check their thinking (note, if time is a problem students can do a similar investigation for homework with this Desmos activity instead of the above steps). 
  8. As a follow up (maybe next day) you can do this physical card sort or this Desmos card sort. The Desmos card sort has some follow up questions to consolidate some of the ideas. 

Activity 2 Graphs in the form  


  1. Students complete the Desmos Investigation. This can be done in class or as a homework assignment.
  2. As consolidation, students can check what they know with this quiz. This can be done using paper and pencil (copy double sided then cut in half) or via this online Google Form. Note that if you use the form, click on this link and then choose advanced options (3 vertical dots) to make a copy. Please do not edit our form.

Activity 3: Desmos MarbleSlides for  

  1. To show if they understand the investigation from Activity 2, students can complete this Desmos MarbleSlides. The first few slides are just to show some possible solutions and then the challenges start. Note that students may have to work with the domain of the function to become successful. 

If you want to know how to create your own Marbleslides, watch this video

Activity 4: Which One Doesn't Belong Assignment

  1. As a review you might choose to assign students to create their own Which One Doesn't Belong (http://wodb.ca/). The goal here is to create four graphs such that each of the graphs could be chosen as the one that doesn't belong based on specific criteria.
Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks 

Monday, 16 January 2017

Equation Strips

In Ontario our grade 7 students are introduced to solving simple equations in the form ax + b = c where the values of a, b and c are whole numbers. We think it's a good idea for them to start by having some sort of visual representation of each equation. In this activity, students are given 16 cards that correspond to 16 equations represented as strips (the top and bottom of the strips represent the left and right sides of the equations). Students solve for x given the strips and then rewrite the algebraic form equation.
  • Grade 7 Patterning & Algebra - solve linear equations of the form ax = c or c = ax and ax + b = c or variations such as b + ax = c and c = bx + a (where a, b, and c are natural numbers) by modelling with concrete materials, by inspection, or by guess and check, with and without the aid of a calculator 
  • Grade 8 Patterning & Algebra - as review

  • Each group gets a set of 16 cards
  • Make several copies of the cards on card stock and laminate them so they last longer. You may wish to copy each set onto a different colour so that if they get mixed up you know each set by their colour.
  • Cut out the cards so that each group gets a set of 16. 

  1. Each group of 2-3 students gets one full set of 16 cards. 
  2. Students are to determine the value of x for each card.
  3. Once determining x then they should then determine the algebraic expression for each card
  4. You can circulate with the solution card to check answers.
  5. Once finished you can create your own cards using this web sketch. This allows you to change the coefficients of a, b & c and it generates all four possible configurations. This web sketch assumes that a, b & c will be whole numbers and will not allow any solutions that have x as negative. Once you put your coefficients in then take a screenshot, use the screen capture software of your choice to copy and paste the version you want to use (For Windows use the Snipping Tool, for Chromebooks use Shift CTRL F5, for Macs use Command Shift 4, or iPad use the Home and Sleep buttons together. You can then paste into the word processor of your choice. 
Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks