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.

Constructing an Understanding of Integer Addition

Making sense of integer addition and subtraction is hard.   The algorithms are complicated and hard to remember.   On their face the results sometimes seem counter intuitive.   Sometimes the answer is positive, sometimes it is negative.   Sometimes the answer gets “bigger” when you subtract and sometimes it gets “smaller” when you add.   Because of all this messiness, students have a really hard time knowing whether an answer makes sense if they haven’t had the chance to build some conceptual understanding before jumping to an algorithm.

I start helping my students make sense of the process using a chip model.    They use black chips to represent positive numbers and red chips to represent negative numbers.    They combine the chips, making zero pairs (one red and one black chip) to find the sum.    I start with the chip model because most students think of addition as combining sets of objects.


After students have mastered the chip model, I move on to a number line model.   The curriculum that I use (Connected Math) does a really nice job introducing the number line model.    It talks about addition as combining sets by providing a context in which two kids each have a number of video games and then talking about the combined number of video games.    It illustrates the problem using a chip model.   Then, it goes on to talk about addition also being representative of a context in which one “adds on”.   It provides a context in which there is a temperature at sunrise and that is “added on” to as the temperature rises over the course of a day.    They model this problem using a number line.

After introducing students to the number line model, I take them out into the hall to walk the number line.   Prior to this, I have had my student aides create a number line for each table group.    Each number line is created using painters tape (for ease of removal when the time comes).   The numbers on each number line range from -10 to 10.    I begin by modeling a couple of integer addition problems on the line.    I walk forward for positive numbers and backward for negative numbers.

After modeling several problems, I have each group complete a set of problems by walking on the number line.    Each group gets a laminated index card with the problems to be completed and a dry erase marker to record their answer.    The first group member walks the number line to solve the first problem.    The other members of the group check his or her work.   The next person in the group walks the number line for the second problem and the others check the work.    Each problem is completed by a new student until everyone in the group has walked the number line.   At that point, the group rotates through again until all the problems  have been completed.

Once students have mastered integer addition on the number line, they use models to construct an algorithm for integer addition.    They begin with a group of four problems.   They solve these problems using a model.    Then, they identify what is the same about the problems and create two more problems to fit the group.   (All of the problems in the group have addends in which the signs match.)    Finally, they come up with an algorithm for the adding problems within the group.   They repeat this process for a second group of problems in which the signs of the addend pairs do not match.    If students have trouble figuring out the algorithm, I remind them that each number has a magnitude (size) and a direction (sign) and suggest that they consider the two parts separately.   This usually helps them get to the algorithm.

After creating the algorithm, students test it on rational numbers (mixed numbers) and verify its efficacy using a number line model.    Finally, they test rational number addition for commutativity.

After students have created an algorithm, I use the foldable shown below to summarize the lesson.

I follow up with this Always, Sometimes, Never exit card.






Every Graph Tells a Story – A Lesson Interpreting Graphs

Every graph tells a story, but can you “read” that story?    Do you see the beginning, middle and end (if there is an end)?   Do you see how the story builds and changes?   These questions are at the heart of a unit I teach my 6th grade students.   (The unit is a Connected Math unit called Variables and Patterns).  They were also at the heart of a lesson I taught recently in which student were asked to interpret a graph, to read its story.

To begin, I wanted students to zero in on the key elements in a graph that reveal the story but I didn’t want to tell them too much.    I decided to try using a chalk talk, which is a technique in a book I have been reading with my PLC,  Making Thinking Visible .   In the chalk talk, students respond to a prompt on a large sheet of paper.    Students “talk” about the prompt by writing on the paper.   This allows them to respond to the prompt and also to the responses that other students have written, hence making thinking visible.     I decided that my prompt would be each of the six graphs that my students would need to interpret.

I drew each of the six  graphs and posted them around the room, each hanging above a blank piece of poster board.  As students entered, I assigned them to a specific graph station. I asked them to look at the graph and write what they noticed about the graph.    After everyone had completed the task, I had them rotate to another graph station.    This time, they could either write about the graph or respond to what someone else had written.

The six graphs were large scale versions of the graphs shown below.


In this phase of the lesson, I was hoping that students would look for markers that tell the story of the graph:   what is the initial value, how does the graph change (e.g., is it a constant rate of change or a variable rate of change),   is the graph increasing or decreasing in value,   does the pattern in the graph repeat,   is there an “end”.   I wanted to draw out student thinking on these ideas as a launch into the lesson.   Since this was the first time that I was using a chalk talk, I didn’t really know what I would get, though.   It was a new process for my students, so they wouldn’t have the benefit of experience with the process to guide them and I wasn’t sure if I was presenting the task well since it was my first attempt with it.   As I looked at their work, I could see the evidence of our inexperience.   The responses varied widely   Some students gave responses  along the lines that I expected.   Some students made up a story for the graph.  A few students responded to other students’ thinking, but most used the thinking of other students to create their own elaborate stories.   Since I got results I hadn’t anticipated, the debrief of the task was a little different than I expected and didn’t really focus on some of the things I had hoped it would.   I didn’t try to force it though because I wanted to give students the chance to make meaning rather than me giving them meaning.  I liked the technique as a launch, I just need to pose the prompt a little better next time.

After the debrief of the chalk talk, I had students work in table groups to match the same set of six graphs to a set of seven different stories.    I gave them  graphs and stories on a set of cards, so it was essentially a card sort.   As students worked in their groups, I told them that they must work silently.   One person would match a story to a graph.   The next person could either match another story to a graph or could change one of the matches on the floor.     I did this to ensure equity of voice in the group, the silence and turn taking ensured that no one could dominate the group and that each must contribute.    I also did this to encourage students to analyze the choices that other students were making.  The card sort used the graphs and stories shown below.

After each group had “completed” the matches, I had the groups do a gallery walk.   During the gallery walk, they went around to each of the other groups and considered the choices that the other groups had made as they matched the stories to the graphs.    During the gallery walk, they could discuss what they saw with their group members but could not make changes to the work that they saw.   At the conclusion of the gallery walk, each group was given time to discuss the choices they had made with their own card sort and make any changes they wished to their group’s card sort.

Finally, we debriefed  the card sort. We returned to the large scale graphs on the walls that we had used for the chalk talk.    I read one of the stories and asked one of the groups to share which graph they believed it matched.   They had to identify the independent and dependent variables and explain how the changes in the dependent variable reflected the story.  They also had to give the graph a title.   Other groups then had the opportunity to comment or question the group’s response.

One of the things that I really like about this lesson is that students grapple with hard questions.   There is the potential for them to wrestle with misconceptions.  As they discuss the graphs and stories, their misconceptions are exposed.   For example, someone always suggests that one of the parabolas is the amount of daylight over the course of time.   Someone always then raises the question of what it means for the amount of daylight to be zero (either initially or in the middle of the graph).   At that point, there is always this lovely “oh!” moment in which students realign their thinking.

Next year, I do want to make a couple of changes to the way I did this.   First, I will re-work my directions for the chalk talk.   In addition to the graph, I will write a prompt asking them to respond to specific components of the graph (initial values,   how the dependent variable changes as the independent value increases, etc).   Second, I will change the way that I do the card sort a little bit.   I think I will give each group a set of dry erase markers and require them to add labels and a title to each of the graph card as they match it to the story.   I am curious whether seeing those things written on the graph will help them see their misconceptions before the whole class debrief.


Identical Triangles Card Sort

Identifying identical triangles was a little bit of a challenge for a few of my 7th grade students.   They could correctly identify a property (Angle, Angle, Side), but then would completely disregard the fact that the sides in the two triangles were not corresponding.    I want to revisit the concept with these students as a quick review activity even though our lessons have moved on to other topics.   To do so, I created a card sort.

In the card sort, each card has a pair of triangles.    Students will sort the cards into categories:   “Identical”, “Not Identical”, or “Not Enough Information To Tell”.   If the triangles are identical, they will be required to identify the property shown.   In order to address the fact that students are not looking to see if the sides are corresponding, I will probably require them to use colored dry erase markers to indicate corresponding sides.



I will pair students who did not master the concept with students who did.   I think I will implement a strategy that I read about on another blog (I don’t remember which one, only that it was a great idea), in which students within the pairing are required to take turns.   The first partner sorts a card into a category.   The second partner then either agrees or disagrees with the sorting.   If he or she disagrees, he or she can move it but must explain why it is being moved.   The second partner then sorts another card and the first partner can agree or disagree.


Making Graphs Anecdotal Records Form

Knowing what my students know is important to me, but remembering who is struggling with what aspect of a given concept from day to day is impossible.   There are just too many students and too many variations.   I absolutely have to write down what they know if I want to act upon it the next day.    Unfortunately, there isn’t a lot of time in a given class period to take those notes.   As I result, I have started creating anecdotal record forms that are specific to a big idea that I am addressing in class.    On these anecdotal records, I have a single box for each kid.   In each box, I have specific items of which I want to ensure mastery.  I use one sheet for each period.   I put a different student’s initials in each box.  Then, I make copies of the sheets so that I have enough to use for more than one day.   Alternatively, I use a different color pencil each day.  I keep them on a clipboard for ease of use.   As I circulate around the room, I circle an item in a student’s box if it is an area that I need to address with him or her.   The next day, I simply look at the sheet from the previous day and I know exactly what I need to address with specific students.

I have just started a unit in which students will be representing and analyzing data in tables and graphs.   During this unit, I will be using the anecdotal records shown below.

I use the” Ind/Dep” category to indicate whether students are correctly selecting the correct axis for the variables (independent variable on x, dependent variable on y).   The “Intervals” category indicates whether the student is maintaining uniform intervals on the axis (a very common error in the early days of making graphs).   The “Continuous/Discrete” category indicates whether a student is correctly determining whether or not to connect the points on the graph.   The “Plot points” category indicates whether a student is correctly plotting points (x,y vs y,x).  The remaining categories are more minor errors, but errors that I want students to clean up.

You can download the form by clicking on the link below the photo.




How far can you ride in a day? – A Lesson Introducing How to Make A Line Graph

What twelve year old wouldn’t want to spend a few days bicycling along the ocean front, spending their days amid sunshine and ocean breezes and their nights under star-filled skies?   Along the way, they get to swing through Cape May, a lovely ocean-side town filled with beautiful Victorian buildings, visit Chincoteague Island to see the annual auction of wild ponies who swim to the island from Assateague Island, and swim in the ocean.   This is the context for my students’ exploration of different ways to represent  and analyze data  (tables, graphs,  and eventually equations).   The series of (Connected Math) lessons center around a set of college students who are setting up a summer bicycle tour business to earn money for school.    In the series of lessons, they explore the question of how long each day’s ride should be, whether the planned route is feasible (they test out the route and collect data), where they should rent bicycles for the tour, finding the perfect price point to maximize their income, how long the drive back from the final destination will take at various different driving speeds, and the cost of taking the tour participants on a side outing to an amusement park.

Yesterday, I started the unit by introducing the problem  context.   I began by showing a short video clip of someone on a bicycle tour through Great Britain.    I chose to begin with a video clip in order to support my English Language Learners and students from lower socio-economic households, in order to bridge language and economic divides that might make the problem context difficult to grasp.   By seeing a bit of a bicycle tour, they would have better access to the problem context.

After students watched the video clip, I introduced the problem – five college students setting up a summer bicycle tour business.    The first question the college students were considering was how long each day’s ride should be.   I asked my students what they thought would be reasonable.   This led into a nice discussion of some of the factors that might influence the answer to that question – the relative incline (uphill, downhill, flat), the terrain (pavement vs sand or gravel), the weather (riding into a wind, no wind, wind at one’s back).   I then asked them if they thought it was reasonable to expect the riders to maintain a constant rate for the entire day.    Some students thought not, but some students thought the riders could pace themselves.    This laid the groundwork for the first part of the lesson.

In order to explore the idea of pacing, I had students conduct a jumping jack experiment.   At each table group, one member of the group would perform jumping jacks for 2 minutes.   Another member of the group would be the timekeeper, marking the time in 10 second intervals.   Another member of the group would be the counter, counting the jumping jacks.   The final member of the group would be the recorder.   (I assigned tasks to group members by their seat position within the group.   If a group had only three members, I had the jumper also do the counting.)   In order to speed the process, I provided the recorder with a pre-made table so that he or she would  spend less time copying out a table and could instead just fill in the table entries.


At the conclusion of the experiment, I had groups collect data from their recorder and complete their own data tables.  Then, I asked students to describe what happened to the rate of jumping jacks as time progressed.   There were a few groups throughout the day that maintained a fairly steady pace, but most groups experienced a steady decline as time passed.   Some groups had a jumper who stopped completely part way through the experiment and then resumed their jumping after a short break.   As the class discussed this, I asked them how they saw these changes in the tables that they had created.  This gave them the chance to  explore the idea of how a change in the jumping jacks compared to a constant change in time. (I had not yet introduced dependent/independent variables.)  At this point, my goal was to begin to tie this lesson back to work they had done with ratio tables in a previous unit.   I wanted them to see that this was not in fact a ratio table because the rate was not constant.

I told students that I wanted them to look at the trends of the jumping jack data in a graph.   In order to do that, they needed to learn how to correctly make a graph.   I began by introducing the concept of independent and dependent variables in a table and talking to them about the mathematical conventions.   Then, I introduced the process for translating data into a line graph using a Flow Map (this is a Thinking Map used for sequential processes).   I provided students with the  advance organizer to complete their notes on creating a graph.  I had a regular version of the organizer and a modified version of the organizer (that is more of cloze activity) to support students with learning disabilities.





After students completed the Flow Map, I had them use the data from the jumping jack experiment to make a graph.   As they worked, I circulated among them, taking anecdotal records on their work.   As they finished, I selected graphs to share with the class.   I was very intentional in selecting graphs with errors.   I then asked the class to examine the graph to see if there were any errors.   This forced the students to think more deeply about the work they had been doing.   When someone found an error, I gave a piece of candy to both the student who allowed us to look at his or her work and to the student who found the error.   I explained that the person who let us see the mistake did as much to help us grow mathematically as the person who found the error did.   (This is a common practice in my class).   I made sure to share multiple graphs with errors in each period to ensure that no one student felt like he or she was the only one still learning how to do this.

After analyzing several graphs, I asked the students to explain how they saw the rate of jumping jacks changing in the graph.   Here, I was laying the ground work for upcoming lessons in which they will be analyzing data in graphs.

After summarizing the lesson, I had students complete an exit card in which they had to find the error in a graph.   After some thought, some of the students were able to see that the independent and dependent variables were on the wrong axis.   For those students who were having difficulty, I told them to go back to their Flow Map and work through each step to see if they could find the error.   Eventually, everyone successfully completed the exit card.


Following the lesson, I posted an anchor chart on the wall of the classroom that corresponds to the Flow Map that they used in their notes.



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.