Different Forms of Linear Equations Card Set with Seven Variations of Use

In order for my students to practice changing forms of linear equations, I created a set of cards that they can use multiple ways.   For each equation, there are four cards:   a version of the equation in y=mx+b form, a version of the equation in x=kx+d form, a version of the equation in standard form, and a scaled up version of the standard form.


I love card sets because I can use them so many ways.   I can use them when I first teach the lesson one way.   I can then use them again a different way for additional practice or as a quick review leading up to a quiz or a test.   I can also use them as part of a differentiated instructional piece of a lesson to work with students who still haven’t mastered the concept.   Finally, I can bring them back out in a month in yet another way as a quick refresher.   (In Make It Stick, the author articulates the power of revisiting a concept after some time has lapsed in order to “make it stick”).

Option 1 – Card Sort

Students work in table groups or in pairs to sort the cards.    They need to match the four different forms of the same equation.

Option 2 – Rummy

Students use the cards to play Rummy.  Students play in pairs or triads.   Each player is dealt six cards.   A single card is face up and the remaining cards are face down in a draw pile.   The player to the left of the dealer begins play.   He or she takes the top card in the draw pile or takes the top face-up card in the discard pile.  The player may lay down any matched sets he or she has face up.  A player may add any one or two cards from his or her hand to a matched set already laying face up on the table.   If he or she does not play any matched sets, he or she must play a card in the discard pile (it can’t be the same card he or she drew from the face up discard pile).  Matches consist of cards that are different forms of the same equation.    The first player to play all of his or her cards is the winner.

Option 3 – Odd One Out

Remove one equation card from the set.    This leaves one equation with only 3 equations.   Since the game play involves making matched pairs, the remaining equation  is the “Odd One Out”    Students play in groups of 3 or 4.   The cards are dealt out.    Players look at their cards and lay down their matches (in pairs).    When it is a player’s turn, he or she draws a card from another player and tries to make matches with the cards in his or her hand.   At the end of the game, the player with the Odd One Out card loses the game.    Of the remaining players, the one with the most matches is the winner.  (This is essentially Old Maid with equations.)

Option 4 – Concentration

Students work with a partner or play individually.   All the cards are laid face down.   The player turns over two cards.   If the two cards represent the same equation, it is a match and the player keeps them.   If they do not represent the same equation, the player turns them face down again.   If two players are playing, the next player takes a turn   The player with the most matches at the end of the game is the winner.

Option 5 – Spoons

You need a set of teaspoons in addition to the cards.   You should have one fewer spoon than the number of players.   Put the spoons in small circle in the middle of the table.   Deal four cards to each player   Each player tries to make four of a kind.   The dealer takes a card off the top of the deck, removes one of his/her cards and passes it facedown to the left.   Each player discards to the person on his/her left.   The last player discards into the trash pile.   This continues until someone gets four of kind and takes a spoon from the center.   Once the player with four of a kind takes a spoon, anyone can take a spoon.   The player without a spoon gets a letter in the word SPOON.    When a player has spelled SPOON, he or she is out of the game.  If the cards run out, reshuffle the trash pile and continue play.


I didn’t have enough spoons on hand, so we played spoons with forks.    Spoons took a far amount of time today, but the students had fun and liked the game.  This option is probably best for a review activity rather than on the first day of instruction.

Option 6 – QuizQuizTrade

Each student is given a card of a particular form.   I direct them to change it into a different form (e.g., they are given a card in standard form and I tell them to change it to slope intercept form).   They solve the problem and then do a Quiz Quiz Trade.  Quiz Quiz Trade is a cooperative learning structure.   The first partner quizzes the second partner to turn his/her equation into a different form.    If he or she struggles, the first partner can give him or her a hint.   If he or she continues to struggle, the first partner can give a second tip.  If he or she continues to struggle, the first partner can tell the second partner how to solve the problem (Tip Tip Tell).   Partners then reverse the process and repeat.   Finally, they trade cards and find new partners.   I have students continue for whatever amount of time I want to allocate to the task.

Option 7 – Concept Attainment Cards

When I make the card activity, I use a different color of card stock for each set.   With this activity, I give each group a set of cards that is all the same color for the desired concept (all the pink cards are in slope intercept form) and a different color for the non-examples (purple cards with equations in standard form or in x=kx+d form).   Students then have to look at the examples and non-examples in order to define the concept illustrated by the examples.

Option 8 – Give One Get One

Students line up in two lines, facing each other.   I give each student in one line a card with an equation in a specified form of an equation (e.g., everyone in the first line has a card in y=mx+b form).   I give each student in the second line a card with an equation in a different form (e.g., everyone in the second line has a card with an equation in x=ky+d form).   I then tell everyone to transform their equation into another form (e.g., standard form).   When they have done so, they step forward into the space between the lines.   They trade cards and solve the other card, discuss their solutions, check each other’s work, and then step back into the lines.    When all the partners are done, one line passes their card down and the other line passes it up.   The second line shifts up by one (the person at the head of the line goes to the end of it and everyone moves forward) and the first line shifts down by one.   Now everyone has a new problem and a new partner.   I repeat the process for whatever amount of time I want to allocate to the task.

You can download the  rewriting-equations-card-sort  file by clicking on the highlighted text.



Solving Two-Step Linear Inequalities Quiz Quiz Trade Cards

I created a set of Quiz/Quiz/Trade cards to use with my students to review solving two-step inequalities.     I wanted something engaging and interactive to use on the quiz review day.   I also wanted something that I can reuse from time to time with students who don’t demonstrate full mastery on the concept on the quiz.   I also wanted to be able to reuse the activity with everyone from time to time because making students revisit concepts after some time has passed helps to ensure that they retain the concept.


Quiz/Quiz/Trade is one of my students/ favorite review activities.    When we begin, I give each student a card.   Each student works independently to solve his or her card.   I laminate the cards before using them so that they can use dry erase markers on the cards as they solve them.   After everyone has solved his or her problem, they erase their answer and then walk around the room to find a partner.   The first student quizzes the second by showing him or her the card.   The second student then solves the problem.   If he or she has difficulty, the first student can give him a tip.   If he or she still has difficulty, the first student gives a second tip.   If the second student still has difficulty, the first student tells the second student how to do the problem.   They then reverse roles and repeat the process Finally, they trade cards and find new partners.   (Quiz/Quiz/Trade with Tip/Tip/Tell if needed).   My students love the chance to move around the room and to interact with different peers.  I love it because it works.   Marzano’s research has shown that movement changes brain chemistry and increases levels of engagement.   His research has also shown that the use of cooperative learning structures increase student outcomes.

You can download the solving-inequalities-quizquiztrade-cards file by clicking here.

Angle Card Sets – Exploring Angle Relationships and Solving Equations with 3 Activities and 5 Variations

I wanted to create an activity that I could use to reteach and review angle relationships with some of my students who did not yet show mastery on these concepts.    I decided to create a card set that I can use multiple ways.    Each card has a picture of angles on a point.   They each include adjacent angles, vertical angles, complementary angles, and supplementary angles (with the exception of one card, which does not have complementary angles).   They also each include unknown angle measures.    This allows me to use the cards for five different purposes:  identifying adjacent angles, identifying vertical angles, identifying complementary angles, identifying supplementary angles, and using angle relationships to solve equations.

img_1758angle-relationship-cards  .

I can use the activity multiple ways for each concept.

Activity 1 – Give One, Get One 

Students form two lines.   Each student has a card.  I direct them to do a task (identify adjacent angles, identify vertical angles, identify complementary angles, identify supplementary angles, or solve for the unknown angle value)   Each student completes the task on his or her card and then steps forward.   When both partners have stepped forward, they discuss their respective cards/solutions.   When they are done, they step back into the line.    When all pairs have returned to their starting position, one of the two lines shifts (down or up) by one so that they have a new partner.   The cards are then shifted so that they also have a new problem.   This repeats until the allotted time for the activity is complete (usually about 5 minutes).

For this task, I have students use dry erase markers to identify the angles.

Variation 1 – Find complementary angles

Variation 2 – Find supplementary angles

Variation 3 – Find adjacent angles

Variation 4- Find vertical angles

Variation 5 – Find the missing angle measures

Activity 2 – Quiz, Quiz, Trade

Students solve their card.    They then find a partner.   They ask their partner to solve the card (find the angle pair or solve for the missing angle measure).   If the partner has difficulty, they may give a tip (hint).   If the partner still has difficulty, they may give another tip (hint).   If the partner still needs help, they show the partner how to do the problem.   (Tip-tip-tell).   The second member of the pair then quizzes the first partner with his or her card.   After the pair is done, they find new partners.

Variation 1 – Find complementary angles

Variation 2 – Find supplementary angles

Variation 3 – Find adjacent angles

Variation 4- Find vertical angles

Variation 5 – Find the missing angle measures

Activity 3 – War

Students each turn over a card.   They find the missing angle measure.   This requires them to use angle relationships to write an equation to find the missing angle measure.   They then solve the equation and use the value of the variable to find the missing angle measure.   The student who has the card with the greater angle measure wins the cards in that round.   The student with the most cards when I call time is the winner of the game.


For this card set, I used images from an EngageNY lesson to create the cards.


Which one is the “orangiest” – A lesson exploring comparisons full of open-middle questions

“Which one is the most orangey?”   The question referred to a set of four different recipes for orange juice.    Students were to determine which recipe was the most “orangey” and which was the least.   What does it mean to be “orangey”, though?   That was the question that I didn’t anticipate the first time that I taught this lesson.   It seemed so obvious.   It was to me, but it wasn’t to my students.   It was especially not so for my ELL students.

As my students entered class, they were greeted with table laden with stacks of paper cups and four different bottles filled with lemonade.   I told them that each bottle had been made with a different lemonade recipe.  In front of each bottle was a post-it with a letter:   A, B, C, or D.   I asked them to come up to the table by table groups and taste test a sample from each bottle to help me figure out which tasted the “lemoniest”.


After everyone had completed the taste test,  I asked students to “Vote With Their Feet”.  “Vote With Your Feet” is a high-engagement strategy that utilizes movement to increase levels of engagement.  (It is taken from Marzano’s The Highly Engaged Classroom.)   In the activity, students were directed to go to one corner of the room if they selected lemonade A as the “lemoniest”, another for lemonade B, another for lemonade C, and another one for lemonade D.   Next, they voted on the recipe that was the least “lemony”.   After concluding the vote, the class discussed what it meant to be the “lemoniest” or the least “lemony”.    At the conclusion of the taste test, the vote, and the discussion, everyone in the room understood what it meant to be the “lemoniest”.

After everyone understood the terminology, I introduced the problem context.    (The lesson is a Connected Math investigation.).   Two girls are at camp and are taking their turn helping to prepare breakfast.    They are making orange juice and need to decide which one of the four recipes to use.   Students are asked to analyze the recipes and decide which is the most “orangey” and which is the least “orangey”.  This seems like such a straight-forward question, but it always produces such rich discussion.


I have students answer these two questions using Kagan’s Numbered Heads Together cooperative learning structure.    In this structure, students work independently.   When they have completed the question, they stand up.    When all of the students at the table group are standing, they discuss their responses.    When they have a consensus, they sit down.    At that point, the class debriefs with a whole class discussion.

I selected which students I called upon to share their thinking very intentionally.    As I listened to students discussing their thinking at table groups, I decided which students I would call upon to share and in what order.   I wanted to be sure to draw out misconceptions and I want to draw out multiple solution paths.   I tried call upon someone who had used  ratio reasoning to share his or thinking first.  Some of them compared the ratios two at a time and reason through which is more or less concentrated to draw their conclusions.   Some of them found equivalent ratios with common “denominators”.      If someone had made the mistake of seeing the ratio of concentrate to water as a fraction instead of as a ratio, they were my first choice.   I wanted to draw out the fact that they can compare with ratios but that there is a significant difference between the ratio and the fraction for a given recipe.   By selecting the ratio pathway as my first presenter, someone who used fractions could catch it if someone using ratios refers to them as fractions.    They could point out that concentrate is not part of the water so it has to be a part to part relationship instead of a part to whole relationship.

After the ratio pathway, I selected someone who used fractions to share their solution.   These students usually seemed to go directly to finding equivalent fractions with a common denominator.   If no one had considered the possibility of converting the fraction to a decimal for the purpose of comparison, I posed the question of whether or not there is another way to compare the fractions.   I wanted to draw out the fact that decimals are also an option without that option being something that I offer.   Finally, I drew out the possibility of using percents.    My intention was for students to see the different pathways and in so doing to help students make connections between ratios, fractions, decimals, and percents.

The following question in the investigation asked students whether recipe B  is 5/9 concentrate of 5/14.   I knew that the authors were posing this question to draw forth the part to part vs part to whole comparisons that are possible.    Since I had spent so much time drawing out different solution paths on the fist part of the investigation, this ended up being something that was addressed in passing because it had been so thoroughly hashed out already.

The final two questions in the investigation focused on the idea of scaling the recipes up or down.

First, students were told that each camper would receive 1/2 cup of juice and that there were 240 campers.    Students were asked to determine how many batches they would need to make  and how many cups of concentrate/water they would need for each of the different recipes.    Students used the Numbered Heads Together cooperative learning structure for this question as well.   I expected to see two different solution paths.   Some of the student would see that they need 120 cups of juice.   They would then divide the 120 cups by the number of cups for a given recipe to find the number of batches.    Some of the students would look at a given recipe and determine how many students it would serve.    They would then divide the 240 students by the number of students served by a recipe to find the number of batches.     I tried to find someone  to speak to each of the two different strategies if I could.   Once students had determined the number of batches, most of them could successfully find the amount of concentrate and the amount of water needed to serve 240 campers.

The final question that student addressed asked them to scale each recipe down to serve only a single cup.

After summarizing the main points of the lesson, I asked students to write a reflection comparing and contrasting the final two questions in the investigation to their experience playing one of the puzzles in the Lure of the Labyrinth.   I wanted students to realize exactly what the math behind this puzzle that they had been playing for a month was, and I wanted them to come to that realization on their own.

I love so many things about this lesson.   The context is real and “relatable” for 11 year olds.   The task is so rich.   It is filled with questions that allow for so many different solution paths and so many connections between ratios, fractions, decimals, and percents.  It is so good at exposing and clarifying misconceptions.


Order of Operations Graphic Organizer


I’m teaching Order of Operations to my 6th graders today.   In preparation, I printed out Sarah’s Order of Operations posters .

I also created an advance organizer for my students.    According to Marzano’s research, the use of advance organizers can produce significant gains in student outcomes (see Classroom Instruction That Works ).   I made the organizer a Flow Map (one of the Thinking Maps ).

You can download the foldable by clicking on Order of Operations Foldable

A MarkUp/MarkDown Foldable for My Twice Exceptional Student


Most problems  have multiple solution paths.   Some solution paths are more efficient and some less so, but forcing a particular solution path down a student’s throat denies that student the chance to make sense of it in his or her own way.   I think that part of the power of exploring problems and different solution paths is the sense-making that is inherent therein.   I also think that some of the power lies in the chance to see how someone else thought about the problem, to have the chance to think about it another way.   In thinking about a problem more than one way and trying to make sense of these divergent paths, there is also a moment when one begins to grapple with the idea of the efficiency of one’s solution path.   At that point, after he or she has had the chance to make sense of something, the student can choose a path that is the most efficient for him or her (a supposedly efficient path is not efficient if a student can’t apply it correctly because it doesn’t make sense to him or her).

This idea of allowing students to find their own best path was one of the things in the forefront of my mind as I was putting together a foldable to summarize markups and markdowns this afternoon.    I like to use foldables as my summary for a lesson from time to time because they bring together a lot of the ideas addressed in class into a single coherent document that students can use to study.   Additionally, I have a twice-exceptional student who has difficulty with the physical act of writing.   Giving him a foldable to complete makes it possible for him to be more successful.

In planning the design of the foldable, I wanted something that would give a whole/part/whole picture of the concept.   This is really important to me because one of the students who will be using it this year is a twice-exceptional student who is pretty extreme on the conceptual end of the conceptual vs sequential continuum of thinking.   He needs to see the whole of a concept to make sense of it and gets lost in the parts if he doesn’t see the whole first  In order to give him the “whole”, I decided to start with the big idea.   Next, I decided to compare/contrast markups and markdowns at each stage in the foldable.   The use of this compare/contrast mechanism seems to really help him to make sense of things.   Marzano’s research in  Classroom Instruction That Works shows that it can have a big impact for all students (27% gain).   Finally, I included two different solution methods for a markup and a markdown.   These are the solution paths that I am anticipating students will have taken as they explored the problems.


MarkUp and MarkDown Foldable

Building Academic Vocabulary Review Games Part Two

I like to use games to review concepts and skills.     They are highly engaging (as documented in Marzano’s The Highly Engaged Classroom ).   They promote student discourse.   As students play, they are constantly evaluating opposing players’ work because they are competitive and want to win.   They are generally fast-paced and can build a lot of review into a little bit of time.

In my last post ,  I shared a set of cards and directions for using them to play several different games (Draw Me and a couple of versions of Charades).    In order to add a little more variety to my vocabulary review, I have also created a version of Taboo.   Students play in table groups.   A “talker” is designated (I will do this by seat position within the group).   The talker tries to get the team to say the name of the word at the top of the card.   However, the talker can not use any of the words below the line.   You can download the set of cards by clicking Taboo


Table groups will compete against each other to identify the most vocabulary words.    I will have them  play for a specified amount of time.   The group that identifies the most words in the game will be the winner (and will each get a piece of candy).