Friday, December 18, 2015

Day 73: Equal Values Method, Graphing Systems & Intro to Residuals

Today's estimation was estimating the height of the 2012 Rockefeller tree. In the picture it's sideways which made it harder for students to visualize it being upright. Most students overestimated and were surprised at the height. I liked when students said the man in the picture is 6 feet, so it's probably 10 men tall. That's sound reasoning.

Class started with me asking students, what were the 3 ways we solved the chubby bunny problem? Give me 1 method. They said one was making 2 tables. I probed students to tell me how the table showed us the answer (7 years, 26 pounds in both tables). Then they told me the second way was graphing and they said the answer showed up when the points lined up at the point of intersection. I asked them what that coordinate was, and they said (7,26). They had trouble recalling the third way because 1 or 2 classes didn't get to that point in the closure of the lesson.

So, at least a few students knew how to combine the two equations into one. The bunny was y=3x+5, and the cat was y=1x+19. I liked how Jonas interpreted it as just take away the y = parts and put them next to each other. Yes, with an equals sign. I think when they hear substitution of the expression that y equals into the other equation, they get a bit confused.

After they told me how to solve x+19=3x+5, I asked how can we find out how much they both weighed at 7 years? They said check your solution. When you check it, you get 26=26, verifying that the equations are equal when they both have x equal to 7.

They then worked on writing a system of equations for two high schools growing and shrinking. At least 2 groups omitted the negative sign for one schools growth, and during closure we discussed and analyzed that mistake. It was a great opportunity to learn from.

Then they graphed the growth of two trees, a ficus, and an oak tree that grew from an acorn, y=2x+0. They also confirmed their answer using the equal values method. The assessment on Friday students will not have to interpret the 2 equations, I'll give them those and they'll have to graph both, find the point of intersection, and confirm it algebraically using the Equal Values method.

In accelerated students revised a MARS task called Scatter Diagrams. I wanted them to be more specific in most of their answers. Then we used the scatterplot from the Candle Lighting Task and then analyzed negative and positive residuals and what they meant. They also worked on a problem involving sugar and the weight of sugar. They analyzed the slope and y-intercept, and then figured out if negative or positive residuals would be better. Students reasoned a negative residual would be better because it has less sugar than what they predicted. 



Thursday, December 17, 2015

Day 74: Pumpkin Time Bomb (after school)

What is a pumpkin time bomb you might ask...? Watch the video.

What measurements of the pumpkin can help us predict how many rubber bands it will take to explode the pumpkin?

To gather more data to make an accurate prediction, go to this Desmos Activity and click the preview screens to see data in table and graph form.

We will submit our data to Mr Orr's Google forms here.

Here are students names, too lows, too highs, estimates, and reasoning.

Here is our video:

The top half of the pumpkin flew nearly 10 feet in the air.
451 rubber bands later... As you could hear from the video, Davin's estimate was off by only 2 rubber bands!! Students concluded that smaller pumpkins take more rubber bands to blow up because they have less empty space in the middle... and their walls may be thicker.

Tuesday, December 15, 2015

Day 72: Chubby Bunny & Candle's Burning Task

Today's estimation is estimating the number of lights on a tree after 17.5 wraps from top to bottom.

I passed back last Friday's assessments. Nearly every student struggled on Skill 13, solving for y and eliminating fractions before solving an equation. We went over it as a class so that they could study it before the assessment this Friday:

Here's an example of the classwork. Megan was able to show her table for the bunny and the cat. Also, her graph shows the point of intersection with the coordinates labeled.

They were first introduced to the Equal Values method. It was nice that the book showed the expressions being put on the left and right side of an equation mat to solve for X. Not pictured here is the check of the solution, which shows that 26=26, proving that after 7 years both animals weigh the same amount.

Students will continue with the same lesson with bigger numbers of writing an equation and using the equal values method.

In accelerated, students will be learning from the Candle Burning 3 act lesson. Kyle Pearce also has a worksheet with extension problems and scaffolded questions. Here is act one:

What questions do you have, what information do you need to answer it?

Once you've shared your ideas, and requested information, here is Act Two:

After student's share their solution strategies, let's see the answer, Act three:

Extension: How long would it take a 26 centimeter tall candle to burn out with same width?

Extension: Draining the Pool Task

After Act one, students told me what information they need and I told them I wanted a neatly labeled scatterplot, line of best fit, equation of it, and the slope and y-intercept interpreted. As you can see, I like 2 different answers students came up with for the slope. -1/20 meant that it shrinks 1/20 of a centimeter each minute or that it shrinks 1 whole centimeter every 20 minutes. That was a great interpretation of the rise and run of slope.

Here you can see that Davin wrote his equation and substituted zero for y and solved for x. He got 340 minutes. The answer ended up being a little more than that. I believe this activity an lead into a discussion about residuals tomorrow.

This is some work from yesterday's Hot Dog eating lesson that Alex D finished that tracked Kobayashi and Black Widows total hot dogs eaten after certain rounds. You can see his parabolic sequence equations at n squared and n squared plus n.

Day 71: Participation Quiz Systems of Equations Kobayashi Hot Dog Task & Differences between two functions: f(x) and t(n) (sneak log in too)

Today's estimation was how many times will Mr. Stadel wrap his lights around his tree. Most students estimated in the teens and some thought about the spool he unwrapped and related it to decorating their own trees.

Here's Dominic's two tables along with his graph from 2nd period! Great work.

I reported out students whose blog URL's I didn't know and whose parents signatures on Trimester 1 finals I didn't have. Then I discussed how I had used direction instruction yesterday when reviewing multiplying a fraction equation by 2 different numbers and choosing a 3rd best number, the LCD. I related that to research that says you learn more in cooperative learning groups following the study team norms, such as: not talking to other groups, explaining and justifying, asking good questions giving hints and not answers, not working ahead, etc.

I then said I was going to record positive and negative conversations and actions reflecting the study team norms on the board, and reviewed it with all students before a student presented and we had our closure.

2nd period Participation quiz:

3rd period participation quiz

5th period:

Students solved a system of equations by using a table, a graph, and using two rules to confirm the correct answer. One student Connor in 2nd period decided to try the Equal Values method before it had even been introduced to him.
He also showed the point of intersection on his neat graph.

In accelerated students decided if sequences were functions. I liked how students reminded themselves by using the glossary and being resourceful. A few students remembered the vertical line test. We also used it after looking at the graph of y=2*3^x. Students reasoned f(x) is a continous graph while a sequence is a function but a discrete graph of whole numbers, as Nuri put it.

I promised my students I would show them a topic from Algebra 2. I asked them in the top right how to solve 700 = 3^x after Gabriel proved that is was not a possible term number because 3 to the 5th power is 243 and 3 to the 6th power is 729. So, 700 is clearly between those. Harrison knew that you had to use the logarithm, which I said was like the inverse of an exponential function, it undoes it. Then the Law of Logarithms allows you to use the exponent, x, as the coefficient of log 3. Then divide by log 3 on both sides for the answer.

in the last 15 minutes we watched the following video and students chose to answer different questions that Robert Kaplinsky posed. Students answered the first few questions.

What questions do you have after watching:

Michael made a table and Sarah explained why there were 36 hot dogs by adding the rounds between the competitors 1 through 8, 8+7+6+5+4+3+2+1. Students also found the 20th round scores for each person by writing arithmetic sequence explicit equations.

K is Kobayashi and B is for Blackwidow. He proved the 20th round using his explicit equations. I didn't get a picture of Alex D's paper where he showed the total hot dogs they had eaten after that round. He came up with those equations also.

Monday, December 14, 2015

Day 70: graphing systems & geometric sequence multipliers

Today's estimation was how tall is mr stadels Christmas tree. Definitely a throwback to day 1 estimating his height. A lot of students thought the tree was 2 feet higher than his height. 

Then we reviewed the answers to the Iditarod activity from Friday. I picked random students' to review the summary statements they wrote. We reviewed the coordinates of the point of intersection and interpreted their meaning.

Then students setup 4 quadrant graphs and graphed 2 linear equations. Only one student all day understood the point of intersections coordinates were solutions to both equations. Some did notice some patterns between the numbers though.

Then with some direct instruction and choral response I guided students to multiplying a fraction equation by one denominator only eliminated one fraction while multiplying all terms both sides by e other denominator only eliminated that fraction. They realized you had to multiply by the lcd or least common denominator to eliminate both fractions in one step. 

They got to practice 2 equations with this and we did not practice solving for y which we will do on Tuesday or Wednesday. 

In accelerated students worked on projected phone sales that increases by 3 percent and some that decreased by 7 percent. Then students started on 5.3.3 which reviewed arithmetic sequences and the differences between sequences and functions. 

Sunday, December 13, 2015

Day 69: Iditarod Line of Intersection & Geometric Sequences

Today's estimation was how many revolutions to unwind the lights. This was difficult because it was hard to see how many sets of lights were wrapped up. They did enjoy watching it get unrolled in the video.

I photocopied the Iditarod scatterplot and reduced it's size to fit 2 on 1 paper, and cut it in half for students to paste into their composition book. Luckily, students had been taught a science lesson based around the Iditarod sled dog race earlier in the year. I have to talk to my colleagues about the lesson they taught.

Students answered the discussion questions as a group. Students thought that the race was 900 miles long because that was the point of Evie's first checkpoint leaving the finish line. Other students said it was 1100 to 1200 miles because Evie was 900 miles from the starting line 30 hours into the race. So, the key was she left the finish line the same time the race started.

Students will graph two lines on Monday to find the point of intersection and wrap up some review of solving for y and eliminating fractions from equations.

In accelerated students worked on writing geometric sequences with a "piPhone"'s projected sales being 15% more each week after selling 100 phones in Week 0. They wrote an equation and then used it to predict the sales for the 4th and 52nd weeks.

I'm not sure if students were working slowly because it was a difficult concept or because they were not as focused and cooperative. I will be speaking to them about this. In the pacing guide this is a one day lesson and I'm concerned that only a few groups got the first two problems done. We will wrap it up Monday and have an additional task on Tuesday involving kobayashi and hot dog eating. 

One of my accelerated students wrote an opinion piece. I'm preparing rebuttals and my goal is to convince him why common core is good for him and all students. 

Thursday, December 10, 2015

Day 68: Fraction Elimination & Comparing Graphs of Arithmetic, Quadratic, & Geometric Sequences

Today's estimation was how many light clips in a box. I liked Charisse's reasoning where she counted about how many groups of 5 were in the pile. This gave her a close estimate.

Today's lesson started with reviewing all the methods they knew to solve a proportion, which lead to the undoing division method, which lead to fraction busters, which lead to the reason why the original method, cross products, works.

Then they converted a decimal equation to a whole number equation and solved it. A lot of students checked their solution in the whole number equation but I prompted them to check it in the decimal equation to make sure that there work in making an equivalent equation was correct.

Then instead of doing a similiar problem from the book, we looked at this clip from Little Big League:

I pointed out that I like this video clip because it shows that they were not afraid to attempt the problem and that it's okay to be wrong. We analyzed why 15 and 8 hours did not make sense as answers. Then when I asked if 4 hours made sense, there was a mix of yes and no answers from the class. Students pointed out if it took 3 hours for 1 person to paint the house by themselves, it can't take more time, or 4 hours, for them to paint it together.

I asked students if it takes me 3 hours to paint the house, how much of the house could I paint in an hour? They said 1/3. So, I defined x as the time it takes to paint the house together. So, 1/3x + 1/5x=1 The 1 represents the whole job getting done. They then applied what they learned to solve the problem. See the attached photos.

In accelerated students graphed the 3 different types of sequences. They then described the growth. I was impressed that Markos got the quadratic equation of t(n)=(3*n)^2. Zoe pointed out the importance of the parentheses to keep the order of operations correct.

Students compared the graphs to see if they represented the growth of a bank account, which one would they want?

Day 67: Solving for Y and Recursive Sequences

The estimation was a box of tree lights. The best reasoning for an estimation was it looks about 30 feet long and there's one light bulb per foot, so about 30 bulbs. That was a close estimate.

When starting a new chapter, I switched up the groups. Chapter 5 starts with changing standard form to y=mx+b form. Students first reviewed the equation from John's Giant redwood tree and identified the starting height and the growth rate. I emphasized that they weren't just numbers from y=4x+5 and that they needed units that reflected the context of the word problem. For example, it's not a growth rate of 4, it's 4 feet PER year and it's not a height of 5, it's 5 feet.

Then they predicted what the growth rate and y-intercept of -6x+2y=10. The most common predictions were -6 for the growth rate and a y-intercept of 10. After using algebra tiles to solve for y, they then identified the actual growth and y-intercept. I asked students how the predictions were different from the actual answers but still good predictions.

In accelerated students investigated the difference between discrete and continuous graphs. Sequences are discrete because you can't have term 1.5, only whole numbers. Functions are continuous therefore you connect the points. They also practiced writing an explicit equation t(n) for an sequence. For example, if the sequence adds 4 each time and term zero is 6, the rule is t(n)=4x+6.  They then were introduced to a recursive equation which starts with t(n+1)=t(n)+4 in this case. What that means is to find the next term in the sequence you take a term and add 4 to it.

At the staff holiday party I got to meet Mrs. Wards baby girl Orla. Isn't she cute?

Tuesday, December 8, 2015

Day 66: sub day

An unexpected family emergency happened and I was absent today. 

Math 8 students were to finish their work from yesterday and complete 4.1.7. Students who finished that worked on chapter 4 closure problems. 

In accelerated students continued with the lesson that was planned and I will review with them tomorrow how they understood the lesson. 

Day 65: differentiate and finish sorting sequences

Today's estimation was the capacity of the bowl. I was off myself because I thought the bowl held more than the large vase. I asked students what they knew about the large vase. They said it was 10.7 ounces. Students yelled no it's 10.7 small vases. Ok. What do we know about small vase? It's 11 ounces. Ok great. Now estimate the capacity of he bowl in ounces. 

Some students still estimated how many small vases it would take to fill the bowl...

In periods 2 through 5 we differentiated. I had students graph y=-2x+4 and label X and y intercepts and the growth triangle. This showed me who understood what the b value 4 told them and who didn't know how to start. I had to remind students to label the intercepts with their coordinates. 

Then students finished graphing for tech 4 rules from Friday. If and when they finished that they could move on to 4.1.7. 

In the above picture I quoted Andrew. His table mate said what's the m value in y=X-1? Instead of telling him the answer he asked what's the difference between x and 1x? The classmate replied nothing and understood. 

We are starting chapter 5 with solving for y fraction busters and systems of equations. 3 big topics.

In accelerated students marched their sequences to the correct tables and graphed their sequences on the graph strips. This helped students see the parabolic sequence easier if they hadn't at first. 

They were to pick a family of sequences with common characteristics and write summary statements to present to the class. Unfortunately this didn't work well because most didn't write down their thoughts or prepare what they were going to say and left it on one group member to talk. Class ended with the new terminology arithmetic and geometric sequences. 

Friday, December 4, 2015

Day 64: Graphing using y=mx+b and Sequence Sorting (Graphing Monday)

Students from 2nd through 5th period should revise their first draft of their blog post and have it published using the button at the top right of the blog where it says Send a tweet. Please send a tweet with a link to whose ever blog you commented on so I can evaluate that.

Today's estimation was how many times would you have to fill and refill the small vase to fill the large vase. For some reason half my 2nd period students thought I meant how many of the actual small vase could fit in... I adjusted my wording on the board. A lot of students estimated between 7 and up to 30. I like that the answer was not a whole number, so I will be seeing how students dealt with the decimal. I was in a hurry so did not calculate percent error with students because they needed 15 minutes for their assessment to conclude class.

Students reviewed what rules they wrote for graphs d, e, and f. They then told me how it grew and the number of tiles in Figure 0 and how that helped them write a rule in y=mx+b form. Then I instructed students to setup a coordinate plane and graph y=4x+3 and y=3x with y=-2x+8 and y=x-1 on another graph. I picked students and volunteers to graph them with Expo marker on the laminated posters from the Algebra walk the first week of school. This was great to revisit a concept and see how graphing themselves as multiple points was the same and different from the new method of graphing the number of tiles in Figure 0 and then using the m or growth factor to find a second point. This goes hand in hand with students blog posts currently at the @joycemathletes Twitter account.

In accelerated students looked at a sequence that was multiplying by 3 each time, which was when the bunny problem was going from 2 bunnies, to 6, then 18, 54 and so on. I liked how students showed off their mental math skills by getting 162 then 486.

Then I passed out the sequences for students to sort. A lot of students figured out the Fibonacci sequence and they all had trouble with the same one that took me a bit of time to process when I had encountered it. Students sorted into groupings. One group sorted a group as +2, divide by 2, and times 2. Others sorted by multiplying and dividing, which they associated with exponential growth and decay from the previous days. A student described how one sequence relied on the previous term to get the next one. This is leading into some topics I look forward to exploring with them.

And I picked up a Christmas tree on the way home from work! This is obviously pre ornaments and lights but here we go holidays. 

Thursday, December 3, 2015

Day 63 Reflection: Blog posts on y=mx+b and Finishing Exponential Decay

The estimation was a small curved vase. Surprisingly a lot of students thought the vase was bigger than the soda can. Then again many students thought it was  a bit smaller.

In 2nd through 5th period I reviewed the Learning Goal for the day and the Success Criteria specific to this day. It is as follows:

The instructions for the blog assignment are below today's post. Students created their own blog and wrote a new post. I asked them to copy and paste the questions from the learning log in the book to guide them in demonstrating their knowledge of y=mx+b. 

Hw tonight 4-59 to 4-63 is a great practice.

In 2nd period Riley saw that when you insert a photo you can click insert from web cam. After you press allow in the window and when the popup appears below the URL part of the screen you can take a picture of your work. I encouraged students to make up a rule with different m and b values to help with their explanations.

To see student work, search for @joycemathletes on Twitter.

Students: not all of you finished or were expected to finish in one class period. I will be checking first drafts next Friday and would like you to comment on someone else's blog post from your class. If that person already has feedback, please give feedback to another person. This project can be finished at school and at home.

Remember students, if you did not share your link, no one can find it, so be sure to click send a tweet in the right hand column and if I don't see it remind me to tweet it out, thanks.

In accelerated we had to finish up exponential decay. Students predicted the rebound height if the ball kept bouncing 6 times from a drop height of 200 cm. They then conducted the experiment and got results that were from 5 cm to 20 cm off from their predictions. There were errors that they said could have been difficulty finding the rebound height, not holding the 2 meter sticks straight, and so on.

Students, such as Arthur, made the connection that the graph kept getting smaller and by a smaller amount each time. This was a great observation. They also saw that they were repeatedly multiplying the rebound ratio each time which lead them to see why an exponent was used. They then generalized it as a formula.

I asked them for the differences between that graph and the rebound versus drop height graph. They then compared it to the months versus bunny problem to see the similiarities and differences between exponential growth and exponential decay. Hans noticed that in the exponential decay a fraction was being repeatedly multiplied while growth a whole number is repeatedly multiplied. 

Day 63: Learning Log Blog Assignment w/ Formative Assessment College Course Info

Update: see this reflection post with links to student work!

Assignment procedure after reviewing the standard, relevant prior experience, building blocks, learning goals, and success criteria:

  1. Go to
  2. Sign in to your school ID Google account.
  3. Create a blog. To see a video about this created by Mr. Sullivan, click here.
  4. Create a new post and copy and paste the following:

Write a step-by-step process for graphing directly from a rule.  A student who has not taken this course should be able to read your process and understand how to create a graph.  It may help you to think about these questions as you write:

What information do you get from your rule?
How does that information show up on the graph?
Where does your graph start?
How do you figure out the next point?
What should you label to make it a complete graph?

Title this entry “Graphing Without an x → y Table,” and label it with today’s date

5. To help you answer these questions, make up a rule and represent it with a table, graph, and tile patterns (Figures 0 to 3). Your "m" and "b" values should be different in the y=mx+b format. You should also show how m and b show up in each representation.
6. Take a picture using the Chromebook camera  or your phone of your graph, table, rule, and pattern work in your composition book to support your explanations. For extra help with this, click here.
7. Click send a tweet and send out a link to your blog post so a classmate can read it and give you feedback in the comments section. Make sure to use your first name and your period number after it.
8. Go to and search for @joycemathletes. Find a person in our class who is also finished so you can read their post and give them at least 2 pieces of math-specific positive feedback and 1 piece of CONSTRUCTIVE feedback that can help them improve their work that could be a suggestion or a question that should lead them to revise some of their work.
9. If time, Graph your example rule in Desmos and provide a link to support your explanation. Read any feedback given on your post and improve your work.

This is a project that will be due next Friday so if you don't finish you can work on it at school or home. You need to at least have your first draft and a comment done by then, and if you don't tweet the link out, I can't give you credit. If you need help consult your notes, look at the building blocks listed below, and of course ask your classmates or myself questions!

The Common Core standard for this lesson is posted below, along with background knowledge you may or may not have in prior grades. I've listed some building blocks that will have more scaffolding involved later on.

Building Blocks of a Standard
Activity 2.15

     Martin Joyce

8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

In the 6th grade students have experience with collections of equivalent ratios and the relationship between tables and graphs. In the 7th grade students worked with the proportional relationship y=kx where k is the constant of proportionality or unit rate.

Earlier in the year students have identified the parameters m and b in y=mx+b with m being the growth factor (rate of change) and b being the number of tiles in Figure 0 (initial value)

Building Blocks of a Standard
     In a tile pattern the number of tiles added from one figure to the next is the growth factor, or m in y=mx+b.
     Prompt students how can you make growth visible in tile patterns? (Shade growth of tiles added on)
     In a tile pattern the number of tiles in Figure 0 is the parameter b in y=mx+b.
     This figure is visible in all the other tile patterns, and especially Figure 1.
     A table can represent a tile pattern with x representing the figure number, and y representing the number of tiles (dependent axis).
     This is like keeping track of hours worked and money paid, except it’s not proportional. It does have a constant rate though.
     A graph can represent the points from your table with the x axis labeled figure number and y axis labeled number of tiles.
     Is it a discrete or continuous pattern? Even so, connect them to see the growth easier.
     Once you know the growth factor, m, and number of tiles in Figure 0, you can substitute these parameters into the equation or rule y=mx+b.
     y=mx+b color coded. M and B are labeled in each of the 4 representations of a pattern
On a graph, the growth factor, m, is represented on the growth triangle between 2 points on the graph while b is the coordinate (0,b) or the y-intercept.
Draw right triangle using ruler between two consecutive coordinates. Label y-intercept as an ordered pair, NOT just a number.

Next up is the Learning Goals and Success Criteria will be posted in relation to this.

Consider a class you are going to teach in the next two weeks. Using what you have learned, write a Learning Goal and Success Criteria for this lesson.

Learning Goal
Success Criteria

Understand that a figure number and it’s tile pattern can be entered into a table and help organize your information.

Understand that the slope triangle tells you the horizontal change is the increase in figure number and the vertical change is the increase in number of tiles (because of how the axes are labeled).

Understand that a tile pattern can be represented by a x/y table, graph, figure, and rule.

Students will construct a x y table of data where x represents figure number and y represents number of tiles.

Students will use this understanding to find other tile amounts for different figure numbers and then graph their data.

Students will use the parameters m, growth factor, and b, tiles in Figure 0, to write a rule in y=mx+b form.

Students will setup a coordinate plane with x and y axes labeled, intervals clearly marked using an appropriate scale (by 1’s, by 2’s, etc.)