Overheard in Math Class

Students were working on solving a system of equations using a graph.  The problem context involved adults and students who had purchased tickets to a performance.   They knew the ticket price for each and the total amount of income.   They also knew the total number of tickets sold.    Students had to write the system of equations and then solve it graphically.

Student 1:   Should the graph be continuous?

Student 2:   No, you can’t have half a person.

Student 3:   Can it go into the negatives?

Student 1:  No, you can’t have negative people.

Student 4:   Yes you can.   Not everyone is an optimist.

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.


Changing Forms of Linear Equations INB pages

My seventh grade classes started this semester with a unit solving systems of equations and inequalities.    While they have a good grasp of linear equations in slope intercept form, I am less confident in what some of them will do when presented with equations in other forms.    Since so much of our work with systems of equations will depend on the ability to work with linear equations in many forms, I am planning to spend some time early in the unit ensuring that everyone can successfully transform equations between various forms.

Today, they worked through a Connected Math lab exploring how to change between forms.   I really like the way Connected Math introduces this topic.   Students are presented with four pieces of student work in which someone has rewritten an equation in a different form that they must analyze.  Students have to determine which pieces of student work are correct, justify the steps taken for those that are correct, and identify the errors for those that are incorrect.   Each of the pieces of student work tackles the task in a different way, so students have to really think about things a little more deeply.   The lesson follows up with a task in which they move equations in slope intercept form into standard form and equations in standard form into slope intercept form.

Tomorrow, I am going to spend a second day on the concept before moving onto the next lesson in the investigation.   I want to take a little more time to solidify the concept.  I I plan to use these Interactive Notebook pages with my students to practice moving between different forms of equations

You can download the   changing-forms files by clicking on the highlighted text.

I will also have them practice with a game.     I will write about that tomorrow after I see how it goes.


A Matter of Belief

What???   That was my first reaction to Sam Shah’s headline revealing this week’s prompt for the MTBosBlogsplosion.   I had absolutely no idea what he was talking about.   Please bear with me on this.   I’m an engineer.   My undergraduate degree requirements were so tightly defined by the university that I could only take three courses in humanities and there were some pretty tight restrictions on what they could be.   That translated to one sociology course and two philosophy courses.   I only took the philosophy courses because I couldn’t take what I really wanted and word on the street was that they were easy As.   My graduate degree in engineering had no humanities.   So, soft skills?  I don’t think I could possibly face a more difficult prompt.    After a week of serious thought, I decided my only hope was to tackle this sideways instead of head on.

Earlier this week, I contacted a parent because I wanted to nominate her daughter for a STEM camp.   I received this in return.

“Thank you for seeing this talent in …. . She’s not been very confident in her abilities to the point that she came home once several years ago saying that she wasn’t good in math… I could not believe my ears and I am glad she persevered and she was so fortunate to have you as her teacher.”

I find the fact that this young woman ever could have thought that she was not good at math absolutely stunning and I’m not really sure what I did to change that.   I didn’t do anything special.   There were no long pep talks.  There was no special “cheerleading”.   There was just normal math class.

After some thought, I think there might be a few things in normal math class that make a little bit of difference for her (and for other girls as well).

  1. I genuinely believe that she is good at math.    (I believe this of all my students.    Everyone can be good at math)   I don’t know if this mattered, but I find that having someone believe in me can be very powerful.
  2. Because I believe she is good at math, there are no softball questions.   I ask her hard questions and she rises to them every time.    That doesn’t mean she gets every one of them right, but she thinks deeply about every one of them and has learned that she can think deeply.
  3. I am lucky enough to use a really good, research-based curriculum (Connected Math).  It’s not a perfect curriculum (nothing is), but it is designed to allow students to construct meaning.   Students leave with a fairly good conceptual understanding of the math rather than a bunch of rules that they blindly follow.   Having the chance to make sense of something means she gets to make it her own, to know that she knows it, to believe in her abilities.
  4. I use Kagan Cooperative Learning Structures a lot.    These structures provide this beautiful interdependence that helps everyone rise.   Students start out working independently (so they have to figure things out, not just ride on someone else’s coat tails).   Then, they work with one or more classmate (depending on the structure), discussing their work and coming to a consensus (the mathematical discourse is so much richer since I started using these structures).   They have a stake in each others’ success because they don’t know who will speak for the group.   I think this interdependence is good for all students, but girls seem to really thrive on it.
  5. I make time for regular review (5-10 minutes every day) of previously learned concepts.   This delayed recall helps students to retain concepts that they’ve learned.   I also use this time to differentiate instruction.   Sometimes students are assigned to work on a station based on their mastery level.   Sometimes students are paired so that someone with mastery is working with someone who is still working on mastery (providing an opportunity for some extra kid to kid discussion that helps to scaffold the concept).   The groupings are ever changing so every student knows the reality that everyone is good at some things and everyone has some things on which they still need to work.   I think knowing that no one is good at everything helps kids see that not being perfect at everything does not equate to not being good at math.
  6. Both of the years that I have had this young woman in my math class, she has been in a class dominated by girls.    It is just a strange fluke in the way that schedules were run, but both years I have had a period in which 3/4 of the class was girls and a period in which all of the class was boys .or in which there was only one girl.    (I try to get a schedule change for the girl when that happens so that she is not alone).   There is a lot of research out there that shows that girls in math classes do better when at least 40% of the population is female because they don’t have to expend energy fighting a stereotype.   My anecdotal evidence is that both years that I have had these girl-dominant classes, the girls have grown mathematically in degrees that I have never seen in any of my other classes.   It’s not that these classes are filled with perfect, compliant students.   It’s more that these classes are filled with strong, smart girls who interact differently than they do when there aren’t so many girls.  .

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.

Minions in Math – My Favorite Thing

Miniature robots zoomed across the floor, lights flashing.    Students bent over computer screens, writing code.      Parents sent emails telling me that their kids couldn’t stop talking about the Ozobots.      My students were reviewing linear relationships, being introduced to systems of equations, and learning some coding skills.   It was the best way I could have hoped to end the semester with my 7th graders.

My students came into this project with a lot of background knowledge.    This is their second year with me.   Last year, I introduced them to some basic coding by having them create video games using SCRATCH.   The Ozobots use Ozoblockly, which is very similar in structure to SCRATCH.   (Both languages are object-oriented visual programming language.  I think this is absolutely the best way to introduce coding.   They allow the user to focus on the big ideas of coding without getting bogged down in the syntax of the language.)    The added challenge this year is having the code control a miniature robot.   This required students to work through a few challenges that had nothing to do with coding, but everything to do with problem solving and making things work.   I documented the journey on my twitter account .   There are videos there showing each step of the projects   (posts dated Dec 8 – 15).  Unfortunately, you will have to go to the twitter account to see the videos if  are interested, because my blog doesn’t support videos.

Building Background Knowledge

InitiallIy, I focused on building knowledge.   Students had to learn how to calibrate the bots, how to write code, and how to load the code onto the bots.


Next, they worked on getting the bots to do what they wanted.   They explored the different capabilities:   getting the bot to move in specific ways, playing with the light features, etc.    During this early phase, I had students work in pairs.   Because there is a certain amount of frustration inherent in the process of figuring out how to make things work, it helps to have someone to share the journey.


The Project

Once I was confident that everyone had a little experience, I assigned a two-part project that would require them to review the work we had done earlier in the year with linear equations and also act as a launch into the work solving systems of linear equations that we will do at the start of the spring semester.

Linear Relationships in Tables and Graphs

For the first part of the project, I assigned each pair of students a different linear relationship.     Each pair was given two ordered pairs, an ordered pair and the y-intercept, or an ordered pair and the slope.    They had to find the linear equation representing their relationship and create a table for the relationship.   Then, they had to create a large scale coordinate plane and write code so that their Ozobot would graph the linear relationship.   The Ozobot had to flash lights on three ordered pairs and perform a different action at the y-intercept (flash different lights or spin on the spot).


Solving a System of Equations Graphically

Next, each pair had to use a second Ozobot to graph a second line intersecting the first line on their coordinate axis.    They had to calculate the point of intersection and time the two Ozobots so that they would not collide at the point of intersection.

While there was plenty of math in the project, the most important piece of math was not what one usually thinks of as math.   It was learning how to make something real actually work.    Students had to learn how to frame a task and then to persevere in the face of challenge (one of the math practice standards).

My Favorite Thing

My biggest measure of success is that the kids dinner time conversations were all about math for the entirety of the project (a number of them ended up getting their own bots for Christmas as a result).