Pythagorean Theorem Interactive Notebook Page

I wanted  something to use in my students’ Interactive Notebooks for the Pythagorean Theorem that would start with the conceptual view and build toward the algebraic formula.

I wanted something that would give the whole, part, whole picture of  the Pythagorean Theorem is and how it is used.   I started with an overview of when it applies, then incorporated some pieces to build conceptual understanding, some practice with the concept algebraically  to solve for missing side lengths, and a final summary of what it is.

Complementary and Supplementary Angles Foldable

As the second part of my lesson on angles, I had my students complete an advance organizer/foldabel on complementary angles, and supplementary angles.   This is a follow-up to the introduction of the idea of adjacent and vertical angles.   As I said in the previous post, I decided to use this as an opportunity to also review solving two-step equations (something they learned last year but that may have suffered from the summer slide a little bit).   By incorporating ratios in the description of the angle relationship and unknown angle sides, I can embed a little additional practice with ratios, writing equations, and solving equations.

I considered putting all four concepts into a single foldable, but felt like I was pushing the bounds of how much information would fit in a single foldable.    I’m not sure how this is going to go, so I may end up re-doing the whole thing for next year.    Since this is the second of two related foldables, I embedded some of the ideas from the adjacent angle/vertical angle foldable into this one as well.   I wanted to show students that these concepts are all related.

You can download the Complementary and Supplementary Angles foldable by clicking on the text.

On Tuesday, I taught my 7th graders about adjacent angles, vertical angles, complementary angles, and supplementary angles.   I decided to use this as an opportunity to also review solving two-step equations (something they learned last year but that may have suffered from the summer slide a little bit).   By incorporating ratios in the description of the angle relationship and unknown angle sides, I can embed a little additional practice with ratios, writing equations, and solving equations.

I created an advance organizer/foldable to use as part of the lesson.

I like the structure of the inside of the organizer but am not crazy about the front.   I tried to have kids incorporate color as they completed it.   It was a valiant attempt I guess, but I think they got better as the week wore on.   Today, they did a really nice job using color to show various angle relationships on a point.   I’m not sure if it is that they got better at it or if it is that I got better at it.    This is my first time with this particular course, so I am finding that there are a lot of things I can do better next time.

A MarkUp/MarkDown Foldable for My Twice Exceptional Student

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

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

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

MarkUp and MarkDown Foldable

What you get when you didn’t get what you wanted – Constructing an understanding of the properties of inequalites

“Experience is what you get when you didn’t get what you wanted.”   Randy Pausch

That certainly held true for me the first time I taught students the properties of inequalities.   I was teaching three different courses that year, each of them for the first time.   I didn’t like it but, I knew that I was going to have to make some conscious choices about where and how I spent my time preparing lessons.

• Course number 1 was a regular 8th grade math course. There were students with significant gaps, students with language barriers, and students with learning disabilities, all of whom needed to be adequately prepared for algebra the next year.
• Course number 2 was a gifted math 6 class that fell into my lap at the 20 day count because the teacher decided to leave a week after school started.   They spent the first month of the academic year with a substitute teacher and then finally landed in my lap because I had the necessary endorsement for the course.   They were great kids with tremendous potential but the loss of the first month of school was definitely going to take work to fix.
• Course number 3 was a high school Algebra class for 8th The students were advanced and would be OK so long as I taught the material.

Given the options, I decided to teach the Algebra course pretty much straight out of the book so that I would have more time (not enough, but more) to address the needs of the students who had the greater need.

Even with that choice, it was all I could do to keep my head above water that year, which brings me to the properties of inequalities.   That first year, I gave my algebra students very clear notes on the properties of inequalities.    I modeled problems for them and gave them practice and called it good.   I called it good, but it really wasn’t.    Some of them “got it”, but some of them didn’t.   Not what I wanted, so I guess I should call it experience.

Constructing the properties of inequalities

This year, I used a set of stations to allow my students to construct an understanding of the properties of inequalities.    Students worked collaboratively at each station to construct inequalities using a pair of dice.    The task card at the station then directed them to perform certain actions on each side of the inequality and decide if the inequality should be preserved (stay the same) or be reversed.

• Station 1 – Students start with an inequality in which one number is positive and one number is negative.    They perform five different problems.   In each problem, they add or subtract a positive or negative value to/from each side of the inequality.    They discover that the inequality is always preserved with addition or subtraction.
• Station 2 – Students start with in inequality in which one number is positive and one number is negative. They multiply both sides of the inequality by -1.   They create a new problem by rolling the dice (one is positive and one is negative).   Again, they multiply both sides by -1.    After repeating this process several times, students discover that multiplying an inequality by -1 results in the need to reverse the inequality.
• Station 3 – Students start with an inequality in which both numbers are negative. They perform five different problems.   In each problem, they multiply or divide by a positive number.    They discover that the inequality is always preserved.
• Station 4 – Students start with an inequality in which one number is positive and one number is negative. They perform five different problems.   In each problem, they multiply or divide both sides of the inequality by a negative number.   In so doing, they discover that the inequality must be reversed whenever one multiplies or divides by a negative number.

Student groups were randomly assigned to an initial station.  Groups rotated through the stations in 6 minute intervals.

I found these stations in EngageNY.   I modified them slightly.   In the original EngageNY version, the inequalities showed -3< 2, (-1)(-3)< (-1)(2), 3>-2.   The middle step in the sequence is not true, so I changed the inequality symbol in the middle step to be a “?”.   I wanted to students to be asking the question of whether the initial state of the inequality was true.   I also don’t want them to establish a habit in which they write things that are not mathematically true.   The “?” mark replacement seemed like a reasonable substitution.   I will provide a link to download the station task cards soon.   (WordPress is not allowing me to attach a PDF right now. I will have to investigate)

After students completed work at all four stations, the class came together for a discussion of their findings.   This was a chance to de-privatize student thinking and make sure that everyone had access to the main ideas in the lesson.  I think it can be powerful to bring these ideas out through student voice.

The last few years, I have made a real effort to conclude the lesson with a summary.   Sometimes, this is something that I give verbally and students put into their own words in their notebooks.   For this lesson, I used a foldable to summarize the ideas. The foldable compares and contrasts the properties for each of the operations, employing a Marzano high-yield strategy.     (I will provide a link to download it soon.)

Applying the properties of inequalities

The following day, I had students apply the properties of inequalities as they solved word problems.    This gave them the opportunity to use the properties in a real context.   It also revisited the work they had done writing and solving equations to solve word problems prior to the work with inequalities.  I spent two days in class working on word problems with inequalities.   I felt my students were pretty good at “solving” the problems but not very good at writing an equation to do so.   This was a chance to revisit that while also working with the properties of inequalities.

On the second day, I required students to also graph the inequality.  They had learned to graph inequalities in a previous unit (without the context of a word problem) and had been quite successful with it.   I expected this to be just a straightforward extension of the word problems, giving them some additional practice with a previously learned skill.   It wasn’t.    Some students correctly wrote the inequality for the problem, solved the inequality and then graphed something completely random.    They randomly pulled numbers from the problem and graphed an inequality using those numbers.    Because I had students work together with Kagan’s Numbered Heads Together Cooperative Learning Structure, the issues they had with the inequality were resolved through their discussions within their table groups.   However, I am still grappling with the “why” behind the randomness of their graphs.   The “model” should add meaning to the solution.   Clearly this handful of students did not recognize that.   I am left with the question of whether I have not done well enough at conveying the “why” behind models or whether I did not write the “question” well enough.   Another bit of “experience” to ponder and improve.

Looking for Imposters – Constructing an Understanding of Similarity

Journey’s end is across the river.    It all starts with a character in a video game.     Specifically, an adorable little guy named Mug Wump.

Mug has a family, all of whom are similar to him.    However, there are also some imposters in this game.   Sneaky little guys who are pretending to be part of the family.  Who knows what nefarious deeds these sly fellows might commit, so it’s important to ferret them out before they can cause too much trouble for the Wumps.

From Ordered Pairs to Coordinate Graphs

In this Connected Math lesson, students are given a set of coordinate pairs for Mug, and a set of rules for creating the other characters.    Students must graph Mug, find the ordered pairs for the other characters using the rules, and then graph the other characters.    Finally, they determine which of the characters are imposters and examine the rules to identify how one might identify similarity using only the rules.

My students love this lesson because Mug is adorable and they like the discovery that goes with creating the other characters in a video game.   I love this lesson because it allows students to build an understanding of similarity visually and then to see it as a scale factor in the diagram and in the rule for the ordered pair.    It also provides practice with graphing and the opportunity to discuss dilation.

Coordinate Graph to Ordered Pairs and Back Again

The next day, students continue to explore how different rules for the coordinate pairs change a figure.    They are given a diagram of a hat for our hero, Mug Wump.    They then use different rules to create different hats.    The activity starts with a graph of the hat.    Students create the list of ordered pairs for the hat and then apply a set of different rules to the ordered pairs to create different hats.    The rules explore how using different coefficients change the hat.    They also explore how adding or subtracting from the ordered pairs change the hat.   Finally, they explore how a coefficient less than one changes the hat.

Students find the idea of making a hat for Mug intriguing.    They also are surprised by how the different rules shift and change the hat.    They sometimes struggle with seeing that different coefficients for x and y don’t create a similar hat.    They seem to see it readily with the Wumps, but then don’t see it readily with the hats.   I think it is because the overall size of the hat is small and the changes are a little more subtle.    This subtlety is something that some kids catch and  others don’t.   It leads to some interesting discussions within table groups as kids decide which hats are similar and which are not.    I always teach this lesson using Kagan’s Numbered Heads Together Cooperative Learning Structure.   I like that this structure gives kids the opportunity to work independently first, a structure for a discussion, and mutual accountability because they never know who is going to be asked to speak for the group.

I like that this lesson deepens students understanding of how rules for coordinate pairs influence similarity.    The discussions are rich and practice with other skills is embedded in the lesson (graphing ordered pairs).    The lesson also gives a nice context to talk a little bit about the idea of dilation and translation of figures on a coordinate plane.    While these ideas aren’t central to the lesson or part of the CCSS standards that I am teaching in this course, it is a nice introduction to vocabulary and concepts that students will see the next year.

I wrap up the lesson with an exit ticket in which I ask students how the rule (2x, 2x+7) will change the hat.   This is a chance to see if students have fully grasped the concept.    Some of them initially say that the shape will be doubled.    Some students are stumped by the combination of a coefficient and an addend in the rule.    Bringing these ideas out as we go over the exit ticket (after I collect them and do a quick scan of them) gives us a chance to solidify everyone’s understanding that the coefficient dilates the figure and the addend translates it.

Exploring Scale Factor and Ratios within Similar Parallelograms & Triangles

With a firm grasp of scale factors in hand, students deepen their understanding of similarity by exploring ratios within similar shapes.    They find the ratio of adjacent sides within a rectangle and then compare that ratio of the corresponding adjacent sides within a similar rectangle.    They repeat this process with parallelograms and triangles.    As they work through this process, I have students use color coding as they identify corresponding sides.    They use the color coding in the diagrams and in the ratios to help them ensure that they are indeed looking at ratios of corresponding sides.   They begin by looking at the angles to find the corresponding sides and then use colors to highlight the pairs of corresponding sides.    This year was the first time that I implemented this color coding.   It was done to support some of my students who are twice exceptional and have visual processing difficulties.    I found that it was really helpful for them.   It was also really helpful for the entire class.    Students grasped the concepts more quickly and with more assurance than in previous years, without me sacrificing the discovery process by “telling” them the answers rather than allowing them to construct their own understanding.

Using the Characteristics of Similar Shapes to Find Missing Side Lengths

Once students have a firm understanding of the characteristics of similar shapes, they use those characteristics to find missing side lengths.    They begin with some similar triangles and parallelograms with missing side lengths.    Then, they proceed on to using these ideas in real-world contexts.    They use the shadow method and the mirror method to create similar triangles and estimate the height of trees, buildings, towers, and basketball hoops.    Finally they usedthe idea of similar triangles to estimate the distance across a river using some trees on one side of the river and some stakes on the other to create similar triangles.   The lessons gives the students the chance to explore how math is used to solve real problems.

Bringing It All Together

Characteristics of similar rectangles parallelograms and triangles

At the conclusion of this series of lessons, I have students complete a foldable summarizing the characteristics of similar figures.    This foldable is designed to help my students who are at the extremes with regard to the conceptual/sequential dimension of cognitive style.    It gives them a picture of the whole and the parts, making explicit connections between the big picture and the details.   This is to help my conceptual learners start with the big picture and then connect the details to that understanding.   It is also to help my sequential learners who tend to compartmentalize information and ideas to see the connections and build the big picture.

I wait until the conclusion of the series of lessons to complete the foldable because I believe students benefit from the opportunity to construct an understanding of these ideas.   The series of lessons are taken from Connected Math.   The premise is that students construct an understanding of concepts and develop a deeper understanding of the math by doing so.   Each lesson includes a summary in which those ideas are drawn out and made explicit after students have had a chance to build their own understanding.    The final foldable is an attempt to summarize the ideas addressed over the course of several lessons.  (Click the link below the photo of the foldable to download it or go to the resources page and click the link below the photo.)

Which Bicycle Shop Should I Use –A Lesson Introducing Proportional Relationships in Tables, Graphs, and Equations

“I’m thinking about purchasing a custom-made bicycle. Bicycle City charges \$160 plus \$80 per day that it takes to make the bicycle. Bike Town charges \$120 per day to make the bicycle. For what number of days will the charge be the same at both bicycle shops?”

I introduced the lesson with this question (taken from the NY Institute for Learning). We began with our usual process of making sense of the problem (What do you notice? What do you wonder? What is the question? What do you know?). Then, students set to work on the task individually.

Once everyone had solved the problem, I had three different students present their solutions (one who had used a table, one who had used a graph, and one who had used equations) to the class using the document camera. This gave us the chance to talk about how you find the solution using different solution paths.

At that point, I told students that a proportional relationship was one in which you could multiply the x value by a constant and get the y value. I asked them to figure out whether either bike shop’s pricing was proportional and how they saw that in their particular solution. This gave us the opportunity to discuss how they see the constant rate of change in each representation. It also gave students the chance to discover that a proportional relationship goes through the origin for themselves.

We wrapped up the lesson by completing a foldable summarizing the characteristics of a proportional relationship and how to see them in a table, graph, and equation. You can download the foldable by clicking on the link.

Proportional Relationships

The lesson is designed to give a whole-part-whole understanding of the concept. This is particularly important with students who are at the extremes on the conceptual/sequential dimension of cognitive style. I have several twice-exceptional students who are at one end of this spectrum or the other. I struggled with how to help them succeed for the entire first semester. I started building in the use of a whole-part-whole structure and the use of graphic organizers for note-taking more after reading Twice-Exceptional Gifted Children by Beverly Trail, EdD. It has had a pretty big impact for these students (test scores moved from the 50-60% range to the 98% range on the last test). I don’t know if that will continue to hold, but with that kind of improvement, I am going to keep incorporating this technique and find out.

The way that the foldable is designed, students can compare and contrast how one sees a proportional relationship in a graph, table, and equation by looking across the foldable.   The can also compare and contrast an example of relationship that is proportional and a relationship that is not proportional by looking down a column.   Marzano’s research shows that the use of comparing and contrasting can result in significant gains in student learning.

I also incorporated the use of color in the completion of the foldable to highlight the important points. This was not as intentional as it should have been. Early in the day, we worked through the foldable summarizing the ideas in the lesson. As the day progressed, I thought it would be a good idea to have kids highlight the key points in each representation (constant rate of change and going through the origin). By fourth period, I finally refined it so that kids used one color to highlight the rate of change and a different color to highlight that the relation goes through the origin. This seemed to really bring out the idea across representations, so I will do it this way from the start the next time I teach this lesson.