The typical class has 6 basic units. This article is a detailed overview of each unit covered in Algebra class. Knowing an overview will familiarize students with the material and help them better grasp the concepts when they are taught in class.
I’ve been teaching Algebra to high school students for years, so I can also let you know about the end-of-year standardized test.
Here are the units covered in Algebra class:
- Number Theory and Expressions
- Equations and Introduction to Functions
- Exponential Functions
- Comparing Functions
- Statistical Analysis
Number Theory And Expressions
You’ve probably heard of “getting x by itself.”
That’s called simplifying expressions in the Algebra world and it’s the biggest part of unit 1.
Simplifying expressions involve:
- combining numbers and variables together,
- knowing when and how to add/subtract, multiply/divide,
- identifying core properties (associative, distributive, commutative), and
- breaking down radicals (taking the perfect square and how to simplify a radical)
Simplifying radical expressions can be a sticking point for a lot of students. Some teachers teach it differently.
I teach it by using “the factor tree.” It’s a visual to help the students grasp the concept and come to the correct answer.
Some teachers teach by breaking down the number inside the radical.
But I find this technique confuses a lot of students. The factor tree is the way I was personally taught and a lot of my students have success with it.
Unit 1 in Algebra covers dimensional analysis, as well. This involves converting from metric unit to US standard unit, converting time, liquid volume, distance, and weight.
Unit 1 in Algebra also covers the classification of numbers. Students learn to identify a real number, rational number, irrational number, whole number, and integers.
This word problem gives you a great idea what your child will see covered in dimensional analysis:
If a cheetah runs 50 miles per hour, how many inches per second is that?
Your child may not know how to solve that now. But by the end of their Algebra course, your student should.
One of the biggest issues I see with conversions is the student struggling to correctly setting up the equation.
This all goes back to the good ole, tried-and-true method for Algebra success:
Successful Algebra students become that way from repetition, becoming familiar with the material, and doing it over and over again.
What about mistakes?
That’s fine. Mistakes are necessary for true learning. Students learn from those mistakes and eventually become better students because of it.
The Number Theory and Expressions unit is typically a student’s first interaction with Algebra.
At first look, this unit doesn’t seem to be difficult to grasp. But I find that students do need some time with the material in order to really digest it.
The problem is most Algebra classes just spend a day on number classification. It’s important for the student to really understand this in order to do well in the other units.
Equations and Introduction to Functions
In this unit, students learn how to solve multiple-step equations and inequalities. They also learn solving systems of inequalities and systems of equations.
A system means two or more.
At this level of math, students see two linear equations/inequalities and solve for both x and y.
Essentially, students are solving for two sets of equations. Students use their simplifying rules to do that along with substitution, elimination, or graphing.
This unit is where Algebra teachers also introduce Functions, with a focus on Linear Functions and Characteristics of Functions.
Parts of Function include:
It helps to think of a function as a machine, like a toaster. Domain is the input, the set of x values that can go in the function – or the food item to be toasted. Functions are a relation where one input has exactly one output.
Range is the set of y values, or the set of outputs. In our toaster example, that would be the actual toasted bagel or waffle or any other food item.
- The x- and y-intercepts
This is when x or y equals 0, where they cross the axis when the function is put on a graph.
- End Behavior
End Behavior describes what happens to y value when you have certain x values. It determines if the function is increasing or decreasing.
Linear functions always make a straight line on a graph.
This unit and the next unit are the biggest topics covered in Algebra. Students use both paper and graphing calculators to learn how to graph linear functions and linear systems of equations and inequalities.
Everything up to this point has been a review of previous math classes.
Unit 2 is a big unit.
The standards take what they already know and dives into a deeper level of thinking in those concepts. A depth of knowledge
Students are explaining their steps along the way. This is a more in-depth understanding.
The difference between functions and equations in Algebra:
- Functions have tables to talk about characteristics and behaviors on a graph. Functions must follow certain rules in order to exist.
- Equations are more about dissecting, going through different steps, and coming to a solution. Students see more word problems where they need to create an equation.
How Students Learn to Use Algebra in Real Life
Besides tests, quizzes, and homework assignments, I also sprinkle in a few math projects when I teach a beginning Algebra course.
I find projects help students see how they can apply math in the real world to solve or analyze everyday problems.
One of my favorite math projects is about world hunger and food supply.
Students learn what people eat in various regions of the world, then students use Algebra concepts to find the cost relationship.
For example, they see how much money can feed a family of six in Asia or Africa for a week and how much that same amount of money can feed a family in the United States or Europe.
Students make the calculations, graph their findings, compare the results, then present their discoveries.
This exercise always catches my students’ attention.
Quadratics and Quadratic Equations
Here’s the meat and potatoes of Algebra.
But, buckle up.
This is the most difficult unit and the most difficult to learn in Algebra class.
But it’s nothing to be afraid of. Just go into it with that understanding to pay even more attention in class and be careful not to fall behind.
Now, quadratics are complex. For the first time, students are seeing a lot of fractions and decimals. And that can throw off some students.
The Quadratics Unit hits these points:
- Factoring basics (Finding the GCF, or greatest common factor),
- Factoring quadratic trinomials;
- Graphing quadratic equations in Standard, Vertex, and Intercept forms (Why so many forms? Each form helps find different characteristics of that quadratic equation.)
Here’s a trick to quickly spot a quadratic function/equation: The exponent will be a 2 in the first term.
Algebra teachers also introduce the Quadratic Formula in this unit.
Depending on the teacher, your student may have homework or classwork where the answer won’t be pretty. Meaning, x won’t be a whole number.
Instead, these not-so-pretty answers end up being a decimal or a number with a radical (so it gets more confusing to the student).
To learn quadratics and be successful, students have to really listen and follow along with the teacher.
Here’s a tip: Encourage your student to develop questions early instead of waiting until the day before the test.
Students should already go in to Quadratics with the mindset that they will have questions about the material.
Questions are a good thing.
I see student questions pop up usually during the lesson, as I’m explaining the steps to solve. While a teacher can sometimes quickly answer these one-off questions during the lecture, there may not be time to answer some questions.
In this case, encourage your student to stockpile their questions as they are learning — make a note of it. If the rest of the lecture doesn’t answer it, make sure your student gets clarity.
That means meeting with your Algebra teacher outside class or talking with a tutor. Either way, make sure your student has a solid understanding or those unanswered questions will come back to bite them later.
Students will also see word problems in the quadratics unit. More practical usability, students learn a little about business by applying Algebra skills to financial rise and fall.
Here’s another example:
Exponential Functions Unit
Exponential Functions are sometimes mixed in Unit 2, where Linear Functions are discussed. Some state standards call for students to compare Linear Functions with Exponential Functions.
For the bulk of the unit, students learn how to create an exponential function based on a given situation, like this:
a = initial value (the starting point of the problem)
b = rate
Simply put, exponential functions are multiplying the same thing over and over and over.
Students learn how to apply Algebra in the real world when using exponential functions to calculate simple interest, compound interest, half-life, and fun applications like spread of bacteria in the body or a zombie apocalypse.
Unit 4 also teaches:
- Geometric Sequences: a series of unending numbers with a pattern of multiplying the same number in each consecutive term.
- Arithmetic Sequences: a series of unending numbers with a pattern of adding the same number in each consecutive term.
- Asymptotes: An invisible line which a graph goes infinitely close to, but will never touch.
An overall concept of exponential functions is that your values grow slowly at first but then quickly increases. It could also decrease quickly but then values slow down the closer the function gets to the asymptote.
After quadratics, the course is pretty much downhill in difficulty.
Teachers aren’t introducing any new standards in this unit. Students are just analyzing Quadratic, Exponential, and Linear functions. Technically, it’s material students should already know. Now, they are looking at it on a higher level.
Comparison methods include:
- Looking at their graphs,
- observing their table of x and y values,
- calculating their rates of change by using the the “Slope formula” (or “Average Rate of Change formula”), and
- analyzing each function’s unique key features.
Here’s an example of a comparison word problem.
See how the student will compare a Linear Function to an Exponential Function?
When looking at graphs, students will be asked to compare y-values and end behaviors.
When looking at tables, students will be asked to compare rates of change.
This is also the unit where students must identify functions simply by looking at a table of values.
- If the y is increasing by multiplication of the same value, then it’s an Exponential function.
- If y is increasing by addition of the same value, then it’s a Linear function.
- If y is increasing then starts decreasing, then this is the rise and fall of a Quadratic function.
Some standardized tests will ask the students to compare a graph of one function with the table of another function.
Statistical Analysis Unit
Algebra teachers choose to teach the Statistical Analysis unit either in the beginning or the end of the course. That’s because Statistical Analysis is really so different from every other aspect of Algebra!
The unit covers statistical information, like mean, median, mode, and range.
Students also learn:
- Data Spread,
- Mean Absolute Deviation,
- Standard Deviation, and
- different types of Graphs (Dot Plot, Histograms, Box and Whisker Plot, and Stem-and-Leaf).
In addition, students will see 2-Way Frequency Tables and will learn how to categorize data.
The most important piece of this unit deals with Central Tendency, Outliers, and The Best Measure of Data.
This is seen the most on standardized tests.
The $150,000 mark is an outlier in this scenario, because it’s so much less than the general $230,000-home values.
This, in turn, causes the neighborhood’s average housing price to drop in value below $230,000, making other potential buyers expect lower prices than the cost of the homes!
The answer: The median
Students should be able to understand the scenario and after thinking about it, answer with confidence.
The goal with this unit is for students to be able to read different types of graphs, understand, and interpret the data.
What’s on the Algebra End-of-Year test?
Functions make up nearly half of the test, at about 40 percent. Equations, then expressions, are the next highest amount of tested material.
With expressions, there’s an emphasis on word problems.
Be aware that students are also expected to develop a certain amount of reasoning. The end-of-course exam will test students’ ability to look at a set of steps and analyzing how it can be better.
This is why students will see some test material with word problems that are already be worked out and solved. Students need to understand the steps backwards and forwards in order to confidently explain it.
Statistical analysis is the lowest amount of tested material at about 10 to 15 percent.
There’s just not a lot of Statistical Analysis on the end-of-year standardized tests. That’s another reason why that unit is so fluid. Teachers are more focused on the tested material that some teachers may just skip the unit altogether.
Now, all these percentages are for the state of Georgia and could only be for the state of Georgia. But other states following the Common Core Curriculum could very likely use the same test percentages in order to prove mastery.
Algebra units can be divided in different ways, depending on the district. So algebra can have up to 8 units. It would be the same material discussed above, just with smaller units.
What skills do I need to pass Algebra?
Students need these skills to pass Algebra class:
- Conceptual Understanding (or Conceptual Knowledge): Learning the unit concepts by hitting it home, so students have a firm grasp.
- Procedural Fluency: Practicing how to do the work with as many problems as necessary.
- Critical Thinking: Exercising individual problem-solving techniques.
Now, some students need drilling and working problems that look like the examples.
Other students know how to “struggle through it” and use the skills and tools they have been given.
The second group doesn’t need to memorize. They don’t need to study as much. They are the best test-takers.
Not a lot of students are in this group.
Based on what I’ve seen, students can develop to this point. But it takes targeted practice.
Students with this goal need work that they haven’t seen before. They need to struggle through it to build up to a level of critical thinking.
When a teacher tries to build critical thinking in their students, these students will often say the teacher didn’t teach and they had to figure it out on their own.
Not to sound like Mr. Miyagi, but that’s precisely the point.
Teaching critical thinking is difficult.
There really has to be a lightbulb moment for student. It has to click.
A math tutor can help build confidence and critical thinking skills.