## Monday, November 30, 2015

### Day 60: deciphering cc standard & table & graphs of exponentials

Today's estimation was a wine glass. I think many students were close. They thought it was less than the soda can and they were very close in capacity. I'm encouraging students to at least make comparisons in their reasonings instead of saying I just guessed.

I had a student read the common core standard we were learning and they were able to relate rate of change to growth factor and initial value to tiles in figure 0. I was glad they weren't all bogged down by the academic language of the standard.

I also reviewed the learning goals and the success criteria. I think next I have to write them on the board so students can refer back to them.

We reviewed the four representations of pattern A from the 2 clues given. Most students finished b and started c. Work progress was slow with it being the first day back from a break.

I met with struggling students and encouraged them to go to homework center and I got a much bigger turn out then I anticipated.

We solved an equation that's answer was an improper fraction so checking it was an epic process of reviewing fractions and integers. I had to take a picture of the results.

In accelerated I passed out weekly assessments because no one was absent when we took it. Then I reviewed their diagrams from Wednesday by looking at the blog post. Then they answered how many rabbits after 12 months.

After they analyzed cases 2 through 4 with different starting values and growth. They graphed case 2 but we didn't have time to discuss their analysis. We'll start class with that tomorrow.

## Wednesday, November 25, 2015

### Day 59: 4.1.5 and exponential bunnies

Today was minimum day 30 minute classes for the day before thanksgiving. The estimation was how much does the turkey cost. It was the same turkey they estimated the weight on. Students were shocked it was only a dollar a pound. I did like that students knew it would be pricier than a chicken.

Students started on 4 different tile patterns that are given only a few clues on each. I had to remind students not to copy the graph from the book and make an accurate graph from their own table. Some students that were cooperating well got a good start on part b. Unfortunately didn't take any pictures. Then we did mathography community building.

In accelerated we started 5.1.1. Some of the higher students wanted to go straight to an equation for the mice and men bunny problem and were getting the wrong answer. Once again I reminded them that as a group I wanted them to make a diagram for months zero to 3. They sometimes struggle with this.

The premise is you start with 2 bunnies. They have a pair of bunnies in the first month. After that each pair has a new pair.

Then they complete a table and find out how many bunnies after a year or 12 months.

I had Jason share his diagram that was organized. After I had nuri and his group share his and I was very impressed. It was color coded to a table, had circles with the month number inside it. It was great. Here it is:

After they came up with tables I asked if they saw any patterns to write a rule.

I labeled the equations with who shared them. We also saw how y=2*2^x is the same as 2^x+1 because you are adding the exponents. Jeffrey also came up with 2^x + 2^x.

Great discussions in 30 minutes. To be continued on Monday.

Oh and how could I forget. Some students made the connection between this problem and the iPhone app 2048. I downloaded it and it's a super fun number puzzle game.

## Tuesday, November 24, 2015

### Day 58: Trimester 1 Final

No estimation. Not much to say. Students took their final and finished at varying speeds. It will count for 10% of the grade. Tomorrow we will start 4.1.5 and in accelerated start Chapter 5 with exponential functions.

## Monday, November 23, 2015

### Day 57: Finish 4.1.4 (y=mx+b) and Around the World 4.3.1

WToday's estimation was estimating the weight of a turkey. Some students' families had already bought their turkey so they had some background knowledge.

In periods 2 through 5, students asked me questions about what would be on the test. We did a brief review of what makes a relationship proportional, making up equations with no solution, infinite solution, and one solution. The review from 2 different periods is as follows:

Then we reviewed what the parameters M and B meant in y=mx+b in relation to a tile pattern. A lot of students forgot this from Friday. Then only a few knew how to write a rule when looking at a graph. Then they worked on writing equations from tables with their groups while I circulated and asked questions, then we reviewed their results. I encouraged students that didn't understand that they need to stop their group and ask clarifying questions. I respect someone more if they are willing to ask another for assistance.

In accelerated I posted 2 copies of 10 problems from 4.3.1 and the students got to pick a person to pair up with and travel "Around the World" solving the problems. Then in the last 10 minutes students presented as many of the solutions as we could. I did not do a trimester final review because all skills from Friday assessments will be represented on the trimester 1 final except order of operations. Here is a list for you to study:

• Graph a parabola using a table with positive and negative x values and mention any y-intercepts, x-intercepts, line of symmetry, vertex, minimum or maximum, and the direction it opens
• Write an equation in y=mx+b form when given the slope and a point, and when given two points, showing all work clearly
• Graph an equation, and graph a line with undefined slope and a slope of zero, and write the equation of it
• Solve linear equations with no solution, one solution, and infinite solutions, and check if possible
• Graph scatter plot data, interpret the data, the y-intercept, and write an equation of your line of best fit (all on your weekly assessments)
• Exponent rules
• Determining if side lengths form a triangle, if so what type, and using pythagorean theorem to find the hypotenuse and missing side lengths (a leg)
• Evaluating cube, square root expressions
• Evaluating absolute value expressions
All students will be given the whole period to take the test, and it will be 10% of the trimester 1 grade.

## Friday, November 20, 2015

### Day 56: Baby News, y=mx+b, Systems

I thought about making an estimation problem based on the baby being 1.39 cm at 7 weeks, and a typical baby is between 40 and 50 centimeters at birth (36 weeks). I quickly realized this relationship is not proportional or linear so I will just show the picture. They were quite happy for me.

The estimation was how many ounces are in a glass? Students quickly said it was a champagne class, and a lot of them offered that they'd seen them with orange juice for mimosas. Surprised how much they knew. Most students estimated less than the can and I liked their reasoning.

I had students start on 4.1.4 with a participation quiz and reviewed up to writing an equation from looking at a line. This is student's first formal introduction to the equation y=mx+b. First students listed the equations from yesterday and were prompted to write down all their observations of their similarities. The best answer I saw all day was y=___x + ___. They remembered that format from the CPM Silent Board game. Some students used words like slope and y-intercept for m and b respectively. I prompted students what does m and b tell you about the pattern? Students thought m was the number of tiles added each time, or the growth. B was the number of tiles in Figure 0 or where the pattern starts. The positive and negative behaviors, cooperation, quotations. Participation Quiz.

The last problem we reviewed was a graph where Figure 0 had 1 tile and it grew by 3. Students said it was y=3x+1. I frequently reminded many students today when they wrote 4x+1, and 4x+3, those were expressions not equations. Then they put y equals. It sounds small, but it is very important.

In every class I did community building with mathography before taking an assessment on Skills 9 and 10. Trimester 1 final is on Tuesday and Trimester 2 will start with skills 10,11,12. Harrison presented his system and elimination work here. They earned 4 cents per capped bottle and lost 2 cents per broken cap. They had earned a total of 6 cents. The second equation is there were 15 total bottles worked on. You can see he multipled the second equation by 2 and then added the first equation to it to eliminate the y variable. He also wrote his answer properly as a point of intersection coordinate pair. I thought this poster's equation was advanced, I just wish they explained how they saw this rule. Some students asked their high school siblings to help them with the rule. I was delighted to hear their curiosity.

In accelerated students solved a system of equations word problem that they solved using elimination and substitution. Then they practiced equal values, substitution, and elimination method by choosing the most efficient strategy.

After school many students retook skills after doing corrections, explaining errors, and showing me evidence of homework.

I put up a larger majority of the CC8 4.1.1 tile pattern posters and made some great observations from them: I marvel at the rule here. They clearly saw it a certain way, I wonder if knowing what Figure 100 looked like helped them. Hard to read but 4 - (x+1)^2 + 2(x+1) +1 is quite a large equation! A couple gaps but that is 16 posters of high quality work up there. Going to fill another wall.

## Thursday, November 19, 2015

### Day 55: Pattern->Table->Graph & More Elimination

During my prep period I started grading students peer to peer feedback post its where I asked them to give a poster 2 stars and a wish. The 2 stars would be positive and math specific feedback. The I wish would be a statement or question that would help the group improve their work. The student on the left said he wishes the "graph was shown in a bigger, clearer way." The student on the write is much more specific and wishes "your intervals on your graph were by something smaller than 10 so the coordinates can be seen more clear." WOW! He really understood the point of being specific in his constructive feedback!!!

I loved today's estimation. Students estimated a tall cylindrical vase. I participated in 2nd period and estimated you could fill 3 cans full of water into the vase. That ended up being close. A lot of students thought it was only double. Only a few noticed the difference in the diameters of the bases.

Class started with observing the similarities and differences between tile patterns 1, 2, and 3. Students said they were all growing by 4 tiles and in different places. They all had a different number of tiles in Figure 0. Then they told me the equations of each pattern. This prompted them to observe any connections between the tile patterns and the equations. They noticed they all had 4x and grew by 4 tiles. They also had plus 1, 2, or 3, which were the different amounts of tiles in Figure 0.

Then they completed the tables based on the tile patterns. Then I had them space out their x axis by 5 square units so that they could see the relationships between the tile pattern graphs. After they color coded the growth to each line, we had a class discussion. Instead of going through parts a, b, and c explicitly, I asked students to tell me what they observed and noticed about the graphs.

They said the first 3 graphs were 1 apart. I asked why that could be. Students noticed that the 3 lines had different Figure 0 amounts that were 1 apart. They also noticed that they were parallel and grew by the same amount. A few students noticed that tile pattern 4 intersected the other 3 lines and only grew by 2. It also shared a point with pattern 3 because they both had 3 tiles in Figure 0.

I asked students how you could figure out the number of tiles in Figure 0 by looking at a graph and not the pattern. They said look at the y axis. I asked if that point had any significance or if we can call it something. They told me it's the y-intercept. So, students made the connection that Figure 0 is the y-intercept. I told some of the classes that that's the beauty of looking at an equation and being able to know where one point is right away.

Then students investigated a graph of 3 tile patterns.

In accelerated I handed back their assessments. We had a nice discussion about common mistakes with negative exponents 4^-3. I asked how we could rewrite it. They told me 1/4^3. So then I asked what is 1/4^-3? Davin had a nice explanation. He said make the 1 in the numerator a 4^0. Then subtract the exponents. 0-(-3) is 3 so it's just 4^3 or reciprocal of 1/4^3. Here is his reasoning:

Students practiced the elimination and substitution method. Then they were presented with a problem that needed to be multiplied by 2 on both sides to get it to cancel. They were also challenged with a problem hat interpreted a no solution. Students reported it meant the two equations were parallel because there is no point of intersection.

The final challenge was a system where they figured each equation would be multiplied by different numbers to get a common multiple that would cancel. I heard groups discuss this problem, but we didn't have enough time for a student to demonstrate it.

The last 10 minutes were student volunteers coming to the board to show how they solved each problem for closure.

## Wednesday, November 18, 2015

### Day 54: Tile Patterns with Figure 100 & Elimination (Bonus: Wheel of Theodorus)

Today's estimation was the dessert dish. It was surprisingly similar to the capacity of the soda can.

We reviewed how the last 2 days students made posters making connections between tile patterns, a table, a graph, and finally a rule. I reminded them that finding a rule is easier when you can see the figure number in the pattern and figure out what Figure 100 would look like. Today is day 1 of my favorite lessons, 4.1.2, tile patterns.

Students completed Figures 0 and 4 for four tile patterns, shaded the growth, and attempted diagrams at figure 100. They worked independently for 8 minutes and then with their groups for 10 minutes, then we had a whole class discussion and I selected students to share that probably wouldn't voluntarily.

When students discussed the growth of the tile patterns, I wanted them to improve the precision of their language. I asked them to use the word diagonal. I also asked what's the difference between column and rows? 7 kids in each class raised their hands, and said columns go up and down. I showed them how rows are like rowing a boat, you go side to side. Columns are like Roman architecture.

Tomorrow we will discuss the similarities and differences in the tile patterns, as well as fill out the tables and graph them on a properly scaled graph.

In accelerated, students were introduced to the elimination method to solve a system of equations. I needed to prompt some students what we do after we find out what x is (plug it in to solve for y). I asked some students if it mattered which equation you used (no). Then I said are you done when you found x and y? How do you properly write your solution? (A coordinate). I asked them why. They said it's a point of intersection between the two lines.

Then I had Michael present his solution method for a problem with a fish catching competition where your given the weight of each fish and the total weight caught. Then it tells you that you gain 2 points for bass but lose a point for trout because they are endangered and the total points earned. He presented a complete solution and we gave him a 3 second clap. They also saw how they could multiply an equation by -1 so that when you add it terms would cancel out.

The students that were not on sojourn were assigned the Wheel of Theodorus from Yummy Math. Some of them turned out really awesome. I especially liked the psychedelic black and white spiral, the snails, and the ancient extinct animal that I forget what my student called.

## Tuesday, November 17, 2015

### Day 53: Gallery Walk & Post it Note feedback & Solving Systems, Participation Quiz

Today's estimation was a wide vase. A lot of students underestimated. I think they are seeing that more width usually has more capacity than more height.

I also passed out the week before last's assessment. Students will have an assessment on Friday and then their trimester final that's 10% of their grade next week.

I reviewed the poster requirements: Figures 0 to 5 of the tile pattern, graph, table, and figure 100 diagram with a rule. The last part was the hardest for most students. I wrote some pocket questions on the board: Where do you see the figure number in the tile pattern? What would it be on Figure 100? Now replace 100 with x to figure out the rule. This brought up some ideas of areas of squares and rectangles. Growth shown, as well as the rule.

Diagram of Figure 100 along with the rule.

With 10 minutes to go in the period, students were given a post it note and put their name on the front. They were to give two stars and a wish on 1 poster. 4 people maximum per table. The post it counts as a classwork grade. The two stars are specific math related feedback that you understood from their poster. The I wish is a statement that moves the poster creators to IMPROVE their poster. I pointed out helpful feedback is NOT: "I wish there was color. I wish there was no white out. I wish you found the rule, etc." Helpful feedback could be "I wish you labeled your axes on your graph. I wish you showed the growth on your table." The cool part I observed was students rushing to improve their posters after reading the feedback. That was the point.

This diagram for Figure 100 makes a lot of sense.

I allowed students to offer a rebuttal to feedback they didn't agree with. This was interesting because one student said "I wish you connected your points on your graph." And the student replied that it was a discrete graph so the points shouldn't be connected.

Another successful rule.

I'll post pictures of students work in progress and later this week I'll take a picture of some of my favorite feedback and get ready to hang the posters all over the walls and ceiling.

One student was so curious what the rule was for his pattern, so he wanted to stay in at lunch and discuss it. So, we had a conversation about how you saw the Figure number 2 in the tile pattern. Then moved to figure 100. Then how to replace 100 with x. Also, how to find the area of the rectangle. Then I previewed a topic from Algebra I with the general rectangle to multiply the 2 binomials. He was pretty satisfied afterwards.

That's why I love this section, because all the patterns are quadratic, and not linear. A lot of students like the added challenge. Some students asked their older siblings last night and came in with rules, which is exciting that they communicated to others about it, but I wanted them to come up with it based on where they see the growth in each figure.

In accelerated students wrote equations from a word problem about yodelers and xylophones. The students had a participation quiz and cooperated together in an improved fashion. There were great discussions about the fact that there were "twice as many yodelers as xylophonists." Students argued whether it was y=2x or 2y=x. Davin reasoned with scissors why the xylophones were 1 pair of scissors and the yodelers were the 2 scissors. Therefore the xylophones were multiplied by 2 to equal the 2 yodelers.

Some groups had to start over because they mixed it up or defined x and y as different variables than their teammates.