Equivalent Numerical Expressions, Day 1 of 2

These two lessons are based on the Laws of Arithmetic lesson that is part of the Mathematics Assessment Project .  

I give the pre-assessment task as homework a few days before I teach these lessons.  I explain to students that they may not know how to do everything on this assignment, but that is okay because they will be working on a similar task in class.  The important thing is that they try their best and explain their thinking. When I review the pre-assessment task I write 1-2 questions on each student’s work, following the MAP Assessment recommendations. I do not give students a grade. 

Teacher's Note: For an extensive list of common student misconceptions and questions, read the Before the lesson section of MAP's Laws_of_Arithmetic.  

Here are a few of the common issues that my students have:

• They do not recognize the function of parentheses
• They do not understand the distributive law of multiplication
• They fail to recognize the commutative property
• They do not see the link between multiplication and addition
• They assume that squaring a number is the same as multiplying by two.
• They do not understand the significance of the fraction bar.
• Student needs an extension.

I use this pre-assessment to create homogeneous groups for students to work in during these two lessons.  I look to pair students up with a partner who has a similar level of understanding. The partners may differ with the specific topics they understand, but they are around the same level.  I want to prevent a student with a relatively low level of understanding being a partner to a student with a relatively high level of understanding.  I have found that high-low pairings often result in the higher student either tutoring the lower student or the higher student completing the task without waiting for the lower student (see my Creating Homogeneous Groups reflection).

Today's Do Now reviews the meaning of the measurement concepts, perimeter and area. In the task Do Now Task my students will representing the area of squares and rectangles with numerical expressions, based on their interpretation of geometric figures.

I plan to ask my students to explain how they found the area of the rectangle.  If a student says he/she multiplied 3 x 2, I ask if multiplying 2 x 3 would also work.  For the square, I ask for a student to explain how they found the area.  If he/she multiplied 3 x 3, I will ask if 3^2 would also work.  I want my students to start to recognize that there are multiple ways to represent the same area, and, to apply the Commutative Property of Addition fluently.

After the Do Now I hand back my students' Pre-Assessment with my questions.  I give them a couple minutes to read my questions and write a response that reflects their thinking on the Review Your Work worksheet.  It is okay if they don’t have a clear response at this time.  My hope is that reading my questions will spark an idea, or, that they will think about the question throughout the lesson.

After students respond to my question, I will ask them to think about Question 1.  I offer that they can use what they wrote about their pre-assessment as a starting place.  It may help some students to find the actual area of the diagram and use that to compare with the expressions.

Once the class works on Question 1, I will ask several students to identify one expression that correctly models the area of the figure. I am looking for students to clearly explain/show their thinking, so I will ask them to explain their answer to the class. Here is what I am hoping to hear:

• I  want my students to recognize that (b) works because it is finding the area of the two rectangles and adding them together. 
• I want my students to understand that (c) works because 3+5+3 +5 is the same as 3+3+5+5 or 3x2 + 5x 2.  
• I want my students to recognize that (d) works because it is finding the area of the larger rectangle that has the dimensions of 2 and 3+5.

After we discuss this task, I again collect the pre-assessments. I will return them to my students after they have completed the post-assessment (see Equivalent Numerical Expressions, Day 2 of 2). 

In order to maximize participation during the next section of the lesson, I pass out Whiteboards and markers to my students.

During this section of the lesson I will be displaying the diagrams included in Whiteboard Practice using a document camera (or LCD Projector). As I display each figure, I ask students to look carefully at the diagram. After they have had time to observe, think and recall, I ask them to write an expression that shows how you would find the area of the figure in each diagram.

I will circle the room as students work observing what they are writing on their whiteboards. If I find a student who is struggling, I will ask him/her to try to find the area first, then write an expression showing how they figured out the area.  If some students easily come up with one expression, I will ask them to write a second expression that also models the area of the diagram.

After a couple minutes I plan to ask students to hold up their whiteboards so I can see all of their responses.  If I see a common mistake, I will make note of it. Eventually, I will write the expression on the board and ask students if they agree/disagree with it. I expect that I will be able to ask a couple students who have written different expressions to explain their thinking to the class.  I am looking for students to share and explain 3 x 4 + 3 x 5 and 3 (4+5).   During this time I tell students to record these expressions on their paper with their pencil.

Next, I will have students look at the next question.  I give them a minute to work on their whiteboards.  When most students have an answer, I ask them to show me their whiteboards.  Again, if I see a common mistake, I will ask students to debate that answer.  I ask a student to explain their thinking and ask another student to add to their thinking.  I ask students to come up with other expressions that would also represent the area of diagram A.

If I have time, I ask students to come up with expressions that represent the area of diagram B and C.  I want them to recognize that they are equivalent because they both have a rectangle that is 4 units by 1 unit and a rectangle that is 5 units by 2 units.

Before moving on, I will have student volunteers collect the whiteboards and markers.

For this section, students work in their homogeneous partner pairs.  I have a volunteer read the rules and expectations from Matching Part 1 to the class.  I ask students what questions they have.  Then I pass out the materials and students start working.

Teacher's Notes:

• Before this lesson, I print and cut out one set of Expressions Cards and one set of Area Diagrams Cards for each partner pair.  I like to print them on card stock and label the sets and place them in envelopes.  For example, for set 1 I put #1 on the back of each card and label the envelope #1.  That way when (inevitably) a card falls on the floor, it can easily be returned to the proper envelope.
• Today, I only give partner pairs an envelope with the following cards: A1, A2, A7, E1, E2, E7, E8, E13, and 3 blank E cards (they will need at least one to create a matching expression for A2).  I do this so students can focus on a smaller amount of cards.  When I gave students all the cards at once, many of them were overwhelmed.  Adjust the amount of cards to meet the needs of your students.

As my students work on the task I walk around and monitor student progress.  I am observing the strategies students are using and commenting on appropriate partner work. Many of my students may struggle at first, and this is okay.  I want my students to engage in real mathematical practices (MP1, MP2, MP3). If a pair is stuck, I will not intervene immediately.  I want students to find ways of applying what they know to find matches.  If students raise their hand and ask for help, I may ask some of the following questions:

• What area card are you working with?
• What is the total area of the diagram?  How do you know?
• How do you know if an expression card matches this diagram?

If students successfully find all the matches, I will ask them to use blank cards to create a different expression that is equivalent to each area diagram.  Once they complete this task, I will have them pair up with another partner pair to compare their matches.

I begin the Closure by asking students to share their matches for A1 with the class.  I want students to show and explain their thinking.  Then I ask if 2 (3 + 4) would represent the area of A1.   I want students to understand that that expression wouldn’t work since the rectangles don’t have the same width.

I will ask students to share out their different strategies for finding matches. 

• Do you find matching expressions first and then match with an area diagram?  Or vice versa? 
• Do you find the area of the diagram and then simplify the expressions?  

As we come to the end of the lesson I want my students to hear about the diverse strategies that their classmates are using. Then, I will have students clean up and organize their cards.  Instead of giving a ticket to go, I will collect and look at my students' work to prepare for Part 2 tomorrow.