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Estimated Time Needed

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Increasing and Decreasing Quantities by a Percent
During the
Lesson
  • Collaborative Activity 1: Matching Card Sets A, B, and C (30 minutes)
      • Organize the class into groups of two or three students.
        • With larger groups, some students may not fully engage in the task.
      • Give each group Card Sets A and B.
      • Use the projector resource to show students how to place Card Set A.
      • Introduce the lesson carefully:
        • I want you to work as a team. Take it in turns to place a percentage card between each pair of money cards.
        • Each time you do this, explain your thinking clearly and carefully. If your partner disagrees with the placement of a card, then challenge him/her. It is important that you both understand the math for all the placements.
        • There is a lot of work to do today, and it doesn't matter if you don't all finish. The important thing is to learn something new, so take your time.

      Pairs of money cards may be considered horizontally or vertically.

      Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving.

      Make a note student approaches to the task.

      You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice any common mistakes. For example, students may make the mistake of pairing an increase of 50% with a decrease of 50%.

      Support student problem solving.

      Try not to make suggestions that move students towards a particular approach to this task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning.

      Students will correct their own errors once the decimal cards are added.

      For students struggling to get started:

        • There are two ways to tackle this task. Can you think what they are? [Working out the percentage difference between the two money cards or taking a percentage card and using guess and check to work out where to place it.]
        • How can you figure out the percentage difference between these two cards?
        • This percentage card states the money goes up by 25%. If this money card (say $160) increases by 25%, what would be its new value? Does your answer match any of the money cards on the table?

      When one student has placed a particular percentage card, challenge their partner to provide an explanation:

        • Maria placed this percentage card here. Martin, why does Maria place it here?

      If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning.

      Students often assume that if an amount is increased and then decreased by the same percent, the amount remains unchanged.

        • The price of a blouse is $20. It increases by ½. What is the new price? [$30]
        • The price of the blouse now decreases by ½. What is the final price? [$15]
        • Now let's apply this to percentages. What happens if the $20 blouse increases by 50%?
        • What happens now when this new price decreases by 50%?
        • What percentage does the price need to decrease by to get it back to $20? [33 ⅓%]
        • What does this show?

      If the whole class is struggling on the same issue, you may want to write a couple of questions on the board and organize a whole-class discussion. The projector resource may be useful when doing this.

      It may help some students to imagine that the money cards represent the cost of an item, for example, the price of an MP3 player at four different stores.

  • Placing Card Set C: Decimal Multipliers
      • As students finish placing the percentage cards, hand out Card Set C: Decimal Multipliers. These provide students with a different way of interpreting the situation.

      Do not collect Card Set B. An important part of this task is for students to make connections between different representations of an increase or decrease.

      Encourage students to use their calculators to check the arithmetic. Students may need help with interpreting the notation used for recurring decimals and in entering 1.3 as 1.33333333 on the calculator.

      As you monitor the work, listen to the discussion, and help students look for patterns and generalizations. The following patterns may be noticed:

      • An increase of, say, 33% is equivalent to multiplying by 1.3.
      • (An increase of 5% is not equivalent to multiplying by 1.5!)
      • A decrease of, say, 33% is equivalent to multiplying by (1−1. 3) = 0.6
      • The inverse of an increase by a percent is not a decrease by the same percent.

      When the decimal multipliers are considered in pairs, the calculator will show that each pair multiplies to give 1, subject to rounding by the calculator.

        • × 2 × 0.5 and 2 × 0.5 = 1
        • × 1.5 × 0.6 and 1.5 × 0.6 = 1
        • × 1.3 × 0.75 and 1.3 × 0.75 = 1
        • × 1.25 × 0.8 and 1.25 × 0.8 = 1
        • × 1.6 × 0.625 and 1.6 × 0.625 = 1

      Extension Activity

      • Ask students who finish quickly to try to find the percent changes and decimal multipliers that lie between the diagonals $150/$160 and $100/$200. Students will need to use blank cards for the diagonals $150/$160.

      Taking Two lessons to Complete All Activities

      You may decide to extend the lesson over two periods.

      • Ten minutes before the end of the first lesson, ask one student from each group to visit another group's work. Students remaining at their seats should explain their reasoning for the position of the cards on their own desk (see the section on Sharing Work that follows for further details).
      • When students are completely satisfied with their own work, hand out the poster template Percents, Decimals, and Fractions (1). Students should use it to record the position of their cards. At this stage, one pair of arrows between each money card will be left blank.
      • At the start of the second lesson, spend a few minutes reminding the class about the activity.
        • Try to remember what we were working on in the last lesson.
        • A mobile phone is reduced by 60% in the sale. Give me an example of what the phone could have originally cost and what it costs now. Ask for multiple examples.
        • [Take one of the examples given above] The mobile phone is not sold. It returns to its original price. What is the percent increase?
      • Return to each group their Percents, Decimals, and Fractions (1) sheet and the Card Sets A, B, and C.
      • Ask students to use their sheet to position their cards on the desk. Working with the cards instead of the sheet means students can easily make changes to their work and encourages collaboration between students.
      • Then move the class on to the second collaborative activity.
  • Sharing Work (10 minutes)
      • When students get as far as they can with placing Card Set C, ask one student from each group to visit another group's work. Students remaining at their desk should explain their reasoning for the matched cards on their own desk.
        • If you are staying at your desk, be ready to explain the reasons for your group's matches.
        • If you are visiting another group, write your card placements on a piece of paper. Go to another group's desk and check to see which matches are different from your own.
        • If there are differences, ask for an explanation. If you still don't agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work.
      • Students may now want to make changes.
  • Collaborative Activity 2: Matching Card Set D (30 minutes)
      • Give out Card Set D: Fraction Multipliers. These may help students to understand why the pattern of decimal multipliers works as it does.
      • Support the students as you did in the first collaborative activity.

      The following pairings appear:

         and
      × 1
      2
      × 4
      3
        and
      × 3
      4
      × 8
      5
       and
      × 5
      8
      × 3
      2
       and
      × 2
      3
      × 5
      4
       and
      × 4
      5
  • Sharing Work (10 minutes)
      • When students get as far as they can placing Card Set D, ask the student who has not already visited another group to go check their answers against that of another group's work.

      As in the previous sharing activity, students remaining at their desks are to explain their reasoning for the matched cards on their own desk.

      • Students may now want to make some final changes to their own work. After they have done this, they can make a poster.

      Either:

      • Give each group a large sheet of paper and a glue stick, and ask students to stick their final arrangement onto the large sheet of paper

        and/or

      • Give each group the poster template, Percents, Decimals, and Fractions (1), and ask students to record the position of their cards. The poster template allows students to record their finished work. It should not replace the cards during the main activities of this lesson as students can more easily make changes when working with the cards, and they encourage collaboration.

      Extension Activities

      • Ask students who finish quickly to try to find the fraction multipliers that lie between the diagonals $150/$160 and $100/$200.

      Card Set E: Money Cards (2) may be given to students who need an additional challenge. Card Sets B-D can again be used with these Money Cards. Students can record their results on the poster template, Percents, Decimals, and Fractions (2).

      • In addition, you could ask some students to devise their own sets of cards.
  • Whole-Class Discussion (10 minutes)
      • Give each student a mini-whiteboard, pen, and eraser.
      • Conclude the lesson by discussing and generalizing what has been learned.

      The generalization involves first extending what has been learned to new examples and then examining some of the conclusions listed above. As you ask students questions like the following, they should respond using mini-whiteboards.

        • Suppose prices increase by 10%. How can I say that as a decimal multiplication?
        • How can I write that as a fraction multiplication?
        • What is the fraction multiplication to get back to the original price?
        • How can you write that as a decimal multiplication?
        • How can you write that as a percentage?
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