Introduction to Solving Equations

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Objective

• Solve one-step algebraic equations that involve addition and multiplication. • Understand that solving an equation as a process of answering the question: which value(s) will make the equation true? • Create an equivalent algebraic expression.

Big Idea

What is going on in the equation 3x + 2 = 17? What does x equal? Students work with the concept of equality and balance to solve one and two-step algebraic equations.

Do Now

5 minutes

Part of my class routine is a do now at the beginning of every class.  Students walk into class and pick up the packet for the day.  They get to work quickly on the problems.  Often, I create do nows that have problems that connect to the task that students will be working on that day.  For this lesson I want students to start thinking about the concept of equality and reviewing algebra vocabulary.

I ask for a volunteer to read number 1a.  I call on student to share his/her answer.  How did you find your answer?  A common mistake is that students put 7 in the blank because they see 15 as the answer to the expression on the left.   I want students to understand that the equal sign indicates that the expressions on both sides of the equal sign must equal the same value when you simplify them.    I ask for students to share out answers and strategies for b and c.  I ask for a student to read 2 and share an answer choice that does not match the description. 

Shopping Problem

7 minutes

After the Do Now, I have a student read the objectives for the day.  I tell students that they will again be solving one and two-step algebraic equations and creating equivalent expressions.   

I have a volunteer read the shopping problem.  Before students start working, I ask a student to make a reasonable estimate of about how much one pair of jeans will cost.  I ask another student whether or not that estimate is reasonable.  I have students work independently for a few minutes to answer the problem and write the equation.  I walk around and monitor student progress.

I ask for a student to share how much one pair of jeans cost.  How did you figure that out?  Why did you do that?  Did anyone solve this problem differently?  I will call on students to share out their equations and post them on the board.  Some possibilities are 56.58 divided by 3 = J, 3 times J = 56.58, J times 3 = 56.58, J+ J+ J = 56.58, or J= 18.86.  A common mistake is that students think that J=18.86 is an algebraic equation that matches the situation, when really it is the solution and does not show what is happening in the situation.  I want to address this misunderstanding before we move on.

Equations, Balances, and Bags of Money

7 minutes

I introduce the concept of an equation having to parts that are balanced or equal.  I connect this to their work on #1 on the do now and to the idea of a balance where there are two items that weigh the same amount.   By using the bag of money instead of a variable and coins, students can connect more easily with the concept of finding the value of the unknown.  We work through 1-2 together on page 4.  Students can usually quickly identify the number of coins in the bag in these examples.  I stress that the equation 2 is b + 2 = 6 (or 2 + b=6 or 6 = b +2, etc). 

Students work on number 2-4 on page 5 independently for a couple of minutes.  I monitor student progress and look at the equations students are writing.  If students are struggling with the equations I put four possible equations up for that problem (2 correct and 2 incorrect).  I tell students to brainstorm with their partner which of the equations match what is going on in the problem.   Students share out ideas and we identify the correct equations and add to them if students have other ideas. 

Equivalent Expressions and Practice

21 minutes

A volunteer reads the definition of equivalent expressions and the examples.  Together we work through problem 1 and 2 together.  I want students to connect 2 + 2 + 2 = 3 times 2 with x + x + x = 3x.  It is using a variable, but it is still repeated addition.  I have students work on 4 and 5 by themselves.  We work through examples together.  For #3 on page 7 I introduce the idea of taking away the six coins on the right side of the balance.  If it is balanced and I take away 6 coins from the right side, what do I need to do to the left side so that both sides of the equation will still be balanced?  I want students to recognize that they need to remove 6 coins from the left side, that way the equation will remain balanced, or equal.  Students work on problems on page 8.

As students work on pages 8- 10, I walk around and monitor student progress.  As students complete problems on a page, they get up and check their work with the answers on a key posted in the room.  See my video, Posting A Key, in my Strategy Folder.  Before they can compare their answers with the key they need to raise their hands and quickly show me their work.  I am looking to see that 1) they’ve completed the problems and adequately shown their work, 2) there are not any glaring problems with their work and they are on the right track.  If I see issues, I will circle them and tell students to go back and try again.  This allows me to give quick feedback and identify students who need extra help.

Questions I may ask students who are struggling:

  • What is happening on each side of the balance?
  • What variable are we using to represent the bag of money?
  • What are two different ways we can represent three bags of money on one side of the balance?  How can you represent the same thing using a different operation?
  • What could we do to get the bags by themselves on that side of the balance?  If we do that, what must we do to keep both sides of the equation balanced?

Students who correctly answer the questions and check their work can move on to the more complex questions on page 10.  I encourage students to guess and check on these problems or work backwards.  I do not introduce the idea of inverse operations in order to solve these problems. 

Closure and Ticket to Go

10 minutes

I ask students to share and compare their answers to question 4 on page 8. I want students to generate different equations that are equivalent, like b+b+b+2= 17 or 3b+2=17.  I ask a couple students to explain how to figure out how many coins are in each bag.  One student may remove the two coins on the left and then remove two coins on the right.  Another student may guess a number of coins and check. 

We skip to #8 and 9 on page 8.  I ask students to share their answers and how they got them.  Why does your strategy work?  How can you prove that your answer works? 

With 4 minutes remaining, pass out the ticket to go.  Students work independently to complete it.