Algebra I : Unit #9 - Understanding Equations : Lesson #1

Understanding Expressions

Objective: SWBAT use variables meaningfully to understand what an expression represents in terms of its context.
Standards: HSA-SSE.A.1 HSA-SSE.A.2 MP1 MP2 MP3 MP8
Subject(s): Math
60 minutes
1 Warm Up - 5 minutes

As the students arrive I will write the following expression on the board: 3x + 4 - 2y - 5x + 2y - 6

Then, I will ask students to work individually to simplify the expression as much as possible.  While students are working, I walk around the room to determine where students are in their understanding of simplifying algebraic expressions.  I carry a copy of the class roster with me so that I can make notes on what I see.  While this particular question is outside of the scope of the lesson, observing how students approach it will give me an indication of students comfort with variables.

When students have completed their work, I will ask, "How many terms do you have in your answer?" Then, I will ask students to hold up either 1, 2, or 3 fingers to signal how many terms.  Based on a quick survey of responses of students, I get a lot of information about what needs to be emphasized over the next couple of days.  To complete the Warm Up, I will have students Turn-and-Talk to discuss how they arrived at their solution, and, to process the information they may have just gained from the quick, non-verbal survey. They may not know the answer, but they know more about it than they did at the start of class. 

2 Opening - 10 minutes

As I open today's instruction, I will share Understanding Expressions with my students. On Slide 2, I will ask students to work with a partner to write down the mathematical translations for each of the statements.  As they work,  I will walk around and take note of the work of individual students. I want to get a sense of how they are working and who might have an interesting idea to share. 

If I come across a student who is struggling, I may make the following suggestion, "With any of these expressions a good way to check your work is to plug in a number as the value for the variable." I will be looking carefully to see where students are having the most difficulty. Coming into class, I expect subtraction may be an issue. My students often struggle with the correct order when modeling a statement like "10 less than z". If student plug in a value for z, like 30, asking "What is 10 less than 30?" often helps them to correct their work. 

Slide 3 requires students to translate in context to understand a situation.  In order to do this they will have to reason about the values abstractly (MP2) because no value is given for x.  Students should be writing the meaning of each of the statements in the context of the question. For example:

  1. Dan has twice as much money in his wallet as I do.
  2. If found a 5 dollar bill or someone gave me 5 dollars so now I have 5 more dollars.
  3. I spent 24 dollars at the store so now I have 24 dollars less than I had before.

3 Independent Practice - 20 minutes

The questions provided for today's Independent Practice are challenging. Students should work with their partner to try to interpret each expression.  In order to do this, the students will have to reason in context, which may require them to read the opening paragraph several times. As they get underway, I encourage my students to think critically about the meaning each sentence and to construct a mathematical explanation that makes sense for each of the expressions (MP3, MP6).  I will also remind students that each person in the partnership has a role. Students should be reading each other's work and making adjustments to ensure that their explanations make sense in the context of the problem. 

If students are having difficulty, then I will use a similar approach as in the opening activity. I will recommend that students substitute numerical values for the variables in order to interpret the meaning of the expressions.  Sometimes, concrete numbers help my students to more easily understand what each expression is saying and the meaning that it holds.


Teacher's Note: For the third expression under Question #1, prompt students to use the word "per" to mean division.  This can be related to a context that students have heard before such as "miles per gallon" and "miles per hour".

Reflecting on Today's Independent Practice
Adjustments to Practice

Students definitely had some difficulty with this independent practice.  The problem was so open ended that, in some ways, I feel that students were over thinking the questions.  As you can see from the student_work, some students bought into the idea of plugging in values for each of the variables. This helped them to visualize what the purpose of the expression was.  

While students were having some definite struggles, I tried to set the expectation that struggling and not knowing are OK.  I know that as a teacher, I have a tendency to be too helpful and so I wanted to be conscious of that.  As a method of giving students attention when they were in need, I tried to ask them questions that would re-direct them and get them thinking. 

Analyzing the student work led me to see that I needed to have students investigate the meaning of simple expressions involving multiplication.  Students were not able to discern the total number of gallons from the total number of orders in this question.  They did not see that if, for example, there were 10 orders at 5 gallons each that would give them 50 gallons not 50 orders.

See my verbal annotation of student work here: understanding_expressions_stu_work.mov.

4 Closure - 5 minutes

The closing activity for this lesson is on Page 4 of Understanding Expressions. This Ticket out the Door will give me a baseline on student understanding from the lesson.  One of the biggest difficulties with a lesson on translating expressions is student's inconsistent understanding of contextual vocabulary.  For example, students can be confused by the fact that "plus", "added to", "sum", "increased by", and "more than" all mean the same thing, as well as have meanings outside of mathematics.  This lesson closing will help me to gauge where students are in their use of vocabulary.  Students who have weak vocabulary may need some remediation in this particular area during the unit in order to bring their understanding up to where it needs to be.