Category Archives: MathMastery

The Standard Algorithm of Addition: Part 3

Algorithm_0In the blog “Standard Algorithm of Addition: Part 1,” we learned how to set up addition problems in a vertical format so that we can manually calculate their sum. We also learned the mathematical terms for the components of arithmetic expressions of additions. In the blog “Standard Algorithm of Addition: Part 2,” I used a {visual model} to explain the concept behind this fundamental algorithm using green square blocks and blue rods to represent the addends. In this blog, I will show you how the standard algorithm works {number model}.


First we set up the arithmetic expression in a vertical format (A). Now we will first focus on the ones column and work our way from right to left (B). We mentally add the digits in the ones (9+7=16). Note the sum is greater than 9. This sum is equivalent to 1 unit of ten (blue rod) and 6 units of ones (green squares) (C). I write the digit for the 6 units of one in the ones column under the underline (D). I place the 1 digit of ten at the top of the tens column (E).


Working our way from right to left, we move to the tens column (F). We add the digits in this column noting that we are actually adding multiples of ten (G). Since the sum is not greater than 99, we simply write the sum of the digits in the tens column underneath the underline (H). Therefore, the sum of the addends, 29 and 47, is 76 (I).  It is the most common technique of addition taught to math students, but it is not the only one.  We’ll look at some other addition algorithms in future blogs.

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Expressing Mathematically – Part 2

In one of my earliest blogs (“Expressing Mathematically”), I introduced the idea that mathematical concepts and skills can be expressed in different ways. Our senses identify natural and artificial objects {world model} that convey math concepts. We use language to identify and communicate our understanding of the sensed math concept written and spoken ways {language model}. We also use geometric curves, shapes and volumes to represent these objects {visual model} in a 2D or 3D mathematical world. We can also plot our visual models on a coördinate plane or volume {graphical model} to generate numerical data about the geometry. Finally, we can express mathematical concepts and skills using symbols, numbers, variables and equations {numerical model}. Keep in mind that all these models are different aspects of what we generally refer to as mathematics. I will use these terms often throughout future blog to explore math concepts and skills through these unique perspectives.

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Word Problems

One of the most common forms of problems in mathematics are word problems.  They are essentially mathematical exercises presented in the form of a hypothetical situation that requires an equation to solve.  They are usually expressed in a narrative form.   I consider this a verbal (literary) form of expressing math (See the blog “Expressing Mathematically”).  There are two insights I want to share with you about word problems:

Hypothetical Situations
First, when I solve word problems, I am actually executing at most five steps, not necessarily in the order given: (1) I translate the hypothetical situation into a visual expression, like a diagram using geometric concepts, or straight into an equation or expression. (2) I find the values that we are given in the  word problem, and (3) I find the variable(s) we need to solve for (the “unknown”) to find the solution to the hypothetical situation. (4) I then decide which concepts and/or equations we need to apply to the hypothetical situation. (5) Finally, I apply the proper skills from our math toolbox to find the solution to the word problem.

Actual Situations
Second, a traditional word problem explicitly supplies the students all the necessary information to do steps (1),(2),and (3).  When I am presented with  a problem in a real-world situation, I usually have to define the situation in a way that I can solve it mathematically and express it as a word problem, or a labeled diagram.  Many times, I have to conduct measurements in order to establish the “given” values.  Usually, the “unknown(s)” end up being those values I need to know that I cannot measure in a practical or direct way.  It usually takes some work to properly define the situation that is usually given in a classroom-type word problem.

Word problems are one of many ways we can test the competency of a math student.  They are also effectively used to help the math student achieve mastery in applying certain skills.  I understand that it is impractical to consistently present students with real-world situations to solve.  However, it is just as important for students to know how to properly generate a problem as it is to solve it.

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Expressing Mathematically

In many ways, mathematics is a language with its own alphabet, words, grammar and syntax.  Mathematics expresses itself in multiple ways. Lets see how.In my home office, I have a coaster illustrating a Siamese cat (A).  I can specify the shape of this coaster verbally, using the proper word (B).  A different word would come to your mind if your native language is not English.  I can also specify the shape of the coaster visually, by drawing a circle of the given radius (C).  By envisioning such a circle on a 2-dimensional coördinate system (D), I can identify specific points on the edge of the coaster numerically (via ordered pairs of numbers).  Finally, I can specify the shape of the coaster in the form of an equation (E).

In math, we must translate a given scenario into one of these translation so that we can successfully generate the requested solution.  We will explore how we specifically do this in future blogs.  This is a source of frustration for math students who have not yet mastered these fundamental, and often overlooked, skills.  It can impede their progress to understanding more complex mathematical concepts.

Do you believe that there are forms of expressing math that does not fall under one of the five categories shown above?

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Hello, World!

Antonio E. Oliva

Welcome! My name is Antonio E. Oliva.

This blog is associated with MathMastery Tutoring, a service I’ve started to help high school students master math via one-on-one tutoring .  In this blog, I will share comments, news, discoveries, insights, on anything math-related.   I’m especially interested in exploring the different ways we learn and apply math all around this 21st century world.

So how about it?  Follow this blog. I hope you will find it interesting and join in the conversation.