Resource Overview Quantile® Measure: 450Q Skill or Concept: Determine perimeter using concrete models, nonstandard units, and standard units. (QT‐M‐146) Determine the area of rectangles, squares, and composite figures using nonstandard units, grids, and standard units. (QT‐M‐192) Excerpted from: The Math Learning Center PO Box 12929, Salem, Oregon 97309‐0929 www.mathlearningcenter.org © Math Learning Center This resource may be available in other Quantile utilities. For full access to these free utilities, visit www.quantiles.com/tools.aspx.
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Set D6 Measurement: Area & Perimeter
Set D6 H Activity 4
Activity
Hexarights
Overview
Recommended Timing
Students continue to investigate relationships between
area and perimeter as they measure and construct polygons called “hexarights” (hexagons with pairs of adjacent
sides that meet at right angles).
Anytime after Set D6 Activity 2
Skills & Concepts
H Measuring Hexarights (page D6.25, half-class set, cut
You’ll need
H Introducing Hexarights (page D6.24, 1 copy on a
transparency)
H determine the perimeters and areas of squares and
other rectangles using formulas and explain why the
formulas work
in half)
H Hexarights, Perimeter = 24 cm (page D6.26, class set)
H Centimeter Grid Paper (page D6.27, class set, plus a few
extra)
H determine the areas of nonrectangular figures that can
be composed or decomposed into rectangles
H piece of paper to mask parts of the overhead
H demonstrate that rectangles with the same area can
have different perimeters, and that rectangles with the
same perimeter can have different areas
H 2 or 3 transparencies and overhead pens
H rulers marked with both centimeters and inches (class set) H solve single- and multi-step contextual problems involving perimeters and areas, and justify the solutions
Instructions for Hexarights
1. Show the top portion of Introducing Hexarights at the overhead, masking the rest with a piece of paper. Give students a minute to pair-share any observations they can make. Then invite volunteers to
share their thinking with the class. Record some of their ideas in the space to the left of the shape.
2. Then reveal the definition below the shape, still keeping the rest of the overhead covered. Read and
discuss it with the class. As you do so, review the meanings of the terms hexagon, perpendicular, and
right angles.
Set D6 Measurement: Area & Perimeter Blackline Run 1 copy on a transparency.
Introducing Hexarights
1
•
•
•
•
•
•
Describe this shape.
has 6 sides
has 5 maybe 6 right angles
has parallel lines
some of the lines are perpendicular
kind of like 2 rectangles stuck together
none of the lines are the same length
This shape is a hexagon because it has 6 sides, but let’s call it a hexaright. A
hexaright is a hexagon in which sides that touch each other is perpendicular
(That is, they meet at right angles.)
2 Here are 2 examples of shapes that are not hexarights. Can you see why?
a
b
Bridges in Mathematics Grade 4 Supplement • D6.21
© The Math Learning Center
3
Find the area and perimeter of the hexarights below.
Set D6 Measurement: Area & Perimeter
Activity 4 Hexarights (cont.)
3. Next, reveal the two counter-examples shown in the middle of the overhead. Can students explain
why neither of these are hexarights? Have them share at the overhead so their classmates can see what
they’re talking about.
Set D6 Measurement: Area & Perimeter Blackline Run 1 copy on a transparency.
Introducing Hexarights
1
•
•
•
•
•
•
Describe this shape.
has 6 sides
has 5 maybe 6 right angles
has parallel lines
some of the lines are perpendicular
kind of like 2 rectangles stuck together
none of the lines are the same length
This shape is a hexagon because it has 6 sides, but let’s call it a hexaright. A
hexaright is a hexagon in which sides that touch each other is perpendicular
(That is, they meet at right angles.)
2 Here are 2 examples of shapes that are not hexarights. Can you see why?
a
3
b
Find the area and perimeter of the hexarights below.
a
b
Students Shape a isn’t a hexaright because there are 2 angles that aren’t right angles.
I thought they were wrong about shape b because it’s all right angles, but then I realized there are 10
sides! A hexaright can only have 6 sides.
4. Now show the 2 hexarights at the bottom of the overhead and briefly discuss strategies for finding the
area and perimeter of each. Then give students each a copy of the Measuring Hexarights half-sheet. Ask
them to experiment with both the inch side and the centimeter side of their rulers. Which unit of measure works best? Students will quickly discover that most of the measurements don’t come out evenly
unless they use centimeters.
5. Solicit agreement from the class that they’ll work in centimeters and square centimeters rather than
inches and square inches, and let them get started. Encourage them to share and compare their strategies and solutions as they work.
6. When most students have finished finding the perimeter and area of at least one of the hexarights,
place a blank transparency on top of the overhead and invite volunteers to share their work with the
class. Move or replace the transparency each time a new volunteer comes up to the overhead to accommodate several different presentations. Here is an example of the sort of work you might expect from
students, although some will divide the hexarights differently.
D6.22 • Bridges in Mathematics Grade 4 Supplement
© The Math Learning Center
Set D6 Measurement: Area & Perimeter
Activity 4 Hexarights (cont.)
Set D6 Measurement: Area & Perimeter Blackline Run a half-class set and cut the sheets in half.
Name
Date
Measuring Hexarights
Find the area and perimeter of the hexarights below. Show all your work.
2 cm
2 x 3 = 6 sq cm
8 cm
3 x 4 = 12 sq cm 3 cm
1 cm
4 cm
1x7
q cm
=7s
7 cm
4 cm
q cm
=4s
2 cm
1x4
3 cm
6 cm
3 cm
1 cm
1 + 8 + 4 + 1 + 3 + 7 = 24 cm
P = 24 cm
A = 11 sq cm
6 + 2 + 3 + 3 + 4 = 18 cm
P = 18 cm
A = 18 sq cm
7. As students share, discuss the methods they’re using to find the area and perimeter of these shapes.
Did they use the perimeter formulas they developed during Set D6 Activity 2? Why not? (Because these
are irregular polygons. All you can do is simply add all the different side lengths.) Did they use the
area formula they developed during Measurement—Area Perimeter Activity 1? How? (To find the area
without covering the shape with centimeter square units or drawing them in, you need to divide each
hexaright into 2 rectangles. Then you can use A = lw to find the area of each rectangle and add these
areas to get the area of the hexaright.)
8. After 2 or 3 strategies have been shared for each hexaright, explain that there is more than one
hexaright with a perimeter of 24 centimeters. Give students each a copy of Hexarights, Perimeter =
24 cm. Review the instructions together and clarify as needed. Place a small stack of the Centimeter
Grid Paper on each table and give students the remainder of the math period to work. Encourage them
to share and compare their strategies for finding other hexarights with perimeters equal to 24 centimeters. What are some of the areas that result? Are they all equal?
Set D6 Measurement: Area & Perimeter Blackline Run a class set.
NAME
DATE
Hexarights, Perimeter = 24 cm
1
Draw 2 different hexarights with a perimeter of 24 cm, and find the area of
each. Then draw a third hexaright with a perimeter of 24 cm. This time, make
the area as large as you can.
2
You can use the space below and the back of this sheet. Or, you can draw your
hexarights on centimeter grid paper, cut them out, and glue them to this sheet.
Use your ruler to help make the lines straight and accurate.
3
Label your hexarights with their dimensions, perimeter, and area. Use numbers, sketches, and/or words to show how you found the perimeter and area of
each hexaright.
4
On the back of the sheet, write at least 2 sentences to describe what you found
out about the areas of hexarights with a perimeter of 24 cm.
Reconvene the class to share strategies and solutions either at the end of the period or at another time.
Note “Hexaright” is not some long-forgotten concept from your high school geometry days. It is a made-up term
borrowed from Measuring Up: Prototypes for Mathematics Assessment (Mathematical Sciences Education
Board National Research Council, 1993. Washington, DC: National Academy Press). You may want to let students know this so that they won’t expect to see, or use it on standardized texts.
© The Math Learning Center
Bridges in Mathematics Grade 4 Supplement • D6.23
Set D6 Measurement: Area & Perimeter Blackline Run 1 copy on a transparency.
Introducing Hexarights
1
Describe this shape.
This shape is a hexagon because it has 6 sides, but let’s call it a hexaright. A
hexaright is a hexagon in which sides that touch each other are perpendicular.
(That is, they meet at right angles.)
2 Here are 2 examples of shapes that are not hexarights. Can you see why?
a
3
b
Find the area and perimeter of the hexarights below.
a
D6.24 • Bridges in Mathematics Grade 4 Supplement
b
© The Math Learning Center
Set D6 Measurement: Area & Perimeter Blackline Run a half-class set and cut the sheets in half.
name
date
Measuring Hexarights
Find the area and perimeter of the hexarights below. Show all your work.
name
date
Measuring Hexarights
Find the area and perimeter of the hexarights below. Show all your work.
© The Math Learning Center
Bridges in Mathematics Grade 4 Supplement • D6.25
Set D6 Measurement: Area & Perimeter Blackline Run a class set.
name
date
Hexarights, Perimeter = 24 cm
1
Draw 2 different hexarights with a perimeter of 24 cm, and find the area of
each. Then draw a third hexaright with a perimeter of 24 cm. This time, make
the area as large as you can.
2
You can use the space below and the back of this sheet. Or, you can draw your
hexarights on centimeter grid paper, cut them out, and glue them to this sheet.
Use your ruler to help make the lines straight and accurate.
3
Label your hexarights with their dimensions, perimeter, and area. Use numbers, sketches, and/or words to show how you found the perimeter and area of
each hexaright.
4
On the back of the sheet, write at least 2 sentences to describe what you found
out about the areas of hexarights with a perimeter of 24 cm.
D6.26 • Bridges in Mathematics Grade 4 Supplement
© The Math Learning Center
Set D6 Measurement: Area & Perimeter Blackline Run a class set, plus a few extras.
name
date
Centimeter Grid Paper
© The Math Learning Center
Bridges in Mathematics Grade 4 Supplement • D6.27
D6.28 • Bridges in Mathematics Grade 4 Supplement
© The Math Learning Center