Equivalent Numerical Expressions, Day 2 of 2

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SWBAT: • Find the area of squares and rectangles. • Simplify an expression using the order of operations. • Write and evaluate numerical expressions from area models. • Explain and give examples of the distributive and commutative properties.

Big Idea

How can you represent the area of a diagram using numerical expressions? Students connect their knowledge of area and equivalent expressions to the commutative and distributive properties for day 2 of this investigation.

Do Now

10 minutes

This two-part lesson is based on the Laws of Arithmetic lesson that is part of the Mathematics Assessment Project. In today's lesson we follow my standard Do Now Strategy to begin the lesson. Although I often create Do Nows that have problems that connect to the task that students will be working on that day, here I want students to review this area diagram that they encountered on the preassessment (see Equivalent Numerical Expressions Day 1).  By analyzing someone’s idea and writing their thinking, students are engaging in MP3 (Construct viable arguments and critique the reasoning of others).

I have students participate in a Think-Write-Pair-Share about Daniel’s idea.  I want students to recognize that according to the order of operations, Daniel’s expression will not match the area diagram.  I ask students how Daniel could add something to his expression to make it match.  I am looking for students to see that by adding parentheses, (5+3) x (5+3), it will work.  If I have time, I ask for students to share other equivalent expressions that match the area diagram.

Commutative & Distributive Properties

10 minutes

At this stage of this two-day lesson my goal is for my students to make connections between the work they have been doing with equivalent expressions and the commutative and distributive properties.  Rather than give students wordy definitions, however, I present two examples of each property and have students come up with their own definition (see Commutative and Distributive Properties). I remind my students that we can use letters, or variables, to represent any number. I plan to ask, "How could we represent this property with variables?" If students are noticing patterns and relationships, they will be able to come up with their own example using variables (MP7, MP8). 

Matching Part 2

15 minutes

As we move on to this activity, I have students join their partner from yesterday's lesson (see the lesson for an explanation of how the groups were formed). To begin our work, I have a volunteer review the rules and expectations on 1.14 Matching Part 2. Then, I ask students to share out the strategies that students used yesterday. We discussed these at the end of the lesson, so I am interested in seeing if students volunteer strategies that their peers' used. Finally, I ask students if they have any questions.  Once the questions are asked and answered, I pass out the materials and students start working.

Teacher's Note:

  • 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.
  • For this lesson, I only give partner pairs an envelope with the following cards: A4, A5, A6, E3, E4, E5, E9, E12, and 3 blank E cards (they will need at least one to create a matching expression for A5).  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.
  • For the Challenge, I give partner pairs a separate envelope with the following cards: A3 and a few blank A cards, as well as E6, E10, E11, E13, E14 and a few blank E cards. 

As students work  on the Matching Task, I walk around and monitor student progress.  I am observing what strategies students are using and I am monitoring behavior.

Many students may struggle at first, and this is okay. If a partner pair is stuck, at first I will not intervene.  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 the matches, I ask them to use blank cards to create a different expression that is equivalent to each area diagram.  If they complete this task, I will have them pair up with another partner pair to compare their matches.  For partner pairs who need an extra challenge, I will give them the Matching Challenge cards.

Making a Poster

20 minutes

I have a volunteer read the directions on the Making a Poster handout.  I set up the card sets, scissors, glue, and poster papers in the room.  I will call partner pairs a few at a time to collect the needed materials.

Teacher's Notes:

  • Before the lesson, I print out one copy of the Area Diagram Card Set and one copy of the Expressions Card Set for each partner pair.  I print them on plain white paper.  Partners will need scissors and glue/tape for their posters.
  • I will make an exemplar poster (creating my own area diagram and expression cards) for students to look at.

As students work I walk around and monitor student progress.  I make sure that partners are working on different matches.  If students struggle, I refer them to my exemplar. 

If students successfully complete the poster, I give them the following choices:

  • Create another poster for a different match
  • Work on the Challenge Matching cards
  • Create another poster where they create their own area diagram and equivalent expressions

Closure and Post-Assessment

5 minutes

I begin today's Closure by asking students to explain the Commutative Property of Addition, the Commutative Property of Multiplication, and the Distributive Property in their own words.  I ask students to share out struggles that they encountered during this two day investigation and how they overcame them.

I have students clean up and organize their cards.  Instead of giving a ticket to go, I collect and look at student work. 

I hand out the HW Post-Assessment.  When I look at the post-assessments I will make comments/questions on student’s thinking.  I will staple students’ pre-assessment to the post-assessment so students can observe and compare their thinking from the Pre-Assessment and the Post-Assessment.