Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
Enduring understanding (Big Idea): Students reason about relationships among shapes to determine area, surface
area, and volume
Essential Questions: :
How do I know whether area or perimeter is involved? What attributes of a shape are important to measure? What
relationships involving area or perimeter, or both, will help solve the problem? How can I find the area and perimeter of a
regular or irregular shape? How can a 3-dimensional shape, such as a book, be represented on a flat page?
BY THE END OF THIS UNIT:
Students will know… In the CCSS, sixth-grade students
solve problems involving area, surface area, and volume.
They know and use the formulas for the area and
circumference of a circle. Students find the areas of
triangles, special quadrilaterals, and polygons by
composing shapes into rectangles or decomposing shapes
into triangles and other shapes. Students reason about the
volume of a right rectangular prism with fractional edge
lengths and then apply the formulas V = l x w x h and
V = Bh to solve related problems. Sixth-grade students use
nets made up of rectangles and triangles to find the surface
areas of three-dimensional figures. Students prepare for
work on scale drawings in later grades by drawing polygons
in the coordinate plane when given coordinates for
the vertices. Students also draw geometric shapes with
given conditions (such as triangles from three measures
of angles).
Unit Resources : Are located below with each objective.
CCSS-M Included: 6G1-4
Vocabulary:
area, vertices,
Suggested Pacing:
20 daysrhombi, surface area, face,
right rectangular prism
volume, base, decomposing, height, edges, trapezoid,
dimensions, three dimensional figures, isosceles, net, right
triangle, vertices, , vertex, rectangles, quadrilateral,
parallelograms, squares, length, width, perpendicular,
coordinate plane, origin, x-axis, y-axis, Quadrant I,
Quadrant II, Quadrant III, Quadrant IV, coordinates, xcoordinate, y-coordinate,
Students will be able to…
I can recognize and know how to compose and decompose
polygons into triangles and rectangles.
I can compare the area of a triangle to the area of the
composed rectangle.
I can apply the techniques of composing and/or
decomposing to find the area of triangles,
special quadrilaterals and polygons to solve mathematical
and real world problems.
I can discuss, develop and justify formulas for triangles
and parallelograms (6th grade introduction).
I can calculate the volume of a right rectangular prism.• I
can apply volume formulas for right rectangular prisms to
solve real-world and mathematical problems involving
rectangular prisms with fractional edge lengths.
I can model the volume of a right rectangular prism with
fractional edge lengths by packing it with unit cubes of the
appropriate unit fraction edge lengths
I can draw polygons in the coordinate plane.
I can use coordinates (with the same x-coordinate or the
same y-coordinate) to find the length of a side of a
polygon.
I can apply the technique of using coordinates to find the
length of a side of a polygon
drawn in the coordinate plane to solve real-world and
mathematical problems.
Mathematical
Practices in Focus:
I can recognize that 3-D figures can be represented by
nets.
Make sense
of problems and
persevere
in solving
I can1-represent
three-dimensional
figures
using nets
made
them
up of rectangles and triangles. I can apply knowledge of
2- Reason
abstractly
and quantitatively
calculating
the area
of rectangles
and triangles to a net.
3Construct
viable
arguments
and
critique
the in the
I can combine the areas for rectangles
and
triangles
reasoning
of others
net to find
the surface
area of a
3-dimensional
4- Model figure.
with mathematics
I can5-solve
andtools
mathematical
problems
Usereal-world
appropriate
strategically
involving
surface
area
using
nets
6- Attend to precision
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
7- Look for and make use of structure
8- Look for and express regularity in repeated
reasoning
Course Name: Math 6 Unit # 5
Unit Plans
Unit Title:
Covering and Surrounding
Investigation
Suggested ACE Questions
Standard 6.G.1
Investigation 1: Designing Bumper
Cars
1.1 Designing Bumper-Car Rides
1.2 Pricing Bumper-Car Rides
1.3 Decoding Designs
Mathematical Reflections
ACE 1-5, 6
ACE 9-15, 28
ACE 16-21, 31
Standard 6.G.1
2.1 Building Storm Shelters
2.2 Stretching the Perimeter
2.3 Fencing in Spaces
2.4 Adding Tiles to Pentominoes
Mathematical Reflections
ACE 1-3, 6, 16-17, 20-21
ACE 7, 18-19
ACE 8-14, 22-25, 27,29
ACE 15-26, 28
3.1 Triangles on Grids
3.2 MoreTriangles
3.3 What’s the Area
3.4 Designing Triangles Under
Constraints
Mathematical Reflections
ACE 1-6, 26-31
ACE 7-20, 32-34
ACE 21-22, 35-38
ACE 23-25, 39-40
4.1 Finding measures of Parallelograms
4.2 Parallelograms and Triangles
4.3 Designing Parallelograms Under
Constraints
4.4 Parks, Hotels, and Quilts
Mathematical Reflections
ACE 1-8, 32
ACE 9-21, 33-35
ACE 22-31, 38-39
ACE 36-37
Standard 6.G.1
Investigation 5: Measuring
Irregular Shapes and Circles
5.1 Measuring Lakes
5.2 Surrounding a Circle
5.3 Pricing Pizzas
5.4 “Squaring” a Circle
Looking Back/Looking Ahead
Mathematical Reflections
ACE 1-4, 47
ACE 5-14, 39-40, 48-49
ACE 15
ACE 16-38, 41-46
Standard 6.G.2, 6.G.4
CC Investigation 4: Measurement
(Common Core Investigationssmall red book)
Standard 6.G.3
CC Investigation 3: Integers and
the Coordinate
Plane
CC-4.1 Nets and Surface Area
CC-4.2 Using Blocks (Volume)
(These titles were made up)
ACE 1-15
ACE 16-25
CC- 3.1/ 3.2 Rational Numbers and the
Number Line
CC- 3.3, 3.4, 3.5 The Coordinate Plane
CC- 3.6 Inequalities
ACE 1-13
ACE 14-55
ACE 56- 69
Investigation 2: Changing Area,
Changing Perimeter
Standard 6.G.1
Investigation 3: Measuring
Triangles
Standard 6.G.1
Investigation 4: Measuring
Parallelograms
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
CORE CONTENT
Cluster Title: Solve real-world and mathematical problems involving area, surface area, and volume.
Standard: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into
rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and
mathematical problems.
Concepts and Skills to Master:








Classify special quadrilaterals: square, rhombus, trapezoid, parallelogram, rectangle, kite
Relate the area of triangles and the area of rectangles
Solve problems in a real-world context.
Identify the relationship between bases and heights in polygons
Determine the area of polygons
Recognize symbolic notation for height (dotted line)
Visually and physically decompose and compose polygons into rectangles and triangles to find area.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Recognize that perpendicular lines form right angles; Define and identify polygons; Polygons have two-dimensions; Identify
square, rhombus, trapezoid, parallelogram, rectangle, kite; The nature of area as an attribute; Since area is a different
attribute it requires a different measurement unit: square units; compose and decompose polygons; Identify right angles in
various orientations; the symbol for right angles.
Academic Vocabulary
Compose, decompose, base, height, right triangle, polygon, special quadrilaterals, perpendicular
Suggested Instructional Strategies:




Discover the area formula for
triangles
-Use a Geoboard to compose and
decompose polygons.
-Use dot paper or grid paper to draw
polygons and find the area.
-Have students decompose paper
polygons by cutting into triangles and
rectangles.
Composing and decomposing
triangles
Resources:
NCTM Activity- Packing the Packages 2002
http://lesage.blogs.uoit.ca/wp-uploads/2010/08/Cerealboxes_Investigation_Junior_-AIMS.pdf (surface area and volume)
MARS TASKS: Being Building Blocks
http://www.illustrativemathematics.org/illustrations/545 - volume activity
(also could be used with surface area)
http://deanadventures.com/htms/geometry.html link to UEN geometry
http://illuminations.nctm.org/LessonDetail.aspx?ID=L577
http://cca6summer2011.wikispaces.com/Tasks+by+Domain%2C+Cluster
%2C+and+Standard
http://illustrativemathematics.org/illustrations/545
Area Formula
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math 6 Unit # 5
Sample Assessment Tasks
Skill-based task
Find the area of the trapezoid
shown using the formulas
for rectangles and triangles.
Unit Title:
Covering and Surrounding
Problem Task
Mario needs to buy sod (grass that rolls out like a carpet) for his
backyard. Here is a diagram of Mario’s backyard.
Determine, using the picture below, how much sod he will need to
purchase.
If sod costs $8.99 per square meter, how much will Mario pay to have
sod put in his yard?
CORE CONTENT
Cluster Title:
Standard: 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of
the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the
edge lengths of the prism. Apply the formulas V = l w h and V = Bh to find volumes of right rectangular prisms with
fractional edge lengths in the context of solving real-world and mathematical problems.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
Concepts and Skills to Master:
Measuring with fractional units requires students to relate volume to multiplication with fractions.
•
Describe the impact of defining volume by fractional factors.
•
Use these formulas interchangeably, V = lwh and V = Bh.
•
Make the connection that when finding volume l x w is the same as B.
•
Composing whole cubes with fractional unit cubes
•
Prove that the volume formula works by creating diagrams of prisms with unit fraction edge lengths and showing
how unit fraction cubes pack them.
•
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Volume is measured with cubic units; the nature of volume as an attribute; since volume is a different attribute it requires a
different measurement unit: unit cubes; prisms are three-dimensional; volume is filling a prism; ability to multiply fractions;
finding the area of polygons – including those with unit fraction edge lengths; substitution for values in formulas; finding
volume of prisms with whole unit side lengths; use of physical models with whole-unit side lengths; find volume using a unit
cube model.
Academic Vocabulary
volume, rectangular prism, length, width, height, base, cubic units, fraction edge length, unit fraction
Suggested Instructional Strategies:
•
Investigation 1 and 2 from Filling and Wrapping
using the appropriate ACE questions and Mathematical
Reflections
•
Review of 5.MD.5:
•
Explore with cubes and arrange them into
layers to create rectangular prisms. Record the
dimensions of the first/base layer, add a second layer,
determine new dimensions, and look for patterns to
predict what will happen when a third layer is added.
Add the third layer and determine if your prediction was
correct. Make connections to formulas.
•
Hold up a cube and explain that the edge
measures one unit and that is the standard for finding
the volume of a solid figure. The volume of a solid figure
is the number of same sized cubes filling the space so
that there are no gaps and overlaps.
•
Make nets of rectangular prisms on graph
paper. Fold and determine volume.
•
Define one cubic unit in order to see fractional
parts. See resource.
•
Apply to formula using fractional edge lengths.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Resources: •
Filling and Wrapping from the 7 grade Connected
Mathematics Project 2 (PearsonSuccess.Net)
Filling and Wrapping
2.3 – volume with whole number dimensions
ACE questions with decimal/fractional dimensions 6(a-b), 13-14, 17
NCTM Activity- Packing the Packages 2002
http://lesage.blogs.uoit.ca/wp-uploads/2010/08/Cerealboxes_Investigation_Junior_-AIMS.pdf (surface area
and volume)
MARS TASKS: Being Building Blocks
http://www.illustrativemathematics.org/illustrations/545
- volume activity (also could be used with surface area)
http://deanadventures.com/htms/geometry.html link to
UEN geometry
Course Name: Math 6 Unit # 5
Unit Title:
Sample Assessment Tasks
Skill-based task
Covering and Surrounding
Problem Task
Build 3 rectangular prisms with the volume of 36 cubic
units.
At least one of the side lengths of each prism is a
fractional unit. What are the dimensions of each of the
rectangular prisms you built?
A flower box is 3 ft. long 2¾ ft. wide and ½ ft. deep. How many
cubic feet of dirt can it hold?
A gallon of water uses 231 cubic in. of space. How many
gallons of water are needed to fill this aquarium?
Draw a diagram to match:
l = 12½ in.
w = 8¼ in.
h = 12½ in
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
CORE CONTENT
Cluster Title:
Standard: 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the
length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the
context of solving real-world and mathematical problems.
Concepts and Skills to Master:
•
Draw polygons in the coordinate plane given coordinates for the vertices
•
Use coordinates to find the length of a side joining points with the same first coordinate or the same second
coordinate.
•
Apply these techniques in the context of solving real-world and mathematical problems
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Graph in all 4 quadrants of the coordinate plane; understand that a line segment from one coordinate pair to another
represents a distance; understand that if two coordinates have the same x- or y-value that they are on the same line; the
distance between two points on a coordinate plane is an absolute value; the units on a coordinate plane define the unit of
distance measure; a coordinate plane can be used to represent real-world contexts (e.g. streets); find the distance
between two points; find the length of line segments BA and BC; plot a polygon in the Cartesian coordinate plane with
given coordinates.
Academic Vocabulary
Coordinate plane, ordered pair, quadrant I, quadrant II, quadrant III, quadrant IV, origin, x-axis, y-axis, horizontal, vertical,
vertex, vertices, coordinates, x-coordinate, y-coordinate, polygons, length, Cartesian coordinate plane, absolute value
Suggested Instructional Strategies:
Have the students explore how to determine the side
length of a polygon. Discuss conjectures and test the
findings. Solidify the two different methods:
1)
Students can find the side length of a
polygon by counting.
2)
Students can find the side length of a
polygon by subtracting x coordinates or y
coordinates.
Archeologists use the coordinate plane to note the
locations of artifacts found. They select a point at the
dig site to be the origin. From that point they mark
the grid with 1-meter squares. When an artifact is
found in a square the ordered pair is recorded. Ask
the students to use coordinate grids to locate the
following objects: a piece of pottery at (5,4), a maize
grinding stone at (5,-5), a doll at (-2,-5), a cup at (2,4). Have the students give you the following
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Resources:
CC Inv. 3
http://deanadventures.com/htms/geometry.html link to
UEN geometry
http://mathlearnnc.sharpschool.com/UserFiles/Servers/Ser
ver_4507209/File/Instructional%20Resources/I-G6.pdf
link to ncscos goals 3.01 and 3.04 address this
http://learnzillion.com/lessonsets/243 videos on plotting
on the plane
http://illuminations.nctm.org/LessonDetail.aspx?id=L280
LearnNC: Archaeology
http://www.learnnc.org/lp/pages/1005
Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
distances:
From the cup to the piece of pottery
From the doll to the cup
Etc.
Sample Assessment Tasks
Skill-based task
Problem Based Task
Given the coordinates A (2,5),B (-4,5), C (4,1), and D (2,1)
Jose says that the distance between A and
D can be found by subtracting 2 from 5.
Prove or disprove. Explain your answer
with words, pictures, and equations
Plot ordered pairs to form a polygon. Determine one of the side lengths
Ex. Plot the ordered pairs: A(2,6) B (2,2) C (-4,4). Find the side length AB
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
CORE CONTENT
Cluster Title:
Standard: 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to
find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical
problems.
Concepts and Skills to Master:
•
The surfaces of three-dimensional shapes are composed of two dimensional faces.
•
Understanding surface area using nets can be used in real-world contexts (e.g., wrapping presents, packaging).
•
The area of two-dimensional shapes can be used to find the surface area of the three-dimensional shape.
•
Transitioning from three dimensions to two dimensions requires spatial reasoning.
•
Use a net to represent a 3-D figure.
•
Use a net to find the surface area of a 3-D figure made up of rectangles and triangles (polyhedron).
•
Compose and decompose a polyhedron using rectangles and triangles .
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Area is covering the surface of a two-dimensional shape; Area is measured with square units; Find the area of a rectangle
and triangle; Polygons can be decomposed.
Academic Vocabulary
Net, three-dimensional figures, surface area, vertices, face, edge, length, width, base, height, polyhedron, prism, pyramid
Suggested Instructional Strategies:
Resources: CC Inv. 4
Make polyhedrons from given nets. Recognize the
rectangles and triangles that compose the polyhedron.
Find the area of each polygon and add together to find
the total surface area of the polyhedron.
Have nets on graph paper to aid in finding the area of
polyhedrons.
Mathematical Task:
Use the real-world problem at Figure This: Real World
Application
http://www.figurethis.org/challenges/c62/challenge.htm
Give the students three small ice blocks and an ice block
the equivalent to the size of all three smaller blocks.
Have the students find the surface area of the block and
each cube. Have them create a hypothesis about which
will melt faster–the intact ice block or the cubes. Have
the students perform the experiment by observing and
measuring the time it takes for the block to melt and for
all three cubes to melt. How does the melting time
compare to the surface area exposed? Generalize the
relationship
Filling and Wrapping 7th grade unit (Pearson Successnet)
1.1- Unit cubes and nets
1.2- Making rectangular boxes
1.3- Testing nets
2.1 surface area
2.2- least surface area
3.1- nets of prisms
3.3- surface area of prisms
Extension: Can you create a formula to show the
relationship?
http://illuminations.nctm.org/LessonDetail.aspx?id=L797
http://mste.illinois.edu/users/carvell/3dbox/default.html
http://studyjams.scholastic.com/studyjams/jams/math/measure
ment/volume.htm
http://deanadventures.com/htms/geometry.html link to UEN
geometry
Interactive Nets:
3D Nets and Surface Area
http://www.learner.org/interactives/geometry/area_surface.htm
l
Graph Paper Nets:
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
Course Name: Math 6 Unit # 5
Unit Title:
Covering and Surrounding
http://www.shodor.org/interactivate/activities/SurfaceAreaAndV
olume/
Surface Area: Painting a Barn
http://illustrativemathematics.org/illustrations/135
Sample Assessment Tasks
Skill-based task Find the surface area
Problem Task
Build 3 rectangular prisms with the surface area of 36 cubic units.
At least one of the side lengths of each prism is a fractional unit.
What are the dimensions of each of the rectangular prisms you
built?
What are the volumes of these boxes?
Belinda had two boxes to wrap for a birthday party. Box A has a
length of 12 in, width of 8 in, and height of 6 in. Box B has a length
of 11 in, width of 9 in, and height of 7 in. Which box will require the
least amount of wrapping paper
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.