## Wednesday, September 30, 2015

### Day 22: Expression Mat "Legal Moves" & Piecewise Graphs

The estimation today was estimating a box of envelopes. Most students overestimated. I liked the estimate of 125 because some students said the box look 25% bigger than the box of file folders.

I passed back last Friday's assessment which was the last chance on Skill 3, Percent error. The photo below is the procedure I want students to follow before retaking the skill. There's no point in retaking it if you still don't understand the concept.

Students finished the 1/2 sheet of paper with two expression comparison mats. I circulated asking students questions and checking their understanding. Once a few students were done I picked a student to show us on the document camera. Prior to this, I reviewed the 3 "legal moves" that I will have students take notes on tomorrow so they can reference it. Then students practiced building expressions, simplifying them, and writing inequalities to compare them.

Since my TV isn't functioning, my 2nd period students helped me inventory the algebra tiles because students were carelessly putting them back in the box letting some spill into the box, making the bags of tiles uneven. Tomorrow we will do some math to balance out the bags. These sets need to be as complete as possible because accelerated will be using them to learn factoring quadratics.

In accelerated, I passed out the 1/2 sheet with the piecewise graph of it. Students had experience in their science class describing distance-time graphs so they were familiar with it. Thanks Mrs. Newiger and Mrs. Bronson! They wrote equations for each of the 3 parts of the graph and wrote the domains of each section. I will hand back their assessments tomorrow. Tomorrow we will review the baby chick problem where students are given a growth rate and a data point and must write an equation of the growth. This is a skill that will not be on Friday's assessment, but next Friday's.

I also want to post this Jon taffer quote and meme in my classroom:

## Tuesday, September 29, 2015

### Day 21: Comparing Expressions & More Rate of Change

Today's estimation was a box of hanging folders. I liked that some students said that they thought the hanging folder was twice as thick as a file folder so the box must be half of a file folder box (1/2 of 100 = 50). Since students overestimated, I took the opportunity to discuss different ways of interpreting percent errors that are 100% or over. Some students freeze up and just write what every the quotient is, as the percent. For example, if the get 50/25 and get 2, they put 2% instead of 200%. Also talked about getting an equivalent fraction with 100 as a denominator. I wanted students to work on their precision of language and not say multiply by 4, but multiply by 4/4, a Giant One. I showed that multiplying by 4 is actually multiplying by 4/1 which changes the fraction.

I dug into students background knowledge of how to compare 58 and 62. What symbol goes between them? They said less than. I asked how they remember the symbol. Some said alligator eats the bigger one and pacman wants to eat more. I took that opportunity to show students my prior blog post, How Inequality Symbols are taught in Shanghai. I asked students to interpret how students learn inequalities using a visual model. Once again, I honored different cultures, and the cultures that some of their parents have.

I then instructed students to build two expressions on the comparison mat. Some students felt better drawing it, but I pushed them to build it, and collaborate on how to properly write unsimplified expressions of each side. They want to avoid the difficulty and simplify then write the expression. I want them to stretch and challenge themselves.

We then reviewed it using the virtual algebra tiles on the projector. They told me how to write the original expressions and told me how to simplify. They saw the relationship between the symbols and the model. Some students remembered to remove the balanced set of x tiles at the end. This can be a confusing point. I thought of an analogy on the spot. Here it is:

Put up 5 fingers on your left hand and 4 on your right. 5 is greater than 4, so my left hand is greater, or more fingers up, then my right, correct? Pretend that x=2, and we are going to put down 2 fingers on each side, or remove x from both sides. I now have 3 and 2. Is the left hand still greater than the right hand? Yes. I think this clicked for a lot of students.

Students then looked at diagrams and described what simplifying moves the person was doing: removing zero pairs, flipping tiles from negative to positive region, and removing balanced sets. They got a half sheet and simplified it and wrote expressions to compare them. We will go over that tomorrow.

Later this week I will mention the Eating Contest analogy, which I will go into more detail later.

In accelerated, we went back to lesson 2.2.2. This was a labor intensive lesson. They had to accurately graph 3 racers in a tricycle race and answer questions about it. A lot of students did not get to the piecewise graph that dealt with domain. Some groups struggled with students working at different paces. I empathize, because it's very hard to be patient when someone in your group is not as focused as the rest and not able to listen, talk, and write at the same time. We will talk about that tomorrow, and I will talk to certain individual students.

## Monday, September 28, 2015

### Day 20: 2.1.4: Making Zero Pairs to Simplify, VIDEO: Finding Slope & Writing Linear Equations

Today's estimation was file folders in a box. The percent error was easy because the denominator (answer) was a multiple of 10. Students continue to struggle with percents that are 100% or greater than 100%. I think some students truly don't see the connection between decimals and percents. I'm going to have an ongoing attack on this weakness all year, and try to find real life scenarios when I encounter them.

Students considered situations where the net gain was zero. A lot of students suggested a variable like x to represent sand taken out of the whole, and the sand x, being added back into the hole, -x, bringing it back to zero. Then they had to make zero 2 ways. Some students just moved a zero pair from one region to the other after asking them how they could make a zero pair with those same tiles. I made mental notes on two different people with the two different ways.

We came together as a class to discuss this. I used an idea from a former colleague, Ms. Gruner to stress the two ways. One is 2 DIFFERENT tiles, or 2 different shadings (1 positive 1 negative) in SAME regions. The other way I ask, are these the same shaded tiles? No. So, DIFFERENT shading. Are they the same or different regions? Different. So, I think choral response helps here. Also, I reinforce how the students must shade them in their sketches. Is negative shaded or unshaded? Etc... This helps and students love to shout out.

Then students build an expression out of tiles and are instructed to write the algebraic expression that is represented by the model. I stressed to do 1 simplification, then write the new expression. Then simplify one more time, and write your final simplified expression. The book mentions "various amounts of tiles" and I showed that I wanted 3 equivalent expressions. Some skipped, and I challenged them to show me the intermediate step.

Then we only had time to represent -(-2y+1) using tiles. This was similar to one of Friday's expressions.

In accelerated math, we did the Big Race Finals. I videotaped my students when they were working on this. We did this lesson, 2.2.3, before 2.2.2 so tomorrow they will do 2.2.2, Rate of Change. I think this lesson will be a chance to reinforce what they learned today.

The really cool aspect of this lesson is that each group is given 6 cards. 2 cards everyone can look at, but each team member gets an individual card that they must share with their group about.

One common mistake I saw was when students were given 2 coordinates of a racer, they connected those points, and then connected the point on the left to the origin. They reasoned that everyone has to start at the starting line. I then pointed them out the rule of the race is that everyone races at a constant speed. Can it be a constant speed if it starts steep and then flattens out? This was a great learning point.

Students end up with a graph that looks like:

## Saturday, September 26, 2015

### Day 19: More expression mats & reviewing y=mx+b

Today's estimation was how many staples in a box of mini staples. There was some good reasoning. Most students underestimated. The ones that were close predicted that each row would have double what you can see because the staples are packed inside of each other to maximize space.

We reviewed how to represent -3x + 4 and showed 4 different ways. Students told me what to do on the virtual tiles on the board. After the first volunteer I had students vote: thumbs up you agree, thumbs sideways you're not sure, thumbs down you disagree. This was great formative assessment feedback for me. I could see who was still confused and made me realize I made a good choice to continue clearing up misconceptions. I called on students to give reasoning for their vote. The best conversation was for the expression -(y-2).

Some students reasoned that -(y-2) is the same is -1(y-2), so use the distributive property. They also reasoned that you could put y-2 in the negative region. I wanted to stress both of these understandings. Basically, if you have a subtraction sign outside the parentheses, you can put the expression inside as it but in the negative region.

This particular expression is where I see the most useful aspect of algebra tiles. Often students will only distribute the negative sign to the first term in parentheses. When both terms are in the negative region, it makes sense to flip them both up to the positive region, therefore taking the opposite of each term.

Students continued with parts c and d of the problem and some got to interpreting the algebra tile models and writing expressions. Then students took a mastery assessment on skills 3, 4, and 5. It seems that students have trouble when a percent error is over 100%. I think it's a lack of understanding of the relationship between decimals, whole numbers, and percents. Skill 4 was worded poorly, so as an 8th grade team we decided to take two different answers, based on the students reasoning for their answer. Finally, skill 5 was combining like terms and overall students seemed pretty successful with that.

In accelerated, students revisited y=mx+b. There was one expression that peaked many students interested that had inputs of 1, 2, and 3, with outputs of 3, 9, and 27, respectively. I didn't want to give them the answer and students reasoned it must be an exponential function like the infection problem from chapter 1. As a class I asked them to notice how we can get the numbers 3, 9, and 27. Harrison came up with y=3^x and Nicholas showed me a method I believe he learned in Kumon that helps you write an exponential expression when you know 2 coordinates, one being the y-intercept. That was a learning experience for me.

Students took skill 5 for the first time, graphing a quadratic and fully describing it using all the vocabulary from when we worked with Polygraph: Parabolas.

I noticed students are confused with what minimum and maximum are. A few remembered that it has to do with the vertex. Also, a few incorrectly wrote the coordinates for their x-intercepts. The ones who wrote just the x-intercept number and not the coordinates avoided this mistake, but I will be stressing you need to write the coordinates, to reinforce that the x-intercept is when the Y coordinate is zero.

A few students stayed after school to show me their homework, corrections on prior assessments, and retake skill 2, solving linear equations and interpreting their solutions. One student was unclear that if you get let's say x + 2 = x + 1 that it's undefined. I discussed that undefined is a concept for slope and for finding the domain of functions. When an equation is not solveable, we call it "no solution" not undefined.

## Thursday, September 24, 2015

### Day 18: Expression Mat Intro & review of slope

Today's estimation was the amount of staples in a mini staple strip. Some students used their own personal staplers to make accurate estimates. I complimented students who used the regular staple strip in the picture that was 210 staples and thought the mini one was 1/4 or 1/5 of the big one. Then they divided. Great reasoning. I gave out raffle tickets to student who did their homework and were self correcting it after the estimation.

In Common Core 8, students brainstormed the meaning of minus: subtraction, negative sign, and the opposite of a number. They reasoned -(-3) is positive 3. I asked them why and they said 2 negatives make a positive. So, I said, so what's the opposite of -3? 3. I wanted them to move away from some rule they heard and think about the meaning.

I introduced students to the positive and negative sides of the tiles. I asked them what the tiles all had in common and they all had a red side. So, they all have a negative side, that red side. Students grappled with the fact that 3 red unit tiles in the positive region is -3 and 3 positive tiles in the negative region is -3. Unfortunately, prior teachers told students the red side was the positive side. I discussed this with the students and the teachers.

One student, originally from China, told me that in China, in the stock market, red means it's positive and going up, and green means it's going down. Apparently red is good luck and a popular color in China, while green is bad luck. It's the exact opposite of our American understanding! I said that's weird. Then I caught myself, told the class I had said this after sharing his story, and said that I wasn't being respectful of the differences between our cultures and that just because I'm not used to it does not mean it's weird.

I brought up the online algebra tile tool and had volunteers show me two ways to show -2x and two ways to show 3x - (-4). Great conversations.

Then I did the community building I learned from math camp this summer with CPM. It had amazing results. For example, I had all students stand up in the middle of class. Stay standing if you have a brother. Stay standing if you like babies. Stay standing if you enjoy penny boarding, and finally stay standing if your birthday is April whatever. Then one student is standing, and I asked him to explain what penny boarding is and why he likes babies. He said it's a mini skateboard and that babies like him and he's good with him. We gave him, and every student in each class that was left a 3 second clap. Students loved it so much they wanted to do it again. I said every week we will do 1 person.

Then students setup expression mat drawings to represent expressions 2 ways. They did not finish and we didn't have time for closure so we will continue tomorrow. I noticed students needed reinforcement on shading for positive and unshaded for negative. They also needed to be reminded the x tiles were rectangles so don't make them squares and make sure you label them x.

Tomorrow is skills 3, 4, and 5 on the assessment. Percent error, proportional relationships, and combining like terms and perimeter of algebra tiles.

In accelerated students once again refreshed on slope. They will refresh on y=mx+b tomorrow. Students are continuing to work on their study team voices, because they can be too darn loud sometimes when working in groups. It's a work in progress.

## Wednesday, September 23, 2015

### Day 17: Pairs check Perimeter of algebra tiles (video) and tilepatterns revisited

Students had a hard time with today's estimation. It was how many staples were in the box. Yesterday's was how many staples in the staple gun strip. One student got really close with their thinking that there were 8 rows of strips that could fit, and they have 2 in each row because 1 can fit inside the other. So, they reasoned 16(78) and got really close to the answer.

Today yogi Berra passed away. One of my students is a true baseball fan because he wore this awesome shirt with his quotes on it today:

I introduced students to one of CPM's study team strategies and modeled it for them.

Here's how the pairs check strategy works:
1. Student A does the first problem talking about what they are doing.
2. Student B watches, does not write, listens, and asks questions if they don't understand or agree.
3. When Student A finishes, they check their work with the other pair at the table (and the board).
4. If correct, Student B completes the problem, and then reverse roles on the next problem.
In the first class I modeled as Student A who needed to be asked questions by Student B. Later on I realized I should model being Student B because I am better at asking good questions (no offense to my students!). The student pretended to not know how to label the sides so I asked what this tile was called (unit tile). What is this tile called (x tile). What do you think this side is then?

I had students check their answers from the board for problems a through d. When the unit tile covered up the x tile they reasoned it was 3/4 of an x tile. That reasoning is sound. I asked them if we knew how much was being covered up and that lead them in the right direction. I was pleasantly surprised that some students with IEP were quick to understand x-1 as a side length! I hope they remember that on Friday when it is Skill 5 on their assessment!

In 5th period two students were willing to present and I filmed them. They did a great job. I added some closure to it to help students understand their methods. Here's the video:

In accelerated, students worked on Figure 0, 4, and 100 of a tile pattern along with the equation. It was a great review from last year. They also identified equations of lines by looking at their graph and their table. Tomorrow is revisiting slope. A lot of students were rusty with it from over the summer so I am excited they get a chance to review it.

## Tuesday, September 22, 2015

### Day 16: Combining like terms. Domain and range.

I taught like this half the last period after our fundraising assembly...

Students estimated the number of staples in a staple gun strip. Only a few students reasoned that the staple gun strips were thicker so they must be less than 210. A lot estimated over 210, the amount in a regular strip. This was a bit concerning so I definitely wanted to open it up to everybody to see if they had compared them.

I passed back the 2nd mastery assessment from Friday. A lot of students improved on the percent error skill. I also showed examples of student work that fully explained why the relationship was proportional (0 bowls for 0 dollars, passed through origin, straight line, constant rate of change, etc.) Friday will be skills 3, 4, and 5, and students will be assessed on combining like terms when given an expression and when given algebra tiles next to each other. I was happy to hear some students wanted to be retaught and retake an assessment, and said I would do so on Friday with about 5 of them.

Periods were shorter so students only got through 4 expressions, matched like terms orally, and setup the first algebra tile picture. Tomorrow they will take out the tiles, build it, draw it accurately, label the sides, write an unsimplified expression, then simplify the expression.

I liked how Yssa put triangles around like terms, squares around other like terms, and used circles. I pointed this out and she said that Mrs. Ko had shown them that last year. I wanted to honor the fact that she was using a strategy she had heard of before. I also encouraged students to color code them to see them similarities in the like terms (same variable and exponent).

In accelerated, I also passed back the assessments and they filled out their skill sheets. I had students figure out where they left off last period, and then checked in on each group. I could see that they were trying to explain to each other and grapple with it. Once they had enough productive struggle, and had asked some good questions that got students in the right direction, we reviewed what they had understood so far.

I asked students how we could write the domain if it was between -1 and 3. I asked what variable the domain represents (x). So, to scaffold the problem, I drew a number line with -1 and 3 on each side, and x in the middle. I asked how we can say that the domain is between these numbers. That got some students thinking of inequalities, and Zoe came up with how to write the inequality. We also discussed range. I asked them how the inequality would change. They said it would be y instead of x.

## Monday, September 21, 2015

### Day 15: New seats, color coding algebra tile perimeters, finishing identifying functions

It being the 15th day of school, I divided 180 by 15 and got 12. I think that is my goal to have 12 group changes through out the year. That way everyone has a chance to work with everyone else.

Estimation 180 was number of staples in a box. Students argued Day 15 staples in a box should have been 5040, (24 strips)(210 per strip), wondered why 5000?? Great % error discussion.

We got a reply from the man behind Estimation 180 himself.. Mr. Stadel:

We started discussing the unit tile after students read 2-2 about the side length being 1 and it's area being 1 square unit. I asked them why this was true and we discussed length times width for area and how 1 times 1 is 1 square unit. Tomorrow lesson's was supposed to be perimeter but Ms. Demailly and I both discussed perimeter at the same time.

When I asked them to measure the side length of the teal skinny rectangle they said 3 and a quarter and definitely not 4 or 5. I basically said it's non-commensurate and no amount of whole units can make a teal side length, so we don't know it's length. What do we call that in math? A variable. The most popular one? x. (choral response)

Once the unit tile, x, x squared, and y were on the board, I introduced color coding side lengths of 1. Then of x, and finally y. Then I had students figure out the last 2 tiles perimeters and areas we hadn't covered (xy and y squared) and color code them.

I projected a picture of tiles and you can see the awesome progression one group went from a description, to an algebraic expression, to a simplified expression below:

We will finish up this lesson tomorrow, self-correct homework after estimation, and then pass back assessment #2 from Friday and enter the scores on the Skills sheet I talked about on Friday. It's a shortened schedule because we have the fundraising assembly as a 6th period double period, 7th and 8th graders going first.

In accelerated I gave students time to finish the Function lesson. I asked questions about each of the first 3 tables and why they were functions or not. One group who thought they were done thought that it wasn't a function if 2 different inputs had the same output. Gabriel pointed out that on the soda machine there were 2 different buttons or inputs for Blast, but they both had the same output, and the machine was still functioning consistently.

I asked what students thought about the graphs. The last question is great because it asks, are all lines functions? One student said x = 0 is not a function because it has points all along the y-axis. This lead right into the discussion of the vertical line test. We took some shortened version of the notes CC 8 kids did last year in a blog post on here entitled "We out here trying to function."

Below I asked if x= y squared is a function. They said no because when x=1, y is 1 and -1. So, the input, 1, has two different outputs, 1 and -1 at the same time. Therefore, it's not a function.

I asked if y=x squared is a function. Is it okay for the input -2 and 2, to both have ouputs of 4? In this case, yes. It passed the vertical line test, twice in this case.

Students got started on the next section that discusses domain. We did not discuss range, yet.

## Friday, September 18, 2015

### Day 14: Intro to SBG Assessment passing back first test, algebra tiles, and functions (soda machines).

Today students estimated how many staples were in a single strip. Students reasoning was a lot of guessing. The most thoughtful one I heard was Yssa saying that she thinks a staple is 1 mm. A stapler is about 6 centimeters long, so she said 10 mm in 1 cm so the stapler is 60 mm or 60 staples long. One student talked about four chunks of 25 pencils.

Last Friday students took their first mastery assessment. For students in Common Core 8, they can download the skills sheet that I handed out to students. The link for CC 8/A Accelerated is here.

Basically last friday students took a 12 minute test with 3 skills on it: Adding and subtracting like and unlike fractions, graphing points on the coordinate plane, and calculating percent error. I assigned students 3 scores, 1 for each skill. They copied these down on to their skill sheet. The last 12 minutes of class today students took Skills 2, 3, and 4 after their first test had Skills 1, 2, and 3. So, students who improve after taking skills 2 or 3 again they will replace their old score. If their score is lower, they will get the lower score. I believe students most current understanding will be represented.

It's ok to fail at something at first, but do something about it before the next two weeks by working on the skills before retaking the skill by appointment. I honestly think students were happy with the new system. Time will tell! These skill assessments are 50% of a students grade, 25% classwork, 15% participation, and 10% final at the end of trimester made up of the skills.

Students who do not understand fractions will understand it after printing out and working to understand pages 2 through 4 at this link. It's a great visual way of looking at equivalent fractions and breaking them down. I actually believe reading this blog post about the multiple personalities of fractions. http://mrreddy.com/blog/2013/08/the-multiple-personalities-of-fractions/. This teacher Mr Reddy explains it more clearly than I currently can. I also like using graph paper to visually show adding fractions. http://wp.me/p4TRVf-1r

If students do not understanding graphing coordinates, they can practice on Khan Academy or re-do the classwork sheet on it. Mathisfun.com has an explanation for percent error, which is Skill 3. Skill 4 is examining situations and determining whether they are proportional relationships or not (pass through the origin when graphed, a straight line, constant rate of growth, you can double one quantity and double the other quantity and still be proportional). This is section 1.2.1 from CPM.org.

For accelerated students, skill 1 is simplifying absolute value expressions. Skill 2 is solving linear equations that may have no solution or infinite solutions while checking the solution. Skill 3 is evaluating algebraic expressions using the order of operations (exponents). Skill 4 is evaluating square root and cubic functions.

### Day 13: Character Education Day

I have first period prep so students previewed some information about the CAMFEL 3-screen presentation. It talked about overcoming obstacles to do amazing things. The presentation ended early at 10 and I had my 2nd period students for 25 minutes before recess because of no KTLR. Students initially did not raise their hands at all. I distributed post-its and made a t-chart and allowed students to post in the middle if they wanted to discuss both. Left side was obstacles we face (height, divorced parents) and must overcome and how we will. The right side was ideas to be amazing people (ex: overcome height disadvantage to excel at high school golf, archery mastery, cross country runner).

I couldn't believe the amount of realness on the left side. Students all came up at once to put the post-its up anonymously for me to ready. Some highlights of obstacles: trouble making friends, anxiety, procrastination, mother passing away at age 10. We discussed the last one, saying we have to value our parents being alive.

The right side received some less thoughtful responses (safe) though I liked reading about a student wanting to help others with their problems, start a Jeans for Teens drive like in the movie, helping a classmate when they don't understand, and using American Sign Language to interpret for the deaf.

After recess with 3rd period we reviewed what it meant to be an upstander. Once again different students anonymously posted ways they have been an upstander or ideas on how to be one. A lot of real stories were shared about negative interactions in P.E.

In 4th period, we talked about being bystanders and being biased and taking the side of the bully. I also pointed out my student from last year about playing the guitar and singing at the talent show. I asked her how she was at guitar when she started and how it made her fingers feel. She said she wasn't good at the beginning and developed the callouses so it didn't hurt to play. This was an example of her having a goal and working towards it.

In 5th period we watched The Bully Project movie. The DVD was skipping over the network KTLR and we had like 30 minutes back in my room before the bell rang at 3. So, I told them we were going to play a math game (next time we will play 4 strikes by Marilyn Burns). This time we played PIG. Students got the hang of it and liked it a lot. Next time I should give them score cards so they are all accountable for the calculations. I wrote it on the white board and asked them to add the totals for me.

If you haven't played pig, here's the rules: All kids stand up. Roll a dice. Lands on 4. If you sit down, you have 4. If you stay standing, you have a chance to get more points. I roll a 3. Now people standing have 7 unless they sit down. Once you roll a 1, and you're standing, you get no points.

Overall, did not do any estimation (no estimating "cheeseballs") but we did a lot of deep digging about some tough to talk about topics.

## Wednesday, September 16, 2015

### Day 12: 3 ways to solve a proportion & function machines

I didn't feel comfortable moving on to introducing algebra tiles until Friday. Students needed thinking time to interpret and understand the 3 alternative methods to solving a proportion: Giant One, Undoing Division, and Fraction Busters, which leads to the cross product method.

Students reported out how to get the number in the Giant One. They applied these methods, while I was still happy when seeing some students successfully execute the cross products method.

Tomorrow is Character Ed Day, watching a Camfel film, debriefing, and "The Bully" movie after lunch.

In accelerated students explored the f(x) notation. Some did not immediately understand that f(x) is the output and represents y. I also let Harrison point out the ability to use the "plus or minus" symbol after you square root both sides of a quadratic equation.

I finished grading all Week 1 mastery assessments and will pass them back this Friday.

## Tuesday, September 15, 2015

### Day 11: Solving Proportions & Finishing Cubic & Absolute Value functions

Today students estimated the number of papers in a ream. We looked at easier strategies to convert the percent error from a fraction to a percent. I also went over the Study Team Norms. Ideally I would have liked to have a conversation with students to construct these themselves, but these are the norms I want to establish and ones that I refer to a lot. Students completed a table by dividing 2 quantities to find the multiplier and using it in the second quantity. I showed students how the Giant One is actually that same method, except written as a proportion (two ratios equal to each other). It's only a one day lesson, so students will tomorrow be introduced to the algebra tiles that will now include xy, y, and y squared.

In accelerated I talked with my students about how disjointed their group work was yesterday and their was too much independent work. Students did not create x y tables for the cube root of x and did not follow the directions in scaling their x axis from -150 to 150. More than one student scaled their Y axis this way. We talked about how and why this was incorrect. They also full described the function, noticing their were no lines of symmetry and the x and y intercepts were at the origin.

The absolute value function was investigated as well. Students saw there was a vertex at the origin, the slope was negative and then positive. They will be working on function machines tomorrow.

## Monday, September 14, 2015

### Day 10: Proportional Situations & Cubic Functions

In CC 8 we reviewed the post it notes from part A of the problem, which said Parvin washed 17 dishes in 10 minutes. Write down various times and dishes cleaned. The post its allowed students to critique each others work and remind them what they did on Friday. Then I let them finish parts b and c, then went over the students results. Then I told them to write down the requirements for a relationship to be proportional. I told them that they were going to tell me what to write. Students told me that when graphed it passes through the origin, is a straight line, grows at a constant rate, and if you double one number you can double the other.

Here in part c, a student deposits \$20 in a bank account that doubles every year. I asked students to notice the differences and which one was correct. They noticed the left one had year 0 at \$20, while the second post it had year 1 at \$20. They thought the one on the left was right because when you deposit the money a year hasn't passed by yet. I also made sure to point out that both graphs showed the fact that the lines were not straight. Students observed they were curved. I asked if it's growing at a constant rate and they said no.

This was part b. Basically, a puppy is born with a weight of 14 ounces. In 10 days it doubles it's weight. It continues growing at a constant rate. The one on the right was a common misconception of a weight of 14 ounces in 10 days. Then they erased and changed it to 0 days. Also, students noticed that 20 days should be 42 days, and not 56. Some students took it literally that it would keep doubling. That was just a coincidence.

A student in another class also said that if it went from 14 to 28 ounces in 10 days, then it grew 7 ounces every 5 days, because that was half the growth. I thought that was a great observation.

Students also analyzed a table that was not a proportional relationship and had to explain why. Students observed that if you found the unit rate for \$17.50 for 5 pounds of ferret food, then that's \$3.50. But when you multiply \$3.50 by 30 you don't get the entry of 30 pounds for \$89. They also saw 5 pounds times 6 equals 30 pounds, but \$17.50 times 6 doesn't equal \$89. Two great explanations. Also, some students graphed it and saw it wasn't a straight line.

In the accelerated class students finished up their square root of x posters, then I gave explicit instructions on how to give positive constructive feedback. They need to work on this. They also worked on graphing the cube root of x. The class didn't combine their ideas in their groups and did too much independent work which left students at all different points. Tomorrow they'll graph the absolute value of x for the first time.

### Day 9: Assessment Friday & red light green light 1.2.1 proportional relationships and Alg I square root of x posters

Today's estimation compared the 10 tissue package to a box of tissues. Using proportional reasoning, students figured 2 fit on the base, and there were 4 layers of that, so 2*4 is 8. So 10*8 is 80. It ended up being quite close to the correct answer.

In CC 8 we started on 1.2.1 which instructs students to create a table and graph for 3 non-routine possible proportional situations. The textbook suggests the red light green light strategy which I love and used but modified it a bit. I basically had 3 posters with the problem and a flap to fold up to check their answer. Once the whole group finished part a, they got a red light. The person who finished first copied their work onto a post it, and brought it up to check their answer, then stick it on when they were correct.

The best part of this study team strategy is that I have a visual aid of how students are progressing. I could see 7 groups had finished part a, 3 had finished part b, and 1 had finished part c. It was awesome feedback. Students didn't get too far because I saved 12 minutes at the end of the period for their first "standards-based" grading assessment. The first skill was adding and subtracting like and unlike fractions, skill 2 was graphing and reading points on the coordinate plane, and the final skill was percent error.

In the accelerated class, students started on their posters for square root functions and finished them up on Monday while giving feedback.

## Thursday, September 10, 2015

### Day 8: newtons revenge day 2 & quadratic functions gallery walk

Today was estimating a small tissue package a lot of students overestimated. One student said they thought that three tissues equals a pencil and it looks 3 pencils high so 3*3=9. It end up being 10 and we talked about 100% error 0% and 20% error.

Cpm guides students through human error in the scatter plot sticky dots  (inaccurate graphing or not adding the 100 cm to the meter stick below.

Students observed from their graph if your reach depends on your height. We discussed students with the same height but different reaches. Intro'd independent and dependent data on the axis. Students are many times unsure of describing the trend of the data. I made the analogy of a science experiment with plants with various hours of daylight and their heights as the dependent variable.

The next part was analyzing the inefficient scaling of the graph. Students understood it wasn't a 4 quadrant graph because there are no negative heights or reaches. Then if you should start at 0. Some told me about the discontinuous graph with the squiggle. Also why did I skip count by 25 instead of listing all the heights. Students volunteered it needs to be even and a pattern or numbers. I tried to stress why you can't have 160 161 then 163.

I asked students if it was safe for all of them to ride newtons revenge. They said yes because all of us are under 200. I asked where. And they said on the y axis. This wasn't visible on the premade graph but students were able to tell me on desmos during closure to make "the death line" as one student put it.

Then they are asked to see if the ride is safe for 228 cm tall Yao Ming. Some students graphed their groups measurement on their own graphs. I shared one under doc camera one of the periods. Though in 5th a student Nicole and Frankie presented their ideas to the class that they subtracted the reaches from the heights to find the differences and then average them. They got 40. They subtracted that from his 228 cm height and concluded that it was safe.

I showed my students how to copy and paste the data into desmos. This was the second representation that proves the safety of the ride.

Note to self. I forgot I could have done regression with y1~mx1 + b (all subscript numbers). Also to use projector mode to make interval text and line bigger.

It was blazing hot in 6th period. Students were tasked with using their textbook and the vocabulary from the desmos polygraph session yesterday to create a poster about a quadratic function.

Students labeled x and y intercepts and some understood the line of symmetry being x equals the x value of the vertex. I want to stress making that connection with better questions about noticing.

Class concluded with a gallery walk with each student getting a post it note to leave 2 stars and a wish. Some kids did not leave positive comments in the stars. It was a huge learning experience I thought too.

Video of the gallery walk:

Here are their posters:

## Wednesday, September 9, 2015

### Day 7: Newton's Revenge Day 1 & Polygraph: Parabolas

Most students were way off on the estimation today. Almost all students underestimated. We highlighted students who used yesterday's fact that there were 28 almonds in a 1/4 cup. Then some estimated how many cups would fit in the big container. It ended up being a staggering 40 1/4 cups. (1/4 of a cup was a serving)

Students had 5 minutes to discuss the Newton's revenge situation before we had a whole class discussion to understand what we needed to find out and how we would. I went over the stations for measuring height and reach. There were a total of 7 stations so all groups could stay busy.

After measuring their height and reach, they put their initials on a sticky dot and plotted it on the class graph. Also, instead of writing down a class list I made a Google form where they submitted their info to. I had students submit on my computer and to save time made a Google URL shortener so students could use their own phone to submit to the form. It makes it SUPER easy to paste it into the Desmos link that CPM provides in the lesson. So we will analyze their scatterplot and the one digitally tomorrow.

In accelerated, I didn't take any pictures or video. We met in the computer lab where we started with estimation. Then I had them play Polygraph: Parabolas. Students once again experienced the frustration of not knowing the vocabulary necessary to accurately describe parabolas.

I did point out some students who asked "Does it open up or open down?" Also another student asked "Does it go through all four quadrants?"

So, halfway through class I wanted to focus our attention on a question: How can we describe a parabola? So we took y=x^2 -1 (y equals x squared minus 1)

I asked them how we graphed equations before we knew y=mx+b. They said plug in points. And how do we organize it? A table. They found values for -2 to 2 and graphed it.

So, we could describe the parabola if it opens up or opens down. Also, they noticed there was an x-intercept at (1,0) and someone else piped in yeah and at (-1,0). They also said there was a y-intercept at (0,-1). They also said the parabola looks symmetric, like a mirror. So I asked where is the mirror. They said the y-axis. I asked what is the equation of the y-axis. One student knew and remembered it was x=0 (Alex D.).

Davin also remembered the vocabulary vertex. It's hard to describe, so I asked the students, is there a highest or lowest point on a graph? The lowest point in this case is where the vertex is, at (0,-1). I asked them what would the graph look like if the vertex was the highest point? They responded that it would be opening down.

Tomorrow each table group will get a quadratic function and have to graph it and describe it using all the vocabulary we went over and to put all their work on a poster.

When they got back to the game, someone said, "What are roots?" I said that was vocab I hadn't gone over but it's a synonym for x-intercepts. Great session.

## Tuesday, September 8, 2015

### Back to School Night 2015

You can view a PDF of my presentation from tonight here:

Presentation 2015

Common Core Math 8 Syllabus

Common Core Math 8 Accelerated Syllabus

### Day 6: Practicing Graphing & Transformations

Today's estimation challenge was a good one. It was estimate the number of almonds in the cup. At first many students said "I guessed 24 because it looked like it." Then some students progressed to counting what they saw, and then doubling it thinking that was how many were on the other side. I highlighted students who discussed counting a layer of 8 almonds on top and seeing about 4 layers total, and multiplying those together. I showed how those students were using their knowledge of how volume works in finding the base layer and multiplying by the number of layers or height.

I distributed textbooks today that can be used as a reference for topics we will be learning. CC 8 practiced graphing points on the worksheet Reteaching 1-10 and to finish for homework.

Accelerated students got textbooks as well and worked on Aaron's Design, a MARS task that reviewed concepts from last year that include transformations like reflections and rotations.

## Monday, September 7, 2015

### Day 5: Desmos Polygraph: Points... and Desmos Match My Pattern/Line with Accelerated

After the day 5 estimation warm-up and students self-correcting their homework, we dove right in to the Desmos activity Polygraph: Points by Robert Kaplinsky. I thought we'd have time to do Match my Pattern after, but boy was I mistaken.

So the premise is this: it's like Guess Who. You pick a graph of a point out of 16 possibilities. You are paired with someone in the class digitally. Then your partner must type you questions with yes or no answers so they can eliminate graphs to narrow it down to the correct one. I can monitor it on my screen.

For my second period class, they experienced the frustration of their peers not using specific language. Sometimes students would ask questions if it was in a certain quadrant, and since some did not know how quadrants were numbered, they'd answer incorrectly. Or it wasn't specific like is the point positive. Some students helped each other and I would repeat questions from the screen aloud to get their reaction.

Halfway through the period, I asked the students to close their laptop screens and get their composition books and pencil ready. I facilitated a discussion about the coordinate plane by relating it to the algebra walk from yesterday. I encouraged students to call out to move along the discussion so they could go back to the game armed with new vocabulary tools.

Me: Where did we all start standing outside during the algebra walk?
S: The x-axis.
Me: What type of line is that?
S: Horizontal.
Me: (discussion about the word horizon being in the word, the sun rises and sets out on the ocean when you look at the horizon)
Me: What's missing from our graph?
S: The y-axis
Me: Where should I put it, left middle right?
S: Middle
Me: OK. What's that point that I jumped on yesterday and asked you about?
S: The origin. (0,0)
Me: What happens to the numbers as I go right? And what type are they?
S: They go up. They are positive.
Me: And backwards?
S: They are negative.
(repeat for y-axis.)

Me: When there are more than 1 axis, the plural is axes. How many sections are there when the axes cross?
S: 4.
Me: What are they called?
Me: See how the root of the word is quad. You go out on an ATV Quad. How many wheels does it have? 4.

Me: So, how do we know how they are numbered? And how can we remember?

So a student Estefany had a great contribution: It starts at 1 at the top right, and moves in a direction of a letter C. We talked about counterclockwise and also roman numerals. Great discussion. I also asked them to identify points. I asked if the order of the numbers mattered. They told me its (x,y).

I asked Justin about a situation. Basically a student eliminated the origin after they had asked is it on the x axis to which he had answered yes. It also brought he question what quadrant is the origin in? Some kids thought all 4. Some thought none. Then some agreed it was on both the x and the y axes which also allowed me to introduce them to the plural form of axis.

Here's video of him explaining it:

Overall this activity really involved a lot of discourse between students between them and myself, and even our digital coach Patrick who admitted to the class he was rusty on how the quadrants are numbered. He asked the class what did they think he did? He said he googled it. Definitely a nice point to make in our Information Age.

Here's Darren describing the difference between 2 points:

In the accelerated class students did not play polygraph. They worked on Jon Orr's Desmos activity called Match my Pattern. Students refreshed completing a table and writing a rule for it to connect the points on the graph with a line.

It also prompted them to describe figure 2 when given 1 and 3 and also how many were in step 73. Once again myself and the students could see each other's answers after submitting.

The second activity was Match my Line which is snuck away in the activity builder  section of the web site. Match my pattern was a great lead in to this activity by Michael Fenton. Students had to remind each other how to find slope between two lines. Another student thought there was an error but then realized he forgot the x variable in his equation.

It was great to see students refresh on slope before we get back into it. I will definitely be doing polygraph: parabolas with this class soon before the textbook introduces them to the vocabulary of vertex line of symmetry and more.

Neither class had time to try Tile Pile, but since I still have the laptops in my room, on Tuesday I can have them do Tile Pile while I assign them textbooks after the warmup.

## Thursday, September 3, 2015

### Day 4: Algebra walk

I setup the xy coordinate with chalk for the 4th year in a row. This lesson plan from CC3 1.1.3 is by far the best learning experience my students current and past have had. Next year, I do want to expand it with more cards and 2 carefully measured ropes with duct tape evenly spaced out.

Using color coded sticky dots back in the classroom to plot their location from outside

I created a video clip on my flip camera that's only two minutes of the highlights. I shortened it down from 20 minutes of being outside. Thank you to our vice principal Miss Macalino for holding the camera.

The beautiful aspect of video is I can reflect on my terrible posture, what my students said, and what I said that sparked their thinking. After watching this, I am so happy I asked my students, "who is easy to find on the graph?" And students would always pick the person standing on the y-axis. It was a great point of discussion.

Choral reading of the equation and the sentence helped. Asking the students if it was decreasing or increasing from left to right lead to an awesome conversation once we got to y equals x squared.

For the accelerated classes, students collected the rest of their data, organized it on tables, and graphed the linear relationship, exponential, and inverse proportion.

## Wednesday, September 2, 2015

### Day 3: Describing Pattern Growth, Collecting Data

The estimation 180 warm-up was Mr. Stadel's son.
I liked that some students in another period said the boy was half of Mr. Stadel's height, and another student said Mr. Stadel was twice the boy's height.

I did not take pictures of students work on the tile pattern, but students pointed out how the bottom row was growing by a total of 2 tiles, a column was added, and the height of the column was 1 more than the previous tile figure. Students justified to the class how their Figure 1 looked, because it was a point of contention at each group.

Using 36 squares to make different hot tub designs.

Algebra walk, 1.1.3, is setup for tomorrow outside... courtesy of green and blue sidewalk chalk. All laminated posters, sticky dots, and color-coded notecards are ready to go. KTLR, out student news network should be by to record one of the classes, and I will bring my Flip camera. I contemplated buying a rope to make it in a bigger area but that will be for next year. I also modify it and use only -3 to positive 3 instead of -6 to positive 6 like the lesson suggests.

## Tuesday, September 1, 2015

### Day 2... more estimation, 4 coordinates, and function machines

I am super proud of the banner I made above my white board to match our Estimation 180 sheet. Today we estimated Mrs. Stadel's height. I liked how Mary Catherine made a safe but accurate too high estimate of 6'4" because that's Mr. Stadel's height. We continued our discussion about percent error with a few of you remembering the ratio is error over actual answer. A lot of you remembered to convert to inches. Right now we are focusing on whole number percent error, but as we get further in the year we will look at rounding to the closest tenth of a percent.

I really liked this groups work because they finished early and accepted my challenge to add more labels to their axes and were quite successful. What can you learn about each student?

The activity above sparked some great discussions in the accelerated class. The above picture is NOT the correct order. Basically, you want to input 15 into one of the function machines at the top, and arrange them in order below it so your final output is negative 6. Hans worded it beautifully. The output for the function becomes the input for the next function. There was some confusion with using the output of the first function as the output of the second function. Also, when I asked groups which function machine can't be last you said the one with the exponent because anything times itself will be a positive number. Great. Some of you also realized it couldn't be the first function because it gave you an output that was way too big. We will wrap up our discussion on this tomorrow.

Tomorrow we will be embarking on 1.1.2 in CC3 for Common Core 8 class and for accelerated 1.1.2 in CC Algebra. Great work ladies and gentlemen.

### day 1: satisfying (sweat and sore feet included)

I had a great first day meeting all my students today. Managed to give all of you a high 5 while taking roll on the way in and completed day 1 of estimation180. Remember: you can go to estimation180.com and make it up when you're absent. And once again, please don't tell the actual answer to later classes, let them make their estimations, thanks.

Great conversations about writing 5 9, 6 feet, 5 1/2 feet, 5 ft 6 in, and the proper math symbols and punctuation as you wrote it 5'9". You transitioned from "little line thingy" to apostrophe and we got our  quotation mark double 2 finger salute to doctor evil.

Reasoning was great. Some of you compared Mr. Stadel's height to average men or your dads height. I also heard references to the fence behind him as well as the bush below it.

Lastly, counting up strategies were heard when finding 3 inches to get from 5'9" to 6 feet and then adding 4 more inches to get to his total height and a total of 7 inches error. Then the conversations about how to make a percent error dividing error by actual answer. We of course shook off the rust and did some choral long division.

As you worked, pieces of 8 graphs were handed out randomly and you found your new group members. After introducing yourselves and figuring out your team roles it was glue and brainstorm time.

I think students had the best thought processes when I asked what's happening in part one and when have you seen a situation like that. The creativity was great. You also thought about how to label the x and y axis to correspond to your story.

I like the vocabulary usage of how the water supply is changing.

Some vivid details here!

Itsy Bitsy Spider... Junior

I like how this group labeled their y-axis with numbers and their y-axis with years that corresponded to their story.

In the accelerated class students used the cc algebra card sorts to match a situation or pattern to its table rule and graph. I could tell some of you didn't think of y=mx+b until I asked what equation did we spend a lot of time on last year that started with y equals... Then your conversations got going.

I like how this group made specific connections to other representations of the linear pattern.

Great day and I am charged up for this school year. Mathography letter to me due Wednesday.