This is supposed to be a quick introduction activity to mixed radicals. Rather than just tell students what mixed radicals are and how they behave, we would start with these and have them explore first and then consolidate afterwards. There are four groups of cards where each group has a different "root" radical. Within each group there are three sets cards that are all equal in value but different in form. Students have to arrange group the cards that are equal and hopefully come up with the connection between a radical in different forms. There is a physical card version as well as a Desmos Cardsort version.
MCR3U - verify, through investigation with and without technology, that √ab = √a x √b, a ≥ 0, b ≥ 0, and use this relationship to simplify radicals (e.g., √24) and radical expressions obtained by adding, subtracting, and multiplying
If you are using the physical version, we recommend printing each set of cards (2 pages) on a different colour of cardstock. This makes it easier to group the cards afterwards since each group will be a different colour. We also suggest laminating the cards before cutting.
You may wish to print out one copy of the full two sheets back to back for yourself as an answer key.
If you are using the Desmos version, make a copy of the activity and create your class code for students.
Put students into groups of 3 (or four). Using larger groups will make this activity go faster but will mean that students do less.
Ask students to determine all the expressions that are equal to each other. Because this is an intro activity and students don't know the properties of mixed radicals, let them use calculators. It is optional to tell them that there are a total of 12 sets of three cards that are equal. If you are pressed for time you might take some of the sets out.
Once students have found all of their matches ask them to further group those sets of three in any way they wish. Hopefully they will put them into four groups with the same "root" radical.
As further extensions
You may wish to ask some of the questions, below, from the Desmos cardsort (there are even more there) to pull out the nature of mixed radicals.
In this activity students are each given one card. The card will either have a fraction, percent or decimal. Their job is to find the two other students who have the same value but a different representation. This shouldn't take too long and could be repeated every couple of days just to solidify conversion between fraction, decimal and percent.
If you wish you can also have students do this individually with this Desmos cardsort.
Gr7 - determine, through investigation, the relationships among fractions, decimals, percents, and ratios
Gr8 - translate between equivalent forms of a number (i.e., decimals, fractions, percents)
MPM1D, MFM1P - As review
Download the cards and cut them out (you may want to put them on cardstock and laminate)
Shuffle the cards and distribute one per student. Note that there are 12 sets of 3 cards so you may want to remove sets to more closely match your student population.
Instruct students to find the two other people that have the same value but a different representation.
Once students find their partners they will be in groups of three,
Group Fractions, Decimals, Percent Cards (Googledoc) (pdf)
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
Put students in groups of no more than three. They can work on vertical surfaces or at a table.
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).
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.
Once complete there is a couple of follow up questions to check their thinking.
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.
Again, once complete, they should walk around the class to see other sets and come to some conclusions.
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).
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
Students complete the Desmos Investigation. This can be done in class or as a homework assignment.
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
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
As a review you might choose to assign students to create their own Which One Doesn't Belong (https://talkingmathwithkids.com/wodb/). 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.
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.
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.
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.
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.
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.
Walk through the front of the handout to show how to create a histogram from data.
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.
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.
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
This is supposed to be a quick introduction activity to mixed radicals. Rather than just tell students what mixed radicals are and how they behave, we would start with these and have them explore first and then consolidate afterwards. There are four groups of cards where each group has a different "root" radical. Within each group there are three sets cards that are all equal in value but different in form. Students have to arrange group the cards that are equal and hopefully come up with the connection between a radical in different forms. There is a physical card version as well as a Desmos Cardsort version.
