I have produced a Beta version of a sequence of algebra 1 videos (up to 1 step equations as of Aug 15, 2021 with more on the way).
My approach is to unpack the concepts before showing the steps so the student understand how the math works. The videos are kept shorter, when possible, and they build on each other.
I will eventually revise many if not most videos based on feedback. Also, I will create a practice worksheet for each. For now I am simply trying to get something out there for the start of the school year. Solving equations is an incredibly important math topic to master and I hope these help.
The orange circle on the right looks bigger, but in fact both are the same size. The deception is based on the additional sensory input.
Similarly, the prerequisites for taking algebra are often considered to be basic skills. This is largely an illusion. I routinely encounter students who are referred to me for help as they have been caught in an infinite loop of working on basic math such as number operations (adding, subtracting, multiplication, and division) before moving on to algebra, with limited progress. I am not suggesting basic math skills are not important but am focused on the context of prerequisites needed to engage algebra. Many of the students I have helped who were in this situation. We worked to quickly move them into algebra where they were successful.
One student worked on half a year of 4th grade math during her 7th grade year. During the spring of that 7th grade year and the subsequent summer, I worked with her on algebraic thinking and algebra topics. She successfully completed algebra 1 during her 8th grade year.
The Common Core of State Standards (CCSS) for Math maps out the prerequisites as seen in the CCSS math domains (below). Throughout elementary school, Operations and Algebraic Thinking topics are covered. The Algebraic Thinking standards establish for the students a foundation for algebra taught in middle and high school. A focus of algebra is to model or represent patterns or relationships in real life situations using equations, tables, and graphs. These include quantities modeled by variables.
Below is a break down of this foundation in elementary school. If you are supporting a student in middle or high school who is taking algebra and has major gaps in his or her math education, look to these standards for the essential prerequisite skills.
First Grade: represent situations in word problems by adding or subtracting, and introduce equations (and equal sign).
Second Grade: Represent, solve word problems, introduce multiplication as groups of objects.
Third Grade: represent, solve word problems, explain patterns
Fourth Grade: Solve word problems, generate and analyze patterns
Fifth Grade: Write expressions (equations are 2 expressions with an = in between), analyze patterns and relationships
I am rolling out a series of videos on math topics from middle school and algebra 1. Here is a link to the folder with the accompanying handouts. This post will be updated periodically.
Let me know if your school is covering a topic that is challenging and students would benefit from an alternative presentation. I will see what I can do. The links to the videos are listed below the photo.
In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)
When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.
To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: “Ability” to execute or perform “tasks” – DOING.
Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.
First I develop an understanding of a balanced equation vis-a-vis an unbalance equation using the seesaw representation.
I then explain that the same number of guys must be removed from both sides to keep the seesaw balanced.
I then apply the subtraction shown above to show how the box (containing an unknown number of guys) is isolated. I explain that the isolated box represents a solution and that getting the box by itself is called solving.
I use a scaffolded handout to flesh out the “mathy” steps. This would be followed by a regular worksheet.
I extend the solving method using division when there are multiple boxes. I introduce the division by explaining how dividing a Snickers bar results in 2 equal parts. When the boxes are divided I explain both boxes have the same number of guys.
The students are then provided a scaffolded handout followed by a regular worksheet.
61 cents per ounce is a rate of change. Graph the line modeled by this (y intercept is 0) and it becomes slope of the line. In referring to algebra we often hear, “when will I ever need this?” My response is “all the time!” Our job as teachers is to make this connection for students.
This is a meaning making approach to introducing equations. I will walk through the parts shown in the photo in the space below this photo. (A revised edition of this handout will be used in a video on this topic.)
First I explain the difference between an expression (no =) and an equation (has =). An equation is two expressions set equal to each other (21 is an expression).
I then develop the idea of a balanced equation and will refer to both sides of the see saw as a prelude to both sides of the equation. I also focus on the same number of people on both sides as necessary for balance.
At this point I am ready to talk about an unknown. Here is the explanation I use with the photo shown below.
I start with the seesaw at the top. The box has some guys in it but we don’t know how many.
We do notice the seesaw is balanced so both sides are equal.
This means there must be 2 guys in the box.
I follow by prompting the students to figure out how many guys are in the box(es) in the bottom two seesaws.
Finally, I explain that the number of guys in the box is the solution because it makes the seesaw balanced.
There are multiple instructional strategies in play.
Connection to student prior knowledge – they intuitively understand a seesaw. This lays the foundation for the parts of an equation and the concept of equality.
Visual representation that can be recalled while discussing the symbolic representation, e.g. x + 1 = 3
Meaning making which allows for more effective storage and recall of information.
I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.
For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math.
This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.
Graphing linear functions and the underlying concept are challenging for many students. The video below shows a scaffolded approach to teaching how to graph. This approach also addresses the concept of the graph as a visual representation of all possible solutions (see photo above). Students often do not realize that the line is actually comprised of an infinite set of points which represent all the solutions. Here is a link to the document used in the video.