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Interpreting Algebraic Expressions
During the
Lesson
  • Interactive Whole-Class Discussion (10 minutes)
      • Give each student a mini-whiteboard, pen, and eraser.
      • Hold a short question-and-answer session.
      • If students show any incorrect answers, write the correct answer on the board and discuss any problems.
        • On your mini-whiteboards, show me an algebraic expression that means:
      Multiply n by 4, and then add 3 to your answer. 4n + 3
      Add 3 to n, and then multiply your answer by 4. 4(3 + n)
      Add 5 to n, and then divide your answer by 3.
      n + 5
      3
      Multiply n by n, and then multiply your answer by 5. 5n2
      Multiply n by 5, and then square your answer. (5n)2
  • Collaborative Activity: Matching Expressions and Words (15 minutes)
    • The first activity is designed to help students interpret symbols and realize that the way the symbols are written defines the order of operations.

      • Organize students into groups of two or three.
      • Display the projector resource P-1, Matching Expressions and Words.

      Note that one of the algebraic expressions on the slide does not have a match in words. This is deliberate! It is to help you explain the task to students.

      • Model the activity briefly for students using the examples on the projector resource.
        • I am going to give each group two sets of cards, one with expressions written in algebra and the other with words.
        • Take it in turns to choose an expression and find the words that match it. [4(n + 2) matches "Add two to n then multiply by 4." 2(n + 4) matches "Add four to n then multiply by 2."]
        • When you are working in groups, you should place these cards side by side on the table and explain how you know that they match.
        • If you cannot find amatching card, then you should write your own. Use the blank cards provided. [4n + 2 does not match any of the word cards shown on the slide. The word card "Multiply n by two then add four" does not match any of the expressions.]
      • Give a copy of Card Set A: Expressions and Card Set B: Words to each small group.
      • Support students in making matches and explaining their decisions.
      • As they do this, encourage student to speak the algebraic expressions out loud.

      Students may not be used to "talking algebra" and may not know how to say what is written or may do so in ways that create ambiguities.

      For example, the following conversation between a teacher and pupil is fairly typical:


      Teacher: Tell me in words what this one says.

      [Teacher writes 3 + n2 ]

      Pupil: Three add n divided by two.

      Teacher: How would you read this one then?

      [Teacher writes:]

      3 + n
      2

      Pupil: Three add n divided by two. Oh, but in the second one, you are dividing it all by two.

      Teacher: So can you try reading the first one again, so it sounds different from the second one?

      Pupil: Three add... [pause] ...n divided by two [said quickly]. Or n divided by two, then add three.


      • Students will need to make word cards to match E10: 3 + n2 and E12: n2 + 62
      • They will also need to make expression cards to match W3: Add 6to n, then multiply by 2 and W10: Square n then multiply by 9.

      Some students may notice that some expressions are equivalent; for example 2(n + 3) and 2n + 6. You do not need to comment on this now: when the Card Set C: Tables is given out, students will notice this for themselves.

  • Collaborative Activity: Matching Expressions, Words, and Tables (15 minutes)
    • Card Set C: Tables will make students substitute numbers into the expressions and will alert them to the fact that different expressions are equivalent.

      • Give each small group of students a copy of the Card Set C: Tables and ask students to match these to the card sets already on the table.
      • Some tables have numbers missing; students will need to write these in.
      • Encourage students to use strategies for matching.
      • There are shortcuts that will help to minimize the work. For example, some may notice that:

        • Since 2(n + 3) is an even number, we can just look at tables with even numbers in them.
        • Since (3n)2 is a square number, we can look for tables with only square numbers in them.
      • Students will notice that there are fewer tables than expressions. This is because some tables match more than one expression.
      • Allow students time to discover this for themselves. As they do so, encourage them to test that they match for all n.

        This is the beginning of a generalization.

        • Do 2(n + 3) and 2n + 6 always give the same answer when n = 1, 2, 3, 4, 5?
        • What about when n = 3246 or when n = −23 or when n = 0.245?
        • Check on your calculator.
        • Can you explain how you can be sure?

      This last question is an important one and will be followed up in the next part of the lesson.

      • It is important not to rush the learning. At about this point, some lessons run out of time
      • If this happens, ask pupils to stack their cards in order, so that matching cards are grouped together, and fasten them with a paper clip.
      • Ask students to write their names on an envelope, and store the matched cards in it.

      These envelopes can be reissued in the next lesson.

  • Collaborative Activity: Matching Expressions, Words, Tables, and Areas (15 minutes)
    • The Card Set D: Areas will help students to understand why the different expressions match the same tables of numbers.

      • Give each small group of students a copy of the Card Set D: Areas, a large sheet of paper, and a glue stick.
        • Each of these cards shows an area.
        • I want you to match these area cards to the cards already on the table.
        • When you reach agreement, paste down your final arrangement onto the large sheet, creating a poster.
        • Next to each group write down why the areas show that different expressions are equivalent.

      These posters will be displayed in the final class discussion.

      • As students match the cards, encourage them to explain and write down why particular pairs of cards go together.
        • Why does this area correspond to n2 + 12n + 36?
        • Show me where n2 is in this diagram. Where is 12n? Where is the 36 part of the diagram?
        • Now show me why it also shows (n + 6)2.
        • Where is the n + 6?
      • Ask students to identify groups of expressions that are equivalent and explain their reasoning.

      For example E1 is equivalent to E10, E8 is equivalent to E9, and E4 is equivalent to E5.

  • Whole-Class Discussion (15 minutes)
      • Hold a whole-class interactive discussion to review what has been learned over this lesson.
      • Ask each group of students to justify using their poster why two expressions are equivalent.
      • Then use mini-whiteboards and questioning to begin to generalize the learning.
        • Draw me an area that shows this expression: 3(x + 4)
        • Write me a different expression that gives the same area.
        • Draw me an area that shows this expression: (4y)2
        • Write me a different expression that gives the same area.
        • Draw me an area that shows this expression: (z + 5)2
        • Write me a different expression that gives the same area.
        • Draw me an area that shows this expression:
          w + 6
          2
        • Write me a different expression that gives the same area.
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