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 

Row Games

We saw this activity in 2010 when I first found @K8Nowak's blog f(t). I don't know if I would call a Row Game a particularly engaging activity but I am convinced that any way we can make doing boring homework questions more palatable for students is a good thing. The premiss is that you pair students up and they get a worksheet of questions. The questions are in two columns. Each person does one column and if they have done things correctly then their questions on the same row should have the same answer. If they don't then either one or both of them are incorrect and they have to work together to get the correct answer. So this is a self checking activity. We made a bunch of them at the time and I just stumbled upon them this week so we thought we would post them. These ones are for ratios, proportion, simplifying expressions and solving simple equations.

MFM1P, MPM1D

• illustrate equivalent ratios, using a variety of tools
• solve for the unknown value in a proportion, using a variety of methods 
• make comparisons using unit rates
• solve problems involving ratios, rates, and directly proportional relationships in various contexts, using a variety of methods
• solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms
• add and subtract polynomials involving the same variable up to degree three, using a variety of tools
• multiply a polynomial by a monomial involving the same variable to give results up to degree three
• solve first-degree equations with non fractional (Applied only) coefficients, using a variety of tools and strategies

•  Just the handouts (see below)

1. Pair students up
2. Have students decide who will be Student A or Student B, and have them complete Problem Set A or B.
3. The answers in each row should match. If they do not match, work together to determine the correct answer.

• See the files in one folder here
• Proportions (Word, PDF)
• Proportions Review (Word, PDF)
• Simplifying Expressions (Word, PDF)
• Adding Polynomials (Word, PDF)
• Simplifying Expressions with Multiplication (Word, PDF)
• Solving Equations (Word, PDF)
• Solving Multistep Equations (Word, PDF)

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

Simplifying Expressions and Solving Equations Tower Challenge

This is a review activity on simplifying expressions and solving equations for grade 9 applied where students answer questions and are rewarded with building materials for each correct answer. The building materials (spagetti & marshmallows) are then used with the goal being the creation of tallest tower. This is based originally on a TIPS activity on quadratics for MBF3C (Unit 3, Day 6).  We have a similar activity for grade 9 academic that can be found here.

MPM 1P

• substitute into and evaluate algebraic expressions involving exponents 
• describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x²; volume, which is three dimensional,can be represented by (x)(x)(x), (x²)(x),or x³]
• add and subtract polynomials involving the same variable up to degree three using a variety of tools
• multiply a polynomial by a monomial involving the same variable to give results up to degree three using a variety of tools
• solve first-degree equations with nonfractional coefficients, using a variety of tools
• substitute into algebraic equations and solve for one variable in the first degree

• 1 bag of spaghetti and 1-2 bags of small marshmallows (or 1 box of straws and 1-inch pieces of tape)  
• a question sheet for each student
• a teacher answer sheet 
• Optional - a whiteboard for each student to work out their solutions
• Optional - prize for the group with the tallest tower

1. Place students in groups (ideally no bigger than 3 per group)
2. Hand out question sheets (and optional whiteboards) to each student.
3. Have students answer questions from their sheet in any order they want. For every correct answer they will get some building materials (eg: 2 spagetti & 3 marshmallows, the amount of each reward is indicated on the student question sheet ). The harder the question the more materials they will get. Eventually the building materials will be used to create a tower with the goal to create the tallest free standing tower.
4. Students work in groups to answer the questions and bring their solutions up to you to be checked. Only one member from each group can come up at a time. Each group can answer each question only once. To keep track of this, use the teacher answer sheet to check off which questions each group has answered as they come up.
5. Leave about 20 min at the end of the class for students to create their towers (students can no longer answer questions)
6. Take lots of pictures and celebrate the group with the tallest free standing tower.

• Gr9AppliedSimplifyingExpression&SolvingEquationsTowerChallengeQuestions (pdf, doc)
• Gr9AppliedSimplifyingExpression&SolvingEquationTowerChallengeTeacherAnswerSsheet (pdf, doc)

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

Solving Equations Balance Method Card Sort

This is a simple activity where students are given a set of cards that represent the steps to take the given equation and use the balance method to solve it. The equations are relatively simple with most being two step solutions. This is not meant to be a big activity but just a warm up or perhaps something to do in the middle of class to break it up. There are 9 equations that can be solved so you could put students in groups or make multiple sets and have them work on them individually

• 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 9 sets of cards to print out and cut. We suggest printing on card stock and laminating. To help, each card will have it's set number just in case you mix up the cards. If you wish to use this with individual students rather than in groups you will need to print out more than one full set.
• NOTE - there are two versions. One with the traditional adding and subtracting of terms on the same line and the other with the adding and subtracting of terms done below. Obviously choose the version you prefer. The equations are the same in both
• We suggest putting each set in it's own envelop or zip-lock bag. 

1. Randomly distribute the envelopes to groups (or individual students). 
2. Have students use all the cards to show the steps to solve their question
3. Students should check their answer on paper or portable whiteboard
4. Once all groups (students) have correctly sorted their cards, they can exchange their set with the next group (since the sets are numbered they can just get the next numbered set - if they have set 1 then pass it to the group with set 2 etc). 
5. This could be done 9 times or stopped when ever you wish.

• Solving Equations Card Sort Horizontal (pdf, doc)
• Solving Equations Card Sort Vertical (pdf, doc)

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

Number Sense and Algebra Tower Challenge

This is a review activity on number sense and algebra for grade 9 academic where students answer questions and are rewarded with building materials for each correct answer. The building materials (spagetti & marshmallows) are then used with the goal being the creation of tallest tower. This is based originally on a TIPS activity on quadratics for MBF3C (Unit 3, Day 6).  We have a similar activity for grade 9 applied that can be found here. 

MPM 1D

• substitute into and evaluate algebraic expressions involving exponents
• describe the relationship between the algebraic and geometric representations of a single-variable term up to degree three [i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional, can be represented by (x)(x) or x²; volume, which is three dimensional,can be represented by (x)(x)(x), (x²)(x),or x³]
• derive, through the investigation and examination of patterns, the exponent rules for multiplying and dividing monomials, and apply these rules in expressions involving one and two variables with positive exponents
• extend the multiplication rule to derive and understand the power of a power rule, and apply it to simplify expressions involving one and two variables with positive exponents.
• relate their understanding of inverse operations to squaring and taking the square root, and apply inverse operations to simplify expressions and solve equations
• add and subtract polynomials with up to two variables
• multiply a polynomial by a monomial involving the same variable
• expand and simplify polynomial expressions involving one variable
• solve first-degree equations, including equations with fractional coefficients, using a variety of tools
• rearrange formulas involving variables in the first degree, with and without substitution
• solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods

• 1 bag of spaghetti and 1-2 bags of small marshmallows (or 1 box of straws and 1-inch pieces of tape)  
• a question sheet for each student
• a teacher answer sheet 
• Optional - a whiteboard for each student to work out their solutions
• Optional - prize for the group with the tallest tower

1. Place students in groups (ideally no bigger than 3 per group)
2. Hand out question sheets (and optional whiteboards) to each student.
3. Have students answer questions from their sheet in any order they want. For every correct answer they will get some building materials (eg: 2 spagetti & 3 marshmallows, the amount of each reward is indicated on the student question sheet ). The harder the question the more materials they will get. Eventually the building materials will be used to create a tower with the goal to create the tallest free standing tower.
4. Students work in groups to answer the questions and bring their solutions up to you to be checked. Only one member from each group can come up at a time. Each group can answer each question only once. To keep track of this, use the teacher answer sheet to check off which questions each group has answered as they come up.
5. Leave about 20 min at the end of the class for students to create their towers (students can no longer answer questions)
6. Take lots of pictures and celebrate the group with the tallest free standing tower

• Gr9AcademicNumberSense&AlgebraTowerChallengeQuestions (pdf, doc)
• Gr9AcademicNumberSense&AlgebraTowerChallengeTeacherAnswerSheet (pdf, doc)

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

I Have, Who Has - Proportions

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 an proportion with a missing value and a "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 24?". Someone else will have a card where their proportion has an answer that equals 24 so they would say " I have 8 over x equals 5 over 15". Who has 56?" That is, they read their proportion that has an answer that equals 24 and then asks their "Who has" statement. Then someone else will have an expression that matches 56 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.

• 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 NOTE - only use the 9 card set if doing this for grade 8;
• MPM1D - solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion
• 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);
• MFM1P - solve for the unknown value in a proportion, using a variety of methods

• 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 answer their "I have" proportion 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 answer is the same 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-Proportions-9cards (pdf) (doc)
• IHaveWhoHas-Proportions-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