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

Geometer's Sketchpad - Trig Ratio Generator

When using the Geometer's Sketchpad (for both computer and iPad) 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 can practice two very specific skills dealing with trigonometry. The first is simply being able to correctly place the names of the sides of a right triangle (opposite, adjacent and hypotenuse). Students drag the side names and then can check their answers and then randomly generate another triangle to try again. The second is one where a random triangle is generated that shows information about two sides and one angle. Students then drag parts of an equation to create a trig ratio equation. They can check their answer and then randomly generate other right angled triangle to try again. 

This is not meant to be something that a student uses for a long length of time but instead just some quick practice to re enforce the basic ideas from trig ratios.

• MFM2P, MPM2D - determine, through investigation (e.g., using dynamic geometry software, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios.
• MCR3U, MCF3M, MBF3C - As review

• 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

• TrigGenerator.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

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

Number Sentences Sort (update)

One of the smaller expectations we have to deal with is the ability for students to interpret algebraic equations. You know: "what does 2x + 1 mean"?. We created this very simple sorting activity where students are given expressions (and equations) and the sentence to describe them and have to match them up. This is meant to be an activity that is relatively quick. We have two versions here. One for grade 7 that only has expressions and one for grade 8 that has equations as well. We also have an Explain Everything version of each so that if you have an iPad (or Chromebook), with that app, you can have your students sort them electronically. This can also be used as review in Grade 9.
Note: This is an update to the same activity posted last year but now with a grade 8 and Explain Everything version
Double Note: This has been updated again to now include a Desmos card sort. So both card sorts are now transferred to this new Desmos feature. You can learn about Desmos Card Sorts by clicking here. Download the Teacher versions (which you can copy) of these activities below in the download section.

• Gr7PA - translate phrases describing simple mathematical relationships into algebraic expressions using concrete materials
• Gr8PA - translate statements describing mathematical relationships into algebraic expressions and equations
• MPM1D, MFM1P - As review

• For the grade 7 version there are four different (but similar) sets. One set per page. For the grade 8 version there are three different (but similar) sets. One set per page. 
• Print each page on card stock (we also suggest laminating). We suggest that each set be printed on different colour card stock for easy sorting. Cut each out and put each set in an envelope.
• Obviously you will have to decide how many sets you will need for your class depending on whether you pair students up or not. 
• Note that in the version with equations, there are some algebraic expressions that do not have matching sentences. In these cases, students will have to write their own.
• If you choose to use the Explain Everything version, then you probably want to download that .xpl file and put it on a server where your students can get easy access to it. 

Explain Everything Screenshot

1. Depending on how many students you have you may want to do this individually, in pairs or in larger groups. The activity is not super complex so we don't recommend anything bigger than pairs. 
2. Students take each set and sort the algebraic expression with the written version. 
3. When they are done their set they can trade with another group that has a different colour of cards. If they are using the Explain Everything version then they can just go to the next slide. 
4. There is a homework sheet for consolidation that includes both expressions and equations as well.

• NumberSentences-Expressions (doc. PDF, XPL, Desmos)
• NumberSentences-Equations (doc, PDF, XPL, Desmos)
• NumberSentencesHomework-Equations (doc, PDF)
• 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

Geometer's Sketchpad - Perfect Square Practice

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). One of the advantages of doing this is that the bulk of the time spent on the software is actually doing math rather than building something. 

In this sketch students can practice recognizing perfect squares up to 144. It is a very simple sketch not meant to take much time but to just familiarize students with the first 12 perfect squares as well as to remind them that perfect squares can also be defined by physical squares.

• Gr7NS1.6 - represent perfect squares and square roots, using a variety of tools (e.g., geoboards, connecting cubes, grid paper);
• Gr8 - could be used as review or see our square root guesser sketch instead

• 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

This sketch randomly selects a number under 150 and asks students whether it is a perfect square. They can make a mental guess and check their answer. Or, before the check their answer,  if they want to test it out they can try to create a square that has area equal to the given number. Once done they can generate another random number and try again. The hope is that this will help them become familiar with the first 12 perfect squares. Watch this video to see a demonstration of how it works.

• PerfectSquares.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

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 - Square Root Number Line Guesser

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). One of the advantages of doing this is that the bulk of the time spent on the software is actually doing math rather than building something. In this sketch students can practice their knowledge of estimating the square root of numbers up to 500. There are several levels of difficulty: perfect squares up to 100, perfect squares up to 500, square roots up to 100 and square roots up to 500. The intent was that this was built as a practice file for grade 8 students but grade 7 students could use it to practice perfect squares.

• Gr7NS - represent perfect squares and square roots, using a variety of tools
• Gr8NS - estimate, and verify using a calculator, the positive square roots of whole numbers, and distinguish between whole numbers that have whole-number square roots (i.e., perfect square numbers) and those that do not

• 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 the sketch works

• SquareRootNumberlineGuesser.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

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