With the earlier lesson, we introduced "zero pairs", but also talked about the "additive identity". "What can we add that won't change the value?" Using this to segway, we can introduce the multiplicative identity. I didn't realize before I used this lesson just how useful the term "multiplicative identity" would be. It became a way to focus our attention during this topic, but also comes up again when solving equations to find a strategy to get rid of a fraction being multiplied with a variable.

In the lesson, you see empty boxes. The have turned out to be a great tool for students while they are working on the boards. 

Normally, I have the students work through an entire lesson before I stop them to discuss what they learned. However, in this lesson, I talk to the whole class to summarize every bullet point. We start by emphasizing that multiplying by one doesn't change the original value. Then we look at the structure of a fraction and what makes that fraction ONE. By the fourth bullet point, we want to keep the fraction to a value of one, but we don't want it to be the exact same number or obviously the same thing. (It sounds vague, but so far my classes jumped right in to filling in the boxes without any more instruction).

It worked very good to focus on one thing at a time and then add on to it. In the fourth bullet point, you see four different fractions with fill-in-the boxes. I had the groups fill them in, then we go over them, and then I address the next bullet point that adds in the units on the first number, allowing us to cancel the units.

In the seventh bullet point, I take away the structure of the boxes and give them the type of problem we want them to be able to do, but with one step. I also introduce them to the equations as a "key" that they can use to help identify the multiplicative identity they will use to convert.

The first progression is meant to bring the students through the idea of "cancelling" while multiplying fractions. The first one clarifies that the fractions don't have to be adjacent (the 3's), and the second one clarifies that numbers can be simplified even if they are not the same number (the 4, 2, 3, and 6). The third focuses on canceling units and the fourth puts it all together. After those four I chose to consolidate to go over all of those concepts. 

For the rest of the problems, groups will progress through figuring out how to set up the fractions they need to convert the units. The first one gives the direct path that they will need more than one "multiplicative identity" to make it happen. The rest they need to find the path themselves. The list of equalities on the right I wrote on the board for the groups to reference if they don't know any of them. The last problem introduces them to a rate, meaning there are two units to deal with. 

Finish with a consolidation of mild, medium, spicy. Write the three problems as a list in a mixed order. Ask the class to help you put them back into order. Then go through the problems simultaneously, one step at a time so that you can compare and contrast the different levels of the problems. 

Today's focus is to convert rates. The process mimics the previous day's focus. Finish with the class writing notes about what they learned. My notes can be found on the main page for Intermediate Algebra. I use the four quadrant notes from Building Thinking Classroom.

I start this lesson with the question "What is multiplication?". I like to tell my students about how, when I was in my second year of teaching, I asked my class this same question and then realized I didn't know how to answer it myself. (It's the beginning of the year, we are getting to know each other, I like to be vulnerable so that they are more willing to be vulnerable with me. Admitting that you don't know something is hard, but so helpful in learning!)

We re-establish the phrase "groups of" to help define multiplication. They say they remember this from when they learned this in elementary. Then we use it, along with mats and centimeter cubes, to explore how negatives affect our answers. Like before, my goal is to learn "why", not just the answer. 

We start with 2(3) and then compare to 3(2) - same answer, but it looks different with the block. 2(3) is 2 groups of 3, where as 3(2) is 3 groups of 2. Then we proceed to introduce negatives. 2(-3) is 2 groups of -3, so all the blocks are in the negative side, so the answer is negative. It gets a little more interesting when the first number is negative: -2(3). 

Here I tell the students to ignore/cover the negative in the front and set up the remaining problem: 2(3). Now we have 2 groups of 3 blocks in the positive side. When we look at the negative in the front, it is telling us the final answer is the OPPOSITE of what we have. So this problem will move all of the blocks to the negative side and result in -6.  We try the same idea with -2(-3). 

The lesson plan doesn't show any more of these type of examples, but I sometimes give more based on how the class is reacting to what we have done. 

The next problems introduce the idea of a variable. We define the yellow "stick" as the variable x. We talk about how some people think the x is really 3 because 3 cm cubes line up to almost be the same length as the stick. However, there is no physical way to make that yellow stick so that you cannot find a matching length with cm cubes, so you just have to commit to a length. The yellow stick could be worth 3 one problem, but then 10 another problem, or -4 another problem. The point is that we have to figure out what it is worth each time.

It is interesting to write 2x on the board and ask students to show you it with the blocks. It brings out their understanding/misunderstanding. Many students will put 2 centimeter cubes and one yellow stick in the same order as the 2 and x are written on the board. A few students will put two yellow sticks. When this first happened, I felt defeated because we had just talked about what multiplication means. But learning takes time and this is another opportunity to teach the concept. Before addressing it, I ask the students to also show me 2+x. For a few students, this helps them realize their mistake with 2x. Without correcting anyone, I ask them again what multiplication is. They tell me "groups of", so then we look at the 2x written on the board and I read it as "two groups of x". I draw the two ways students represented this on their mats and talk about which one is showing "two groups of x". 

We then move through the rest of the multiplication problems in the lesson. Making sure to cover distribution and fractions. It is difficult to show fractions with the cubes. I encourage them to just have that fraction of the cube in the mat and the rest outside of the mat.

Division goes much the same. During this lesson, I lead them to use the idea of splitting into n groups and the number in the group is the answer. If the number you are dividing by is negative, you ignore that negative to begin with, and then use it to do the opposite of your answer in the end. 

I like to focus on adding and subtracting as verbs. I want them to know that adding is when they bring more blocks to the mat and subtracting is when they take blocks away. 

Their curriculum in their JH years has the student change subtraction problems into addition problems by writing the opposite of the subtrahend. Using the blocks, I try to emphasize why both problems will give them the same answer and how the math compares. I have them do both versions a couple times and then tell me how they are similar and different. 

Note: when subtracting with the blocks, if there aren't enough to "take away", you will need to create zero pairs to "add" more. 

I again use dice to create problems to practice. 

End the lesson with numbers too big to use the centimeter cubes. This pushes the students to think about it without the aid of the blocks. But the numbers are easy enough to think about (no borrowing or complications) that they are still able to focus on zero pairs and signs. 

Students are often missing a fundamental understanding of negative vs opposite vs subtraction. They also often do not understand the interaction between positive and negative numbers (zero pairs). So this lesson is focused on developing those understandings.

I decided to use an old fashioned duel to explain many of these things. I had two students pace off  3 paces (from zero). We then talked about how both students were 3 units from zero (absolute value), but on opposite sides (introducing the word "opposite"). 

I like to use Algeblocks to work on integers and solving equations. During this lesson, I just use centimeter cubes (you could use any small item: beans, cheerios, skittles...) and the basic mat from Algeblocks. 

As I add negatives, I have students move the blocks back and forth between the "two worlds". My purpose is to get them to internalize that there are only two world and so it takes two moves to get back to where you started. Hence "two negatives make a positive". (I really want them to understand why, we spend a lot of time talking about this).

Zero pairs is a term we use to talk about the additive identity. We use the term constantly throughout the year. To introduce this idea, I use a version of Hansel and Gretel. In the story, there is an old lady in a cottage in the woods with a big black pot of water. She happens to have magical red cubes (centimeter cubes in the demonstration) that make the temperature of the water increase one degree for each cube added to the water. There are also two kids in the cottage. They have magical blue cubes that make the temperature of the water decrease one degree for each cube added to the water. "If the old lady and the kids each add three cubes (demonstrate), what happens to the temperature of the water?" This is zero pairs. 

I then show students how to represent zero pairs on the Basic Mat by lining them up together at the line. 
Use the rest of the period to practice doing this on the Basic Mats. I find it tedious to come up with problems, so I like to roll dice to get values (it also adds a LITTLE fun to the activity).

Prerequisite Knowledge:
- Can graph a line in slope-intercept form
- Can solve single-variable equations

Objectives:
- Know how to find intercepts from an equation
​- Can graph a line from the intercepts

Lesson:
Although students are familiar with the y-intercept for graphing lines in slope-intercept form, I still spend time discussing the details of intercepts. Specifically, I want them to identify that y equals zero for an x-intercept and x equals zeros for a y-intercept. So I start with a notice/wonder.

For this image, I purposely chose lines with positive and negative slopes. I also made sure one of the intercepts was a fraction. However, I should have varied the intercepts from 2 and 3. (My students call me out about using 3 too much).

Once students have identified that respective values of zeros, I send them to their boards. I start with the equation 2x + 3y =6. (Notice it is one of the lines from the notice/wonder. I make sure to remove the image so that none of the students are using that as a tool to answer this). I ask the students to find the x- and y-intercept from the equation. I also give them the instructions that they can't turn it into slope-intercept form and graph it to find the intercepts.

There is often some creative solutions to this problem. Some still turn it into slope-intercept from and use the graph, even though I tell them not to. Some create a table to see where zero happens (a valid way, but not efficient). Although this can be frustrating, I remind myself to celebrate that they are trying something and not waiting for me to tell them what to do. There have been some times that I have to  plant a seed with a group to get them to use evaluation to find the intercepts. I have wondered if I should create the thin slicing a little different. Maybe have them start by doing a table (like some groups do) and then tell them we just want to focus on the intercepts. Then follow that up with another equation and say "just find the intercepts". I am not sure how that would go, but it might be worth a try. I also wonder if we use the equation and give them an x-value to use to evaluate. And then another x-value, and then maybe a y-value. Follow that up with asking for the y-intercept (they will probably need reminding about the notice/wonder they did at the beginning).

Something like this: 
Using 2x + 3y = 12 
Find y when x = 3
Find y when x = 9
Find x when y = 6
Find the x-intercept
Find the y-intercept

Once they are using evaluation to find the intercepts, you can continue on with the thin slicing:
2x-4y=12
-2x-4y=6
y=2x+5
etc

Consolidation: (using the thin slicing consolidation)
As a class, put in order from easiest to hardest: 
-x-3y=8
2x+4y=8
y=-4x+9
Once in order, find the x- and y-intercepts simultaneously 

​

I love my lesson on introducing negative and zero exponents now that I added in the visuals. Check out the lesson here. I also like the lesson I do after where I use smudge math to get them to think about what the exponents should be. BUT, I have come to realize that I need a lesson between the two. My students need more time to work with negative and zero exponents, to play with the patterns, to really get a feel for how the exponents work. I haven't done the lesson with them yet, but following is my plan.

I am guessing that if you are reading this you are a middle or high school math teacher. Before I share my lesson with you, I want to share a lesson from the primary classroom that inspired me. ​

While scrolling through Twitter, I have come across the fun activities that others have shared of puzzles from the hundreds chart where students have to fill in missing numbers in scattered places of the chart. I love how the activity increases the thinking and adds some play to the learning of the numbers from 1 to 100.

Depending on where we left off the day before, this lesson may begin with consolidating from the intro lesson. Also, it may need to include talking about what to put in the meaningful notes (this is something I don't normally do, but it will give me a good idea of what they did and didn't learn from the day before).

Ultimately I want the students to become familiar with this chart, with how it moves horizontally and vertically. One extension at this point could be to ask them to continue the rows and columns (if it can be done).

Some things I have learned from practicing this lesson on groups of teachers:
1) There is more than one way to fill this in. I was going to fix that by putting another clue in a box, but now I like it because it gives me an extension for the groups that finish quicker than others ("what's another way you could have filled this in?")
2) Students will take a long time reconstructing the grid of the puzzle on the whiteboards. You will save a lot of time if you print this off, put it in a plastic sleeve and tape it to the board 

I really liked how this lesson went with teacher groups I have practiced it with. There were rich discussions amongst math teachers who are experienced with exponents, so I imagine it will be even more beneficial with students learning about the exponents. I also think I will print several different puzzles and use them as quick tasks at the beginning of the class for continued learning and review.

I would like to start with a disclaimer. I spent a ridiculous amount of time decorating my classroom door/cove this last week. I wasn't going to decorate at all, but then the elementary teachers challenged us (high school) to a contest and I decided I was going to win. Once I got started, I couldn't stop. I found it to be very therapeutic and fun. It was nice to take a big break from household chores, school work, and basic busy-ness. Plus I got to spend a day with my sister who came to help.

I started planning by searching google for ideas. There are not many math ideas out there! So I decided to take what I saw and make it my own. One of the things I really liked was the 3D Serpinski's triangle Christmas Tree. (Video) However, I am decorating my door, so this was going to be difficult to incorporate. My door is in a cove of sorts with another teacher's door. So I figured I should be able to hang it from the ceiling safely, but why would I do it? And the idea of having the Grinch stealing the Christmas tree was born. I made two Serpinski Triangles and taped them together to give a tiered look. For the star, I decided to glue 3 Koch Snowflakes together. ​

~Objective: 

• Be able to algebraically convert the standard form of a line into slope intercept form

~Prerequisite Knowledge:

• Know the structure of the slope-intercept form of a line
• Know how to solve for a variable

 
Lesson:
I started right away in groups at the boards. We quickly reviewed the structure of slope-intercept form. I then gave the groups a line in standard form and asked them to turn it into slope-intercept form. When they were done, they were supposed use Desmos to check their equation versus the original equation. (I like to use Desmos as an "answer key" whenever I can to connect the algebra with the graph in a natural way).

The first equation was   2x+4y = 8

This first one took a while. Many groups had various strategies and I roamed around and visited with them about their ideas. I chose to consolidate right after this problem because there were a few different strategies and I wanted to discuss the pros and cons of each one. There were also some mistakes that we needed to address as a class. (We found that the most efficient strategy was to move 2x and then divide everything by 4).

I had the groups then do a 2-3 more problems:

• -3x+2y=6
• -x+2y=8
• 5x+3y=7 (I only did this for the quicker groups while they waited for the others to finish the two before this).

I found this set of problems to be enough, but you could do more in the groups.

I next wanted to get the students working more individually while I was around to help. I decided to add a little movement and fun with a snowball fight. (I think I got the original idea from Sarah Carter).

I put a large set of equations of lines in standard form on the board and gave each student a piece of paper. They chose one equation and wrote it down. Then they crumpled the paper into a ball, I set a timer for 20 seconds, and they threw the snowballs around until the time ran out. Then they each found a snowball and worked out the problem on it so that it was in slope-intercept form. Again, I had them check their work on Desmos. 

I would have liked to do another snowball fight, but this brought us to the end of the class period. I used Delta Math for Check Your Understanding problems. 

I have made some big changes this year! I am finally applying all 14 of the Building Thinking Classroom principles and loving it. I did as many as I could before from my research on websites, blogs, podcasts, and interviews. But there were a few principles that I just couldn't grasp until I finally got to read the book. (If you are interested in using BTC, I highly recommend you get the book: BTC). I decided Thanksgiving was a great time to reflect and write about what changes I am making this year and how it is going. I am specifically writing about check understanding, build autonomy, and formative assessment.

Most of these are based off the work I did this summer in preparation for the changes. I went through each chapter and decided what I wanted to assess for basic, intermediate and advanced (My grading is 3, 4, 5 since they will be translated into letter grades like the rest of my school). I created a table for each chapter and included it in the meaningful notes packet I make for students at the beginning of the year. You can see an example from Algebra below.

1. Check understanding. 
I have given homework for a grade my entire career. I have never graded it based on being correct, but instead on completion and showing of work. Doing it this way cut down on how much time I spent on grading since I didn't have to grade each problem for correctness, but I still found myself often wishing the students would go away so that I could get my work done. I needed to grade it all and get it into the gradebook. I hated that part of teaching. I felt like only part of my job was educating students, but the majority of it was correcting and recording work. 

This year I am no longer grading homework. I still assign problems, but I do not ask them to turn it in. I rarely give problems from the book, but I really like using Delta Math (deltamath.com) for problems. The students get immediate feedback and can see the solution if they get it wrong. I have seen my students put more value into doing their problems correct rather than just getting the problems done with this program. Because I am not correcting/grading their work, the points system doesn't matter, so the students can do as many or as few problems as they would like. Which is another bonus of DM, they can do lots of problems if they would like, where a textbook has a limited number of problems to work on. 

Reflection: As expected, fewer students are doing the problems than I would like. But, as we do little formative assessments each day, students find out where they are in their own understanding and I see them go back to the DM to practice. They are learning the new system and finding out that they need the practice in some places and not in others. Ultimately, if they are successful in the formative assessments, it doesn't matter to me if they do the DM. (BTW: I just realized have not worried about Photomath once this year!)

I created a table for the students to reference which DM assignments align to the topics they will be assessed on (see below). I also include this in the meaningful notes booklet. I have found this table to be useful when helping a student figure out what they can do to get ready for more assessment problems to help improve their grade.

2. Build Autonomy:
Managing flow! How do I get my students to help themselves instead of waiting for me to get to them? I was the teacher that would tell my students to steal a problem from another board if they finish and are waiting for me. But then I would leave a group and see two groups just sitting there! 

Finally, after reading the chapter, I have figured it out. Instead of giving the next problem to the waiting groups, I point at the board that already has the next problem and tell them to take it. (I was enabling them when I would give it to them). Now, my students have started helping themselves instead of waiting for me. Thank Goodness! (This is also true if a group is stuck. I will often tell them to go peek at a certain board to get an idea instead of giving them any help)

Another thing I am working on this year is what I do BEFORE groups need help. I used to wait idly and just watch the groups work. Now, I walk to each group, point at a specific item on the board and ask them to tell me about it or to convince me it is right. I am finding the lesson goes much more smoothly than before, I am keeping more involved in their learning, and I am having much richer observations of each and every student.

3. Formative Assessment:
I have been trying to figure out standards based grading for years. I have tried a few different things, but none of them felt quite right or worked the way I wanted. I am using the assessment and grading from BTC and I love it!

Quizzes. I don't give quizzes any more. I have to admit, when Peter says that you will find you give too many quizzes already, that wasn't true for me. In the first two units this year, I found that we got to the end of the unit and I have very few data collections. I am finding my way though. I try to give one or two problems every day at the beginning of the hour and we call is an "understanding check" instead of a quiz. I also like to use "My favorite no" for these. I make sure the students put their name on them so I can record the data, but it lends itself well to having a purpose in our lesson instead of just a quick check.

Quiz reflection: I learned through this process that many students see a quiz as potential punishment. If they aren't ready for the quiz then any problems they do incorrectly lose them points that they can never get back. It is taking me a long time to get them to understand that there are no points taken away for a wrong solution; there will always be more opportunities to show me they can do it. An x at the beginning followed by check marks means they learned, which is the goal. Not all of my students have grasped this yet, but we keep working on it. 

End of unit test. I hate writing tests. It really is the worst part of my job. Since I am changing how I assess, I have to rewrite all my tests. I am not enjoying it, but it isn't as bad as writing a test from scratch. I use my table (see above) as a guide and make two problems for each cell of the table. I am not completely sure this is what I should do, but it makes sure that a student can possibly get two check marks if they haven't gotten any yet. I do like separating the test into all the basic problems first, followed by all the intermediate problems, and finishing with all the advanced problems like Peter suggests. I find it gives students a reference to where each problem fits in the table. 

The second worst thing about teaching, after writing a test, is grading a test. However, this has improved with BTC grading. Now I just give it a check mark, x, or S. I don't have to debate how many points to give! Transferring their marks onto their progress tracking sheet is time consuming, but it is easy enough to do that I haven't minded it yet. 

Test reflection: I was worried the grading would be "inflated" (is that the right word)? At the beginning, students would finish a test and say they failed as they turned it in. But then they would end up with a 20 out of 22. They felt like they failed because they couldn't answer four of the problems in the advanced section. In the past, if they didn't answer four problems, they would have lost a lot of points. But with this, they only lost two points, getting 4 out of 5 on two of the topics. However, when I look over the final grades each time, the letter grade seems very appropriate for each student. So I don't think grades are inflated, they seem to be more appropriate than before. (Side note: this is my 19th year of teaching. Quarter 1 was my very first quarter in my career that I did not have a student fail my class. FIRST TIME EVER! Reflecting on why, I think there were just some students before that found homework to be too big of a workload and could not/would not do it all. I also think that there are so many valuable things we do with all 14 principles that they are just learning better too).

Below are a couple examples of the progress tracking sheets for a unit. I put them on a google doc and update it as we go. Students and their parents have access to this sheet at any time.

I still have one question about grading. If a student has x's all the way across a row, I will give them a 0 out of 5. But, what if they have a couple check marks sprinkled throughout  the row, just not two in a row? Do you give them a 1 or a 2, or do you still give a 0?  

Improving grades after the test. This has also changed my teaching life. I used to have students stay after school to get help on their homework. Their focus was on getting the assignments done, not necessarily learning the math. I hated it! I do not have the patience for homework help. But now, when a student needs/wants to improve their grade, they stay after with me. We look at their progress tracking sheet and their test to see what they need to learn. The time we spend going over problems is in an attempt to learn, not complete problems. I usually send them on their way with problems to practice. Then we find another time for them to do "assessment" problems to add to their data. Since they only need to do a couple, they can usually get this done during a part of class, at the end of lunch, or during their resource hour. It feels so much more worthwhile now. The students are more focused and aren't coming to me with the stress of a mountain of homework and a checklist to complete. 

I have also found this useful for a particular student that has extreme test anxiety. We now have scheduled him to come to my room during my prep (his study hall) every Wednesday. We use the time to do some old or new problems and then I can add to his data with my observations instead of just formal assessment problems. It has been a huge help in making sure his grade reflects what he knows. Below is an example of his data after a few Wednesday's of going over a previous chapter. 

Final Reflection:
After writing all this and rereading it, I realize that there were more things about teaching that I hated than I thought! I really love my job, so I am a bit surprised. But it has also made me realize why I have been leaving the school building earlier than usual and with less weight on my shoulders the last couple weeks. The things that I disliked the most about teaching have changed drastically. I still don't like writing tests, but I don't hate it. Grading them is no big deal. The paperwork of homework and grading is pretty much nonexistent. The time I spend with students is actually teaching rather than assisting with a checklist. I finally feel like the majority of my day is spent educating.

I was in love with BTC before because my class changed for my students. They were engaged and THINKING. I have fallen in love with it all over again this year because my class changed for me. I am more engaged, less overworked with tedious tasks, and enjoying how I assess the learning of my students.

Objectives:

• Students can use the area model to factor a quadratic into two binomials

~Preknowledge:

• Can use the area model to multiply binomials/polynomials

Lesson:
I start this lesson at the boards right away. I tell the groups they will be doing the opposite from the lesson before this. I want them to tell me what the two bubbles are that would multiply to give me the quadratic. Their answer should be the two bubbles written out as factors (multiplied).
The thin slicing for this is structured that I give them some "clues" but gradually take them away. The series of problems I go through are below. (Please forgive my crude notes, I haven't taken the time to make them more professional). 

This pretty much takes up most of the class. I consolidate with the class and often give them class time to work on  check their understanding problems. I want to make sure they take the time to work on it individually right away.