## Thursday, October 29, 2015

### Day 41: Guess my Number & Taco Cart

Today's estimation was how high is the giant wheel? It's a huge ferris wheel, with Mr. Stadel standing at the bottom. We confronted the mistake of multiplying his height 6 foot 4 inches as 6.4 feet. It still ends up helping you make a reasonable estimate. Some students reasoned to use 6 foot and 1/3 then use distributive property. Others said round up to 6 foot 6 inches which is 6.5 feet.

I passed back assessments and went over the grading rubric. Some students were out on a drama field trip. Then we played guess my number. This is a great activity because it starts off easy with one solution. Students were eager to convert them into equations. Some great conversations were had with the equation with infinite solutions. Some classes told me one answer. Then I asked if anyone had a different answer. So, we proved that there were two solutions and more. So, they said it must be all solutions. I asked them how many and they told me infinite solutions. It leaves you with x=x and eventually 0=0 which means that there are infinite solutions. The next example had no solution. They said there was no solution because the variables end up getting canceled out and you're left with a false statement.

Then I did community building. It was entertaining and we got to learn about one person in more detail in each class period. Then they worked on some practice equations. I instructed students to write it down and ask the group, so what do you think we should do for our first step? I wanted them to work together and most did. I wrote up the first 4 problems and had volunteers come up and solve them on the board. Some were brave enough to come back up and explain their steps. I had to explain some people's steps that were too shy.

I told students that I was going to a funeral tomorrow and that they would be completing a sheet called Mystery Numbers then would put up the privacy folders and be given 25 minutes to work on the weekly assessment.

In accelerated, I spent a bit more time going over the answer key to their test. Some students are having trouble writing a linear equation given slope and a point, and making some mistakes when only given two points. A lot of students forgot what strength meant when drawing a line of best fit through data.

After discovering the patterns of side lengths that form triangles, and the area of squares that formed acute, obtuse, and right triangles, I introduced Dan Meyer's Taco Cart task. I explained the format of the 3 act lesson below:

Here are all of the questions they came up with. They didn't end up getting the main question until later:

Why do you think you walk faster on the ground?
What’s the distance between the person’s starting point and the taco cart?
How fast do you go on the street and how fast do you go on the sand?
Are they taking any breaks? (no)
Are they walking the same speed?
Are they walking at a constant speed? (yes)
Can you run?
Is the taco cart going to move? (no)
Are there any obstacles in between you and the taco cart?
What is the distance between the starting point and the road?
Can I see a scale map and layout?
Does the road have vehicles on it? (I hope not)
How much time did it take for each person to finish?
How far is it from the point from where you get to the side walk to the cart?
Who gets to the taco cart first, Ben or Dan?

I made the questions that I would answer a bigger font, and answered some as I wrote them on my Google docs document. Then, in Act 2 I gave them information they asked.

All students were truly engaged in the process. I had one student who got it help me walk around confirming if others had understood it.

The main goal was to ask students questions that helped them improve the detail of their work with units and labels so that anyone who didn't understand the problem could get it from looking at their work.

One aspect I overlooked was letting students keep their answer in seconds. I should have asked them for minutes because that brings another conversation about converting decimal minutes into minutes and seconds. I will revisit that at some point.

## Wednesday, October 28, 2015

### Day 40: Checking Solutions & Summary of Solving Equations (Sub day for PD with SVMI)

My students had a former Taylor math teacher as their sub today, Mrs. Seffens. She taught at Taylor for 30 years and I was lucky enough to work with her a bit and take over her job. Luckily, all the classes behaved except one with two particular students. Those students will be writing an apology letter on their free time.

Common Core Math 8 classes worked on solving equations and checking solutions. The accelerated class solved a variety of equations. They solved multivariable equations for a variable, absolute value, and equations where one side was two binomials being multiplied such as (x+3)(x+2) = x^2.

Our school is a member of SVMI, the Silicon Valley Math initiative. This picture shows what percentage of the Smarter Balanced tests are DOK levels 3 and 4, compared to the old CST or standardized tests.

This was an example of two tasks that assess student's knowledge of perimeter and area. The one on the right has a much higher cognitive demand because your answer must be justified and is not as straight forward as the one on the left.

We looked into the TruMath Suite on the Math shell web site. We focused on Cognitive demand. I had some experience learning about this at my summer professional development (PD) with CPM. We did a task sort today, to see how the level of cognitive demand of a task was. We related Bloom's Taxonomy to the new Webb's DOK or Depth of Knowledge levels. It's on levels 1 through 4. I took a picture of our poster below:

And I also took a picture of a great example poster where they cut out the example worksheet with the keywords for each level which was a brilliant idea.

I graded all students assessments tonight and will turn them back tomorrow. They will have a substitute tomorrow and will be given the last 20 minutes of class to complete this week's assessment, after working on a worksheet.

I wanted to point out a student who was successful on a skill on their test and added some humor that I thought was funny also. This was testing understanding of scatter plots.

## Tuesday, October 27, 2015

### Day 39: Reteaching Integers & Using Squares to Make Acute, Obtuse, &Right Triangles

Today's estimation was total blow pops in a box of bags of them. The video reveal was quite entertaining.

I'm not happy with my students understanding of integers so I've created the following document. It will involve some direct instruction and I want to get as much student participation and choral response as possible.

If you can't view it, the link is here. A fellow teacher on twitter suggest I only introduce one strategy in a day, and I took his advice and only discussed plus and minus tiles. Next week I'll review how to use a number line to visualize adding and subtracting integers.
Hearing students report out what they knew about integers was a bit uneventful. I did like that some students related it to how the x and y axes are labeled on a graph before you graph points. One student thought that decimals and fractions were not integers. I asked if you could have -2.3? They said yes. One student mentioned a tic taco to with a single diagonal of plus tiles and the rest minuses. A teacher had taught them this was the rule for multiplying integers. In the next period, a student brought it up, and said, "I was taught these are the rules for adding. A negative plus a negative is a positive." A student raised their hand and interjected "that's not true" and gave an example.

It basically backed up the need for the PDF file that I've read, and that all teachers should read: Nix the Tricks. It's created by teachers for teachers, parents, and students. It's great. Of course I also heard the dreaded same change change which they said is the reason when you minus a negative it's a positive. I once again said that that rule could easily be confused with multiplying integers. A student also said a teacher had taught them when they see 3 - (-2) that the 2 minuses and the left hand parentheses look like a plus so make it a plus by connecting it. Some students also said that integers go on forever.

When I asked what questions they wrote down that they had about integers, I didn't hear many courageous or honest responses besides one student in 2nd period who said "I want to know what you're allowed to do when adding, subtracting, multiplying, and dividing!" Other students did not want have questions to share or did not feel like sharing them.

We discussed adding integers chorally. I also said what is 4 + (-10) sort of the same as? Luckily a few students said that it's like 10 minus 4, and then make it negative. This gave me the opportunity to show them when adding different signed integers you are subtracting the absolute values. The difference takes the sign of the integer with the greater absolute value. Students also liked learning the words addend, minuend, subtrahend, and difference.

After that, students wrote down 4 example integer problems. One that was a negative plus a negative. One that was adding integers with different signs. One with a smaller number minus a bigger number, and finally one where you were minusing a negative. Then they exchanged post it notes and solved someone else's. Then they checked their partners work and talked to them if they got it wrong.

The last 15 minutes were spent on the Math Shell formative assessment lesson pre assessment called Building and Solving Equations. We will begin this lesson Thursday and finish it on Monday.

In accelerated we got to do a lesson we skipped in Course 3 yesterday and one of my favorite lessons (9.2.1): using side lengths of squares to discover patterns about acute, obtuse, and right angles. Students were quick to say that it was the pythagorean theorem. They also were reminded and proved again what must be true for 3 sides to be a triangle. Aquiles, our ASB president, and new student to the school and the class was able to report out to the class that the sum of the 2 smaller sides must be greater than the longer side. He did not learn that at his last school. That was a great moment.
Homework:

## Monday, October 26, 2015

### Day 38: Discrete & Continuous & Solving for Celcius in Fahrenheit Equation

Today's estimation was how many blow pops in a bag. I was impressed that a student saw that the corners were triangles, and the middle was a diamond made of 4 triangles. So the bag is a total of 8 triangles. Students estimated about 10 blowpops in each triangle then multiplied. Ended up being quite accurate! We also discussed ways to change a fraction with a denominator of 80 into a percent using good strategies.

Since my colleague Ms. Demailly fully covered discrete and continuous graphs with her classes and wanted it as the last part of our graphing situation skill question on the test, I had student volunteers read the Math Notes box. I then asked them to paraphrase the definition of discrete. It's basically a fancy way to say the points on the graph are not connected because the values in between do not make sense. Conversely, a continuous graph has points connected because there's growth in between the points. For example, you can find out how tall John's Giant Redwood Tree is at 2.5 years by looking at 2.5 years on the x axis and then going straight up and seeing where the line lines up with the y-axis.

Then students reminded me how to get the perimeter of the dented square from Friday. They reminded me of the two ways. Then I asked what values x could and could not be. They said it couldn't be negative and it couldn't be zero. They also said it couldn't be 1 because the unit tile's side length is 1 so it can't be the same as the x value. So, we made a table of x values 1 through 4 and they told me the outputs or y values. Then I asked how to scale the graph, and graphed the points. About 4 or 5 kids in each class remembered how to graph the inequality x>1. I asked them how we could show this on our coordinate graph. A few kids came up with an open dot. Then I asked if this was a continuous graph from there? They said yes, because x could be decimal numbers.

Then students worked on Goofy Graphing. They looked at a table, extracted the (x,y) coordinates, and then analyzed the mistakes students made in setting up the graphs. The end result was 3 huge mistakes to avoid when setting up a graph:

1. Figure out how many quadrants you need to you can be efficient. If it's all positive numbers, only use a 1st quadrant graph.
2. When graphing your coordinates, make sure you graph x first then y in an (x,y) coordinate.
3. Always include the origin, (0,0).
4. Even intervals so the graph is efficient and spaced out properly.

In accelerated I had volunteers review the absolute value equations. Then they started changing an equation from standard from into y=mx+b form. They showed me two different ways to do that. Then I had volunteers solve for force in the work formula: Work = Force * distance or W=Fd. This was difficult for some students. Then Alex David showed us how to solve for Celcius in the formula F=9/5C + 32. Some students still didn't understand so I showed the intermediate steps that he did mentally.

Then a student tried and got close to solving the hardest problem that involved an exponent:
I= W(12.6r^2)

Davin ended up doing it on the board and Nicholas confirmed it was correct. What I loved was that the first student was outside my room kneeling down on the ground and fixing the problem. Talk about perseverance! I was very proud of him, especially after when we gave him a 3 second clap for his effort in class and as he walked back to his desk he said "I bet everyone is happy to see I got it wrong."

## Friday, October 23, 2015

### Day 37: Dented Square & Absolute Value Equations

Today's estimation was how many vines in a full container. It's a great video and students loved the anticipation. I liked one student's reasoning who said they thought there were 10 handfuls in a container, so since 1 handful is 18 red vines, then 18*10 is 180. This was close to the correct answer. We also talked about estimating the percent error.

I had students grapple with the perimeter of a dented square. I asked probing questions as students worked on it. Once half the groups figured it out I had students explain how they got the perimeter. Some students were able to explain it two different ways. You can label all the sides, or label the right side x and erase the left side of the unit square that's labeled 1. Or, label that side 1 and the right side is x-1. Then we discussed the boundaries of x. They realized it had to be positive, and it couldn't be 1, so x>1. On Monday we'll finish talking about how this is shown on a graph. They took their assessments on skills 6, 7, and 8. I can tell students remember nothing about discrete and continuous graphs to we will review that Monday and write some definitions. Also, on Tuesday, I will be doing a crash course on integers using number lines and integer tiles. Then they will complete a pre-assessment that will set up a Formative Assessment Lesson for Thursday and Friday.

In accelerated, class didn't start well. I had to deal with a student who was disrespectful and also talk to my class about how they told a sub that another teacher (Miss Wong) refuses to sub their class. I told them that that is nothing to be proud of. I also said the fact that the best teacher on campus doesn't want to sub your class, should be something you're ashamed of.

Students worked on distributive property and absolute value equations. It was a bit frustrating because once again some students were not working cooperatively, leading to many students at different paces. One student came up and gave a great explanation of how to write 2 equations to solve from 1 absolute value equation, which is pictured below.

Next week, accelerated will learn Pythagorean theorem, a topic every 8th grader will learn later on in the year during chapter 9.

Also, a former student who I taught in 7th and 8th grade, Jorge came to visit me at lunch. He was on campus after coming back from a week at Outdoor Education being a cabin leader. He's a sophomore at Mills taking Geometry now. Great to see former students, especially seeing progress and growth both physically and mentally.

## Thursday, October 22, 2015

### Day 36: Complete Graphs, Graphing & Generic Rectangles

Today's estimation was how many red vines in Mr. Stadel's hand. I liked how one student grabbed a handful of pens from his supplies, counted them, and then rounded up. I thought that was a creative strategy. They will use that clue to estimate how many are in the whole package.

Today started with a Silent Board game. It was nice to discuss how students saw the rule after seeing students mistakes. I commended students willing to make a mistake and also how that mistake provided a valuable clue for them and the rest of the class.

Then they analyzed three graphs with errors on them. This lead to them understanding 3 aspects that make a complete graph:

1. The x and y axes are labeled with number intervals or else you don't know the coordinates.

2. The line must extend all the way left and right and also have arrows on each end. (we also discussed what it was called without arrows. At least one student knew it was called a line segment. Then I offered a bonus question. What if it ends on one side and extends with an arrow? They knew it was called a ray)
3. The x and y axes must be labeled.

Then students were instructed to graph y=-x+1, y=0.5x+2, and y=x^2-4. I asked students how you could read y=-x+1 after seeing mistakes in 2nd period. They only knew one way, so I asked them when we brainstormed the meaning of minus, how do we properly say -(-3)? Students remembered that that is called the opposite of -3. So, the equation can be read, "y equals the opposite of x plus 1". I had them copy that. I asked them what is your first step before you graph a rule? They said make a table. I showed them a horizontal table, but also made a T symbol and showed you can make a T chart, which is a vertical table. I also said you always want to include 0, but include two numbers to the left and right of zero for a complete graph, so x values of -2 through positive 2.

Evelina's graph was nice and compact, no wasted space!

Then students graphed, I circulated and asked clarifying questions, especially those who numbered axes in the wrong order. They also analyzed how the graphs were the same and different. They also labeled their x and y intercepts with coordinates. I also did some community building with mathographies with 15 minutes to go.

Jarren was moving along quite nicely on all 3 graphs! Next stop: labeling x and y intercepts.
In accelerated we discussed the pattern for fractions with a denominator of 9. I threw at them, what would 4/9 be then? What about 9/9? They said 1. I said is there another way based on the pattern? They said .9 repeating. This proves that .9 repeating is equivalent to 1.

We also read the math notes box aloud that introduced the vocabulary polynomials and binomials. They worked on distributive property, and then were introduced to generic rectangles to multiply binomials.

So each week I build community in my class with letters they wrote to me at beginning of year. I said stay standing if you are helpful in groups, if you love classic video games like Zelda, stay standing if your birthday is August 3rd. Then we honor that student with 3 second clap and I ask them to elaborate about themselves. Well my accelerated class is same 32 kids from last year so one said how come we don't get to find out more about you? So I gave them all a post it note and said sign their name on it and ask me one appropriate question and I'll read it and answer it to the class each week. One question was why do I love game of thrones so much? This week was can you beat Harrison in an arm wrestling match? With 2 minutes to go in class I had to oblige. Very fun. I won but not easily. Haha.

Here is a link to an arm wrestling match between myself and my student Harrison:

## Wednesday, October 21, 2015

### Day 35: Good Tipper & Distributive Property Using Algebra Tiles

Today's estimation was a quick one. How many diaper in whole box? Students estimated 4, 5, and 6 packages of 30.

Today started off with another silent board game. I was surprised that out of the whole class, roughly 5 to 8 in each class figured out the rule and could contribute. I think the issue is students can multiply by a negative, but then when adding a positive to a negative they are not fully proficient. We will definitely be spending a day some point soon on adding and subtracting integers using plus and minus tiles along with a number line.

The lesson then continues with Mr. Wallis' tip table. I asked students what background knowledge they knew about giving tips at restaurants and how much they should be. Some students knew that it is money you give to the waiter based on how good the service is. They also knew that it was a percentage of the dinner bill. Students contribute 8 typical prices for a dinner for two at a restaurant. This gave me a chance to share with students that at the dinner where I proposed to my wife it cost me \$180. They calculated and were amazed that on a 15% tip I payed the waiter \$27. I told students in small groups that the wait staff was very attentive and helpful and I gave them more of a 20% tip.

Students calculate the various tip amounts, estimate some tips, and then re-familiarized themselves with dependent and independent variables and how the y axis is always the dependent variable. We related it back to the height of John's giant redwood tree depends on what? The years it's growing they answered. I also related it back to the Newton's Revenge problem where the dependent variable was their reach, which depended on their height.

Students graphed it, and then didn't get much further than that. It was Wednesday, short day.

In accelerated, we discussed the first problem with getting the dimensions of an algebra tile collection. Haley caught a mistake in my drawing and the class gave her a hearty clap for noticing.

Here students were only given the drawing of the algebra tiles. They told me the dimensions and the area as a sum of it's parts. To the left of the exit sign is the unsimplified version and below it is where the like terms were combined.

Then students worked together on building the collections. They worked together pretty well. I must say, I walked around and didn't interrupt much because they were busy the entire time. I've taken some pictures of some of their collections as they follow:

Here students were given (2+x+y)(x+4) and came up with this figure! You can see that the sum of it's parts is what you get from multiplying it, which is x^2+6x+xy+4y+8
Here they are in the process of building a collection...
And the finished product with all tiles turned over to the positive side. They were given (x+xy+2)(2x+5) and came up with 2x^2+9x+2xy+5y+10.

Students really enjoyed using the manipulatives. They did complain that it takes awhile to set it up with the tiles, but it makes it quite easy to combine the like terms. It also helps students who are kinesthetic learners and like hands-on learning.

## Tuesday, October 20, 2015

### Day 34: Completing Tables to Graph a Rule & Factoring Using AlgebraTiles

The estimation was a package of diapers. I asked students if the sticker in the middle helped them at all. Some students said the sticker was about 5 diapers wide, and the package was about 7 stickers wide, so 35 diapers. Some said up to the sticker is 10 diapers, so double that and the total is 10 diapers. Both were sound reasoning. I think some were surprised how thick they were.

In lesson 3.1.4 I gave each table group 2 sticky dots and 2 x-values. I structured it better after first period. I wrote the rule y=-5x+12 and modeled it for x=0, and how you showed your work with substitution. They then got an output value which created an ordered (x,y) pair. They then wrote that coordinate on the sticky dot and graphed it on the board. I asked what patterns they noticed with the graph and the points. As the graph was going left to right, as x increased by 1, y decreased by 5. I asked where they could see that in the rule. They said it's after y= and before the x. Also, I asked what point was easy to see. They said the one on the y-axis. They said it was (0,12). Same question. They said it's the plus 12 part of the rule.

Then I had students copy the table that had x values from -4 to 4 and they had to figure out the y values for the rule y=2x+1. This reinforced that I need to reteach integers to the class, seeing how students struggles with negative values. I also encouraged them to calculate 2 and try to see a pattern. They then setup their own x and y axes, graphed the points, and connected the points to form a line. They also practiced on a parabola, and we discussed how to properly say it. Also, what a vertex is, and what the vertex of y=x^2 is. I related a lot of this lesson back to the first week when they did the Algebra Walk on Day 5.

In accelerated, we got to a lesson I was really excited about. They were given a jumbled mixture of algebra tiles in a picture and were told to build a rectangle. Then to label the side lengths and set it equal to the area of the sum of the algebra tile's individual's areas. All the students that do enrichment math work at Kumon finally understood what factoring meant with algebra tiles. I encouraged students that already knew procedures to use the tiles to challenge and push themselves. I took a bunch of pictures of configurations students made and also some of the creative arrangements along with some jokes of is this one okay? (the really long 1 unit wide rectangle). One student joked that people that work at Kumon have to eat too. I got a good laugh out of that one.

Oh, and this student was a bit off task, but brought up an interesting point. He claims he now knows what y equals in algebra tiles (purple is y squared). He says it must be 5 and 1/2 because 2 y's are 11 unit tiles long. He's trying to prove that the tiles are non-commensurate, and I'll remind him that they are. Y is unknown!

## Monday, October 19, 2015

### Day 33: The Big C's & Equation Mat w/ Alg Tiles

Today's estimation was how long is the new paper towel roll? They already know how many sheets a new paper towel roll has, so I will ask all the classes what do we already know about a new paper towel roll?

Today will be a recap if what they learned on Friday when I was out with the Big C's pattern from lesson 3.1.3. I want to show students these awesome posters that I plan to have my TA make for my class to help operate a graphing calculator:

Also, students will be inputting their x,y tables into the graphing calculator, and then graphing their rule and noticing what happens. Here are the instructions from my blog post from last year:

1. First off, press STAT, then press Edit...
2. Here you will see a screen with 3 columns labeled L1, L2, and L3.
3. L1 represents your x values, and L2 represents your y values. Enter in a list of points, in this case it was (1,9) (2, 15) (3, 21)
4. When you press GRAPH in the top right, you'll get 3 points.
5. For some students, no points showed up. I troubleshooted this problem and found out that you need to press "2nd" "STAT PLOT" (where the Y= button is) and make sure that your Plot 1 is on so that the L1 and L2 values show up.

(end of anticipation lesson plan, start of reflection on lessons)

So, estimation went alright. I liked that some students estimated the length would be 10 inches, so 10 *80 is 800 inches, then divide that by 12 to see how many feet long it was. I also asked students to see how Mr. Stadel calculated that the paper towel was 93.3 feet long. This took some careful noticing.

I discussed with two of my classes in particular the sub report from Friday. I wasn't happy with 2nd and 3rd period's performance. I talked about how a student can be one of 5 people when a substitute is teaching:

3. A positive follower
4. A negative follower
5. A bystander
I discussed how it is easy to be a negative leader or negative follower, but much harder to be a positive leader and positive follower. A bystander at least does their work, but does not regulate on other students. A positive leader and sometimes a positive follower are the only people that direct their peers to make better choices, and are clearly the most difficult roles to play.

I took the chance to discuss this with students outside, before giving them their new teams. I had students discuss for 5 to 10 minutes Figures 0 and 4 for the Big C's, the table, the rule, and the graph. Some students thought the rule was y=x+6, and some thought it was y=6x+3. They also had difficulty describing what Figure 100 looks like, but some were able to input 100 for x and find that Figure 100 had 603 tiles. I tried to get students to label sides, and notice any patterns related the figure numbers. The beauty of finding Figure 100, is it is a nice transition to finding "Figure X," which leads to the rule. As I guided students through that while asking questions, they were amazed that it ended up with the rule, and also gave a chance to review combining like terms. I asked where does 6 show up in our pattern? It's the growth. Where does 3 show up in our pattern? It's the number of tiles in Figure 0.

Unfortunately I only had 4 graphing calculators that had working batteries so I had to demonstrate the table and rule using Desmos on the projector. Then they reviewed their answers to the last problem that analyzed ideas about how many tiles were in certain figure numbers.

In accelerated, it was a 2 day lesson in 1 day. The perimeter of algebra tiles was a review, but students were definitely confused about some of the legal moves and solving equations with tiles in the negative region. I'm not too worried about this because students showed their proficiency of solving equations on assessments prior. I will make sure they are tested again on equations that have the distributive property like 2(x+3) and -(x-3).

Students were able to show their ideas of finding the perimeter of tiles, and student volunteers showed and explained how they found the perimeter of two complex algebra tile figures.

Tomorrow will be brand new, and show an intuitive way of factoring quadratics using algebra tiles.

## Thursday, October 15, 2015

### Day 32: Sub Day - The Big C's and Negative Exponents

Periods 2 through 5 will work on the Big C's problem, drawing figure 0 and 5, graphing it, completing a table, and writing a rule. We will work with graphing calculators for the first time on monday with this task.

Accelerated will work more on negative exponents.

### Day 31: Silent Board Game, Giant Redwood Tree & Exponent Mistakes

Today's estimation asked for the number of sheets on a roll of paper towels, when compared to a toilet paper roll that has 435. I liked that students discussed thickness, and that the paper towel was 3 times as big, so they divided 435 by 3, and rounded down.

Lesson 3.1.2 starts with a Silent Board game. This is where you can see the students who can think quicker than others, and figure out the rule by telling the missing output values when given just inputs and a few correct inputs. Take a look at the Silent Board game A for yourself, and students practice it if you want more practice. Students don't tell me the rule until all outputs are filled in. This was also practice from the previous nights homework.

Then students move to analyzing a table to figure out how the tree is growing. They are also given its height in years 3, 4, and 5, and are asked to explain how they get the height in years 2 and 7. Then they are asked to figure out the height of the tree the year it was planted. A lot of students assumed the answer was 9 feet, because that's the height for 1 year. I asked them, "is that the height when it was planted or after 1 year of growth?" That made them rethink their answer. Some students assume it's a proportional relationship especially when they predict how tall it it is in 50 years. Some estimated 250 feet because in 5 years it's 25 feet, so they multiply 5 by 10 and 25 by 10. Some estimated 205 feet because it grows 4 feet each year, and 50 times 4 is 200 and add the 5 feet for when it was planted. They test these predictions at the end of the lesson.

After the estimation, they completed the rest of the table. They are asked if it makes sense to connect the points. It ends up being a continuous graph, so it makes sense to connect it because it's growing in between each year at a constant rate. Students did mention it grew at a constant rate. Thankfully, they did say it was not a proportional relationship because it did not pass through the origin. I asked students to improve their answer if they said it didn't start at 0. James made a great comment, that yeah, it does start at 0, it's 5 feet at 0. I took that opportunity to mention he was displaying the common core standard of mathematical practice of using precision of language.

We then had some closure talking about the rule of the graph, and seeing if their estimation was correct or not and why.

In accelerated, I finally got to deploy a lesson I've been dying to try, which is Exponent Mistakes by Andrew Stadel (maker of our warmups, Estimation 180). It has 8 problems on it and they are all incorrect. They are instructed to explain the mistake, correct it, and justify why your new answer is correct.

Students work in groups almost daily in this class, so working for 20 minutes independently was a nice change. I then asked them to discuss what they came up with their groups after that. I then saw some lively conversations, some erasing, some changing of answers, and finally I prepared for volunteers to go up to the board.

So, I asked any student to come up and analyze any of the questions on the board. Alex showed why 2^5 does not equal 10 with a great explanation. Zoe attacked the 7^-2=49 mistake and said that 7^2 is 49, so 7^-2 can't be. She said 7^-2 is like 1/7 * 1/7 so it's 1/49. I then asked Harrison to discuss the 37^0 problem, and relate it to why negative exponents are fractions. He said as you increase the exponent you multiply by 7 again. As you decrease the exponent you divide by 7. This explains why 7^1 is 7 because 49/7 is 7. Then it shows why 7^0 is 1 because 7/7 is 1. Therefore, 7^-1 is 1/7 because 1 divided by 7 is 1/7.

One student attacked the problem 100^1/2 and said it was 1000. Another student at his group came up and wrote 100^1/2 is 10 because anything to the 1/2 is square root.

I wanted students to move away from the answer is this because of a certain rule. I wanted them to prove it through patterns and reasoning. I basically explained that what they were doing was algebraic proofs in a way. Prove it and convince the class why you are right, in a way that more than a few people could understand.

I asked the class, are you convinced? Can anyone prove that? So, as Mr. Stadel suggested, I asked students to calculate 100^1, 100^2, and 100^3. Alex D told us the answers. I then asked, how could this help us understand 100^1/2? Markos said that since each time you increase the exponent, there's 2 more zeros, then when you decrease the exponent you take away 2 zeros. But between 1 and 0 that's 1/2, so you take away 1/2 the amount of zeros, or 1 zero. There were a lot of a ha's. Then I asked for factors of 100. I said how else can we rewrite 10*10? They said 10^2. So I substituted that and it showed (10^2)^1/2. A lot of a ha moments and hands raised. Davin explained that you multiply the exponents so 1/2 of 2 is 1, and 10^1, is 10.

Nicholas had time to prove x^3/x^8 is 1/x^5 because he broke up x^8 as x^5 * x^3. Then canceled out the x^3 in numerator and denominator. I asked students what other way could we write that? They said x to the -3. I asked what rule does this prove? Gabriel stated that when you divide exponents with the same base you subtract the exponents. Great discussion!

We got through all the problems except for 1. This was a lot of fun and students genuinely enjoy coming to the board.

Tomorrow students have a substitute, so behave your best!

## Wednesday, October 14, 2015

### Day 30: Solving Equations on Whiteboards, Patterns & Frequency Tables

So yesterday was a PD day so students had a 4 day weekend.

The estimation was a good one today, what is the total length of a roll of toilet paper if we know there are 425 sheets. I liked students that estimated how long one sheet is in inches (3 or 4 inches) and then multiply that by the number of sheets. I even had one student use centimeters which gave a great opportunity in 4th period to discuss centimeters in a meter, because the answer was given in feet or meters.

After looking at some of their assessments I was very concerned with their understanding of solving equations. It almost seemed like an all or nothing, not much middle ground. So, I asked students to practice solving equations a through f of problem 2-75 over the 4 day weekend. I put the answers on the board and picked a simpler equation and a harder equation to go over. Basically I wanted students to use an equation mat if they didn't know how to start. Raise their hand when they got an answer and I can say yes or no. If yes, turn it over. If no, re-work it, or I gave specific feedback on what they did.

I think students are motivated when using whiteboards and I can instantly see who gets it and who doesn't. It's great formative assessment. Then students copied down figures 2, 3, and 4, of a tile pattern, and drew figures 1 and 5 based on the pattern's growth. They then described what Figure 100 would look like. I was happy to see that students were able to work backwards.

Homework is 3-4 to 3-9.

In accelerated, we took a survey to see any association between playing a musical instrument and playing an individual or team sport. Instead of starting with a hypothesis, I should have had students come up with their own conclusions based on the data. The hypothesis students came up with is if you play a musical instrument you are more likely to also play a sport. This hypothesis ended up being supported with the data. I got this lesson idea from illustrativemathematics.org.

Then students worked on making a relative frequency table based on the heights of corn stalks in sandy or clay soil. In the future, I want students to come up with their own 2 yes or no questions for their classmates to see if they can make an association between two questions like we did in class.

## Friday, October 9, 2015

### Plan for Desmos PD with our math department

Math-specific Technology Professional Development

Goals: Share what I learned at the CPM Best Practices Academy regarding Desmos. Demonstrate lessons with teachers acting as students with Desmos activities that can be implemented this year, for each specific grade level. One of the LCAP goals is to infuse technology in our classroom lessons.

Background: Eli Luberoff, CEO of Desmos, presented to 32 teachers this summer at CPM’s professional development on the last day. I implemented what I learned below:

I blogged about a summary of everything I learned at the week long professional development here:

I reviewed how Polygraph: Points (Math 8) and Match my Pattern and Match My Line (Math 8 accel) to review equations of patterns along with slope of lines:

Here is how Polygraph: Parabolas went with the accelerated class:

Agenda: I will start up the activity and show what it looks like from the teacher perspective. Then all math teachers will join in using the class code. I’m anticipating 20 minutes per grade level.

Tile Pile (proportional reasoning using tiles and a table)

Polygraph: Shaded Rectangles (fractions vocabulary review)

Central Park: (algebraic thinking)
Polygraph: Systems of Equations: (fully aligned to CCSS 8th grade)
Polygraph: Points (necessary at beginning of school year)

Extensions:

http://teacher.desmos.com (search bar at the top for Desmos & user-created content)

Math 8 Accelerated (Algebra I content)

 Intro: Match My Line Visual Patterns Match my Parabola

### Day 29: Patterns in Tables & Writing Equations from a Scatterplot

Today's estimation was a fun one because it had a video reveal. Students estimated the number of sheets on a smaller roll of toilet paper that was next to a full roll. Discussed student's reasoning for finding 1/6 of 425. They said divide by 6. I should have asked them why are we dividing by 6?

Students started lesson 3.1.1 by doing the first 5 patterns as a class. To get ready for silent board game, I had students tell me missing entries before telling me the rule, so students who didn't get it had a chance to figure out the pattern. Then they add 2 entries that fit the rule. I had students finish the rest with their groups. Then we did Mathography community building: everyone stand up, stay standing if this, stay standing if that, etc... Students love it.

Then students took a 15 minute assessment. They were assessed on Skills 5, 6, and 7. The new skill was 7, solving equations. Students asked if they needed to draw an equation mat. I said it was not necessary, and only if you thought it would help you. I'm interested to see how the model helps them. It will definitely earn them a little more credit in the effort department when I score it.

In accelerated I passed back their post it note feedback. I think they were surprised I graded it, and I think that students will take giving constructive feedback more seriously next time.

Since I have the same accelerated students as I did last year, I decided to start an extra step in community building when a student asked, "when do we get to learn more about you?" I had previously thought about this and now was prompted by a student. So, I gave every student a chance to write a question on a post it note that was personal but appropriate for school. Every week I will read the question and answer it.

Students took their assessment. I have a feeling a lot of students will be doing makeup homework because a lot of students made mistakes on graphing an absolute value function. To retake skill 5 they will be required to show evidence of homework, correct and analyze all mistakes on skill 5 (graphing parabolas, cubic, and absolute value functions) and complete a practice sheet to qualify for retaking skill 5. I will outline this again in a future blog post. Unfortunately, I did not take any pictures today.

## Thursday, October 8, 2015

### Day 28: Subtraction Number Talk, Solving Equations & Equations of Lines of Best Fit

Today's estimation was how many sheets on a roll of new toilet paper. Some students estimated a 1000. I took that opportunity to do a number talk regarding subtraction, because the first step in calculating percent error is finding the error by finding the difference between the estimate and the answer, then dividing that error by the answer.

As you can see in the picture, I asked students to volunteer how you can subtract 425 from 1000. I was delighted when students offered repeated subtraction, 1000-400=600, then 600-25 is 575. Then some offered counting up, like counting back change at a register. A student said add 75 to 425 to get 500, then add 500 more to get 1000. That's 500+75 which is the difference, or the answer.

Some students offered the old fashioned way of crossing out and regrouping after they knew what the answer was. I asked if it's easy to make an error that way. They agreed it was. I then wanted to know them what exactly the 9, 9, and 0 meant. It was 900 + 90 + 10. A lot of "ohhhhs." Another student offered 1000-500 to get 500. Then since you subtracted 75 to much, add 75 back on. I tried to stress that when you put -500 and +75 together you get your -425. I finally suggested 1 of my favorite strategies that no students mentioned, which I call take one give one or sometimes with money, "take a penny, give back a penny." I asked what's 1000 -1 everyone? 999. Then you subtract. I commented that notice I regrouped NOTHING here. Then what do I do after subtracting? Add the 1 back on. I circled the -1 and +1, and showed we basically just used a zero pair, relating to our work with simplifying.

In class, students simplified equations, determining their answers (x=-3, x=0, and x=any solution). With 15 minutes to go in class, for closure, different student volunteers operated my computer with the virtual tiles while other volunteers. Students aren't used to seeing an x tile and no tiles on the other side, but some picked it up quickly. If you see nothing in math, it still has a number, zero, because you can add zero to anything. Some students thought infinite solutions problem was no solution or 0. If they thought 0, I asked them to pretend -x was 0. if -x=-x, is it true if x=0? Yes. Try it for another.

Finally, there was a great problem, -x+2=4. Students were able to tell me using their background knowledge how to interpret -x=2. I took a picture of it. Basically, students said to multiply or divide by -1 on both sides, which involves physically turning over all tiles both sides. I challenged students to show me another way. They said to add balanced sets of positive x and then add balanced sets of -2 to make zero pairs on the right. It was great.

In accelerated, we reviewed how to interpret y=3 and x=4 equations. Basically, I asked them what does y=3 mean? It means that the y-coordinate is 3. So I said give me a couple points that are on y=3. They gave me let's say (1,3) and (4,3). I asked them to calculate the slope. Even when they saw 0/3, some students said undefined. So we continued to discuss this. Then graphing it, and relating it to rise over run. If you have a pizza, and you ate 0/3 of it, you ate no pizza. It's possible.

For x=4, same routine.They calculated it, and there was still a bit of uncertainty. Some once again said undefined, which is correct. How can we prove it? They related it to rise over run. The graph rises towards infinity, or any number, and it does not run, or move horizontally left and right. We then summarized with y=3 is a horizontal line, and x=4 is a vertical line.

We did not get to start the next lesson, because many had not finished from yesterday. So we analyzed a volunteers scatter plot after reviewing why time was the dependent variable for the biking data and distance biked was the independent variable. They then wrote equations of the lines of best fit and then confirmed their predictions with it. Instead of starting the section I had planned, they worked on the last problem of the section instead.

Oh. 1 more thing. On a students assessment they don't understand the easier way to graph this slope but mathematically it is 100% correct. I thought it showed a lot of ingenuity.