Andrea Palmer PROSPECT HILL ACADEMY CHARTER SCHOOL, SOMERVILLE, MA
6th Grade Math : Unit #6 - Expressions, Equations, & Inequalities : Lesson #6

Writing and Solving Equations

Objective: SWBAT: • Write an algebraic equation that fits a given situation. • Solve a one-step algebraic equation.
Standards: 6.EE.A.2 6.EE.A.2a 6.EE.B.5 6.EE.B.7 MP2 MP3 MP4
Subject(s): Math
60 minutes
1 Do Now - 7 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to review writing expressions and creating and solving equations from the previous lessons.

Problem one can be difficult for some students.  If students struggle to write an expression,  I ask them what they know and what they are trying to model.  I want students to understand that twice the amount of Elizabeth’s crayons would be 2e, and three more than that would be 2e + 3 or 3+2e. 

For problem 2 and 3 I ask a student to share and explain their thinking.  I call on other students to share if they agree and disagree and why.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

2 Problem - 10 minutes

I ask students, “What is the difference between an algebraic expression and an algebraic equation?”  I want students to recognize that an expression shows the relationship between a variable and numbers.  An algebraic equation sets two expressions equal to each other, like the balance we used in the previous lesson.  Solving an equation is the process of figuring which values make the equation true.  I also review that if a variable is right next to a number, the two are being multiplied together.  I remind students they can also use the dot to show multiplication, but that we should avoid using the x symbol because it can be confused as a variable.

I tell students that they must read through the scenarios and decide whether equation a or b fits that scenario.  Students participate in a Think Write Pair ShareI call on students to share out their thinking.  I ask students, “How do you know that equation works?  Why can’t the other equation work?”  Even though both scenarios include the same values, I want them to recognize the difference between equation a and b. 

Then I ask students to work with their partner to figure out the value of x for equation a and b.  What value must x be in order to make the equation true?  Students are engaging with MP2: Reason abstractly and quantitatively.  Some students may guess a value and check the answer and then readjust as needed.  Other students may use the inverse operation to figure out the value of x.  I ask two students who used these two strategies to show and explain their work.  I do not explicitly teach students to use the inverse operations step by step.  In the past when I have taught this way, students have memorized the procedure oftentimes without having a deep understanding of what they were doing and why it worked. 

3 Writing and Solving Equations - 23 minutes

Notes:

  • I use the data from the previous lesson’s ticket to go to Create Homogeneous Groups of 3-4 students.
  • I give each group a Group Work Rubric to track individual and group behavior.
  • I Post A Key so that groups can get up and check their work as they finish each page.  I have groups check in with me when they finish a page, so I can make sure they are on track.

We work on Part A together.  I have a volunteer read the situation out loud.  I ask, “What is going on?”, “What do we know?” and “What does the variable represent?”.  If students struggle to see that the expression would be 2w, I substitute 11 in for w.  What if the plant grew for 11 weeks, how tall would it be?  Sometimes it’s easier for students to replace the variable with a number to understand what operation they would use.  I ask students how to set up the equation and how to solve for w.  I require students to use substitution to prove that their value for x works.  I ask students, “Are there any other values that would also work for x?”.  I want students to realize that for this particular equation, x can only be 8.

Students move into their groups.  As students work, I walk around and monitor student progress.  Students are engaging in MP2: Reason abstractly and quantitatively and MP4: Model with mathematics.

If students are struggling, I may ask them the following questions:

  • What is going on in the situation?
  • What do you know?
  • What does the variable stand for?
  • What operation is going on between the value and the variable?  How do you know?
  • What is the difference between an expression and an equation?
  • How can you prove that your value for x works?
  • Are there any other values for x that would also work?  Why or why not?

If students are correctly working through the examples, they can move onto the challenge questions.

Addressing Student Mistakes
Developing a Conceptual Understanding

During group work time, I walked around to monitor student progress.  A couple groups struggled with Part E.  One group misread the problem and created the expression “m + 10”.  I asked them to reread the situation and explain their expression.  I asked them, “If I had some money and then gave them 10 dollars, would I have more or less money?” They were able to recognize that the operation had to be subtraction.  I asked them whether the expression should be “m -10” or “10 – m”.  They explained that it would be “m – 10” since Lisa started with m dollars, not 10 dollars.

A few groups struggled with part H.  At first a few groups wrote “3n” as their expression.  I asked them, “How many lines does the poem have?” and “What would 3n mean in this situation?”  I wanted students to recognize that 3n would represent each student reading the entire poem.  Students realized that they needed to use division.

4 Closure and Ticket to Go - 10 minutes

I ask students to turn to Part G.  I ask students to share their thinking about the expression, equation, and the number of points Kevin Garnett made.  I ask other students to share if they agree or disagree with the other students’ thinking and why.  I want students to understand that if Paul Pierce made 26 points, then Kevin Garnett must have made 12 points since 12 + 14 = 26.  As a challenge question I ask, “Garnett and Pierce made a total of 50 points.  How many points did each player make?”  Students participate in a Think Pair ShareI want students to be able to extend the pattern and see that Pierce still has to make 14 more points than Garnett.  Some students may guess and check, while others may work backwards.  If students are still working on it, I tell them they can continue thinking about it after they complete their ticket to go.

 I pass out the Ticket to Go and the HW Writing and Solving Equations.

Extending Student Thinking
Rigor

After we talked about Part G, I asked, “What if Paul Pierce still made 14 more points than Kevin Garnett, but together they scored a total of 50 points?”  I had students talk about it with their partner for a minute.  I started a table on the whiteboard and recorded student guesses.  With each guess, I asked other students to use the information to find the other player’s points and the total to see if it matched the problem.  It was interesting to see student guesses.  They were able to keep the pattern of Paul Pierce’s points being 14 more, but they struggled to get the total to 50 points.  Eventually a few students said that if Kevin Garnett scored 18 points, Paul Pierce would score 32 points and 18 + 32 = 50.  Students were excited to take on the challenge and figure out a complex problem.

Looking at Exit Tickets
Exit Tickets

I collected the tickets to go to see what students understood about translating algebraic expressions and equations as well as what gaps in understanding they had.  I corrected the tickets to go and grouped them in the following way:

Novice:

These students struggled to model a situation with an expression.  Often they were able to recognize that the answer to problem 3c was 7 hours, but their expression and equation did not match.  This particular student created the equation “12 + h  = 84” and then added 12 seven times to get 84.  She did not see the disconnect between her answer of 7 hours and her own equation.  Another common struggle was that students could create an expression, but did not know how to make it into an equation.

Approaching Mastery: 

These students were able to create an expression to model most situations, but they usually made one mistake either writing their equation or answer 1c.  Some students just copied their expression without creating an equation.  Other students created an equation including the answer 7, rather than “h”.  Other students made a mistake answering problem 1c.  This particular student made a mistake dividing and said that 9 x 12 = 81.  It is clear that she understands the algebra concepts in this lesson, but needs extra practice on her multiplication and division facts.

Proficient: 

These students were able to correctly model situations with expressions and solve equations.  They also understand the difference between an expression and an equation.

For this situation, I did not include an “advanced” category because the content did not require students to explain or analyze their work.   Most students were in the proficient and approaching mastery category.  For the few students who were novices, they will be in a group together during the next lesson.  I will work with their group to address the issues I’ve noticed from this ticket to go.  I will also pass back the tickets to go to students so they can see and correct their work.