Saturday, February 27, 2016

Day 109: Scientific Notation FAL Day 2 & Solving for A in Quadratics

Todays estimation was a Steve Jobs biography. It also gave us a chance to remember how influential of a person he was. Students knew he founded Apple and was also behind the iPhone.

Most of the class finished their posters yesterday, so I started today with analyzing the posters. I asked students what they matched the thickness of a dollar with. They said 1 x 10^-4 and 0.0001. We had a discussion how to properly say a number in scientific notation. I asked students to confirm why the decimal was equal to the scientific notation expression. Students told me the exponent of -4 means to move the decimal point 4 space to the left on the 1.

A common mistake yesterday was writing 0.012 in scientific notation, since it was a blank card. Some groups wrote 12 x 10^-3 and others wrote 1.2 x 10^-2. I asked them to confirm whether these were equivalent to the decimal. They said both were. I asked if both were in scientific notation then. They said that 1.2 was the correct one because it was less than 10 and greater than or equal to 1.

Finally I asked students what they matched the height of a door with, which was 2 x 10^0. It was matched to a blank card. Some students wrote 20, and some wrote 2 meters. I asked students to convince myself and the class which was right and why. Some students explained that the exponent of 0 means not to move it left or right, just leave the decimal where it is. 

Matthew in 2nd period and Andrew in 4th period, had the same amazing alternative explanation: they said that since 10^3 is a 1 with 3 zeros, then 10^0 is a 10 with zero zeroes. My mind was blown and I verbalized how impressed I was to the class. I have NEVER thought of it that way and it is SO intuitive.
Analyzing the matches on their poster.

The thought that blew my mind.
Then students were given arrows that had multipliers on them. They had to find relationships between two objects to see how much bigger one object was than another. I think that some students used our prior work with scale factor for this, used calculators, and mental math to reason about it. As students worked I wrote down matches they made that we would analyze before the Friday assessment.

Two of the relationships found.
As a class I asked students how to say the number 80,000,000. I was happy that many quickly shouted out 80 million. So, I asked raise your hand and tell me how to write it in scientific notation. Students said it was 8 x 10^7. I noticed students counting by pointing so I asked how they were figuring it out. They said they were counting the number of zeroes after 8 because the number of zeroes would be the exponent of the 10.

Now here's where the magic happened. I asked students what they noticed about how the 2 factors multiplied to get the product. At least five students noticed that all you do is multiply 1 times 8, and then when you multiply the powers of 10 you add the exponents. So they said -4 plus 7 was 3, so the answer has an exponent of 3. I asked volunteers to rephrase what the volunteer had said.
When I asked students to tell me 4,000 in scientific notation, more hands raised when I said, guys, we have a clue on the board here of how 8,000 is written in scientific notation.
Students verbalizing what the exponent did to the number being multiplied.
Another interpretation of an exponent of 0.
A relationship students found using mental math.
Here students reasoned you multiply 4 by 2 to get 8, and add the exponents 1 and 2 to get 10 to the third power.
In this class I threw in the challenge of saying 0.0001 properly instead of "zero point zero zero zero 1." There were at least 5 different ways students thought you said it. This reminds me that I should review place value at the beginning of the year, and try to spiral it in more often.
Here's an example of what the MARS FAL posters looked like. I took a picture of this one because they used some creativity to make the arrow rise off the poster so another arrow could go under it.
In accelerated we started with reviewing what could be on the trimester 2 final. It's pretty much most of the skills from the Friday weekly assessments. Some students wanted to review it in class. I may do this next week, and if so I'll have to find a fun way to do it with stations to get students out of their seats.
Trimester 2 final topics
Today students saw how an equation in factored form could be written when you see the 2 x intercepts. A graph was given so they could clearly see where the vertex was. They solved for a, by substituting the coordinates of the vertex (4,32), and then substituting the a value back into the equation to properly write the equation. Then students wrote it in standard form by multiplying it out.

Zoe told us the answer for part d of the problem, but we didn't have time to hear an explanation so we will pick up with that on Monday.

Solving for a
Then I had some private tutoring with a 7th grader. We went over the 5 important numbers needed to make a box plot. I took the idea of tracing his hand from Sarah over at Math Equals Love. We also discussed that the whisker in a box and whisker plot is the length of the range, because range is maximum minus minimum. This wasn't the best example because the minimum was the same as the lower quartile so it looked a tad strange.

Working on box plots.
Then at dinner I got a text from a student I tutor in Algebra 2 that he nailed his quiz. Pleasant surprise.
And to end the night, I met up with some teachers I met at CPM's academy of best practice math camp this past summer in seattle. They were in town for the CPM conference and we ate some delicious dinner at Kincaids in Burlingame.

Thursday, February 25, 2016

Day 108: Scientific Notation Intro & Table->Equation for Quadratics

Students estimated the number of pages in the book Life of Pi. Although it looked 3/4 or 2/3 the size of the book Where the Sidewalk Ends, it was surprisingly deceiving more pages than students estimated.

Today students worked on the MARS FAL Estimating Length with Scientific Notation. The lesson starts with quite a bit of whiteboard. It gives students 3 expressions that all equal the same number. Some did not know the number. Students explained that 3 x 10^3 is like 10*10*10 which is a thousand and multiply that by 3. Then they had to justify why the other 3 expressions were equivalent. Students generalized that the exponent told you how many places to move the decimal, and which direction. If it were negative, the decimal moved to the left. I was pleasantly surprised that some students knew 10^-1 would divide it by 10, and 1 student knew it was equivalent to the reciprocal of 10, so he said it was 1/10.

Then students decided which of the expressions were in scientific notation. Most students thought they all were, some thought the one with the negative exponent was it because it was more complicated. Then students saw that 3 x 10^3 was the simplest looking, so it must be the most efficient. That was solid thinking. One student actually knew that it must be like a single digit before the decimal point. He had seen scientific notation in, where else, his science textbook.

Then we wrote down the definition below with the inequality and the plain English translation. Then students lastly decided which of 2 numbers in scientific notation was greater. They decided the number with -1 as an exponent was greater because -1 is greater than an exponent of -3. Then students showed me how to write the numbers in decimal form.

Students matched decimal notation to scientific notation. One match had it's decimal representation missing and another had its scientific notation missing. Students had trouble with 2 x 10^0 and thought it was 20. When I asked what the exponent tells you they said how to move the decimal. Well I asked, what would an exponent of 0 tell you to do? They said do nothing. So, what does that leave you with? 2. It actually made me deepen my understanding of raising to the 0 exponent and this was an intuitive way of looking at it.

Lastly, students matched real life objects to the measurements and glued it to their poster. Students that finished early were instructed to justify their choices. Students worked at varying paces. Tomorrow they will place arrows between the objects to show how you can multiply one quantity by a multiplier to get another.
Instead of working in partners I had students work in groups of 4.
Here were two of the equivalent expressions equal to 1,000.
Here's the definition I had students copy down in their composition notebooks.
In one class 2 students made the same mistake for 8x10^-3. They thought the 3 was in the numerator. Interestingly, he correctly interpreted 10^-1 as 1/10.
In accelerated, students worked on writing a quadratic equation from a table of values. Some graphed it, and some realized they could write the quadratic in factored form knowing its x-intercepts from the table. We didn't get time to go over the problem where they solve for the a value in a quadratic when given the x-intercepts on a graph. We will review this on Monday.

Students took a placement test for Mills High school and asked me questions about what the letter i meant. Apparently the placement test was adaptive because it gave them harder and harder problems if they were getting them right to see what questions were out of their range.

As you can see below, we did a participation quiz, and it definitely had students cooperating more than usual.

Tomorrow students will be assessed on graphing a quadratic in standard form. They'll start the last section on completing the square, finish it on Monday, and I believe we will work on Desmos Marbleslides Parabolas on Tuesday. I have to reserve the chrome carts, and I think I'll only reserve 1 so I can have a 2 to 1 ratio. I'd love for students to record themselves solving one of the challenges and uploading it on a blog, I'll just have to find time to pull them away from it, or just do it the next day.

Participation quiz. Recording good behaviors, conversations, quotes, and some groups not cooperating.
This was a hilarious quote. One student was falling behind, but he hilariously came up with this poetic quote. I had to give him props and write it on the board because it was so funny.

Wednesday, February 24, 2016

Day 107: compound interest for exponents & solving for x in vertex form

Today's estimation was the number of pages in Where the Sidewalk Ends. We then discussed the difference between the simple and compound interest data tables. Kids were able to describe the growth of the account balances, but I wanted them to really think about the question, "How was the INTEREST growing?" They realized simple interest had a constant interest based on the initial balance. Compound interest had a growing amount of interest because the interest was based on a percent of the previous year's balance.

Below was a common mistake on some students where they multiplied the total interest earned by 1.05 instead of the balance.
One of the mistakes I encountered.
Then students graphed the total interest earned and saw that it was slightly curved. They then analyzed the patterns they saw in the table. They saw that you multiply by 1.05 as many times as the years you've had the money in the bank. They articulated that (500*1.05) represented the year 1 balance and was equal to $525.

This lead to a need for an easier way to write the calculation without writing 1.05 so many times. They remembered that repeated multiplication can be written with an exponent. This was in anticipation of our Scientific Notation FAL tomorrow. We didn't have time to complete the pre-test, but I'm not worried, I'll use it as a post-assessment classwork assignment.

In accelerated we reviewed a set of 4 homework problems to start. Nicholas showed us how a factored equation can lead to writing 2 separate equations to solve for x. In part B Hailey described how to factor using a generic rectangle and a diamond. Then solve for x. Part C Davin showed the class how to complete the square before we got to that part of the chapter, and another student showed the factoring way. While completing the square is an important skill, students saw that it wasn't the most efficient method for this particular situation. Finally they factored 4x squared -1. It was a difference of squares problem, and one student actually added 1 to both sides, divided both sides by 4, and took the square root. I was impressed that Alex N showed this method because I hadn't thought of it that way and it was a great way to solve it, and get positive and negative 1/2 as the roots.

Homework review explained by students.
Students were at different points of the lesson, and I had different students come up and show the class how they interpreted certain problems. Tiffany showed us how to properly write an equation in vertex or graphing form in standard form. I graphed it in Desmos and asked how I knew it was correct. They said it was right because it overlapped the original graph.

In the second equation, Alex D correctly interpreted the vertex coordinates as (-3,2). He said it was the opposite of 3 and the y coordinate was positive 2. Robert demonstrated how to solve for the x intercepts when the equation was in vertex form, and finally Aaron showed us how to do the same procedure except showed what happens when you have a non-perfect square or irrational number as part of your answer.
Reviewing classwork.

Tuesday, February 23, 2016

Day 106: Compound Interest & Vertex->Standard Form

The estimation today was a Yellow Pages phone book. Most students underestimated it. They liked watching the video, and there's always comments about "How much time does he have?" I asked them if they had somewhere to be at the time. Haha. Also, I pointed out that if they were watching carefully, the pages don't start getting numbered until the beginning.

I passed back assessments and reviewed how they were graded. I pointed out common mistakes. A lot of students forgot the negative sign on the decreasing line. Then we went over a homework problem, y=-1/2x+6. I structured our discussion around what format the equation is in, what B meant. We originally learned it as Figure 0, but we now call it y-intercept. M used to be growth factor, and still is in y=mx+b, but we know it's synonym is slope. Students knew slope is the ratio of vertical change over horizontal change. The most concise explanation by students was to go down 1, and over to the right 2. I asked them if 1/-2 was the same as -1/2. They said it looked different, but was still the same because a positive divided by a negative is negative. I showed how slope can get you a point to the left of a point by going backwards and using rise of 1, and run to the left of -2.

Then students analyzed a simple interest and compound interest table. They noticed that simple interest had a constant rate of change, while compound interest had a growth that kept on increasing. They saw this when the graphed a compound interest situation. First they filled out a table, seeing that multiplying by 1.05 would get them 5% interest on the current account balance. It ended up being a curved graph.

Tomorrow we will analyze how this can be written as a function, and will be their reintroduction to what an exponent is used for.

Reviewing how to graph y=-1/2x+6. Also showing how to use the equation to solve for the x-intercept by substituting 0 for y.
Illustrating using slope to go down and up a line.
In accelerated we first reviewed how to find the vertex of a quadratic when the x-intercepts were given. I wanted to review the first step to prove what the x-intercepts were. Students are forgetting to factor out any common factors before they start the factoring process.

Then students worked on converting vertex form to standard form. Many students did not expand (x-2)^2 correctly. They said it was x^2-4. About 70% of the class made this mistake. So, during closure we discussed the mistake and a few students explained how instead of using a generic rectangle to multiply (x-2)(x-2) you can just square the first and last terms to get x squared and positive 4, and the middle terms are x times -2 then double it. That gets you x^2-4x+4.

A lot of students saw how the vertex form gave you the vertex. The number added or subtracted was the y coordinate of the vertex, and the number next to x was the opposite of the x coordinate.

Tomorrow students will practice this more, and also learn a new method for solving for the x intercepts when the equation is in vertex form.

Homework review. Zero product property.
Students saw that the vertex was (2,-5) when in this form. They also saw that in standard form it was the same graph as the vertex form.

Monday, February 22, 2016

Day 105: Simple Interest & Zero Product

Today's estimation was an Everyday Math workbook. Most students overestimated. It was deceiving smaller than it looked. I did like some students that figured with 180 days in the school year, maybe 2 pages per day, so 360 pages. Others said it was 5 times bigger, so 78 times 5.

I gave students new seats and we started chapter 8. To make the lesson begin quicker, we discussed 4 graph situations to sketch. The first one was the relationship between the months of the year and average hours of sunlight per day. Students knew there were less hours in winter, more in summer, and less again in winter. I told students this is what they'll later study as a cosine wave, a trigonometric function. The second situation was months and a bank account with simple interest. I had them think pair share and then share out what interest is, and what simple interest is.

Some confused it with compound interest, but realized that simple interest takes a percentage of what you first put in and adds it each time period. I was surprised that one student knew the answer to my question, "How do banks make money if they give you money for having money in the bank?" He said that they give out loans and charge a higher interest. So, it was mini lesson on economics.

The third situation was selling an item for 50% of it's price multiple times. 2nd period suggested a popular item is a hover board. I made up that I work part time at a hover board company and got an employee discount and got one cheap. I said that I'm greedy, so I want to increase the price by 50% and sell it. Students reasoned that half of 100 is 50, so the new price is 150. One student who usually doesn't participate was totally zoned in on the discussion and was giving all the answers. The topic of money always gets students attention! After graphing it, students noticed it was a curve. I told them this was called an exponential curve.

The last situation was the relationship between the length and height of a rectangle with an area of 24 square units. This also produced a curve, but I told classes this was called a rational function that they will investigate their junior year.

The lesson was basically about the 2nd and 3rd graphs, but in greater detail. Students realized that simple interest had constant growth and the graph was linear. It was also proportional since it went through the origin. The selling of an $8 baseball card repeatedly for 50% more was an exponential curve, or non-linear. They also said it was not proportional because it did not go through the origin. I was pleased that a few students in each class saw that instead of adding the same number, it multiplied by 1.5 or 1 1/2 each time.

The 4 graphs to start class.
In accelerated we reviewed the first problem from Friday about how many points you need to graph a parabola. Then students worked on factoring to solve the roots or x intercepts of quadratic equations. They saw that the factors setup 2 simple equations to solve that are their x intercepts. This will be Friday's assessment skill.

Then students worked on a pre assessment. I only gave them 10 minutes which was not enough time to finish, but they will work on group posters later this week or next week that will solidify their understanding and allow them to improve their thinking for a final grade later on that pre test.
Using a generic rectangle, then a diamond, and then factoring to find the side lengths.
Here are the two simple equations to solve it once they've factored.
Oh and I have to share this. A few of my students had a science project where they created a planet that had it's own periodic table of elements. As you can see below, I made it on there as "Joyceium." I have an atomic mass of 5 and a melting point of 1050. Awesome! Great job guys.
And in there sample element you can see they called it Hu yon. This is an ongoing inside joke from the CPM Core connections course 2 curriculum, where they use a super giant one to divide by a fraction. They call it Huy's method. Any time students use a giant one they say I just used Huy's method. It's hilarious.

Friday, February 19, 2016

Day 104: Equation from 2 points, Scatterplot & X-intercepts

Todays estimation was how many pages in a Rolling Stone magazine. It was more than most students expected. Yesterday's estimation was Harper Lee's To Kill A Mockingbird. Coincidentally and sadly, she passed away today. One student actually already knew it. A little eerie.

I loved the homework problem I reviewed with students today. It was so relevant to the assessment and what we've been working on: slope. Students were instructed to graph (-6,3) and (-3,-1). I graphed it and asked students how to find the slope. They said first draw your slope triangle. To find the height, they said subtract 3 and -1. I reviewed 3-(-1) with a plus and minus drawing. Students said that gets you the vertical change because it's the y coordinates. Then to get the base of the triangle, -6 -(-3) which gets you -3. They told me these 2 facts combined to make the slope 4/-3. I asked them if -4/3 was the same? They thought about it and agreed.

From here, they saw on their graph that the y-intercept was (0,-5). I asked them why this makes it easy to write an equation. They said that in y=mx+b, m is the slope, so put -4/3 in, and b is the y-intercept, so put in -5. So, the equation is y=-4/3x+(-5).

I wanted to preview a method from Algebra 1 that they'd probably see next year. I asked them how they could use algebra to find the y-intercept, especially since I just sketched my graph and I couldn't count any tick marks. I wrote the equation y=mx+b and substituted -4/3 for m. I said how can I use this? y=-4/3x+b? In each class I got 1 or 2 hands to raise. That wasn't enough for me. So, I went over to the coordinates and selected one. I asked them what the first digit represented. They said x. And the second? They yelled out Y!!! So.. I underlined that with my hand, and walked back over to the equation and underlined the whole equation.

To my delight, at least 6 more hands raised up and they said to substitute the coordinates for x and y into the equation. I asked them how to multiply fractions, then they chorally told me the steps to solve for b. They liked seeing this and were amazed that algebra got them the answer.

One curious student asked, could we have used the other coordinate? I told her to try it out. If it's also on the line, shouldn't it work?

Writing an equation from two coordinates. Solving for B.
Students had a limited amount of time for the classwork. They revisited the Newton's revenge problem. Students realized that starting at the origin made all the data clump together making it hard to draw a line of best fit. Some students rescaled their graphs so they could make a line of best fit easier.

This student made a big graph, then realized he wanted to rescale it.
This graph at the bottom was properly scaled and shows how easier it was to draw the line of best fit.
A few students wrote an equation of the line of best fit and used it to test if the roller coaster was safe for Yao Ming to ride on.

In accelerated we went over graphing y=x^2-8x+7. I asked what we could get right away with the choice of vertex, x-intercepts, and y-intercepts. They said it was (0,7) because if you plug in 0 for x y would be 7. I asked them what we could do with this quadratic. They said factor it. I asked them how this could help us. Some said to plug in 0 for y. Therefore, the x intercepts are (7,0) and (1,0). They said the vertex is halfway, so 7-1 is 6, half of 6 is 3, so 1+3 is 4, so the x coordinate of the vertex is 4. I asked how to get y, and they said plug 4 into either equation. I plugged it into the factored one which I always think is the easier way. That got us -9. Therefore, all the requirements were there to graph it completely.

Students worked on the zero property, but there were a lot of disruptions with students coming back from the music field trip, so we will finish up this lesson on Monday and then take a pre test on a formative assessment lesson on quadratics.

Homework problem review.
I was impressed with how this student solved this quadratic without factoring the common factors out first. I'll review this a bit on Monday, to make the students life easier.

Thursday, February 18, 2016

Day 103: Equation of Trend Line & Quadratic Reps

Today's estimation was the pages in To Kill A Mockingbird. Students thought it was about one and a half times as long as Charlotte's web. There's a LOT of book page estimating coming up, so I feel I'm going to skip some so I can squeeze in some Which One Doesn't Belong Warmups.

We discussed a great homework problem where 4 students got 4 different answers for the slope of a line. They got 3/4, -3/4, 4/3, and 3/4. I started with asking students what the answer couldn't be. They saw, probably with help from the Slope Dude Says video, that it was a line with negative slope, so it couldn't have a positive slope. Then students described where they saw the lattice points and pointed out that the vertical change was negative 3, and the horizontal change was positive 4.

Then we reviewed the topic from yesterday because it was a bit rushed at the end. I asked a random student who I thought may not have calculated their slope yesterday for their 2 coordinates for number of times they wrote their name in 1/2 minute and their prediction for 4 minutes. I asked how we could find slope? They said draw a triangle. Then I reminded them that we could count squares, but my example had no axes drawn, so how could we do it more efficiently? They said that you can subtract the y coordinates to find the vertical change, and subtract the x coordinates to get the horizontal change. After getting an equivalent ratio without decimals, then dividing, they saw their slope was their unit rate, for how many times they wrote their name in 1 minute.
Students reminded me to find 4 minutes, you multiply 10 by 8 because there are 8 1/2 minute chunks in 4 minutes.
Then, I introduced them to the slope formula, that they had used without really knowing it. I told them they would see it again next year in Algebra I.

I then introduced the Learning goal which was to make a scatterplot, write an equation of the line of best fit, and use it. The success criteria was to use what you know about m and b to write the equation in y=mx+b form. Use substitution to make a prediction. I think I may have given too much info to use substitution, but they had to articulate what number they were substituting for what variable, which was good later.

The main problem was a 30 mile charity bike ride. A girl, Aurelie, had been training, and wanted to predict how long she would take to finish the ride. They convinced me that the dependent variable was the quantity we wanted to predict, so that was on the y axis. I related it back to Newton's revenge where we wanted to predict Yao Ming's reach because we knew his height.

Here Dominic had a nice graph, showed how he got the slope. We talked about the substitution, but he didn't show his work for that yet.

Here the substitution was shown.
The really cool part about this lessons progression, is that you make a prediction with your line of best fit first, and it's not that accurate. Then when students write an equation and use substitution, they get a super accurate prediction that is within the range they thought it was on the graph. A lot of a hah moments.
My new favorite strategy: showing a dashed line to show exactly how you used your line of best fit to make a prediction.


Modeling the substitution work.

An easier substitution.
More substitution.
In accelerated students finished their graphs of the 4 people in the water balloon throwing competition. Before they continued, we reviewed 4 homework problems. Two of these type are on the assessment. The first one was difference of squares and students saw that both terms are squares, so you put the square roots in each factor and have one plus and one minus sign, HENCE "difference" of squares.

I forgot to label the second official method, but it's called perfect square trinomials. It's basically doing a diamond with mental math. Problem c was also a perfect square trinomial but with a positive middle term. The final one was a difference of squares problem that had to be factored completely first. The student divided all terms by 5, factored, and left it at x squared minus 9. Another student raised their hand and said that's not right because they forgot to put the 5 outside the parentheses. I complimented this student for politely showing how you respond to an incorrect answer.
Homework review.
Students found the x intercepts and the coordinates of the vertices of each parabola. There were great conversations about how they found it. I made a point to talk about the situation to graph where the water balloon was launched from the 10 yard line and landed at the 16 reaching a height of 27. They interpreted that as x intercepts of 10 and 16 and the y coordinate of the vertex was 27. I asked them how we could find the x coordinate. Another student raised their hand and said it's halfway between the x intercepts because the vertex splits the parabola in half. I was impressed. Therefore, there's a distance of 6 between them, half of that is 3, so 10 plus 3 is 13, so 13 was the vertex. Here are their results:

They also showed me how they got 8.5 for the x coordinate of the vertex, as well as plugging it in for x to the equation to get the y coordinate of 30.25.
We also discussed proper way to write domain and range.

To finish off, I was super impressed with this students work for the graph where only 3 points were known. He went ahead and did some math work and got an equation. I said well, let me put it in Desmos and see if it's right! And it was. We got some oooohhhhs and aahhhhhs when we saw this. I had to take a picture of how he did it. I haven't taught him any of this but he literally wrote a system of 3 quadratic equations. You can see he subtracted a pair of them to eliminate the C variable. Then he eliminated the b variable, to solve for a. He plugged a into one of them to solve for b, and then solved for c. He got y=-3x^2+78x-480. I was floored and it's probably the most excited I've been by what a student did through their own determination. I did not teach this student any of these methods. He learned them in Kumon, and I am super impressed that he applied his techniques properly and in the context of our CPM textbook!


Systems of quadratics!!! (not an Algebra I topic, to my knowledge....)

Confirmation with Desmos... very impressed.