Here is a matching activity on a Google Slides file for various representations of a set of linear functions: verbal, symbolic (equation), graphical, and tabular (or data). The students use gallery view of the slides and sort them by function. Then they can change the background color with a different color for each function. This invokes their analytical skills to decipher key elements of the function and of each representation, for example they may identify the value of the y-intercept in the equation and find a graph with the same value.
Three ways to represent perimeter: I taught a lesson on perimeter to a 5th grade class. First I had them create a rectangular pen for their animals and they counted the number of fence pieces. Then we drew a rectangle to represent the pen. Finally we looked at the formula. This allows a deeper conceptual understanding of the concept. This is known as Concrete-Representation-Abstract – representing the concept at all three levels.
Here is a link to the handout. This approach uses the other representations to lead into the symbol representations – the equations. You can follow up by asking them the total for 0. This allows you to highlight the intercept.
Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions.
Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.
Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.
The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a previous post with these handouts.
Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).
A question was posed recently that I find intriguing and important. The question was, “what is the difference between slope, constant of proportionality, unit rate, and rate of change?” I researched the answer to this to evaluate my own understanding of these terms. Here is what I found (and am open to feedback as my understanding evolves).
First, I found a credible website that provides a glossary of math terms, the Connected Math Program page on the Michigan State University website.
In this glossary I found the definitions of the terms in question, along with the term rate.
I then found examples from a Google search that provided more of a visual image of each term.
Note that when we identify slope in an equation, we are identifying the slope of a line from the equation of the line. Slope is a measure of steepness of a line so the number likely should be thought of in that context.
The constant of proportionality can be found in different representations, but it is a constant while the slope is a measure of steepness – a special type of constant. (See definitions from MSU).
A rate, as seen in the definition above, involves units. A rate of change is the change of input and output variables so this is a little different than a simple rate as units may not be cited but could be cited (see last two images below).
The confusion for many of us is that there is overlap between these terms, depending on context. This is similar to the man on the right who can be referred as father or son, depending on the context. Slope is a rate of change for a line. A constant of proportionality is the slope for a linear representation of a proportional relationship.
The constant of proportionality is a constant but can be interpreted in a given context.
Slope is a ratio that can be interpreted in a given context. In the example below, slope = 100/1 and is interpreted as a rate of change of $100 per month.
This post was inspired by some posts on teacher Facebook pages by new teachers asking for ideas. Classroom management is a common, if not the most common, issue that arises among teacher candidates and new teacher. It is challenging to some degree for almost all teachers. As a long time teacher trainer for the states of Connecticut and South Carolina, I had compiled presentation ideas about classroom management I used to share with the candidates I was helping. I hope this post can serve as a cheat sheet for new teachers.
First, I identify 3 stages of classroom management:
Prevention – actions taken to avoid common undesired behaviors, and shaping desired behaviors
Intervention – actions taken to address problems as soon as they arise
Remediation – actions taken to address full blown problems.
The use of the image below left was inspired by a teacher candidate who lamented that during a school observation of a class there were no behavioral issues that would allow her to see classroom management in action. I explained that in fact she saw effective classroom management because of a preventative process implemented. Similarly, there are a couple retail chain stores I have observed with parallel issues with some process, such as how used dishes are cleared out or how bathrooms are maintained (at least the men’s rooms).
I will focus on prevention, which involves being proactive as opposed to being reactive. Like the man waiting for the leaf to fall, some teachers fall into the trap of waiting for a problem to occur. There are two books I recommend that help with being proactive: The First Days of School and Every Minute Counts.
Being proactive involves helping students understand what they are to do. This involves creating positive “Norms!” which is short for what normally happens in class. These can be positive or negative (often assumed to be positive). An example of a negative norm could be students standing at the door the last couple minutes waiting to leave. To establish positive norms, a teacher can set expectations and procedures for various situations in daily class functioning: how to enter and leave the classroom, ask a question, sharpen a pencil, work through problems in classwork etc. For example, when asking a question, are students allowed to blurt out answers or do they raise their hand and wait to be called upon?
Norms will NOT be established by simply posting rules or by verbally explaining our beloved syllabi. Posting rules (bottom left) is akin to a speed limit sign. Many if not most cars routinely drive over the speed limit on an interstate. The drivers do not respect the sign because they know there is an unwritten actual limit that one must cross before getting a ticket. Similarly, students know that often classroom rules are malleable as well.
Here is an example. Years ago, during the first day of class for my freshmen, one student was barely doing any work, despite my prompts. I pulled him aside to ask why he wasn’t working. He replied, “If I do all of my work now, you will expect me to do all my work all year.” He understood that often classroom expectations are conditional.
This Garfield cartoon (below) speaks to such expectations. Most students will toe the line once they understand what the line is. Again, not by what is posted but the teacher’s actions, day in and day out. In other words, their behaviors are shaped.
The image, bottom left, shows one of my classes at the start of a class during a first week. I was projecting a daily point sheet I use to provide them immediate feedback on how they are meeting expectations. The first expectation is that they are to work on the Do Now within 2 minutes of the bell. I have that part enlarged on the projection in the photo. To help shape behavior, I use a “Gotcha” ticket in which I am catching students doing something right. I do this to provide feedback on my 3 classroom expectations by writing a short blurb about what they did RIGHT. Every Monday I would start class by collecting the tickets I gave them and conduct a lottery with a handful of small prizes from Dollar Tree type stores. Some students would not turn in their tickets because the written praise was reward enough!
An organized classroom can support establishing norms. The images below are examples from my classrooms. (Elementary school teachers are routinely doing this!)
Left: I color code each course (e.g., green for Algebra 2). Each class has a file box in which each student has a folder in which I return papers and share individual information as necessary. (e.g., I researched some content on welding for a student interested in that vocation.) The white paper next to some file boxes is the Do Now for the day.
Middle: Each student is assigned a number and I use a shoe caddie to keep track of their respective calculators.
Right: This is a learning wall with notes for the current topics. When students would ask a question about a problem, I would direct them to find the matching problem on the wall and explain the first step (8th grade is coded red and 7th grade was coded blue.)
Bottom: On the right is a board with the daily objective and do now for each class – again, color coded. The posted photos of leaders like MLK Jr. are representations of my expectations: Responsible, Respectful, Resourceful as seen in the Gotcha Ticket. They learn these expectations by our daily activities and feedback – shaping.
As a complement to a set of classroom norms for effective classroom management is engaging instruction. Math teachers are often guilty of standing in front of the class going through multiple examples. To many students, this sounds like the teachers in the Peanuts cartoons, “wah waaah wah ….” Oral communication of information is far less effective than student centered learning. If students are not engaged by your instruction they will find something else to engage them – often an undesired behavior.
It is very challenging to differentiate while standing at the front of the room. Engage students with hands on work and walk around to differentiate. Parents do this with their children. Below my son is having a snack and watching a fire truck parade on YouTube and I was free to attend to other matters, like helping his brother.
Also, make the initial instruction accessible to all students using a Universal Design for Learning (UDL). The cartoon explains this approach. I use scaffolded handouts, color coding, manipulatives, meaning making strategies, and multiple representations. I refer to this as reverse differentiation. In lieu of waiting to provide specialized instructional strategies after an issue arises, I provide the strategies to all in the initial instruction and allow them to opt out, e.g., one student would repeatedly tell me, “Mr. E, you know I don’t use highlighters.” and he didn’t need them. An example is a lesson on perimeter and area (below) I taught to a group of elementary students (image below). I started the lesson with them building a rectangular pen for their animals and then they counted the number of pieces to determine perimeter. The formula was the LAST thing I showed them.
Finally, we can be responsive to student needs. In the image below, the female circled in red was having serious family issues. She could not focus so I offered her an alternative of playing math games on the computer instead. Often, I hear teachers warn that other students will complain about different treatment. I have almost never encountered this (see all the students on task in the photo). If the students see a teacher differentiating and attempting to meet a variety of needs for all students, they are very accepting.
Math is a language with words and other symbols that also makes sense of the world around us. We consume and know more math than we realize or allow ourselves credit for.
When buying the latest iteration of an iPhone, we may call forth algebra. How much will you pay if you buy an iPhone for $1000 and pay $80 a month for service? Well, that depends on how many months you will use this iteration before moving on to the next iPhone. The number of months is unknown so algebra gives us a symbol to represent this unknown number of months, x (or n or whichever letter you want).
Just as there is formal and informal English (or other language), we can engage algebra formally or informally. You don’t need to write an equation such as y = 1000 + 80x to figure out how much you will pay. You can do this informally, compute 80 times 10 months + 1000 on the calculator. Then try 80 times 12 months etc.
Math provides us a means of organizing and communicating ideas that involve quantities like the total cost for buying an iPhone.
The difficulty in learning math is that it is often taught out of context, like a secret code. In contrast, a major emphasis in reading is comprehension through meaning, such as activating prior knowledge (see below).
In fact, math absolutely can and, in my view, should be taught by activating prior knowledge. My approach is to work from where the student is and move towards the “mathy” way of doing a problem.
Without meaning, students are mindlessly following steps, not closer to making sense of the aspects of the world that involve numbers.
Effective instruction is effective because it addresses the key elements of how the brain processes information. I want to share an analogy to help adults (parents and educators) fully appreciate this.
Our senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
We have a filter that allows only some of these stimuli in. We focus on the ones that are most interesting or relevant to us.
Our working memory works to make sense of the stimuli and to package it for storage. Our working memory is like a computer, if there is too much going on, working memory will buffer.
The information will be stored in long term memory.
Some will be dropped off in some random location and our brain will forget the location (like losing our keys)
Some will be stored in a file cabinet in a drawer with other information just like it. This information is easier to find.
Here is the analogy. You are driving down the street, like the one shown below.
There is a lot of visual stimuli. The priority is for you to pay attention to the arrows for the lanes, the red light and the cars in front of you. You have to process your intended direction and choose the lane.
There is other stimuli that you filter out because it is not pertinent to your task: a car parked off to the right, the herbie curbies (trash bins), the little white arrows at the bottom of the photo. There is extraneous info you may allow to pass through your filter because it catches your eye: the ladder on the right or the cloud formation in the middle.
Maybe you are anxious because you are running late or had a bad experience that you are mulling over. This is using up band width in your working memory. Maybe you are a relatively new driver and simple driving tasks eat up the bandwidth as well.
For students with a disability that impacts processing or attention, the task demands described above are even more challenging. A student with ADHD has a filter that is less effective. A student with autism (a rule follower type) may not understand social settings such as a driver that will run a red light that just turned red. A student with visual processing issues may struggle with picking out the turn arrows.
DISCLAIMER: This is a very mathy, math geek post but it also has value in demonstrating instructional strategies and multiple representations.
We all understand speed intuitively. Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling. Both cars below are traveling at equivalent speeds.
The velocity can be graphed (the red curve below). Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left. The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed). (I will address the blue line at the end of this post as to not clutter the essence of what is shown here for the lay people who are not math geeks.)
Below is an example of using instructional strategies to help make sense of the graph and of velocity, acceleration, speeding up and slowing down.
As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. As the car moves towards a stop sign it will slow down. When a car moves away from a stop sign it speeds up.
The concept and the graph analysis are challenging for many if not most students taking higher level math. This example shows how instructional strategies are not simply for students who struggle with math. Good instruction works for ALL students!
It is counterintuitive that when acceleration is negative the car can be speeding up. The rule of thumb is when acceleration and velocity share the same sign (+ or -) the object is speeding up. When the signs are different the object is slowing down. This rule is shown in the graph but the stop sign makes this more intuitive.