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 

Sunday, 4 December 2016

Introducing Histograms


In grade 8 in Ontario histograms are one of the new data management topics. Below is a series of activities that range from paper & pencil to virtual manipulatives. They start by a simple comparison of bar graphs and histograms to identify characteristics, then moves to some consolidation of ideas using a Desmos Sort. Then some data collection of student heights to formally introduce creating a histogram followed by some practice making histograms by hand and then with Google Sheets. If you did everything here it should take about 2-3 classes.

Grade 8
  • collect and organize categorical, discrete, or continuous primary data and secondary data (e.g., electronic data from websites such as E-Stat or Census At Schools), and display the data in charts, tables, and graphs (including histograms and scatter plots) that have appropriate titles, labels (e.g., appropriate units marked on the axes), and scales (e.g., with appropriate increments) that suit the range and distribution of the data, using a variety of tools (e.g., graph paper, spreadsheets, dynamic statistical software); 
  • select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph (i.e., from types of graphs already studied, including histograms and scatter plots);
  • read, interpret, and draw conclusions from primary data (e.g., survey results, measurements, observations) and from secondary data (e.g., election data or temperature data from the newspaper, data from the Internet about lifestyles), presented in charts, tables, and graphs (including frequency tables with intervals, histograms, and scatter plots);
  • demonstrate an understanding of the appropriate uses of bar graphs and histograms by comparing their characteristics
  • Could also be used as a review in MBF3C or MDM4U
  • If you are doing the physical card sort then print graphs to card stock (we suggest each set is a different colour so that if students mix them up they are easy to separate). We also suggest lamination. There are two pages for a total of 16 cards. 
  • If you are doing any of the online activities then chromebooks/laptops/computers/iPads will be needed. You will also need to make copies of the Desmos version of the card sort and/or the Desmos card sort consolidation.
  • Copies of the handout
  1. The purpose of the card sort is to start to distinguish both histograms and bar graphs and continuous vs categorical data. But first have them do an open sort. Hand out cards to students (or give students the code for the Desmos Sort). Ask them to sort them in any way they wish. The only stipulations are that there should be at least two groups and each group must have at least two cards. Students will sort them in all kids of ways (by the numbers, by the topics, by the looks). Circulate and encourage them to explain how they were sorted.
  2. Once sorted it is likely that most will not have them sorted how you wish. Have them describe their sorts then ask them to sort them in a way so that there are only two groups. 
  3. By this time some may have them sorted into bar graphs and histograms. If not show one bar graph and then one histogram stating that you want those cards to represent characteristics of each group (you can use the slideshow to show the graphs). Use these graphs to develop the difference between continuous and discrete data.
  4. Time to change the pace and have students collect some data. Have students measure their heights and put them on a dot plot on the board. Use this to create a histogram (without creating a tally). You might want to collect their heights in a spreadsheet so that you can create a histogram with it later. 
  5. Walk through the front of the handout to show how to create a histogram from data. 
  6. Likely you will be at least done one class at this point (if not more). At the beginning of the second class, start with the consolidation Desmos card sort (with some stuff on average) to remind students of continuous and discrete data. Another possible way to start is to have students use this Google form to enter some examples of continuous and discrete data. You can take up the results. 
  7. Once done taking up the worksheet you can then introduce students to creating histograms using Google Sheets. Share this sheet with continuous data. This video will tell you how. 
Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks 

Monday, 20 June 2016

Developing Derivatives Graphically

In this activity students develop the visual connection between a quadratic function and its derivative. Each student is given a quadratic function and an x value. They are to determine the point and the instantaneous rate of change of their function at that x value. Once they do this they plot their point and then also plot their rate of change (as a point at that x value) on a piece of chart paper. Since everyone is given a different x value, a graph of both the original function and its derivative should be constructed. This would be a good activity to introduce the concept of the derivative shortly after students know how to find the instantaneous rate of change by using a limit.
Once students are done the graphs on the charts, they can then move to extend their thinking with this Desmos activity where they visually estimate the slope of the tangent to eventually build the derivative (based on a original sketch by @eluberoff). As they slide the sliders to estimate each tangent, once they get them all close to where they should be, the derivative appears (see animated gif below)They do this for several simple quadratic functions and (hopefully) see the beginnings of how the power rule works as they move towards simple cubic functions.


  • MCV4U-A2.2 - generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f ’(x) or dy/dx , and make connections between the graphs of f(x) and f ’(x) or y and dy/dx
  • A3.1 - verify the power rule for functions of the form f(x) = xn , where n is a natural number
  • Grid chart paper (one per function - write a different function on the top of each piece). You may wish to draw the grid on each piece before hand. Because the scale can get disproportionally large on the y vs x axis you may want to spread the scale out on the x so that it's not so crowded. Note that if you choose to us a disproportionate scale on the chart paper then the kids must use a similar scale in their mini graph papers ( I've done this activity a few times and only now, as I created this post, did I figure out that this is why the slopes of the tangents didn't quite match the graphs physically). 
  • Copies of each mini graph for each student via the handout (see image to the right for a sample). There are four functions (f(x) = 1/4x2, f(x) = 1/2x2, f(x) = x2, and f(x) = 2x2) and potentially nine x values for each function (-4, -3, ....3, 4). You may not want to give out the -4 or 4 for the f(x) = 2x2 function as it requires a big y axis scale. Depending on how big your class is, you may not want to hand out all of the mini graph sheets. For example, if you had 25 students you might want to use 3 of the functions. Cut these out ahead of time. You may wish to have more than you need ready so that if someone finished quickly, they can be given another one. 
  • Scotch tape for students to stick their graphs on the chart paper.
  • Markers. Preferably different colours (one colour for the function points, derivative points, function line, derivative line and tangent lines).
  • Laptops, Chromebooks, iPads if you will be extending this to the Desmos version
  • Place chart paper on the walls and distribute one mini graph to each student. Make sure there is enough of each function so that the shape will be visible
  • Each student then does the following:
  1. Determine the point on that function at that x value: A(     ,      )
  2. Determine the slope of the tangent at that x value for your function: Slope of tangent = _____
  3. On your mini graph, draw a line with the same slope as the slope you just calculated and going through the middle point on the mini graph
  4. Plot the point from #1 on the large graph. Stick your mini graph on the large graph paper so the point on the mini graph paper is on top of the point you just plotted (be careful with your orientation)
  5. Plot a second point that has your x value and its y value is the slope of your tangent: B(   ,   )
  • As students plot their points by sticking their mini graphs on the chart paper, you may wish to use the markers to draw over their points and lines to make them more visible. Note: If a student makes a mistake, do not correct it right away (or make it permanent with a marker). Instead ask students if they think anything is out of place either at the time or during the consolidation
  • Once all the graphs are done consolidate the ideas
  • Once consolidated then move to the Desmos activity. If you have not done a Desmos activity before you might like to watch some of these tutorials first: Navigating, starting an activity, and teacher dashboard.
  • Note that it is not necessary for students to get all the way through the activity. As long as they get through the quadratics. The cubical are the extension. The whole idea behind this is they discover the power rule.
Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Monday, 6 June 2016

Derivative Matching Cards

This is a very simple matching activity for Calculus. Students are give a set of cards with either a linear, quadratic or cubic function on them. Their job is to pair them up so that one is a function and the other is its derivative. There are a total of 12 functions with 12 derivatives. The first six are all linear or quadratic graphs and the second six are either quadratic or cubic graphs (if you wanted to give students an easier set you could only give them the first six). This is not meant to be a brain buster of an activity but it does help to solidify thinking in terms of the characteristics of the connections between a function and its derivative.
NEW: Desmos has turned this activity into one of their new CardSort activities. You can get that version here

  • MCV4U - A2.2 - generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f ’(x) or , and make connections between the graphs of f(x) and f ’(x) or y and dy/dx
  • MCV4U - B1.1 - sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes)
  • As mentioned above, there are 12 cards and their derivatives but you could break them up into sets of 6 cards and their derivatives where the first set was made of linear and quadratic functions and the second set is made of quadratic and cubic functions (or you could just put them all together). On each page there are six graphs. The first column are the functions and the second column are the matching derivatives.
  • Print the sheets out on card stock (and laminate if possible). We tend to print each set out on different colours. This way if they get mixed up all you need to do is collect 24 cards of one colour and you will know you have a full set
  • You may also want to print a copy of the teacher answer key which has all 24 graphs on one page so you can easily check student's answers.
  • Put students in groups of two or three
  • Distribute cards and tell them they have to pair the cards up in terms of a function its derivative. 
  • Instruct them that every card is paired up and they will likely be correct if they have no cards left over
  • Encourage them to use properties of functions and derivatives (zeros, max/mins etc) to speed up the process 

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks