Friday, September 29, 2017

9/25-9/29 Problem Solving w/ Bar Models, Integers, Graphs

Sam's 4-6th Math Class

With the 5/6s gone on Thursday and Friday the workload was a little smaller this week. The kids should still try to have their Singapore assignments completed for Monday 10/2. On Tuesday and Wednesday we learned about using the bar model strategy to solve problems with whole numbers, particularly problems involving the idea of consecutive multiples. It is important for the students to develop a way to represent these ideas graphically, as many of them are not developmentally ready to approach these problems abstractly using algebra.

For the problem, "The sum of three consecutive even numbers is 36. Find the greatest," the students would make a drawing akin to this:
We can then talk about the idea of maintaining equality by removing the same amount from the bars as from the total, such that this problem can be reduced to the three boxes being equal to 30, which then allows the student to figure out that one box is 10, and then carefully re-read the problem to figure out how to answer the question asked. 
We will continue to work with this technique for problems with percent, fractions, and ratios in coming weeks. 

If you'd like more background on the technique or the rationale take a look at these websites:

Pre-Algebra and Algebra

The blue and green groups both got some lessons on the need for orthopraxy [our unofficial vocab. word for the week] in the creation of coordinate graphs, which is a fancy way of saying I told them exactly how I want them to make their graphs every time. The key features of a coordinate graph are: clearly labelled origins and axes, consistent and clear numbering of the grid, and in my case I add the requirement that the axes are only labelled with arrowheads in the positive direction, which is a convention I learned to use in physics classes. 

The (Blue) Pre-algebra group completed chapter 1 in the text and had their first formal assessment (that's a traditional test in this case). The major topics covered in chapter 1 were variables, the order of operations, exponents, basic absolute value concepts and the coordinate plane.

Maddy uses the Algebra Tiles to
investigate inconsistent linear equations.
The (Green) completing Algebra group is digging into a chapter on systems of linear equations. We've talked about graphing and estimating solutions, using substitution and the elimination by linear combinations method. 

The (Red) beginning Algebra group is working on linear equations in one-variable which is partially a review from last year, but now we are adding in the additional complication of working with fractions and decimals. We spent most of Thursday talking about problems that are "weird", where the solution is that the problem is always or never true; things like 2x + 4 = 2x - 3.

Thursday, September 21, 2017

9/18-9/22 Fractions, Books, Khan Academy and Blackjack

Sam's 4-6th Math Class

This week has mostly focused on fractions. We've been practicing a technique using partially shaded arrays to represent fractions; two arrays (carefully constructed to be the correct sizes) can be overlapped to create a set of boxes, depending on how the shaded boxes are counted, the sum, difference and product of the fractions can be determined. 
Here is an example of a fraction graph that is incorrect. We used this example to check for student understanding of the process. 

How many mistakes can you find in the above problem?

After the discussion about what was wrong with the above grid we started to talk about how this technique often gives you an answer that is an unreduced fraction. We spent Wednesday examining what aspects of the problem determined whether or not this technique would produce a reduced or unreduced fraction, eventually arriving at the idea of whether or not the denominators shared any common factors or were relatively prime.

On Thursday with some faculty and students out for a holiday, our group met with Wendy's math class and played a variety of teacher selected board and card games that have some mathematical or logical component--Tsuro, Castle Panic, Forbidden Island, Poison, and Gimme the Brain!. 

Pre-algebra and Algebra:

This week the Pre-algebra class is working on completing chapter one in the text; they had a lecture covering the general topics--arithmetic with integers on the number line--on Monday, and had time to work and ask questions during the middle of the week. 
The Beginning Algebra Group finished up the activities with the lab gear, including thinking about like terms and some discussion of how to use the Lab Gear to factor polynomials, though that will mostly be covered later on. 
The group of students already in the middle of the Algebra curriculum spent the week reviewing for and then taking a traditional test on inequalities in one and two variable including with absolute value. 
On Friday, Ed Feng returned for his first visit of the year. Ed is continuing to focus on probability & statistics, and Python programming. This week the students have begun investigating the probabilities in Blackjack. More about Ed's experiences at SK can be found on his blog:

Friday, September 15, 2017

Week of 9/11 - 9/15

Sam's 4-6 Math Group:

This week the class was led by guest teachers on Monday and Tuesday. On Monday, Mary Perrin assisted the students in creating a tessellating tile from which the students are making a larger art piece. On Tuesday, Josh got the class started on their Singapore books, though a few students were still finishing assessments this week. On Wednesday, one small group of students had a lesson about using estimation to help with multiplication and division problems when decimals are involved, while the other kids had time to work. On Thursday, we reversed that, except instead of a lesson I met individually with each of the kids who had not had the lesson on Wednesday. Friday the class had time to finish up the tessellations and Singapore assignments, and then got to play math games if they finished.

Working on tessellating art.

Some of the finished Tessellated Art with the original cardboard tile.

7/8 Math:

After being at Tiller's International on Monday and Tuesday, the 7/8s fully split into their small groups for lessons on Wednesday and Thursday. 
The pre-algebra group is still getting used to working in the textbook and some of the introductory ideas and notation used; mostly we are working in chapter 1 in the text. 
The group just starting Algebra this year has been doing an enrichment about algorithms and exercises using the Algebra Tiles to solve single variable equations. They decided they needed negative versions of the blocks and we figured out that we could put black stripes on the side of a block to make it a "negative". They then derived the idea that a black block and a regular block of the same shape eliminated each other. Essentially, they were able to derive the inverse property of addition from working with the blocks. 
The other students have been learning or reviewing linear inequalities in one dimension with absolute value, and at the end of the week linear inequalities in two-variables.

Friday, September 8, 2017

Welcome to the 2017-2018 School Year

Hello Parents, Students, and Friends, 

Welcome back from the summer. For those of you who have followed this blog in past years, you may have noticed a title change. My role at Summers Knoll continues to evolve; this year I will be primarily focusing on teaching math. I will also be working with Rachel in the 7/8 homeroom. This blog will solely report on the activities of my math classes.

Week 1: 

Sam's 4-6th Math Group

This week on Tuesday and Wednesday our math group met with Wendy's group as we did some sorting out (some students are working on an assessment) and getting reacquainted. We began instruction with a number guessing game and then talked about factors and divisibility rules, using factor trees.

On Thursday we split off from Wendy's group and spent the time getting familiar with our space (upstairs) and some of the tools we will be using this year (rulers and protractors) and then began an investigation of tessellations and tilings that will lead to an art project that should be finished next week.

Oliver, Mark and Manon work to figure out which shapes can gaplessly tile the plane.

7/8 Math

The 7/8s are working on a couple of different things depending on what they were working on last year. The pre-algebra group is working with the order of operations, exponents and variables. The group starting algebra this year is doing an investigation of algorithms with the algebra tiles. The students who were well into, or done with, algebra had some time to get reacquainted and find their place in the textbooks and are now working on reviewing or learning about absolute value equations and inequalities in one variable.

Miel and Niko work through and algorithm with the Algebra Tiles.