Optional Lesson 7

LESSON 7, Optional Kids ActivityAll, Some, or No SolutionsIn previous lessons, students have mostly worked with equations that have exactly one solution and have solved those equations by a sequence of steps that lead to an equation of the form x=number. In this lesson they encounter equations that have no solutions and equations for which every number is a solution. In the first case, when students try to solve the equation, they end up with false statement like 0=5. In the second case, they end up with a statement that is always true, such as 6x=6x. In preparation for the next lesson, where students will learn to predict the number of solutions from the structure of an equation, students complete equations in three different ways to make them have no solution, one solution, or infinitely many solutions.WARM-UP: 5 minutes7.1: Which One Doesn’t Belong: EquationsThe purpose of this warm-up is for students to think about equality and properties of operations when deciding whether equations are true. While there are many reasons students may decide one equation doesn’t belong, highlight responses that mention both sides of the equation being equal and ask students to explain how they can tell.LaunchArrange students in groups of 2–4. Give students 1 minute of quiet think time. Ask students to indicate when they have noticed one equation that does not belong and can explain why not. Give students time to share their thinking with their group. After everyone has conferred in groups, ask the group to offer at least one reason each equation doesn’t belong.Student-Facing Task StatementWhich one doesn’t belong?
1. 5+7=7+5
2. 5⋅7=7⋅5
3. 2=7−5
4. 5−7=7−5
1. Sort these equations into the two types: true for all values and true for no values.
2. Write the other side of this equation so that this equation is true for all values of u.6(u−2)+2=
3. Write the other side of this equation so that this equation is true for no values of u.6(u−2)+2=
Student Response
1. True for all values: n=n, y⋅-6⋅-3=2⋅y⋅9, 2t+6=2(t+3), 5−9+3x=-10+6+3x True for no values: 12+x=13+x, 3(n+1)=3n+1, 14(20d+4)=5d, v+2=v−2
2. Answers vary. Sample response: 6(u−2)+2=6u−10
3. Answers vary. Sample response: 6(u−2)+2=6u
1. Complete each equation so that it is true for all values of x.
1. 3x+6=3(x+⎯⎯⎯⎯⎯)
2. x−2=-(⎯⎯⎯⎯⎯−x)
3. 15x−105=⎯⎯⎯⎯⎯−2
2. Complete each equation so that it is true for no values of x.
1. 3x+6=3(x+⎯⎯⎯⎯⎯)
2. x−2=-(⎯⎯⎯⎯⎯−x)
3. 15x−105=⎯⎯⎯⎯⎯−2
3. Describe how you know whether an equation will be true for all values of x or true for no values of x.
Student Response
1. 2
2. 2
3. 3x
1. Answers vary. Any number other than 2 will give an equation with no solution.
2. Answers vary. Any number other than 2 will give an equation with no solution.
3. Answers vary. Any expression of the form (3x+a number other than 0) will give an equation with no solution. Note: A numerical answer will yield a linear equation of one variable which has one solution.
1. Explanations vary. Sample response: Equations which are always true for any value of x have equivalent expressions on each side. Equations which which have no solution for any value of x simplify to a statement of two unequal numbers being equal, which is always false.
Activity SynthesisDisplay each equation with a large space for writing. Under each equation, invite students to share what they used to make the equation be true for all values of x and record these for all to see. Ask:
• “What did all these answers have in common?” (There is only one possible answer for each equation that will make it be always true.)
• “What strategy did you use to figure out what that answer had to be?” (The solution had to be something that would make the right side equivalent to the left.)
Next, invite students to share what they used to make the equation true for no values of x and record these for all to see. Ask:
• “Why are there so many different solutions for these questions?” (As long as the answer wasn't what we chose in part 1, then the equation will never have a solution.)
• “What was different about Equation C?” (We had to be careful to make sure that the variable coefficient was 3 and we added a constant so that the equation wouldn't have a single solution.)
Ask students to share observations they made for the last question. If no student points it out, explain that an equation with no solution can always be rearranged or manipulated to say that two unequal values are equal (e.g., 2=3), which means the equation is never true.Lesson SynthesisAsk students to think about some ways they were able to determine how many solutions there were to the equations they solved today. Invite students to share some thing they did. For example, students may suggest:
• tested different values for the variable
• applied allowable moves to generate equivalent equations
• examined the structure of the equation
Ask students to write a short letter to someone taking the class next year about what they should look for when trying to decide how many solutions an equation has. Tell students to use examples, share any struggles they had in deciding on the number of solutions, and which strategies they prefer for figuring out the number of solutions.COOL-DOWN: 5 minutes7.4: Choose Your Own SolutionStudent-Facing Task Statement3x+8=3x+What value could you write in after 3x that would make the equation true for:
1. no values of x?
2. all values of x?
3. just one value of x?
Student Response
1. Answers vary. Sample response: 7. The equation 3x+8=3x+7 has no solutions. If you triple a number and add 8 to it, and triple the same number and add 7 to it, the results will never be equal, no matter what number you choose.
2. 8. The equation 3x+8=3x+8 has many solutions. If you triple a number and add 8 to it, and triple the same number and add 8 to it, the results will always be equal, no matter what number you choose.
3. Answers vary. Sample response: x. The equation 3x+8=3x+x has one solution. Students should add some variable term in order to create an equation with one solution.