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

Polygraph - Distance Time Graphs

Have you ever played the game "Guess Who?". In that game you have a grid of people to look at and you choose one. Your partner has the same grid and gets to ask you yes or no
questions. As they ask questions and get their answers they start to eliminate people until they have only one left. Well the crew at Desmos have created a gave called Polygraph that works the same way except for graphs (and other topics). So far they have created one for parabolas, lines, rational functions, quadrilaterals, advanced quadrilaterals and hexagons. The way it works is you start a new session and get the 4 character code. Your students then go to  student.desmos.com and enter that code. They will the get a brief (optional) training session on how to play the game and then the software will randomly select students to play against each other, taking turns as to who asks and answers the questions.

But that is not the best part. The best part is they have created the Polygraph editor so anyone can create their own Polygraph. So that is what we did. This one is about the characteristics of a Distance-Time graph (or more generally a piecewise linear graph). There are no scales on these graphs so students will have to use general terms but we still think they will be able to relate to these with appropriate questions.

• MFM1P, MPM1D - describe a situation that would explain the events illustrated by a given graph of a relationship between two variables 

• Each student needs their own device connected to the internet. A phone will work but the images will be small. You may want to group kids together if the number of devices is limited. 

1. Go to this link and click on Start a New Session. You will have to log in to start a session but you can do that with your Google account or create your own. This will keep track of how your students responded and you can look at this any time you wish. You will be give a 4 character code that your students type in at student.desmos.com.
2. Once students log in they will try a test round with faces (they can skip this once they know what to do) and then as students complete the test round they will be paired up automatically with other students who have finished the test round. 
3. Instruct students to ask questions as if they were distance-time graphs 
4. Circulate through your class if there are questions. 
5. Note that you will need an even number of players otherwise there will be always someone sitting out.

No downloads needed, just the link https://teacher.desmos.com/polygraph/custom/56251492a23c2d7208dfe072

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

Geometer's Sketchpad - Practice Line of Best Fit

When using the Geometer's Sketchpad it is often better to "start from sketch, not from scratch". That is, give students a premade sketch rather having them build something from nothing (as many textbooks would have you do).
In this activity, students practice placing the line of best fit on a linear set of data. It's not meant to be really difficult but just to reenforce the idea of what the line of best fit is. Students can check their answer and try as many as they like. Clicking the Medium or Hard buttons will spread the points out more randomly to make the line a bit harder to determine. This is not meant to be really hard but just a quick way to determine if students have the basic concept of what a line of best fit is

• MPM1D, MFM1P - construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools (e.g., spreadsheets, graphing software, graphing calculators, paper and pencil), for linearly related and non-linearly related data collected from a variety of sources
• MAP4C - D1.4 - create a graphical summary of two-variable data using a scatter plot (e.g., by identifying and justifying the dependent and independent variables; by drawing the line of best fit, when appropriate), with and without technology
• MDM4U - D2.4 - generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)

• All that is needed is the electronic download (below)
• Note that this really works well on an iPad using the Sketchpad Explorer App (which is free)
• You can also use this on any web based computer (or Chromebook) with this Web sketch

Watch the video below to see how to use the sketch

• Line of Best Fit.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my look at the dynamic web sketch tab above

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

Linear Scavenger Hunt

The linear scavenger hunt is an activity where there are sets of four representations of linear relations on cards (graphical, tabular, algebraic and word problem) that students have to search for and match.

• MPM1D, MFM1P, MFM2P
• determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation
• determine other representations of a linear relation, given one representation
• describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation 
• determine the equation of a line from information about the line

• Linear Scavenger Hunt Cards (printed 1 sided)
• Linear Scavenger Hunt Question Cards (printed back to back)
• Tape/stick tack
• Beach Ball (optional)
• Whiteboard markers (optional)

1. Each group of 3 students gets 1 representation of a linear relationship  - either a story, equation, table or graph.  Students will complete a scavenger hunt by roaming around the room to find the other 3 representations of their linear relationship. The other three cards in each set are randomly distributed around the class before hand.
2. Once all four cards are together, each group must fill in missing information on the graph (title and axes labels) and in the table (column titles and table values -  CAUTION:  yellow rows are out of step in the pattern). Teacher can check missing information quickly using reference card.
3. Each group will tape their equation in the middle of their desks.
4. Each group will now tape their story to the beach ball.  Students will throw the ball and each group will select a card (not their own).  That group must now find the equation of the new story they got off the beach ball.  One member will then circulate to check if the equation is correct with the original group for their new story and return it to that group.
5. Repeat step 4 with the tables.
6. Repeat step 4 with the graphs.
7. Each group should now have the original 4 representations of their linear relationship.  Students will now answer 5 questions related to their relationship found on the Question Cards.  

Other possible uses

• give a story or equation to each student and have them find their partner
• give each group of 3 an equation and have them describe several situations that could be modelled by the given linear equation.
• give students a card and have them identify as a direct or partial variation
• give a student a table or graph - have them find the rate and explain its meaning

The video, below is only visible in the WECDSB domain. That is, only teachers in our school board can see the video if they are logged into their MyTools2Go accounts.

• Linear Scavenger Hunt Cards (doc) (pdf)
• Linear Scavenger Hunt Question Cards (doc) (pdf)
• Note that if you print these out on card stock and laminate them, students can use whiteboard markers or whiteboard crayons to do their work on them and can be reused.

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