Resource Overview Quantile® Measure: 560Q Skill or Concept: Determine perimeter using concrete models, nonstandard units, and standard units. (QT‐M‐146) Use grids to develop the relationship between the total numbers of square units in a rectangle and the length and width of the rectangle (l x w). (QT‐M‐191) Determine the area of rectangles, squares, and composite figures using nonstandard units, grids, and standard units. (QT‐M‐192) Excerpted from: Gourmet Learning 1937 IH 35 North Suite 105 New Braunfels, TX 78130 www.gourmetlearning.com © Gourmet Learning This resource may be available in other Quantile utilities. For full access to these free utilities, visit www.quantiles.com/tools.aspx.
The Quantile® Framework for Mathematics, developed by educational measurement and research organization MetaMetrics®, comprises more than 500 skills and concepts (called QTaxons) taught from kindergarten through high school. The Quantile Framework depicts the developmental nature of mathematics and the connections between mathematics content across the strands. By matching a student’s Quantile measure with the Quantile measure of a mathematical skill or concept, you can determine if the student is ready to learn that skill, needs to learn supporting concepts first, or has already learned it. For more information and to use free Quantile utilities, visit www.Quantiles.com. 1000 Park Forty Plaza Drive, Suite 120, Durham, North Carolina 27713 METAMETRICS®, the METAMETRICS® logo and tagline, QUANTILE®, QUANTILE FRAMEWORK® and the QUANTILE® logo are trademarks of MetaMetrics, Inc., and are
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Unit 1 – Lesson 3
Measurement
Student Expectation: Students will disprove common misconceptions relating
perimeter and area in rectangles
Enrichment
Finding Area with Concrete/Pictorial Models
“Around the Area”
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Teacher note: In this activity, students will address 2 questions. Question 1: If two
rectangles have the same area, do they have the same perimeter? Question 2: If two
rectangles have the same perimeter, do they have the same area?
Group size: no more than four students
Materials: graph paper; 36 square inches in a bag for each group (from the Cooperative
Learning Activity); poster boards; markers; sticky notes; index cards; markers
Before class: If you did not prepare the square inch paper bags for the Cooperative
Learning Activity, you will need to do this (see Cooperative Learning “Before Class”
section). Create one large poster with the word “True” on it and one with the word
“False” on it. Write the following “LENGTH/WIDTH” combinations on index cards (one
per index card): LENGTH 11, WIDTH 1; LENGTH 10, WIDTH 2; LENGTH 9, WIDTH 3;
LENGTH 8, WIDTH 4; LENGTH 7, WIDTH 5
Directions:
• Write this statement on the board or overhead: “If two rectangles have the same area,
then they must have the same perimeter.”
• Hang the “True” and “False” posters on the wall.
• Poll the class by giving each student a sticky note. Each student will write his/her
name on the note. Then when you say “go” (after about 2 minutes), have students place
their notes on either the “True” poster or the “False” poster. Do not ask them why they
believe the statement is correct or not, and don’t give hints as to which is the correct
answer.
• Tell the students that they are able to change their sticky note to the other poster/
answer at any time during this activity.
• Hand out a bag of square inches to each group, and ask each group to create a square
using all 36 square inches inside.
Questioning Technique
Instructional Strategy
Ask: What is the area of your square? (Ask this to each group, to which each should
respond 36 square inches.)
Ask: What is the perimeter of your square? (Ask this to each group, to which each should
respond 24 inches - 4 sides of 6 inches each.)
Teacher note: Some students may opt to change their sticky notes at this time; allow
them to do so, but do not say anything. Continue the activity!
Gourmet Curriculum Press, Inc.©
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Unit 1 – Lesson 3
Measurement
Student Expectation: Students will disprove common misconceptions relating
perimeter and area in rectangles
Enrichment
Finding Area with Concrete/Pictorial Models
“Around the Area”
Questioning Technique
Instructional Strategy
Say: Create a rectangle using all of these square inches. (Allow students time to create
rectangles. As a teacher, you are hoping that they will be different dimensions, so do not
make them all create the same one.)
Ask: What is the area of your rectangle? (Ask this to each group, to which each should
respond 36 square inches.)
Ask: What is the perimeter of your rectangle? (Answers will vary. If it is 1 x 36, the
perimeter will be 74 inches. If it is 2 x 18, the perimeter will be 40 inches. If it is 3 x 12,
the perimeter will be 30 inches. If it is 4 x 9, the perimeter will be 26 inches.) (Have the
students explain how they arrived at their answers. They could even come to the overhead
and show the class.)
Teacher note: At this time, if the students used different rectangles, they should see that
rectangles with the same area can have different perimeters. Most of the students will
want to move their sticky notes to the “False” poster. If the students all made the same
rectangle on this second attempt, then you will want to have a 3rd round of creating
rectangles, at which time you should assign certain rectangles to different groups so that
you will get different perimeters. (In reality, you have actually already disproved the
statement by comparing the first rectangle with the square, but depending on their level
of understanding of squares and rectangles, the students might not realize this.)
If there are any students who still have their notes on the “True” side, please ask them to
justify their reasoning at this time. Help them come to the realization on their own by
questioning them and having them analyze the situation.
Ask: Did we see a situation in which everyone made the same shape? (Yes, the square.)
Ask: Can anyone rewrite this statement so that it IS true? (If two squares have the same
area, then they have the same perimeter.)
Ask: Why does it work with squares but not all rectangles? (The sides of a square are all
the same length.)
• Erase the statement from the board, have the students retrieve their notes, and write
the following on the board: “If two rectangles have the same perimeter, then they
must have the same area.”
• Poll the class again using the same process. No comments or sharing are permitted.
• Have the students create a square with a perimeter of 24. (They should use all their
square inches. The area will be 36 square inches.)
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Gourmet Curriculum Press, Inc.©
Unit 1 – Lesson 3
Measurement
Student Expectation: Students will disprove common misconceptions relating
perimeter and area in rectangles
Enrichment
Finding Area with Concrete/Pictorial Models
“Around the Area”
Questioning Technique
Instructional Strategy
Ask: What is the perimeter of your square? (Ask this to each group, to which each should
respond 24 inches.)
Ask: What is the area of your square? (Ask this to each group, to which each should
respond 36 square inches.)
Teacher note: Some students may opt to change their notes at this time. Allow them to
do so, but do not say anything. Continue the activity!
• Have each group of students draw an index card (prepared before class) with a length
and width on it.
• Have each group create the rectangle as described on the card using their square inch
tiles. (They will not use all of them.)
When everyone is finished, have each group calculate the perimeter and area of its figure.
Ask: What is the perimeter of your rectangle? (Ask this to each group, to which each
should respond 24 inches.)
Ask: What is the area of your rectangle? (Ask this to each group. The answers will vary.
A 1 x 11 rectangle will be 11 square inches. A 2 x 10 rectangle will be 20 square inches. A
3 x 9 rectangle will be 27 square inches. A 4 x 8 rectangle will be 32 square inches. A 5 x
7 rectangle will be 35 square inches.)
Teacher note: By this time, all the students should have realized that the areas will not
be the same just because they have the same perimeter. All of their notes should be on
the “False” side. If this is not the case, help the students come to a realization through
questioning and discussion that this statement is not true.
Ask: Was there ever a case when directed that you all got the same area when given the
perimeter? (Yes—with the square)
Ask: Is there anyone who can reword this statement so that it is true? (If two squares
have the same perimeter, then they have the same area.)
Ask: Can anyone explain why this works for squares and not all rectangles? (Answers
will vary, but it has to do with the fact that the four sides of a square are always the same
length. The number of squares in each row and column are the same, making it the same
area. With other rectangles, the number of squares in each row and column are different,
giving them each a different area.)
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