Visual Pattern Cards

This activity was inspired by @JoBoalar's TED talks were she talks about people using colours to show how patterns grow. It was also inspired by @FawnpNguyen's awesome VisualPatterns.org site. The premiss is simple, students are each given a card with three terms of a linear pattern on it. They are to recreate the pattern (and add two more terms) using colour to show how it grows. Then they use what they see to help come up with a general term for their pattern.

For this activity there are 32 cards made that have linear patterns in the form Ax, x + B and Ax + B. For grade 7, if your focus is developing the general term, then you should just use the cards with general terms in the form Ax & x + B (patterns that have expressions involving only one operation).

• Gr7PA – represent linear growing patterns, using a variety of tools (e.g., concrete materials, paper and pencil, calculators, spreadsheets) and strategies (e.g., make a table of values using the term number and the term; plot the coordinates on a graph; write a pattern rule using words);
• Gr7PA – make predictions about linear growing patterns, through investigation with concrete materials;
• Gr7PA – develop and represent the general term of a linear growing pattern, using algebraic expressions involving one operation (e.g., the general term for the sequence 4, 5, 6, 7, … can be written algebraically as n + 3, where n represents the term number; the general term for the sequence 5, 10, 15, 20, … can be written algebraically as 5n, where n represents the term number); 
• Gr8PA – represent, through investigation with concrete materials, the general term of a linear pattern, using one or more algebraic expressions (e.g.,“Using toothpicks, I noticed that 1 square needs 4 toothpicks, 2 connected squares need 7 toothpicks, and 3 connected squares need 10 toothpicks. I think that for n connected squares I will need 4 + 3(n – 1) toothpicks, because the number of toothpicks keeps going up by 3 and I started with 4 toothpicks. Or, if I think of starting with 1 toothpick and adding 3 toothpicks at a time, the pattern can be represented as 1 + 3n.”);
• Gr8PA – represent linear patterns graphically (i.e., make a table of values that shows the term number and the term, and plot the coordinates on a graph), using a variety of tools (e.g., graph paper, calculators, dynamic statistical software);
• Gr8PA – determine a term, given its term number, in a linear pattern that is represented by a graph or an algebraic equation 

1. For this activity there are 32 possible cards to use. Print them out on card stock (and laminate them if possible). Cut out each card to create a class set. If you are doing this with grade 7s then you may want two sets of the last two pages of cards since they deal with general terms in the form Ax & x + B.
2. Print out the Answer card so that you can circulate easily giving help or advice. 
3. Students should have enough connecting cubes or colour tiles of various colours to create the patterns. Or students could use this virtual colour tiles app from Mathies.ca. 

1. Hand out one card per student (you could also do one card per pair and have extra cards for students that finish early)
2. Instruct students to recreate their pattern using colours (connecting cubes or colour tiles) to show how the pattern grows. You may also ask them to restrict to two colours and show it that way as well. 
3. Once students have recreated their pattern, have them create the next two terms using the same colour distinctions. 
4. Once you are satisfied with their five terms, have them re arrange their tiles so that they create a line of each set of tiles for each term (like a bar graph).  While groups are getting to the same place you might ask quicker groups to determine the number of tiles needed for the 15th term
5. Next have students use their tiles to help determine the general term 
6. You might want to take several different examples to consolidate creating the general term. 

As an alternate procedure you might consider the progression of steps seen in this video using some of these cards to more systematically develop the a method for generating the algebraic expression for a linear pattern. 

• Visual Pattern Cards (Google Doc, Google Slides, PDF)
• Visual Pattern Cards Answer Key (Google Doc, PDF)
• This is the app we used to make the cards

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

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). They are more commonly called bar models but we have always used the name equation strips. Students solve for x given the strips and then rewrite the algebraic form equation. We originally got this idea from the 2011 Solving Equations Gap Closing resource (pg14).
[Updated Mar 6, 2018 - now both the printable cards and the dynamic websketch have equations in the form 2x - 5 = 19. This puts it beyond the grade 7 expectation but could be an extension or just used for grade 8]
[Updated Dec 19, 2019 - now including a single dynamic Desmos version where you can switch all the versions using sliders as well as a Desmos Activity where students first do some static practice problems and then it finishes with a Challenge Creator where students make their own question for the others to solve]

[Updated May 9th, 2022 - I added a card sort to the Desmos Activity and included a pdf version]

• 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 (24 if you use the cards with minuses)
• 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 or this Desmos 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. 
6. If using the Desmos Activity, pair students so that they can have conversations about the
   
   strips. Be sure to explain how when they create their challenge, they will first have to solve it before they can submit it for the rest of the class to do. When using the Desmos activity, note that the "x's" are missing to make the connections to explicit equations less visible. This is so that students are free from the stigma of actual equations while still solving them. You might want to start with this. 

Note that if you want to modify the Desmos activity, you might want to watch this video on how to do it:

[Added July 2024] And here is @Howie_Hua doing some explainer videos of how to use Equation Strips (Bar Models)

Introducing one-step and two-step linear equations with bar models (Part 1) pic.twitter.com/eGlrAe0FNX

— Howie Hua (@howie_hua) July 18, 2024

(Part 2) pic.twitter.com/pyxqew4G7h

— Howie Hua (@howie_hua) July 18, 2024

• Equation Strip Cards Print version (Google Doc, PDF)
• Equation Strip Solution card (Google Doc, PDF)
• Equation Strip Card Sort (Google Doc, PDF)
• Equation Strip Solution template
• Equation Strip Web Sketch
• Desmos Sketch
• Desmos Activity

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

I Have, Who Has - Equations

An I Have, Who Has game is not a new concept. The premiss is that each person gets a card that has two statements. One is the "I have" statement and the other is the "Who has" statement. In this case the "I have" statement is a simple equation "Who has" statement which is the answer. The way the game works is that a person starts by reading their "Who has" statement. For example, someone might say "Who has 7?". Someone else will have a card where their equation has an answer that equals 7 so they would say " I have 2x = 14". Who has negative 4?" That is, they read their equation that has an answer that equals 7 and then asks their "Who has" statement. Then someone else will have an equation that matches -4 and the game continues. If done correctly, it should end up with the person who started giving their "I have" statement. It works really well as a warm up and one of nice things about this is that you could do it multiple days and kids will likely get different cards.

• Gr7PA - 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 NOTE - only use the 9 card set if doing this for grade 7 as there are a few multi step equations;
• Gr8PA - solve and verify linear equations involving a one-variable term and having solutions that are integers, by using inspection, guess and check, and a “balance” model 
• MPM1D, MFM1P - solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);

• There are two sets of cards that you could download here. One set (pictured here) has only 9 cards in it (you can see that the card on the top left has the "I have" to match the "Who has" of the card on the bottom right). Depending on the size of class you have you might want to use this set multiple times (ie groups of 9) or use the larger set of 27. Either way, in order for the game to work, all cards need to be passed out. So some students may need to have more than one card.
• Regardless. Print out the set you want (ideally on coloured card stock) and we also suggest lamination to lengthen the lifespan of the cards.
• Be sure to print out a set for yourself that you don't cut out so that it will be easier for you to check as students play the game.

1. Distribute the cards one per student. All cards must be handed out so some students might need more than one card.
2. Tell each person to solve their "I have" equation and check their answer with at least one other person. 
3. Once students are confident with their answer all students should stand and then you choose one to read their "Who has" statement. The person who's equation has the same answer should read their "I have" statement followed by their "Who has" statement and then sit down. Eventually the last person standing should be the person who started. 
4. A variation might be to have students walk to the front and stand next to the person who they were matched with and eventually form an entire loop around the class.

• IHaveWhoHas-Equations-9cards (pdf) (doc)
• IHaveWhoHas-Equationss-27cards (pdf) (doc)
• IHaveWhoHas-BlankTemplate (doc)

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

Connecting Words and Algebraic Expressions

Students are given a set of cards that have either an algebraic expression or a word sentence. They then have to match up the expressions with the word sentences. This is not meant to be a big activity but just a quick one to help kids connect the different forms. You could use this in grade 7, 8 or 9.

NOTE: This activity has been updated and a newer version can be found here

• Gr7 - translate phrases describing simple mathematical relationships into algebraic expressions, using concrete materials
• Gr8 - translate statements describing mathematical relationships into algebraic expressions and equations
• MPM1D, MFM1P - As review

• There are four sets. Print each on a different colour card stock (there is no real reason for this except that if the cards get mixed up it is easier to get them back into their groups) and laminate (if possible) then cut them out.
• Mix each set up and put each into an envelope
• Because there are only four sets, you will have to make up multiple sets depending on how big your class is. 

1. Put students into pairs or triads
2. Give each group one set of cards
3. Students are to match up the word sentence with the algebraic expression
4. If time permits, have groups trade sets for a different colour.

• Number Sentences (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