Do you feel like you are giving up your entire life trying to teach linear equations? It is a lot of content and can be very overwhelming. Today I want to give you some tips to make it easier for you and your students.
The Big Picture
It is helpful to get the big picture in mind when talking about linear and quadratic functions. Things like inverse functions, slope intercept, slopes, quadratics, and translating linear graphs using functional notation can seem to take up all of your school days and hours.
Common Core Math says...
This unit is much more comfortable for me as the teacher. The focus feels clearer, and the goals are more familiar than unit 1. As is stated in the Common Core State Standards in the Algebra introduction:
An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.
Keep in mind that just because it is a linear equation chapter, you do not want the students to lose the information you have given them in unit 1. It is a wonderful thing to force students to think outside the units to solve problems.
Standards for Mathematical Practice
4. Model with mathematics8. Look for and express regularity in repeated reasoningBig Picture Lesson Planning for the Common Core
Big Picture Lesson Planning forces us as teachers to answer the question, "What do I want my students to be able to do in life with the skills they obtain from my class?" Or, "Why am I teaching this?"
The student will understand patterns in business, economics, environment, and behaviors to predict future outcomes to make knowledgeable decisions for success. This includes researching and formulating the models and understanding the target audience and how that affects the results.
This goal is the same as Unit 1, but will be much more specific to linear equations.
Can the student find a finish date for a project?Is the student able to take a situation and create a business proposal complete with reasoning and graphs showing a future profit?Can the student decide how much to charge for a job done, such as painting a home?What skills will they need to achieve this goal?
There is a lot of information in the linear functions unit that the students need to master to be successful in coming units. As far as quadratics go, the Common Core standards do not require mastery of that content yet. So while we will get into it, we won’t be teaching to mastery at this point.
Here is a look at my breakdown:
Week #5 Slope
Find the slope of a function from a graph or table (F.IF.6)Find the slope given an equation, including solving for y (F.IF.6)Based on the context of a situation, explain the meaning of the coefficient (slope) and intercepts in a linear function (F.LE.1a, F.LE.1b, S.ID.7)
Week #6 Linear functions and their inverses
Graph linear functions and intercepts (F.IF.7a)Create a linear function given a table, graph or situation (F.LE.5)Construct arithmetic sequences (F.LE.2)Find the inverse of a function (F.BF.4a)
Week #7 Translating Graphs
Identify the effect on the graph of f(x) by f(x) + k, kf(x), f(kx), and f(x+k) (F.BF.3)Given the graphs above, find k (F.BF.3)Lesson Ideas
(The complete list of standards is at the end of this post)
Remind Your Students of What They’ve Already Learned
Before I begin teaching my students any difficult concept, I always start by reminding them of what they’ve already learned.
The great thing when I was teaching 8th grade was that I was in the same building with my students. I knew what the 7th-grade math teachers were doing and what kind of language they were using with the kids. But when I moved to teaching 9th grade, I lost some of that closeness and the ability to communicate with those teachers.
I now had kids coming from three different middle schools, with different demographics. I also had students coming in from several charter schools and several inner-city schools that our district had at the end of the bus line. That meant that I was dealing with anywhere from 6-8 different schools and the various curriculums and past experiences from those students. We also had ESL students, as well as refugees who were excellent at math, but not great at speaking English.
Ask them what they remember.
The way I dealt with so many different backgrounds from so many different students was a simple one.
I just asked the students how they were taught and what worked well for their learning.
I gave them examples of vocabulary and asked them which strategy their teachers had used. For example, there are many different ways to talk about slope. So I might ask, “Did your teacher talk about "rise over run" or about a staircase? Did your teacher use the example of filling the bucket?”
I would give 4-5 examples of different ways teachers usually teach it. Then when I began to talk about slope, they had a frame of reference. The important thing is to make those connections with the kids and use the language they understand.
Remind students of previous units.
I used the concepts they had learned about notation and functions in the first unit I had taught to connect them to the current unit. Go back to anything you have already covered and found a way to make the connection between it and what you are about to start on.
Compare to other classes.
The majority of students at my school took Algebra 1 along with Biology, so I took a look at what they were learning in Biology and what kind of language the Biology teacher was using. It turns out, the students were learning about domain and range with linear functions, but they were using terms like input and output.
I believe that using a common language with the students gives them a much more complete and better understanding of the concepts than if we each use whatever terms we are comfortable with for a subject. When I make connections like this, it makes it easier for the kids to study for tests and to feel more at ease with the material.
How to Prevent Losing Struggling Students
If a child was struggling in class, I tried to connect the concept to something outside of school, something that they were successful at previously.
Remind them of how their brain works
Reminded them how their brain works.
We talk about what it feels like when you are struggling and how your brain behaves when it is struggling to grasp a concept. We talk about how things feel easier the next day after the brain has had a rest through sleep.
Some productive things you can do when struggling with a concept are to read about it, look at a website that explains it, ask a neighbor, etc. You, as the teacher, need to give them some strategies so that when they do get stuck, they know how to get unstuck. You can also tell them, “When you struggle, it’s because I haven’t taught you ____ yet.” Fill in the blank with what is appropriate.
Make Connections on the Paper
Even in Algebra 1, some of the content on linear functions should be a review. I love to use a worksheet that is available in my membership site called Cricket Activity, which lets the student map out on the table how many times a cricket chirps in a minute at a specific temperature.
Inverse functions always tripped up the kids in my classrooms, and even if they learned it, there was zero retention rate. I began using an activity sheet that showed them an easy way to understand this concept.
This activity sheet uses technical writing, which is required for English classes that follow the Common Core standards. The technical writing often looks like writing directions. The English teacher and I decided to tag-team when it came to technical writing to benefit the students. She had them sit at their desk and write directions to the school office from their desk.
Once they finished this assignment, she had them pull out another piece of paper and told them to write the directions on how to get back to their desk from the office. What the English teacher found was that the return trip was very often not correct. So she would walk it with them and have them fix their mistakes as they walked it.
Once they understood that every right turn became a left on the way back and every upstairs downstairs, they could then come in my class the next day, and when I told them that to do an inverse function you switch the x and y, it made sense to them why.
I gave them a worksheet with directions to a friend’s house and asked them to write out how to get back home. They had to use the same skills they had learned in English the day before to fill out this worksheet. We then used it on some number problems.
And guess what? My students never forgot inverse functions.
After we have done several connection types of exercises, we go to a sheet where we work with the inverse functions themselves. Some of my students used to completely freeze up when they saw problems like these with function notation, but since I started making so many outside connections, they can not only do the problem, they understand why.
With each step, tell them what the tricky parts will be and where they might struggle and remind them that you are there to help them understand it. Remind them that nobody is left to figure it out alone.
How to Save Time as the Teacher
We have covered a lot of ways today on how to save time for your students when it comes to learning and retaining linear functions, but what about you as the teacher? You need to save time when it comes to preparing for class, too.
My biggest suggestion is to automate everything you possibly can. If your school allows you to use tools such as Socrative, you can significantly automate your grading with this program. This system allowed me to work with the students who were falling behind, but not to the detriment of the students who were ready to move ahead.
The second thing is to have a routine for your classroom. The routine is not optional. Your students need to know what to expect every day when they come into the classroom. Don’t permit your students to be passive in your class; use things like exit slips so that they pay attention throughout the entire class.
I hope all of this has been helpful to you. If you would like my FREE lesson plan on Inverse Functions, along with nine additional Algebra 1 lesson plans, just put your name in the box below and sign up and I would be happy to send you those.
And as always, please comment below and let me know how you are implementing these ideas in your classroom and what is working for you and your students.
Common Core Standards addressed within this unit
*Note that this chapter only deals with the linear part of each standard. The exponentials are dealt with in later units.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*
F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F.IF.2 Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* (Emphasize quadratic, linear, and exponential functions and comparisons among them)
F.IF.5 Interpret functions that arise in applications in terms of the context. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*
F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, give a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
F.BF.1a Write a function that describes a relationship between two quantities. (Emphasize linear, quadratic, and exponential functions). Determine an explicit expression, a recursive process, or steps for calculation from a context. Video explanation
F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Video explanation
F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2 x3 or f(x) = (x + 1)/(x - 1) for x ≠ 1.
F.LE.1a Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.
F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-?-output pairs (include reading these from a table).
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.