Solving Real-World Problems with Algebra (Multiplication & Division)

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SWBAT solve real-world and mathematical problems arithmetically and by writing and solving algebraic equations

Big Idea

Equations can model real-world problems in an abstract way by using numbers and symbols to make meaning of and define problems.

Think About It

8 minutes

Students work in partners to complete the Think About It problem.  

The questions I use to talk about this problem are familiar to students:

  • What do we start with?
  • What happens next? 
  • What's our unknown value?

I want students to really make sense of the problems, and to understand what's happening in each.

Students may decide to use 12.30 ÷ 4.10 = x as the equation to represent the scenario in the Think About It problem.  While this is how students will solve for the unknown, it is important that students represent the 'what do we start with, what happens.'  The equation I want students to use is    You start with 12.30, you divide it between a certain number of people, x, and each person got $4.10, so we write it as 12.30 ÷ x = 4.10

I frame the lesson by telling students that today we are going to continue on with the same work we were doing yesterday- translating equations that represent real-world scenarios and solving them in the context of the problem.  We’ll be using all four operations today, with a focus on multiplication and division.  

Guided Practice

15 minutes

Like in the previous lesson, there is not new material in this lesson.   Instead of an Intro to New Material section, I facilitate a few guided practice problems.  

The questions I use most often while guiding students are:  "what do we start with," "then what happens," and "what's our unknown?"  I want students to really make sense of the problems, and to understand what's happening in each.

All of these problems also present an opportunity to talk about the reasonableness of an answer.  

Students have access to a visual anchor as they work.  This visual anchor is lengthy, so I copy it on colored paper and pass it out to all students as a resource. 

I guide students through as many problems as I feel they need my support with.  I pull back my level of support as we work, and push more of the work and thinking on to the kids. 

Partner Practice and Discussion

17 minutes

Students work in pairs on the Partner Practice problem set.  As they work, I circulate around the classroom.  I look for:

  • Are students annotating the problems?
  • Are students choosing the correct equation to represent each scenario?
  • Are students solving the equation correctly?
  • Are students checking to see that their answer makes sense?

I am asking:

  • How do you know which equation to pick?
  • Why does this equation represents the problem? 
  • How can you check that your equation is reasonable?
  • Why isn't it (pick the equation that uses multiplication over division, or vice versa)

If students finish early, they can write scenarios that could be represented by the equations that have not been used in the partner practice.

After 10 minutes of work time, students work independently on the check for understanding problem.  We have a whole-class conversation, once students have work time.  During the conversation, I expect it to come up that students could write P = 4x or P = x + x + x + x   This is a great chance to reinforce the idea of combining like terms to simplify equations.    


Independent Practice

15 minutes

Students work on their own for the Independent Practice problem set.  This is a challenging problem set, because the problem types are quite varied.  I circulate around the room multiple times while students are working, so that I can both check their work and also encourage persistence.

After 12 minutes of work time, I bring the class together to talk about what equations they wrote for problems 4b/c and 5b/c.  This is another opportunity to talk about equivalent expressions.

I have students turn and talk with their partners about how they chose to work on problem 8.  I want students to engage in conversation about how to solve a more novel problem.  Problem 8 is not straightforward, and it is a chance for students to strengthen their problem-solving skills. 


Closing and Exit Ticket

10 minutes

If, as I was walking around during independent practice, students are successfully solving problems on their own, we talk about the first extension problem (#9) as a class.  However, if I determine that students need a bit more conversation to cement mastery in this lesson, we discuss problem #3 from the independent practice set.

My first step in discussing problem 3 is to have students articulate what we're trying to find.  Thus, students define the variable, m.  We then look at each answer choice and reason about what the equation means.  For example, for choice a, we talk about how the equation would mean the movie length divided by three would give us the length of the bonus material.  I have students verbalize why this can't be the correct equation.  I want students engaging with the material in a meaningful way, and I want to model good thinking and problem solving strategies.

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