What are the key concepts and skills that students need to master in algebra?

continued from below.... The Kindergarteners think it's hysterical to give me 5000-5000+9=9 and then 1000000-1000000+9=9 and then 999999-999999+9=9. So I write down the first one and the second one. Then when they give me the third one I say something to the effect of "Yes! you're right. Let's say that any number (write an x on the board) minus itself (write - x on the board) plus nine equals nine (write +9=9 on the board)." Not only do they understand my use of variables but it gets them out of the loop of reproducing the same equation with a new number.

Variables are abstract, so it’s a great idea to start with something more concrete. Some a variable is an unknown quantity, students can think about it as “A box of apples”. Have them think about shorthand, and how they could write “5 boxes of apples” more succinctly. This concept of a variable comes quite nicely from this.

One thing I have found helpful is to point out that symbols used as variables may include Greek letters. I proceed to give a list of these letters to help those who may end up taking future lessons that relies heavily on these. Imagine my confusion hearing "epsilon", "phi" and others during first year of undergraduate study and not sure what those were and how to spell or "represent" them.

I teach variables to my lower elementary school students so that by the time they are "doing algebra" the idea of a variable is neither strange, nor abstract. We do an exercise where we choose a number for the day and work as a team to come up with as many ways as possible to get to that number using any numbers and functions and any number of numbers and functions.... see next comment for the rest

El álgebra es en definitiva, muy parecido a las operaciones combinadas. De hecho, cuando trabajamos con el álgebra lo que hacemos son operaciones aritméticas con números y valores desconocidos. Por ese motivo es fundamental dominar muy bien la prioridad y las propiedades de las operaciones. Si esto se domina, el álgebra es mucho más sencilla en general.

The ability to work with equations and inequalities is a powerful tool in problem-solving. Representing solutions on number lines and coordinate planes adds a visual dimension to algebra. It's a fantastic way to help students see the real-world implications of their solutions.

So many students use the equals sign as a way to show "what they did next" instead of representing two quantities that have the same value. It's one of the downsides of showing work horizontally. For example: 2-4+7+8-14-2 = 2-4 = -2+7 = 5+8 = 13-14= -1-2 = -3 Encouraging students, especially younger ones, to use mathematical symbols accurately helps so much with their understanding in the long run!

I have noticed that students face challenges when working with equations for the first time. The issue lies in a lack of understanding of the operations involved. We must ensure that students grasp the concept that applying the same operation to both sides of the equation maintains equality. 2x + 3 = 11 ⟹ 2x + 3 - 3 = 11 - 3 ⟹ 2x = 11 - 3 2x = 11 - 3 ⟹ 2x / 2 = (11 - 3) / 2 ⟹ x = (11 - 3)/2

Functions are really important concepts, not just in algebra but in other parts of math as well. The definition of a function must be clearly established and it's elements must be well understood. Students should be able to check whether the function is defined or not, domain and range of the function and also be able to plot them

In the case of systems of linear equations, I believe that their representation in matrix-vector form can be of great help, since it allows us to directly see the problem in an orderly fashion. However, many times this can add an extra degree of abstraction that hinders the student, so it must be used with care.

In my experience, teaching multiple equation-solving systems simultaneously tends to create unnecessary confusion. Most often, the reduction or elimination method proves the safest and simplest, simultaneously reinforcing the concept of combining different equations with the same variables (Gauss crunching the numbers from 1 to 100 is a great story and example). While the substitution method holds considerable power, often utilized in complex problem-solving and integral calculus, I find its introduction at this stage counterproductive. As a student, I didn't grasp its relevance till much later, and even now as an adult, its inclusion continues to puzzle me.

Regarding word problems, I am quick to add they pay attention to key phrases and vocabulary that they come across in the narratives. Key words like consecutive, odd numbers, even numbers, twice , among others offer students a unique perspective on mathematical representation for the given word problems.