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.



Ratio 3-2-1 Exit Card – Exposing Misconceptions and Engaging Metacognition

The expectation regarding student depth of understanding of equivalent ratios in middle school  has risen significantly since the advent of the Common Core.   A few years ago, students just scaled up or scaled down ratios to find equivalent ratios.   Now, they are expected to utilize ratio tables, tape diagrams, and double number lines to solve fairly complex problems.   I think, though, that oftentimes students don’t see the connections between these ideas.   I think that a lot of the time, they see them as completely disparate concepts, just one more thing to know about ratios.    As we wrapped up today’s lesson on double number lines (the last of the three representations for my class), I wanted to explore this idea.   Do they see the big picture, that these are all different ways to represent equivalent ratios and that the whole point of the representation is to visualize things so that one can solve problems?    I gave my students this exit ticket to find out a little bit more about their thinking.    img_1772


Their responses were illuminating, as always.    Student responses to the first question seemed to fall into two groups: those who saw the big picture (like the one shown) and those who responded with three ways to write a ratio.   I think those who were mistaken on this question zeroed in on the word three and went straight for the ways to write a ratio without closely reading the question.   However, I also think that if they had made the connections between the representations, they would not have honed on that word.   So, I have some more work to do.

Student responses to the second and third question showed that most of them are still figuring out how to think about their thinking.    I am reading Making Thinking Visible  and am hoping to gain some insights into how to develop stronger metacognitio in my students.  .

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.