Sunday, September 1, 2019

Friday, March 8, 2019

February -- Inequalities and the Train Tracks

February 8, 2019

Still re-adjusting from the missed days, two of the normal math classes this week were used to work on the lingering second projects (in both ⅚ and ⅞). This meant we only had math two times this week. The two days were quite productive--a lot of work on inequalities, reviewing of the distributive property, learning standard form of a linear equation, and finishing up assessments did happen.

February 22, 2019

This week the Pre-Algebra class has been working more closely with linear equations and introducing linear inequalities in one-variable. We have been especially focused on how to work in an organized fashion (the “train tracks” technique) using the reverse order of operations as a guideline for sequencing the multiple steps that may be required. On Thursday and Friday the class worked on an investigation related to these concepts called “Who is Justice Equilibrator”.

The Algebra and Geometry students have been working independently, finishing up diverse topics from linear equations in two-variables, to geometric concepts of quadrilaterals.

Thursday, September 6, 2018

Welcome to the 2018-2019 School Year


You'll find weekly updates about middle school math classes here.
For information about 7/8 Homeroom and Projects look at the 7/8 blog I share with Rachel.

General Overview:

Class Times and Groups:
As in past years, I will be working directly with all of the 7-8 students in math. However, this year, our classes will also include some fifth and sixth grade students who are working on the same material. Every student will have math class for 50 minutes each morning. The students are split up based on on there progress in the pre-algebra/algebra/geometry sequence, with all the students working on pre-algebra content on one group and the remaining students in the other group.   

Textbooks and Content:
We will be continuing to use the same books we’ve been using for the last several years, McDougall-Littell’s Pre-Algebra, Algebra: Concepts and Skills, and Jurgensen Geometry.

This year, I am asking that students work on their core (textbook) math at home at least three times per week. This is a guideline, not a strict requirement, and the exact amount will vary depending on prior familiarity/comfort of the individual with the specific content and the amount of math content that is part of the current project.

A document called “<Algebra/Pre-algebra/Geometry> Suggested Assignments” will be shared with students and parents. This document has the book/Khan Academy assignments broken down into discrete chunks roughly corresponding to one section in the text. The exact problems to be completed are suggestions; if a topic is truly review, or if something is especially difficult, a student may end up glossing over some of the problems, or choosing/being asked to complete more of a specific type.

Checking the Answers and Homework Feedback:
I will not be collecting and correcting book assignments. Students will have an opportunity to ask for assistance about specific concepts or problems during whole group or one-on-one sessions. The majority of the suggested problems have answers in the back of the textbook and students are expected to make an attempt and check their own answers before deciding to move on from a topic or asking for help. Students should endeavor to complete three to five sections (a row in the assignments document) per week, expecting to do one to two in school and two or three at home.

Assessments: (Quizzes, Tests, and Other)
The goal is always comprehension of concepts and mastery of skills. To that end, assessments are not solely summative checklists, but also used throughout the course of study to refine and direct the next steps in the learning process.

End of Unit:
At the end of each unit (usually one chapter in the text), students will complete a formal assessment (test). These end of unit assessments are untimed and may be worked on outside of school. These are meant to identify any areas that need further instruction before moving on to new content. Frequently, after completing an assessment, a student will be asked to review a specific topic in greater depth using any of a variety of handouts, online programs, or direct conversation/instruction from me. The goal, again, is always comprehension of concepts and mastery of skills.

Snapshot/”Pop” Quiz:
Every so often students will be asked to complete a short, timed assessment during class. These assessments are meant as a snapshot of current progress/status and as an opportunity for students to gain familiarity working on traditional assessments in a relatively low stakes setting (with an eye toward high school preparedness).

Thursday, October 12, 2017

Counterfeit Coin Puzzles -- What's the Real Value?

There is a great tradition of mathematicians sending each other puzzles with simple guidelines and potentially deep mathematical content. One category of these are the counterfeit coin problems.

The basic phrasing is given N coins, a simple double-pan balance, and the knowledge that one coin is counterfeit, how many weighings is the least that will be required to be guaranteed we can determine the identity of the bogus coin.

We began looking at a few simple cases:
--often in math it is useful to consider a simple cases and then build up to more complex ones, even if the ultimate goal is to solve the more complex problem. (In this problem it is very useful to know how many weighings it will take with 2, 3 and 4 coins if we want to analyze larger numbers because a situation often arises in the middle of solving for a larger number that results in needing to identify the coin from a remaining group of 3, 4 or 5 coins. 

Case of 3 coins -- how many weighings will it take?

This one is pretty straightforward and the kids all figured it out. I'll leave it as an exercise for the reader.

Case of 9 coins, one is heavier. How many?
(If you work through this note how useful it is to already know how to do 3 coins.)

The added restriction makes this problem a little simpler. Again, I will leave it to the reader.

What is the most than can be done in 2 weighings?

This is not a trivial question, though it can be solved exhaustively. 

Throughout these exercises I emphasized the importance of accurate data recording and the value of finding useful ways of displaying the data. For this type of problem we learned about branching (tree) diagrams and how to work back through or data to check the validity of each conclusion.

It is doubly important for the children to see math as not always an open and shut, algorithmic process. While this problem does have an algorithmic solution, developing that algorithm is as much art as an exhaustive search for every possibility. Through these kinds of exercises the students can come to see the value of trying out different tactics because sometimes what doesn't work for one situation actually informs the next step and leads to a process that does work. Interestingly, in class we also tried to figure out the maximum number of coins that could be done in three weighings, given that we didn't know if the bogus coin was lighter or heavier; I hadn't looked at the solution in a while and we tried to re-derive it in class, but failed to arrive at the correct answer. Sometimes, 55 minutes just isn't enough time to try out all the possibilities.

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.

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.

Tuesday, December 6, 2016

Authenticity in Projects: An Example with Ethnography

Our class recently completed our major project for the "Exploration" theme--an ethnographic study of the kindergarten class. (Click here to see the presentation.) 

One of key features that makes Summers-Knoll different is our commitment to project-based learning. Authenticity in education, especially in projects, is a common refrain of progressive education. The whole faculty has been investigating more closely what this means to us and how it impacts your children. 

Walter recently forwarded an blog post from Buck Institute for Education to the teaching staff here about PBL, or Project Based Learning. Quoting from the blog:

“Fully authentic” means students are doing work that is real to them—it is authentic to their lives— or the work has a direct impact on or use in the real world.

A project can be authentic in four ways, some of which may be combined in one project:
1. It meets a real need in the world beyond the classroom or the products students create are used by real people.
Observing and reporting on the behavior of the kindergarteners is of immediate value to the SK community, especially the teachers who directly work with that class. 

2. It focuses on a problem or an issue or topic that is relevant to students’ lives—the more directly, the better—or on a problem or issue that is actually being faced by adults in the world students will soon enter.

While the problems of kindergarteners are not identical to the issues faced by my 5/6 students; the larger concept of how students interact in a classroom is extremely and directly relevant to my students.

3. It sets up a scenario or simulation that is realistic, even if it is fictitious.
4. It involves tools, tasks, standards, or processes used by adults in real settings and by professionals in the workplace. 
The general scenario--making observations about behavior, drawing conclusions from those observations, and submitting the data and observations for review by peers--is the process used by social scientists.  Additional, Heidi Ganzen, a UofM graduate student in sociology came to our class and talked about the types and techniques for making observations and helped us think about how to turn those observations into supported conclusions.  In meeting with Heidi, my students got to hear first hand about the real tools, tasks and processes used by someone actively engaged in the same kind of research.

Authenticity in projects:
Supports and promotes intrinsic motivation through clarity and meaningfulness of task--the students know WHY they are doing what they are doing.
Enhances comprehension and life skills through interaction with the actual tools and processes used by professionals in the field. 

Authenticity in projects is core to what we do at SK, and one of the reasons I love teaching here.