Fifth Grade Suggested quarter 1or 2
Idaho SySTEMic Total time needed __90 minutes__
Area and Perimeter Modeling
Short Description of Lesson: In Option 1, Children use factors, multiples, squares, addition, and division as they find the area and perimeter of rectangles and other polygons. In Option 2, children are challenged to build polygons which are congruent and/or similar to an original model.
MSD Standards addressed:
Area RT number Description Evidence?
Math standards RT 4, 6, 7, 10, 11 Area and Perimeter Model
Science standards Nature of Science
Background information for teacher: In various versions of this lesson, students look for the pattern that is created when the area and perimeter of a rectangle are doubled and tripled. In addition, they attempt to create polygons of given areas with varying perimeters, to show the relationship between perimeter and area.
Math resources: Investigations Unit 5, Lessons 2 and 3. Suggested connections to curriculum topics:
Students will find patterns by analyzing doubling, tripling and other multiplying events (RT 7) In Area and Perimeter 2 (another lesson), students will develop more understanding of similar and congruent shapes (RT 10).
Materials needed:
1. 40--2x2s for each working group, (Optional: 2--8x16 brick plates for each group)
2. Vocabulary wall: square roots, factors, products, multiples, similar, congruent, doubling, 3. Paper materials: multiplication chart for individuals, optional--grid sheet (Bricklab Grid,
attached), Evidence RT 4, 6 on a chart, transparency or downloaded to smart board, student worksheets for whole class lesson
4. Pencil, highlighter Grouping of Children:
Partners or small groups, depending on the bricks available Classroom Procedure:
1. Preparation: pre-sort the 2x2s in baggies if desired, copy worksheets for individuals, make a transparency or download to smart board.
2. Direct instruction: (with the entire class)
1. Class Definition: a 2x2 brick represents 1 unit (1x1)
2. If one dimension is doubled, what would the area and perimeter equal? (A=2 square units, P=6 units) This would no longer be a square, because all sides would not be equal. 3. Are these shapes similar? Congruent? How do you know? The shape that is two times as
large should be similar to the original shape. This shape is not similar because it is only two times the width.
4. To make a similar or congruent figure it must be two times the original in length and width. (in all dimensions)
3. Children in groups: Continuing with this pattern: make the square 3 times as big, 4 times as big, etc, students build squares until they have completed a square at least 6 times the original shape (This will require 36 bricks).
4. Children chart results on the attached worksheet and look for a pattern which works for producing the perimeter and the area of the squares they have produced.
Sample start for class chart:
Unit Area Perimeter
1 1unit squared 4 units
4 4 units
squared
8 units
Inquiry based questions:
1. What pattern do you notice in calculating the area?
2. What pattern do you notice in calculating the perimeter? What is the relationship between the area and perimeter?
3. Can you predict what would happen if the unit were 10 times its original size?
4. Using the models made what would the area be if you counted the posts as units? (For instance on a 2x2 brick, the perimeter of one brick would be 8 units and the area would be 1 square unit.) Children can use this value, rather than the previous unit, to re-calculate their perimeters and areas.
5. Does the relationship between area and perimeter stay the same? Why or why not?
6. Optional: Students transfer models to grid paper (brick lab) for formal evidence piece, this piece could be used as an “anchor” lesson to refer back to with future doubling and tripling lessons.
Discussion, Whole Class:
1. Bringing the class back together, chart patterns on overhead or Smart board
2. Discuss patterns, what did students notice? What was the relationship between the area and perimeter to the original shape?
3. What would the area and perimeter be if the figure was ten times the original shape? (100 units squared and 40 units) How do you know?
4. Could we come up with a formula that would represent the area that would work every time? A = l x w
5. Perimeter? P = length x number of sides . Each number you got for perimeter is a multiple of 4, why? Four sides
6. Using your multiplication chart, highlight the area of each shape (1, 4, 9, 16, 25, 36) What do you notice? They are all square numbers. Why are they called square numbers? A 3x3 makes a perfect square.
Extension options
Rectangles instead of squares:
What happens if you use a 2x3 as the original unit?” Children would follow a procedure similar to the first part of the lesson, and fill out their worksheets with information for 2x3 bricks: doubling and tripling. Their results would all be similar rectangles.
The PCS holiday card diagrams
can also be doubled and tripled as an extension of this lesson.
How can perimeter vary even when the area stays the same?
1. Using four 2x2 bricks, children attempt to form the 2D structure with the smallest perimeter, counted in studs (16 studs2, with 16 studs on perimeter)
2. Using the same four bricks, children attempt to form the 2D structure with the largest perimeter, counted in studs (zig-zag with 26 studs on perimeter)
3. Children continue this exercise with four 2x4 bricks, an area of 32 studs2. When the bricks are made into a rectangle, the perimeter has 24 studs. When made into the “biggest possible perimeter”, the perimeter has 42 studs.
4. The attached chart shows some arrangements of bricks which can cause this big difference among possible perimeters with a given area.
Inquiry questions for this section:
1. When might it be an advantage to have a larger perimeter for a given area? (Think about a living or non-living thing moving through water or space, absorbing things like food or air, contractors making small spaces out of large ones in buildings)
2. Why would it be a disadvantage? (Think about a living thing or a space staying warm or not losing water, contractors buying fencing for yards or wall board for rooms)
Necessary Additions for all of these parts of the lesson: 1. Student data sheet for multiplying dimensions
2. Bricklab grid for recording possible perimeter variations
Name ____________________ Date _____________
Student Data Chart for Area and Perimeter of Squares and Rectangles
You will use a 2x2 brick for your original unit. Its length is one unit, its width is one unit, and its area is ONE Square Unit. When you double the original shape, what happens to the area and perimeter? Triple? What would the area and perimeter be for all possibilities? Use the chart to show your thinking.
Dimensions Area Perimeter
2 units x 2 unit 4 “square units” 2+2+2+2=8 units
3 units x 3 units
1. What would the area and perimeter be if the dimensions for the figure were ten times the original dimensions on each side? How do you know?
______________________________________________________________________________ ______________________________________________
2. Could you come up with a formula to represent the area that would work every time?
______________________________________________________________________________ ______________________________________________
