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.

Thursday, October 19, 2017

10/16 - 10/20 -- A Math Classic

Sam's 4-6th Math Class

This week in addition to our usual work in the texts, we looked at a classic math puzzle from antiquity. The common phrasing of the puzzle comes to us (translated from Latin) through the author Metrodorus:

'Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old:
'God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage,
After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.

We spent a good portion of the class decoding--discussing the phrasing of the problem and how to parse the words into mathematical notation--a skill critical for real world problem solving. The students rapidly found fractions they could extract from the text and with a little direction were generally able to make some kind of model to represent the situation.

Most of the students choose to treat the problem as they would a word problem from the Singapore texts and drew a model something like this:

But what do the orange regions represent?

The difficulty with this approach was how to think about the extra 5 years between his marriage and the birth of his son and the four years at the end of his life. Eventually, we got to the idea that the orange regions in the above picture must be those 9 (5 +4) years. By figuring out what fraction of his life these 9 years represented, the students could then figure out his age.

One group of students who have some experience with algebra and variables was asked to express and solve the problem algebraically. They, correctly, arrived at:
x/6 + x/12 + x/7 + 5 + x/2 + 4 = x 
which can be solved for the correct answer.

At the end of the lesson we spent some time talking about the similarities between the algebraic and pictorial representations. One of the foundational principles of the Singapore Math program is the idea that student learning should progress from concrete models (physical objects) through pictorial representations (bar models, other drawings) before students are asked to work on fully abstract, algebraic notation. Explicitly showing the students the connection between the bar models and algebra facilitates this transition.

Pre-Algebra and Algebra

The 7/8s had a number of special activities this week that preempted or cut into math, you can read about those in the 7/8 class section. Nevertheless,  all the students were at the end of a unit of study and are independently reviewing or working on a assessment (in this case a chapter test).