Monday, February 19, 2018

Winter 2017/8 -- A Very Overdue Update Posting (Dec 18 - Feb 9)

Note: You'll notice it hasn't been updated in a while. My bad; however, members of the SK community have access to all the information here through the Google Doc attached to the weekly newsletter.

December 15 and December 21, 2017
Sam’s 4-6th Math Class
This time focused on getting everyone to a good place to take a vacation. Many of the students worked on assessments. The students who did not have an assessment to work on (because they’ve already done at least one this year) took a look at a foundational problem that links geometry and number theory--specifically, the number of regions created by n lines inside a square. They focused on looking for patterns in the maximum possible number of regions for a given number of lines and developing first an iterative and then explicit formula for the number of regions based on the number of lines. One student commented on a connection between this problem and the famous (if apocryphal) story about Gauss and the sum of the first 100 whole numbers. The last days before break we also took a look at what happens when you try to build Pascal’s triangle in binary--surprise, the result is the Sierpinski Triangle with 1s and 0s.
Algebra and Pre-Algebra
These groups are also working toward completing a chapter or unit before break. The Pre-Algebra class is working with factors, multiples, and fractions concepts including variables.
The Beginning Algebra group continued work on writing linear equations based on points, a data set, or other relevant information. The Completing Algebra group was introduced to the Quadratic Formula and mostly worked on finding solutions (zeroes) of quadratic equations.  

Also, as part of their presentations on electronics components, the students looked at direct applications of Ohm’s Law, and more general proportional reasoning as with a variable resistor.

January 12, 2018
Sam’s 4-6th Math Class
This week we worked on a common math puzzle for the beginning of a new year. The goal is to write expressions that evaluate to all the numbers from 1 to 100 using just the digits of the current year. We played around with different sets of restrictions, beginning with very prescriptive rules: the digits 2, 0, 1 and 8 must appear in that order and the only other symbols (operations) allowed were: parentheses, +, -, X, /, !(factorial),(square root) , ^ (putting things in exponents). We were able to get about 25 with this set of restrictions. We then loosened the rules to allow for concatenation (so 2 and ) could be used to make 20, etc), this got us quite a few more of the numbers. On the last day, we dropped the order restriction which was sufficient to get us most, but not all of the 100 numbers.
As a teacher, I love puzzles like this because they allow for students with a wide variation in knowledge to still all access the same problem, all the while emphasizing analytic reasoning, order of operations and accuracy in calculation.
Algebra and Pre-Algebra
The pre-algebra class learned about rules for exponents this week, including the meaning and use of negative and zero exponents. The Beginning Algebra class looked at the relationship of the slopes of parallel and perpendicular lines. The Completing Algebra group is working on memorizing the Quadratic formula and did a lesson where they researched and analyzed the quality of youtube videos on that topic. My favorite video they found is here.
Following on our visit to AVL, we encountered concepts related to pressure, so we talked about the proportional relationships between pressure, volume and temperature (Charles’ and Boyle’s Laws).

January 19, 2018
Sam’s 4-6th Math Class
The week for this group was a little staccato. On Tuesday, I was home ill, and the class worked with Wendy’s group on problems about symmetry and geometry. We used Wednesday to catch up on Singapore work that was planned to get done last Friday. Thursday was move up day; the 6th graders were travelling with the 7/8s and, as the 4’s were visiting with the 5’s, they all met together for a math activity with Wendy and Lisa. On Friday we got going on Singapore assignments for next week. Most of the students are working on concepts with place value: unit conversions, percents, decimals.

Algebra and Pre-Algebra
The Pre-Algebra group are all working on the chapter 4 assessment, covering topics with exponents and powers. The Beginning Algebra group continued talking about equations of lines and writing equations of parallel and perpendicular lines. They also spent one class period goal setting; planning out their workflow with the optimistic goal of completing chapter 6 (on inequalities) before Mid-Winter break. We got out the planners and talked about how to decide what times would be useful to work during and how much you might expect to get done. In Completing Algebra we’re wrapping up a chapter on graphing quadratic equations, including an introduction to the quadratic formula. This week we especially looked at how to use the quadratic formula and real world meanings of “finding the zeros of a parabola.”

January 26, 2018
Sam’s 4-6th Math Class
This week had us going further with geometry and measurement in the form of letter dissection
puzzles. Building off the T and H puzzles we’d worked on before, students used grid paper
to make their own letter puzzle then tried them out with friends and made revisions.
The activity culminated with the students learning to use Tinkercad, 3D modeling software,
to create a makefile for their letter puzzles, which are then being printed on the Makerbots.
There wasn’t enough time to print all of the puzzles last week; everyone should have a
completed and printed puzzle to take home by the end of the day on Monday January 29.

Algebra and Pre-Algebra
The Pre-Algebra group all wrapped up their work in chapter 4 and have moved on to chapter 5, covering fractions and rational numbers more generally; for the next few couple of weeks, they’ll be solidifying their calculation skills with fractions and applying those concepts to solving single variable equations that include rational numbers. The Beginning Algebra group is starting a series of lessons on inequalities and absolute value; we had a lecture and discussion about the basic concepts on Thursday and will dig more into those concepts next week. The Completing Algebra group is working on finishing up their initial study of parabolas; some of the students are still working on applying the Quadratic Formula to get the zeros of parabola, others have moved on to thinking about factoring quadratics more generally.

February 2, 2018
Sam’s 4-6th Math Class
This week we began working with some concepts in pre-algebra, in particular, variables and solving equations in one-variable (with whole number coefficients). We are making connections between concrete models, using the Algebra Lab Gear; pictorials models, like the Singapore bad models, and fully symbolic algebraic notation.  

Algebra and Pre-Algebra
The Pre-algebra group is continuing their work on chapter 5, now looking at expressions with rational (fraction/decimal) coefficients, the Beginning Algebra group is beginning to discuss inequalities and absolute value while the Continuing Algebra group wrapped up the chapter on graphing parabolas and the Quadratic formula.  
February 9, 2018
Sam’s 4-6th Math Class
On this snow shortened week, we continued working with the lab gear, this week introducing rules for working with negative numbers, and the distributive property. The students noted that the lab gear blocks (and by extension discrete manipulatives in general) do not do as good a job of modeling fractions and decimals as integers (though we aren’t using all of that vocabulary, yet).

Algebra and Pre-Algebra
The Prealgebra group is motoring through chapter 5 (on fractions), some would have finished if not for the two snow days. The Beginning Algebra class had more time to develop concepts around absolute value, particularly thinking of absolute value equations and inequalities as something about distance from a designated central point. The Continuing Algebra class has mostly wrapped up graphing parabolas and is now studying arithmetic with polynomials, especially factoring quadratic equations. Two students also have completed Algebra and begun work in Geometry. This week we met to talk about the basic underlying “rules of the game” of Euclidean Geometry with points, lines and planes as the “playing pieces” and a few very basic rules to get us started (Euclid’s first four postulates).

Friday, December 8, 2017

Week of December 8

December 8, 2017

Sam’s 4-6th Math Class

Etienne and Ishan work on Sierpinski Carpet fractal.
Counting boxes to make the fractals.
This week we continued to look at fractals. The class spent a significant amount of time working on a fifth iteration Sierpinski Carpet, which is now on display in the MS across from the elevator. It was a technically challenging experience for many of the students that required following an iterative process with a high degree of attention to details. If even one square was miscounted the whole image looked wrong. (If you get a chance to see the finished product, you can see a few places where someone had to correct an error.
Algebra and Pre-Algebra

Nick and Niko explore the relation between batteries (voltage),
 resistance (lightbulbs) and current (brightness of bulbs)
Math this week largely was combined with science activities. As we are working with the Arduinos, we’ve begun to learn some of the basic concepts surrounding electricity and currents. Tuesday’s batteries and lightbulbs investigations emphasized proportional reasoning and we formalized that connection with the introduction of Ohm’s Law on Wednesday.

The d=rt and V=IR graphic organizer pyramids visualize
the proportional relationship of those quantities.
In more formal math classes, the Completing Algebra group has been working on graphing parabolas; especially investigating establishing a relationship between changes in the coefficients of the terms and the appearance of the graph. The Beginning Algebra group continued to work with linear equations and the Pre-Algebra group is going into more depth on single variable equations and inequalities.

Friday, December 1, 2017

Week of December 1

December 1, 2017

Sam’s 4-6th Math Class

This week we dug into concepts related to ratios and fractals. We watched the thought provoking Nova episode: Fractals the Hidden Dimension and then talked about what it means to be “self-similar” and how an object can have a finite area but an infinite perimeter.
It was also very interesting that in their Morning Meeting presentation about Wildlife in Michigan, they mentioned the length of the coastline, which led us to another discussion about fractals and measurement related to “How Long is the Coastline of Britain.”
Algebra and Pre-Algebra

The 7th and 8th graders have mostly been completing assessments this week. All of the groups also had some new material introduced. The pre-algebra group started a unit on factors, fractions and exponents; the beginning algebra group is learning how to write the equations of lines from points or graphs and the completing Algebra group has started working with square roots and quadratic equations.

Friday, November 17, 2017

Week of November 17

November 17, 2017
The highlight of this week was the AMC 8 contest on Tuesday. All of the students in my math classes participated. We spent Monday looking at sample problems and talking about test taking strategies, especially that the goal is to get as many correct as possible in a short time frame and NOT to do every problem correctly and in order. We also talked about identifying which problems might be solvable, but would require such a large investment of time as to not actually be worth completing.

Sam’s 4-6th Math Class

In addition to the AMC contest and the usual work in Singapore math we also started to look at some basic problems in combinatorics, including a lengthy discussion about strategies to stay organized while counting and enumerating possibilities. Specifically, the students worked on these two problems:
  • You have 3 pairs of jeans (we called them X,Y and Z), 7 T-shirts, (called one, two, three, four and five) and 5 pairs of socks (A, B, C, D, and E). Make a list of all of the possible outfits you can wear.
  • What are all the possible outcomes when three normal six-sided dice are rolled, and how likely are each of the possible sums of the three dice.
Algebra and Pre-Algebra

All of the students in these classes are working on finishing up the chapters we’ve been working on with a goal of being done by Thanksgiving break. Some of the students got to the assessments on Thursday or Friday but most will be studying this weekend and working on the assessment on Monday and Tuesday.

Week of November 10

November 10, 2017

Sam’s 4-6th Math Class

On Wednesday we began a new recurring exercise--the timed quiz. This first timed quiz was 3 minutes to complete 60 single-digit multiplication problems. We were practicing quick factual recall; everyone in the class can do these problems, speed, dealing with anxiety and the educational concept of automaticity, were the focus of the discussion post-quiz. On Thursday and Friday the class worked on their Singapore assignments while meeting with Wendy’s class,
Algebra and Pre-Algebra

Because of the aforementioned short week and absences (mine and the students) the pre-algebra group didn’t meet with me this week. They should all be continuing work independently in chapter 3 of the text. The Beginning Algebra group did have a lesson on Wednesday going over the concept of slope--using the “staircase” analogy and as a ratio. The Completing Algebra group also met formally on Wednesday to begin our discussion of exponents and exponential functions, though so far we are only looking at the properties of exponents and vocabulary.

Friday, November 3, 2017

November 3, 2017 -- Coordinate Graphs

November 3, 2017

Sam's 4-6th Math Class

This week we focused on the coordinate plane. Many students were familiar with number lines; introducing the full coordinate plane requires introducing negative numbers (which most students had not worked with formally before) and moving from one-dimensional to two-dimensional thinking. This can be a rather dry conversation, so instead of lecturing on the topic, we talked about the basic ideas and then transitioned right into an activity: making a picture from a set of points--basically an advanced form of connect the dots, but the students had to put the dots in the correct place. After they finished the initial exercise, everyone made a coordinate picture of their own by reversing the process used to make the kitty cat--first make the drawing, locate the important points (places where the line changed direction--vertices), and then right out the list of points as line 1, line 2, etc, so that someone else could recreate their picture. IMG-2990.JPG

Here is one a child made in class if you want to try it at home.

Algebra and Pre-Algebra

We have also been doing quite a bit of graphing on the coordinate plane in the ⅞ math classes. The Completing Algebra group worked on some basic exercises in linear programming as an application and review of graphing systems of inequalities. The Beginning Algebra group is just starting to work with linear equations in two variables and this week talked about graphing lines from an equation by finding points and a little bit about intercepts and horizontal and vertical lines. They also had a brief “pop” quiz, which will be a recurring feature of the class. [One of my professional concerns about timed assessments is that the goal of any timed assessment is to get as many points as possible, not to answer the all questions correctly, our students need practice switching their mindset before being asked to take many timed assessments in high school.] The Pre-algebra group continued their work with linear equations in one-variable.

Friday, October 27, 2017

Melons to Market, Reviewing with Khan Academy

Sam's 4th-6th Grade Math

This week in addition to the usual work in the texts we spent quite a bit of time investigating this situation:

A boy has 45 watermelons in the desert. He needs to get them across to the Oasis fair, 15 miles away. He can only carry 15 watermelons at a time, and he eats one watermelon every mile he walks, including walking back to where he started from. He can also leave watermelons at any mile he has walked, but no fractions of a mile. How many watermelons can he possibly take to the fair? How did you arrive at your conclusion?

The class spent a good amount of time, in groups of 2 or 3, working out how to even begin to approach this problem. Much of the class initially believed it to be impossible. By the end of the first day, everyone agreed that it was possible to get at least 2 melons to the market; the task for day two was how to determine the maximum number. Investigating this problem required much more than straightforward arithmetic; it's more of a logic puzzle and while working through it our class ended up discussing ideas like resupply/refuel depots for Antarctic exploration, note-taking/tracking techniques (how do you keep track of where all the melons are, and how many are left) and the important strategies of using manipulatives, working backwards, and looking at simpler cases.

Evie, Jacob and Ben used a number
line and manipulatives (beads) to help
them think about the problem.
Ishan and Juliana also used beads
to represent the melons. 

Algebra and Pre-Algebra

The seventh and eighth graders had a lot going on this week--two days with field trips and an extended OWL class that cut into our math time. Almost all the students completed an assessment this week and are either working on reviewing concepts using Khan Academy or are moving on to the next major unit of study: solving single-variable linear equations for the Pre-algebra group; linear equations in two-variables for the Beginning Algebra group; a detour into linear programming for the Completing Algebra Group.