Solving Multi Step Equations: Special Cases (Day 3 of 4)

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SWBAT solve equations that have one solution, no solution, or an infinite number of solutions.

Big Idea

Students will deepen their understanding of the algebraic property of equality by solving equations that yield no solution or infinite solutions.


10 minutes

Students will complete the Multistep Step Do Now. Six students will come up to the board to show their work and solutions on the whiteboard for the class to review.

Next, a student will read the objective to the class: SWBAT solve equations that have one solution, no solution, or an infinite number of solutions.

I will refer to the do-now to illustrate an equation that has one solution:

For the problem, 8x - 1 = 23 - 4x, we calculated that x was equal to 2. If I evaluate the equation for x = 2, we end up with a true statement: 15 = 15. If we plug another number in for x, let's pick 3, we end up with an untrue statement: 23 = 11.

I will then challenge the class to test a few more numbers, to see if they can find another value for x that will make both sides of the equation 8x - 1 = 23 - 4x equal.

After about a minute, I will tell students that all of the equations that we have solved have only had one solution up to this point, and that we will spend today's class examining some special cases that are encountered when solving equations.

Guided Notes + Practice

25 minutes

For today's Guided Notes I plan to introduce students to several different algebraic equations using the no solutions infinite solutions PowerPoint. Below I highlight some of the key elements I plan to discuss.

Slide 2: We will solve the first equation as a whole group to remind students of the idea that in their experience, an algebraic equation typically has only one solution.

Slide 3: I will begin by asking students to solve this equation on their own. Most of the class will realize that something is "wrong" with this equation. I will play along with their intuitions and solve the problem aloud on the board to see if I am able to catch any mistakes. Once we examine the equation together and analyze our work, we will discuss whether 10 = 25 can be considered a true statement.

Since 10 will never equal 25, this equation has no solution. I explain to the class that since x refers to a particular solution to the equation, there is no way to create a true statement for this equation using a single value for x.  A number, multiplied by 5 and added to 10, will never equal the same number multiplied by 5 and added to 25. 

Slide 4: I will tell students to solve this equation on their own. Many students will immediately assume that this equation is another "no solutions" example. I then ask my students if 7 = 7 is a true statement. Since this statement is true, this equation does not result in the same type of situations as the previous example. 

I tell my students to pick their favorite number, and to evaluate the equation with that number. I will ask students on the count of three, to put a thumb up if they finished with a true statement, and a thumbs down if they finished with a false statement. Once we see that everyone had a thumb up, I will ask the students to share the numbers that they chose. 

This equation has infinite solutions because there are an unlimited number of x values that will yield a true equality statement. To solidify the concept of infinite solutions, I will plug x = 0.1, x = 1000, and x = -50 into the same equation.

Art Activity

35 minutes

The whole class will complete the create a face activity to practice solving equations that have a special solution. Students will solve each equation and draw the corresponding facial feature on the blank circle provided on the second page of the documents.

After the guided practice section many students began overusing “no solutions” and “infinite solutions” for each equation that was solved thereafter. The create a face activity is a good balance of equations with zero, one, or many solutions. Students will be forced to analyze each equation to decide the number of solutions based on the definitions provided in class. Since students wanted their end picture to be correct, I noticed them slowing down and taking their time to solve each equation and check their work for errors.

My students love any opportunity where they can express their creativity, so this activity was well received by all. This activity was also fantastic from the teacher standpoint, because I was able to quickly see which students needed extra help based on the facial features students drawn on their paper. I have provided two samples of student work, here and here.




10 minutes

I will ask one student to summarize what we did in class today. Students will then complete the exit cardThe exit cards should be graded directly after class. Students should be grouped by the percentage of correct answers to differentiate the next lesson.