# Determining Solutions

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## Objective

SWBAT use substitution to determine whether a given number in a specified set makes the equation true. SWBAT determine whether an equation has one, none, or many solutions.

#### Big Idea

How can we prove equality? In this lesson students determine if a given number is a solution to an equation. Skill mastery is a focus.

10 minutes

The purpose of this Think About It problem is to have students use reasoning to determine if x=6 can be a solution to the equation.  They don’t know about inverse operations, so they’re relying on number sense to figure this out.  I want them to feel success with this problem, and keep their enthusiasm for algebra high.

I chose to have students work with their partners to figure out this problem.  The come to the conclusion pretty quickly that x cannot equal 6.  Most of their 3 minutes of talk time is spent explaining how they know.

Some students think in terms of ‘some number’ – that is, ’12 plus some number equals 15.’  They know that 12 – 3 = 15, so it doesn’t make sense to suggest that x is 6.

Other students will substitute the 6 into the equation and see that 12 + 6 does not equal 15.

Both lines of thinking are important for all kids to hear.  When we share out whole group, I cold call a student to share out what (s)he talked about with his/her partner.  Once the initial idea is shared, I ask if any other groups thought about the problem in a different way.  It’s important that students have multiple pathways to the solution.

I frame the lesson for kids, letting them know that today we’re starting our equations unit.  I let them know that they’ll get to practice many of the skills we learned in the expressions unit we just finished, and that this means we’ll keep becoming stronger algebra students.

## Intro to New Material

15 minutes

In this lesson, we look at equations and decide if a particular number, when substituted into that equation, will make the equation true.  Students have substituted to evaluate expressions before, but this is their first time working with equations. Today, students master the idea that an equation is a statement of two equal expressions.  Students also learn that equations can have one, no, or many solutions.

Before filling in the notes, I guide students through the problems in the Intro to New Material section (the completed notes can be found in the Visual Anchor, which I leave up as students work in partners and independently).

As we work through the examples, we spend time on the 3rd and 4th boxes.  We complete the 3rd problem twice:

Solution 1: First, we substitute, then we add within the parentheses, finally we multiply by 4

Solution 2: Distribute the 4 first.

For the 4th problem, we again solve twice. I want students to see multiple pathways to the answers.  I also want them to love algebra and the enjoy the puzzle-like nature of the work.

## Partner Practice

15 minutes

Kids work in partners on the Partner Practice problem set.  I circulate around the room, listening to every pair.  I’m hoping to hear kids say the equations in words.  I am looking to see that students are simplifying expressions on either side of the equation, if needed.  I’m checking that students are substituting the correct value and simplifying the expression correctly.

Questions that I ask groups as I circulate:

• How do you know x can’t be 4?
• Are there any other numbers that x could be?
• What operation did you use first to evaluate?
• Is 4 a solution for this equation?  How do you know?

## Discussion and Check for Understanding

5 minutes

At the end of partner practice, students are evaluating solutions for an equation that has an infinite number of solutions.  The purpose of our whole-class conversation here is two-fold:  students talk through the process for determining if a number is a solution and they explore what happens when an equation has more than one solution.

After they prove that two numbers are solutions, I ask them what they notice about the equation.  I call on hands, and once someone shares that the expressions on each side of the equal sign are the same, I have the students turn and talk with their neighbor about what that means for the solutions of the equation.

For the final CFU (at the end of Partner Practice), students have to evaluate the same equation with two different possible solutions.  I give students silent, independent work time to decide which student has the right answer.  Students then use a physical movement to share which student they think is right (hands on head if it's Maria, finger on nose if it is Tabitha).  This way, I can quickly see where students are with evaluation before they move on to Independent Practice.

## Independent Practice

15 minutes

Students work on the Independent Practice problem set.  As students work, I circulate around the classroom.  I am looking to see that:

• Students are putting the equations into words.
• Students are simplifying expressions on either side of the equation if necessary.
• Students are substituting in the correct value and correctly evaluating.
• Students are correctly determining if a value is or is not a solution to a problem depending on whether both sides of the equation are true.

## Closing and Exit Ticket

5 minutes

After independent work time, I bring the class back together and we discuss Problem_11.  I pick this problem to talk about because I want to use it to highlight for students that they need to stick with a problem.  I am working to build strong habits, along with content knowledge. Students need to test each of the answer choices using substitution, to see if it is a solution.

Students work on the Exit Ticket independently to end the lesson.