# Proportional Relationship Anecdotal Records

You just can’t hurry some things.   They take time.    In middle school, grasping the big ideas of proportional relationships seems to be one of those things.    Students need to build an understanding of a constant rate of change and how it presents in tables, graphs, equations, and verbal representations.   They need to build an understanding of what it means to have no “start-up” value in each of these representations as well.   They need to explore these ideas in a little of different ways in order to make sense of it.

Because there are so many pieces to this idea, there are a lot of different places that understanding can break down.   In order to know where that breakdown is happening for each student, I like to take anecdotal records.   At the end of the day, I can go back and see who knows what and make decisions about what I need to do the next day to help each of them move forward.   The thing is, I don’t have time to write a paragraph about each kid as I walk around the class looking at their work and listening to their discussion.    I need something that just takes a second or two for each kid at any given time.    To that end, I decided to create an anecdotal record form specific to proportional relationships.

I started with what I want to know my students can do.

• Recognize a proportional relationship in a table
• Recognize a proportional relationship in a graph
• Recognize a proportional relationship in an equation
• Recognize a proportional relationship in a verbal representation (word problem)
• Be able to connect proportional relationships represented in a table and a graph.
• Be able to connect proportional relationships represented in a table and equation.
• Be able to connect proportional relationships represented in a table and word problem
• Be able to connect proportional relationships represented in a graph and equation
• Be able to connect proportional relationships represented in a graph and a word problem

If students can’t identify a proportional relationship in a given representation, I also want to know if the break down is the rate of change or the y-intercept.

I came up with this Proportional Relationships Anecdotal Records form.    I have one box for each student (I can use more than one page for a given period).   I put the student’s initials in the box and then circle the place where a breakdown is happening.   If the student is not recognizing a proportional relationship, I can write in k if the breakdown is the rate of change or the letter b if the breakdown is at the y-intercept.

I will probably use this to drive a brief review activity the following day.    I will pair students to do a card sort or to use the cards to play rummy.   Initially, I will pair them so that a student who has the idea down is working with a student who doesn’t.  After working with the idea for several days, I may place them so that kids who are struggling with the same issue are working together.     At that point, I can work with that small group to address whatever disconnect is still in place.   You can get the card sort here.

# Accommodating Imperfection – Proportional Relationships Cards with Multiple Variations For Play

Successful design is not the achievement of perfection but the minimization and accommodation of imperfection. – Henry Petroski

I know that I will not ever design the perfect lesson any more than I will ever create the perfect design as an engineer.  I’m not sure that perfect exists in this world.   While Dr Petroski (a civil engineering and history professor at Duke University) focused his work on failure analysis in an engineering context, the underlying principle he espouses applies to my work as an educator as well.    On any given day, I know that some students will walk away  not having fully mastered the concepts addressed in class.   So, I plan ways to revisit concepts in small chunks of time until everyone does “get it.”  When I plan ways to revisit concepts, I try to create activities that I can use a lot of different ways because I want to be able to use them more than once rather than having to create an endless array of materials.

Proportional Relationships – What I Want Them To Know

I want students to know that proportional relationships are linear and go through the origin.

Proportional Relationships – What I Want Them To Be Able To Do

I want to ensure that students see proportional relationships in tables, graphs, equations, and word problems.   In each representation, I want them to see the constant rate of change and that there is no “start-up” value (the y-intercept is 0).     I want to incorporate Trail’s work (Twice-Exceptional Gifted Children) to support conceptual learners, so I envision a whole-part-whole instructional sequence:    what is a proportional relationship; how do you see it in a table, how do you see it in a graph, how do you see it in an equation, how do you see it in a word problem;   how is the constant rate of change shown in each representation, how is the “no-start up” shown in each representation (compare and contrast these in the different representations).

Proportional Relationship Cards

I created a set of 48 cards.   There are twelve cards for each of the four representations.   The cards can be used separately or together.   That is, I can use just the cards relating to a single representation if I want to focus on that representation.   Alternatively, I can use cards from all four representations if I want students to make connections across representations.   You can download the cards and game instructions by clicking on the link below the photos.

Proportional Relationship Card Sort and Game

Quiz-Quiz-Trade is a Kagan Cooperative Learning Structure.   In this structure, students partner and quiz each other.   Then, they find a new partner and repeat the process.   Marzano’s research (Classroom Instruction That Works) shows that using cooperative learning structures produces gains of 27%    It also shows that incorporating movement increases levels of engagement (The Highly Engaged Classroom).

Activity Two – Give One, Get One

Like Quiz-Quiz-Trade, students work with a partner and quiz each other.   I will use this activity when I want students to make connections across representations.   It incorporates movement and a cooperative learning structure.   It is outlined in Marzano’s The Highly Engaged Classroom.   I have students form two lines facing each other with about three feet in between the lines (the structure does not specify this, but I find it works well this way).   Each student is given a card.   I will give one line a single representation and the other line a different representation.   The cards for partners will be different representations of the same problem.   Each partner will have to find the rate of change and the y-intercept using their card.   When a partner has found them, he or she steps forward.   When both partners are in the middle, they quiz each other on what the rate of change is/how it is shown on their card and on the y-intercept/how it is shown in their card.  When they are done, they step back into the line.   When all the pairs are done,  have one line pass their card to the next person and the other line shift (line one passes the card down one, line two shifts up one).

This is an activity that I do for 5 minutes at the end or start of class.   I end it based on time.   I don’t try to have every student do every problem.

Activity Three – Rummy

Students play in groups of 2 to 4 players.   They use the entire set of 48 cards to match the table, graph, equation, and word problem.

This is an activity I will use so that students compare and contrast the different representations.   Marzano’s research (Classroom Instruction That Works) shows that finding similarities and differences can produce gains of 45%.   His research (The Highly Engaged Classroom) also shows that using a game increases levels of engagement.

I may have students play the game in heterogenous groups as a general review activity.

I may use this as a differentiated instruction activity.  I will have students play the game in groups according to their level of mastery.   Students who have not attained mastery play the game.   Students who have attained mastery play a different game reflective of their own skill gaps.

I will have students play the game for 5-10 minutes at the end of class.   If the game is not over, the player with the most sets wins the game (and a piece of candy)

Activity Four – Card Sort with Three Variations

Students work individually, in pairs, or in table groups to sort the cards.   In the first variation, they work with a single representation and sort them into proportional/not proportional categories.   In the second variation, they work with a mixture of representations to sort them into proportional/not proportional categories.   In the third variation, they work with a mixed set of representations and find the matching cards (same situation represented in a table, graph, equation, word problem).

I may have students work in table groups as a general review activity.

I may have students work with intentional pairing.   In this scenario, I pair a student who is struggling with the concept with a student who has mastered the concept.    As they sort the cards, the discussion is scaffolded for the student who has not yet attained mastery.

I may have students work individually and use this as a formative assessment.

# Timber! DI Decimal Division Jenga

When I first started teaching middle school, I guess you could say I was a generalist.  I tended to see my students as a single body.  I thought about what they knew or didn’t know as a group.   I would look at mastery levels on tests and then decide that I needed to do some reteaching on specific concepts based on the performance of the whole group.

Over the last few years, I have shifted my focus to the specific.   Now, I track each student’s performance on each of the standards or skills that I address in my course.  I use an excel spreadsheet with the various standards/skills as the column headings and the kids (grouped by period) as the row headings.   At the start of the year, I give a pre-assessment for the course so that I know where everyone is starting.   After each quiz or test, I update levels of mastery in the spreadsheet based on each student’s demonstrated mastery.   I use this data on a daily basis to ensure that each student is working on things at his or her level.    My goal is to have every single student with full mastery by the end of the year.   I don’t always completely meet this goal, but I come a lot closer to it than I did before I started doing this.

As I said in yesterday’s post, I use the last five to ten minutes of class every day for review and reteaching.   I try to make the review focused and fun.   Since it is a review/reteach, I have already taught the concept conceptually.   Sometimes, the review continues to be conceptual.   Sometimes, it  is just working with a skill.

One of the activities I use is Decimal Division Jenga.  I like the use of a game to practice.   Everyone knows a game is more fun than a worksheet and Marzano’s research supports this in The Highly Engaged Classroom.

I have decimal division problems on Jenga blocks.   Students pull blocks from the Jenga tower and have to do the decimal division problem.   They check their work with a calculator.   If they got the problem correct, they keep the block.   If they made a mistake, they put the block back on top of the tower.   If a player topples the tower, he or she must put all of his or her blocks back and rebuild the tower (the other players keep their blocks).   Their are several free blocks in the tower.   No player is allowed to have more than 3 free blocks.   If he or she does, the free block must be replaced and another block drawn.   The player with the most blocks when I call time is the winner and receives a piece of candy.

Decimal Division Jenga

I have three versions of this game.   The first version has problems in which a decimal is divided by a whole number.   The second version of the game has problems in which a decimal is divided by a decimal.   The third version of the game has problems in which a decimal is divided by a decimal resulting in a solution with zeros in the quotient.   I assign students to the appropriate version of the game based on their pre-assessment results initially.   I circulate between the games and monitor their work.   I work with students individually on their specific errors.   As students demonstrate sufficient levels of mastery over time, I shift them to a more difficult version of the game.   Eventually, everyone is playing only the most difficult level of the game.  You can download the file with the problems for all three levels by clicking on the link below the photo.

I let students know which version of the game they are playing using an index card system. I have an index card for each student with his or her name on it.   I write the name of the game they are playing in colored ink.   The color of the ink matches a piece of paper under the Jenga tower.   They match the color on their card to the color of the paper and go to the correct game pretty seamlessly.

# Not a Moment To Spare

A few years ago, I read Steven Leinwand’s Accessible Mathematics – 10 Instructional Shifts That Raise Student Achievement.     The idea that small instructional shifts can have a big outcome really resonated with me.   My first reaction was that this is all so “do-able”. I found myself checking them off – I could do this and this and this to incorporate these shifts.

I picked the book up again this weekend.   I wanted to revisit and check my progress.   Exactly how was I doing in bringing about these changes?   I thought I would take each of the ten shifts and look at what I have done or haven’t done to make that a part of my regular instruction.

The first shift is to incorporate ongoing cumulative review into every day’s lesson.

I started tackling this shift by building specific time into my lesson for review every day.      I decided that I wanted to do a very quick review activity at the start of class to get students engaged as soon as they walked in the door.  I also wanted a slightly larger chunk of time to review ideas that take more than a minute or two.

I call my opening review the Math Minute because I wanted it to last only a minute or two.   I could use this time to do quick review on number operations skills (fraction, decimal, and rational number operations), order of operations practice, and estimation. These are usually problems that I post on the Promethean Board.

I call my larger review activity the Flashback.   I have it as the final 5 -10 minutes of the period.  (This is after I do the lesson summary.   Originally, I had it before the summary, but found that sometimes that meant the summary was too hurried.   I want to have a solid summary to make sure that everyone walks away with the main ideas of the day so I switched the order.   It’s easier to cut off the review early if I need to do so).I use the Flashback to review more time-consuming concepts.   When I first started this a few years ago, this was a whole class activity.   Over the last few years, though, I have begun to also use it as a chunk of time to differentiate the review to fill gaps in learning.

• Sometimes, I use tiered instruction where students progress through a series of tasks or games as they demonstrate mastery.   An example of this would be the decimal division game that I use early in the year.   I have three versions of a Jenga game.   The simplest requires students to divide a decimal by a whole number.   The second version of the game requires students to divide a decimal by a decimal.   The final version of the game requires students to divide a decimal by a decimal but there are zeros in the quotient.
• Sometimes, I use intentional pairing where I pair a student with mastery with a student without mastery to work on a problem together.  An example of this would be having students work on a more complicated word problem in which they need to make a double number line to solve a ratio problem.
• Sometimes I use a series of games that address different concepts, with each student playing a game that corresponds to a gap or weakness he or she has demonstrated. An example of this would be when I have some students who are still struggling with decimal division play the Jenga game, some students who still struggling with decimal multiplication playing a Zap game, some students who are struggling with fraction addition working with a fortune teller, and some students who are still struggling with fraction division playing Fraction Flip It.

Since these two review chunks of time are a regular part of an instructional day, I have added specific spaces for them in the Interactive Notebooks that we use.   I use the first page for the day for the Math Minute, In, and Flashback.   The lesson starts on the second page for the day.