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.

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).