Rate of Change Continuum

A continuum is something where the level difficulty increases in incremental steps. In this case the continuum is dealing with calculating the rate of change (slope) of a linear relationship.
We have previously posted a continuum for solving equations here and here but this one is a bit different. This one has five levels of determining the rate of change from a graph (in context) for a linear relationship. The first level shows lattice points, a rate triangle and the calculation of both rise and run (super basic) and the difficulty increases with each level (see below) until the last level where there is only a scale with no grid lines (so the answer is more of an estimate).

Each page has 6 graphs and students (once they choose the level to start with) choose to answer any three. If they do so correctly then they can move to the next level. The To make things a bit more fun, rather than check the answers with you, we suggest using a UV pen and ink written on the question cards for students to check.
This activity is probably best meant as a consolidation. Note that the expectation is about investigating so hopefully students will have had a chance to develop their own strategies for determining the rate of change. This activity just helps to scaffold it a bit in case they are having trouble (Eg a common mistake that students make when determining the rate of change when the line is in context is to just count boxes for the rise and run without considering the scale).
Note that we also have an Explain Everything version if you have students who have iPads (you may even want to try out the new Explain Everything Collaborative Whiteboard app to have students work in groups from different devices).

• MPM1D, MFM1P - determine, through investigation, connections among the representations of a constant rate of change of a linear relation.

• 20 copies of each of the question cards in different colour cardstock for each level,  laminated (use colours that allow seeing the magic pen writing). Note that you may not need 20 copies of each. Perhaps fewer of the first couple levels and last level as most kids will probably be starting in the 2nd or 3rd level
• 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). To help distinguish the answer cards to the question cards you should put a stamp or sticker on the back.
• 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.

1. For this activity to be successful, students must 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. 
2. Place the questions in piles in order of difficulty and set up three stations for the answer cards. Students will get a card and answer any 3 questions. 
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 3 and then check. Students are monitoring themselves so they decide. If they get the first 3 right, they have a level of mastery to move themselves to the next level. If not there are more questions on the card until they master that type. 
4. As they move through the continuum, the hope is that they reach level 4 which matches the grade 9 curriculum. Since our goal is to get them to level 4, students should solve ALL equations on that card instead of just three. 
5. The fifth level is set up to challenge students who are moving forward quickly. They should solve all questions on this card. They require some estimation and so answers that students get should be approximate. 

Note that for the Explain Everything version, there are still 6 possible graphs for each level but only two on each page. And to check the answer, slide the black ellipse to either the bottom left or right corner. 

• Graphical Rate Continuum (Google Doc, PDF, Explain Everything)
• Entire Folder

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

Sort students into groups using Quadratic Representations

In this activity students are each given one card. The card will either have a graph, table of values or equation of a quadratic relation. Their job is to find the two other students who have the other two representations of the same quadratic relationship. This shouldn't take too long and could be repeated every couple of days to solidify conversion between representations.
New: Alternatively, you could have students just work individually on this Desmos card sort 

• MPM2D - A3.3 determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts (i.e., the zeros) of the graph of the corresponding quadratic relation, expressed in the form y = a(x – r)(x – s);
• MBF3C - A1.8 determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts of the graph of the corresponding quadratic relation
• MCF3M - A1.5 determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the graph of the corresponding quadratic relation
• MCR3U - As review

• Download the cards and cut them out (you may want to laminate them)
• If you are doing the Desmos Cardsort instead, students should have devices to do the sort on (note that phones have screens that are too small)

1. Shuffle the cards and distribute one per student. Note that there are 11 sets of 3 cards so you may want to remove sets to more closely match your student population.
2. Instruct students to find the two other people that have different representations for the same quadratic relation. 
3. Once students find their partners they will be in groups of three

If doing the Desmos Cardsort instead, have students (or pairs of students) complete each page of the cardsort. You may wish to consolidate on the last page of the card sort.

• GroupQuadratics (Googledoc) (pdf)
• Desmos CardSort 

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

Sort Students into Groups using Linear Representations

In this activities students are each given one card. The card will either have a graph, table of values or equation of a linear relation. Their job is to find the two other students who have the other two representations of the same linear relationship. This shouldn't take too long and could be repeated every couple of days to solidify conversion between representations

MPM1D, MFM2P  - 

• graph lines by hand, using a variety of techniques
• identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b

• Download the cards and cut them out (you may want to laminate them)

1. Shuffle the cards and distribute one per student. Note that there are 11 sets of 3 cards so you may want to remove sets to more closely match your student population.
2. Students will find the other people who have the same linear relationship

• GraphTableLineMatch (doc) (pdf)

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

Data Management Scavenger Hunt

This is an activity that we adapted from a TIPS activity (Grade 8 Unit 11, Day1, Pg 3). In this activity student are given a set of data (with no labels) and asked to find their matching graph and title (which are randomly placed around the room). Once finding their cards they are given a question card to use with their graph and data. It's actually a surprisingly difficult activity but it helps bring out conversations about types of data and what they should look like.

• Gr7DM - select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph; read, interpret, and draw conclusions from primary data and from secondary data presented in charts, tables, and graphs (including relative frequency tables and circle graphs);
• Gr8DM - select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph; read, interpret, and draw conclusions from primary data and from secondary data, presented in charts, tables, and graphs (including frequency tables with intervals, histograms, and scatter plots);
• MBF3C - D1.5 identify different types of one-variable data, and represent the data, with and without technology, in appropriate graphical forms (e.g.,histograms, bar graphs, circle graphs, pictographs)
• MAP4C - D1.4 create a graphical summary of two-variable data using a scatter plot , with and without technology
• MDM4U - D1.5 interpret statistical summaries to describe the characteristics of a one-variable data set (note that this certainly can be used as review of all types of data)

Data Management Cards Prep work: Print, laminate (optional) and cut the cards out. Separated out the sets of data from the graphs and Titles. Each page contains one set of data, graph and title (note that one set has two possible matching graphs). Also included in the set are a page with extra questions and a page with the answer key on it.

1. Prep the classroom by randomly sticking the graphs and the titles around the room for students to find. Note that this activity was originally made for grade 8 but if you wanted to do it for grade 7 you should probably consider taking out any graphs with scatter plots.
2. Put students into groups of 2 or three (there are 17 different sets of cards)
3. Each group gets a set of data and are told to wander the room looking for their matching graph and title. Some of these are not easy so students should work together to decipher the information. You should probably circulated to give hints (with your answer key).
4. Once a group finds their matching cards, they are given a question card to answer regarding their graph.
5. If they finish early give them another set of data to try.

Data Management Scavenger Hunt (Doc) (PDF)

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

Sort Graphs into Types of Data

This is a simple and quick sorting activity for graphs to help introduce the idea of types of data. And although that is the intended purpose it is done as an open sort so that students can (hopefully) start to see that there are different types of data without being formally introduced to them. This is not intended to be a long activity but just a short Minds On type to begin a class. Keep in mind that if students don't sort them by data type then you may have to ask some leading questions to help move them that way. Though this was originally developed for the MDM4U class, by removing some of the graphs, it could be used for grades as low as 7 (for example, grade 7 students will likely not have seen histograms).
[Updtae - Now a Desmos Card Sort version exists]

• Gr7DM, Gr8DM - collect and organize categorical, discrete, or continuous primary data and secondary data
• MBF3C - identify different types of one-variable data (i.e., categorical, discrete, continuous)
• MDM4U - distinguish different types of statistical data (i.e., discrete from continuous, qualitative from quantitative, categorical from numerical, nominal from ordinal, primary from secondary, experimental from observational, microdata from aggregate data)

• One set of graph cards per group

1. put students in groups
2. distribute cards with graphs 
3. ask them to sort in any way they see fit. Use the results to steer towards types of data. Students should focus on both the look of the graph and the characteristics of the data being shown.
4. possible leading questions: "Can you arrange these so that there are only two groups? If so can you arrange each of those two groups into sub groups?" 
5. As an extension or alternate activity, students could sort by types of graph.

Note that the graph of the number of friends is a histogram which might lead some to think it is continuous data. This could be a good talking point as the number of friends a person can have can only be a discrete amount. 

Types of Data cards (doc) (pdf)
Desmos Card Sort (with self checking feature for numerical and categorical)

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