Research—Best Practices
Putting Research
into Practice
From the National Research Council Report
on Mathematics Learning in Early Childhood
•Shapes with right angles: rectangles, squares, right triangles,
including isosceles right triangles
•Shapes based on equilateral triangles: equilateral triangles, and
rhombi, trapezoids, hexagons
•Shapes based on parallelograms: non-rectangular parallelograms
and the acute and obtuse triangles made by their diagonals
Together these experiences include the most important shapes and
properties in geometry. Properties of shapes and reasoning about
these properties play central roles in the study of geometry. Some
questions about properties that are important for students to
investigate include:
•Is the shape closed or open? convex or concave?
•Are the lines straight or curved?
•What is the number of sides? of angles?
•Do the sides have equal lengths? all or some?
•Is an angle a right angle, or larger or smaller than a right angle?
UNIT 6 | Overview | 651O
Research
Experiences with three kinds of geometric shapes are important as
bases for the formal treatment of shapes and their properties in
Grades 3 through 5.
UNIT 6
Dr. Karen C. Fuson,
Math Expressions Author
The National Research Council Report Mathematics learning in early
childhood: Paths toward excellence and equity shows the importance
of students analyzing, discussing, and composing or decomposing
right-angled shapes. Right-angled shapes are the foundation of
many important mathematical concepts: the simplest quadrilaterals
(rectangles and squares), the right triangles that compose these
shapes, and the unit squares that make up area. Equilateral triangles
are also simple shapes, but the shapes they compose (a rhombus,
a trapezoid, and a hexagon) create fewer rich mathematical ideas.
Parallelograms are the beginning of student experiences with
parallel lines.
Contents
Planning
Research & Math Background
From Recent Research
A recent National Assessment of Educational Progress report
showed that many third graders did not understand perimeter and
area, could not differentiate between the measures, and could not
compute the measures. Because poor performance in upper grades
resulted from the lack of understanding in lower grades, there is
a need to improve understanding of these concepts in the lower
grades. Conceptual development of properties of perimeter and area,
use of appropriate units, and counting strategies should occur in early
grades and be followed by formulas only in later grades.
Other Useful References: Geometry and Measurement
Batista, Michael T. “Learning Geometry in
a Dynamic Computer Environment.”
Teaching Children Mathematics. 8.6
(Feb. 2002): p. 333.
Nitabach, E. and R. Lehrer. “Developing
Spatial Sense through Area
Measurement.” Teaching Children
Mathematics. (Apr. 1996): pp. 473–476.
Fuson, Karen C., Clements, D.H., and
Beckman, S. Focus in Grade 2:
Teaching with Curriculum Focal
Points. Reston, VA: National Council
of Teachers of Mathematics, 2011.
Whitin, Phyllis. “Promoting ProblemPosing Explorations.” Teaching
Children Mathematics. 11.4 (Nov.
2004): p. 180.
National Council of Teachers of
Mathematics. Principles and Standards
for School Mathematics (Number and
Operations Standard for Grades 3–5).
Reston: NCTM, 2000. pp. 97, 103–105.
651P | UNIT 6 | Overview
Wilson, P.S. and R.E. Rowland. “Teaching
Measurement.” Research Ideas for
the Classroom: Early Childhood
Mathematics. Ed. R.J. Jensen. Old
Tappan, NJ: Macmillan, 1993.
171–194.
Getting Ready To Teach Unit 6
Using the Common Core Standards
for Mathematical Practice
The Common Core State Standards for Mathematical Content indicate
what concepts, skills, and problem solving students should learn. The
Common Core State Standards for Mathematical Practice indicate how
students should demonstrate understanding. These Mathematical
Practices are embedded directly into the Student and Teacher Editions
for each unit in Math Expressions. As you use the teaching suggestions,
you will automatically implement a teaching style that encourages
students to demonstrate a thorough understanding of concepts, skills,
and problems. In this program, Math Talk suggestions are a vehicle
used to encourage discussion that supports all eight Mathematical
Practices. See examples in Mathematical Practice 6.
UNIT 6
Mathematical Practice 1
Make sense of problems and persevere in solving them.
Teacher Edition: Examples from Unit 6
MP.1 Make Sense of Problems Ask
students to draw and label a rectangle
on their MathBoards as you draw one
on the Class MathBoard. Explain that the
perimeter of the rectangle is 20 units, one
side length is 4 units, and the other side
length is unknown.
?
4 units
Then use Solve and Discuss
to solve all the problems in the lesson.
Point out that deciding first whether
the answer to a problem is an area,
a perimeter, or a side length helps in
planning how to solve the problem.
M AT H TA L K
Perimeter = 20 units
•How can you use what you know about
perimeter to find the unknown side
length? 4 + 4 + ? + ? = 20, so subtract 8
from 20 and divide that number by 2 to
find the unknown side length, 6 units.
Lesson  6
MP.1, MP.4 Make Sense of Problems/
Model with MathematicsDraw a
Diagram Discuss the first problem on
Student Book page 331 with the class.
Ask students whether they can figure out
the meaning of dimensions from reading
the problem. If necessary, explain that
the word dimensions means the same
as “sizes” and that it is often used with
measurements.
Lesson  9
ACTIVITY 1
ACTIVITY 2
Mathematical Practice 1 is integrated into Unit 6 in the following way:
Make Sense of Problems
UNIT 6 | Overview | 651Q
MATH BACKG ROUND
Students analyze and make conjectures about how to solve a problem.
They plan, monitor, and check their solutions. They determine if their
answers are reasonable and can justify their reasoning.
Contents
Planning
Research & Math Background
Mathematical Practice 2
Reason abstractly and quantitatively.
Students make sense of quantities and their relationships in problem
situations. They can connect diagrams and equations for a given
situation. Quantitative reasoning entails attending to the meaning
of quantities. In this unit, this involves exploring characteristics of
polygons, using different methods to find the area of figures that are
rectangles and figures that are not rectangles, and examining area and
perimeter to see how they are different.
Teacher Edition: Examples from Unit 6
MP.2 Reason Abstractly and
Quantitatively Connect Diagrams and
Equations Draw this figure on the Class
MathBoard and ask students to copy it
on their MathBoards or Centimeter Dot
Paper (TRB M18).
MP.2 Reason Abstractly and
Quantitatively Ask for Ideas Use these
questions to consolidate student thinking
about the perimeter and area concepts
they have been studying.
•How are perimeter and area different?
Perimeter is the distance around a
figure; area is the number of square
units inside a figure.
•How is perimeter related to the
side lengths of a figure? To find the
perimeter, you add the side lengths.
Ask for Ideas Use these questions to elicit
ways to find the area of the figure.
•Since this is not a rectangle, you cannot
find the area by multiplying the side
lengths. How might you find the area?
Draw unit squares and count them,
count the number of unit squares in
each row and add them.
•How is area related to the side lengths
of a rectangle? To find the area of a
rectangle, you multiply the side lengths.
•What are some ways to find the area of
figures that are not rectangles? You can
decompose the figures into rectangles
or you can count the unit squares inside
the figure.
Lesson  9
•You know how to find the area of
a rectangle. Does anyone see any
rectangles in this figure? Allow students
to share their observations.
Lesson  8
ACTIVITY 1
Mathematical Practice 2 is integrated into Unit 6 in the following ways:
Reason Abstractly and Quantitatively
Reason Abstractly
651R | UNIT 6 | Overview
Reason Quantitatively
Connect Diagrams and Equations
ACTIVITY 1
Mathematical Practice 3
Construct viable arguments and critique the reasoning of others.
Students use stated assumptions, definitions, and previously
established results in constructing arguments. They are able to analyze
situations and can recognize and use counterexamples. They justify
their conclusions, communicate them to others, and respond to the
arguments of others.
Students are also able to distinguish correct logic or reasoning from
that which is flawed, and—if there is a flaw in an argument—explain
what it is. Students can listen to or read the arguments of others,
decide whether they make sense, and ask useful questions to clarify or
improve the arguments.
is a conversation tool by which students formulate ideas and
analyze responses and engage in discourse. See also MP.6 Attend to
Precision.
MATH TAL K
UNIT 6
Teacher Edition: Examples from Unit 6
•a triangle with a right angle and three
equal sides
• a triangle with three equal sides and an
angle larger than a right angle
Encourage students to share their
observations. For example, they might
mention that a triangle with three angles
smaller than a right angle cannot have
a right angle or an angle larger than a
right angle.
Lesson  1
What’s the Error? WHOLE CLASS
MP.3, MP.6 Construct Viable
Arguments/Critique Reasoning of
Others Puzzled Penguin Give students
time to read the letter from Puzzled
Penguin. Then ask for volunteers to tell
how they would respond. Ask a volunteer
to come to the board and write the
correct solution to the problem.
Lesson  4
ACTIVITY 2
ACTIVITY 2
Mathematical Practice 3 is integrated into Unit 6 in the following ways:
Construct a Viable Argument
Critique the Reasoning of Others
Puzzled Penguin
Compare Representations
Justify Conclusions
UNIT 6 | Overview | 651S
MATH BACKG ROUND
MP.3 Construct a Viable Argument Justify Conclusions Challenge students to
try to draw these triangles and explain
why they cannot exist.
Contents
Research & Math Background
Planning
Mathematical Practice 4
Model with mathematics.
Students can apply the mathematics they know to solve problems that
arise in everyday life. This might be as simple as writing an equation
to solve a problem. Students might draw diagrams to lead them to
a solution for a problem. Students apply what they know and are
comfortable making assumptions and approximations to simplify a
complicated situation. They are able to identify important quantities in
a practical situation and represent their relationships using such tools
as diagrams, tables, graphs, and formulas.
Teacher Edition: Examples from Unit 6
MP.1, MP.4 Make Sense of Problems/
Model with MathematicsDraw a
Diagram Discuss the first problem on
Student Book page 331 with the class.
Ask students whether they can figure out
the meaning of dimensions from reading
the problem. If necessary, explain that
the word dimensions means the same
as “sizes” and that it is often used with
measurements.
MATH TA L K
Then use Solve and Discuss
to solve all the problems in the lesson.
Point out that deciding first whether
the answer to a problem is an area,
a perimeter, or a side length helps in
planning how to solve the problem.
MP.1, MP.4 Make Sense of Problems/
Model with Mathematics Draw a
Diagram Have students describe the
strategies that they used to draw the
gardens in Exercises 7 and 8 on Student
Book page 342. Students’ explanations for
drawing shapes with the same perimeter
should include putting the lengths of
adjacent sides in a table. The lengths of
two adjacent sides would be half the
perimeter, or 14 feet. There are six pairs
of whole numbers whose sum is 14. For
drawing shapes with the same area,
the lengths of adjacent sides are put in
a table. There are four pairs of whole
numbers whose product is 42.
For Problem 2, students may need to
review the properties of a hexagon.
Lesson  9
Lesson  11
ACTIVITY 1
Mathematical Practice 4 is integrated into Unit 6 in the following ways:
Model with Mathematics
651T | UNIT 6 | Overview
Write an Equation
Draw a Diagram
ACTIVITY 1
Mathematical Practice 5
Use appropriate tools strategically.
Students consider the available tools and models when solving
mathematical problems. Students make sound decisions about when
each of these tools might be helpful. These tools might include paper
and pencil, a straightedge, a ruler, or the MathBoard. They recognize
both the insight to be gained from using the tool and the tool’s
limitations. When making mathematical models, they are able to
identify quantities in a practical situation and represent relationships
using modeling tools such as diagrams, grid paper, tables, graphs, and
equations.
UNIT 6
Modeling numbers in problems and in computations is a central focus
in Math Expressions lessons. Students learn and develop models to
solve numerical problems and to model problem situations. Students
continually use both kinds of modeling throughout the program.
Teacher Edition: Examples from Unit 6
• What can you do to show how you
broke the rectangle into two parts?
You can draw a line that shows the two
parts and change the side length labels.
3 units
+
4 units
3 units
Ask students to draw two rectangles on
their MathBoards that show two different
ways to break the rectangle into two
parts. Circulate and check drawings as
students are working.
Lesson  6
MP.5 Use Appropriate Tools Unit
Squares Explain that students will again
use unit squares to tile a rectangle just as
they did with the rectangle they covered
with self-stick notes. This time they will
use a unit square that has an area of
1 square inch. Point out that they can
use sq in. as an abbreviation for
square inch.
Ask students to complete Exercises 13–18
independently but provide guidance as
needed. You might suggest that students
draw outlines of the 1-inch unit squares
as a record of their work. When students
complete the page, let volunteers present
their results.
Lesson  5
ACTIVITY 2
ACTIVITY 1
Mathematical Practice 5 is integrated into Unit 6 in the following ways:
Use Appropriate Tools
Class MathBoard
Unit Squares
Category Diagram
Paper Model
Concrete Model
Tangram Pieces
Straightedge or Ruler
Cutouts
UNIT 6 | Overview | 651U
MATH BACKG ROUND
MP.5 Use Appropriate Tools MathBoard Ask students to draw this rectangle on
their MathBoards.
Contents
Research & Math Background
Planning
Mathematical Practice 6
Attend to precision.
Students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning.
They state the meaning of the symbols they choose. They are careful
about specifying units of measure to clarify the correspondence with
quantities in a problem. They calculate accurately and efficiently,
expressing numerical answers with a degree of precision appropriate
for the problem context. Students give carefully formulated
explanations to each other.
Teacher Edition: Examples from Unit 6
MP.6 Attend to Precision Describe
Methods Ask groups to think about
different sorting rules they might use to
sort the figures they cut out.
Read aloud the directions on
Student Book page 311. Have students
work independently to complete the
activity. Review the results, asking students
to provide explanations for their answers.
MATH TALK
•How can you sort the quadrilaterals
using the lengths of their sides? equal
lengths and different lengths
•Is the figure in Exercise 1 a
quadrilateral? yes
•How do you know? It has four sides.
•Is it a parallelogram? yes
•How do you know? Both pairs of
opposite sides are parallel.
•How else can you sort the quadrilaterals
using descriptions of their sides? parallel
sides, sides that meet at right angles
Lesson  4
ACTIVITY 3
Lesson  4
MP.6 Attend to Precision Explain
a Representation Draw the six
quadrilaterals below on the board. Ask
a volunteer to circle the figures that the
class names for each category.
L
K
K
L
M
N
O
N
M
in ACTION
O
ince the dots on your MathBoard are
S
1 centimeter apart, how can you find
the actual perimeter of the rectangle
you drew?
Zena: I put new labels on all four sides of the
rectangle
and then I added the lengths shown on
P
the four labels. So I added 5 cm + 3 cm + 5 cm
+ 3 cm. That equals 16 cm.
Jason: After I fixed the labels, I added 5 cm +
3 cm to get 8 cm and then I multiplied 8 cm by 2
to get 16 cm. That’s because the opposite sides
have equal lengths.
P
Lesson  2
MATH TALK
Lesson  5
ACTIVITY 3
Mathematical Practice 6 is integrated into Unit 6 in the following ways:
Attend to Precision
Describe a Method
Verify Solutions
651V | UNIT 6 | Overview
ACTIVITY 1
Explain an Example
Explain a Solution
Explain a Representation
Puzzled Penguin
Explain Methods
ACTIVITY 1
Mathematical Practice 7
Look for structure.
Students analyze problems to discern a pattern or structure. They
draw conclusions about the structure of the relationships they have
identified.
Teacher Edition: Examples from Unit 6
MP.7 Look for Structure Identify
Relationships Ask students to place their
seven tangram pieces on their desks.
Remind them that they compared the side
lengths in the first activity. Say:
Lesson  10
Lesson  11
ACTIVITY 2
ACTIVITY 2
Mathematical Practice 7 is integrated into Unit 6 in the following ways:
Look for Structure
Identify Relationships
Use Structure
UNIT 6 | Overview | 651W
MATH BACKG ROUND
•Suppose the area of the blue square
is 1 square inch. Can you exactly cover
this blue square with any of the other
pieces? The two small triangles fit
together to cover the square. Since
the area of the square is 1 square inch,
what do you think the area of 1 green
triangle is? one-half square inch So, the
two green pieces each have an area of
one-half square inch.
After students identify the quadrilateral
that does not belong in the same group as
the others, ask them to justify their answer
by giving the sorting rule they used.
UNIT 6
•You have used your tangram pieces to
solve some tangram puzzles. As you
worked, what did you notice about how
the areas of these pieces are related?
Accept any reasonable observations that
students make.
MP.7 Look for Structure Identify
Relationships Draw the quadrilaterals
below on the board. Ask students to find
the shape that does not belong in the
same group as the others.
Contents
Research & Math Background
Planning
Mathematical Practice 8
Look for and express regularity in repeated reasoning.
Students use repeated reasoning as they analyze patterns, relationships,
and calculations to generalize methods, rules, and shortcuts. As
they work to solve a problem, students maintain oversight of the
process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
Teacher Edition: Examples from Unit 6
MP.8 Use Repeated Reasoning Conclude Ask students to examine all of the rectangles
and their completed table. They should see
that as the lengths of the adjacent sides
become closer to the same length, the area
increases. Next, have students complete
Exercise 4. Ask several volunteers to share
their rectangles with the rest of the class.
Ask students to complete Exercises 5 and 6.
Then discuss their answers. They should
recognize again that the rectangle with the
least area is long and skinny. The rectangle
with the greatest area has side lengths close
to the same length.
Lesson  7
MP.8 Use Repeated Reasoning Conclude Ask volunteers to demonstrate
each of the three ways to use the blue
and green tangram pieces to find the
area of one of the purple triangles. Then
guide students to discover that the area
of a purple triangle is 2 square inches
by adding the areas of the three pieces:
1 square inch plus 2 half-square inches
equals 2 square inches.
Lesson  10
ACTIVITY 2
ACTIVITY 1
Mathematical Practice 8 is integrated into Unit 6 in the following ways:
Use Repeated Reasoning
Conclude
Student EDITION: Lesson 11 pages 341–342
Focus on Mathematical Practices
Unit 6 includes a special lesson
that involves solving real world
problems and incorporates all
eight Mathematical Practices. In
this lesson, students use what they
know about finding perimeter and
area to design a garden.
6-11
Name
Date
6-11
Class Activity
Use the dot paper below to draw a different
garden that has the same perimeter as Yoakim’s
combined garden. Beside it, draw a different
garden that has the same area as Yoakim’s garden.
Possible drawings are shown.
8 feet
B
A
3 feet
1 ft
2 feet
3 feet
6 feet
Look at the drawing of Yoakim’s garden.
It is divided into two quadrilaterals.
10 feet
7. What is the area of your garden
that has the same perimeter as
Yoakim’s garden?
© Houghton Mifflin Harcourt Publishing Company
3. Will Yoakim need more fencing to enclose the two
parts of his garden separately or to enclose the
combined garden? He will need more fencing to
8. What is the perimeter of your
garden that has the same area
as Yoakim’s garden?
enclose the two parts of his garden separately.
4. What is the area of part A?
36 square feet
What is the area of part B?
6 square feet
Possible answer: 26 feet
42 square feet
6. How does the total area of the two parts of the garden
compare to the area of the combined garden?
The area for corn is 12 square feet.
The area for beans is 25 square feet.
The area for tomatoes is 20 square feet. Possible drawings are shown.
The areas are equal.
3_MNLESE824536_U06L11.indd 341
beans
9. Use the centimeter dot paper at the
right to draw separate areas within a
garden where you would plant corn,
beans, and tomatoes.
5. What is the area of the combined garden?
UNIT 6 LESSON 11
corn
Possible answer: 40 square feet
tomatoes
24 feet
What is the perimeter of part B?
Focus on Mathematical Practices
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©PHOTO 24/Getty Images
1. What is the perimeter of part A?
2. What is the perimeter of the combined
28 feet
garden?
651X | UNIT 6 | Overview
Date
► Design a Garden
► Math and Gardening
6 feet
Name
Class Activity
341
3/27/12 3:12 AM
342
UNIT 6 LESSON 11
3_MNLESE824536_U06L11.indd 342
Focus on Mathematical Practices
28/03/12 2:42 AM
Getting Ready to Teach Unit 6
Learning Path in the Common Core Standards In
this unit, students study the attributes of triangles, quadrilaterals,
and other polygons. They find perimeter and area of various shapes
and delve deeply into concepts of area of rectangular figures. They
explore the relationship between perimeter and area by investigating
rectangles with the same perimeter and different areas and
rectangles with the same area and different perimeters.
Math Expressions
VOCABULARY
As you teach this unit, emphasize
understanding of these terms.
•triangle
•quadrilateral
•area
See the Teacher Glossary.
Visual models and real world situations are used throughout the unit
to illustrate important fraction concepts.
Help Students Avoid Common Errors
Math Expressions gives students opportunities to analyze and correct
errors, explaining why the reasoning was flawed.
UNIT 6
In this unit, we use Puzzled Penguin to show typical errors that
students make. Students enjoy teaching Puzzled Penguin the correct
way, why this way is correct, and why the error is wrong. The
common errors are presented as letters from Puzzled Penguin
to the students:
MATH BACKG ROUND
→ Lesson 4: drawing a hexagon to represent a
quadrilateral with parallel sides that is not a
rectangle, square, or rhombus
→ Lesson 8: decomposing a figure correctly into three
rectangles, but only labeling and finding the areas
of two of the rectangles
In addition to Puzzled Penguin, there are other
suggestions listed in the Teacher Edition to help you watch
for situations that may lead to common errors. As a part of
the Unit Test Teacher Edition pages, you will find a common
error and prescription listed for each test item.
UNIT 6 | Overview | 651Y
Contents
Research & Math Background
Planning
Lessons
Geometric Concepts
1
2
3
4
The Grade 3 Common Core State Standards ask that students reason
with shapes and their attributes in two ways.
1.Understand that shapes in different categories (e.g.,
rhombuses, rectangles, and others) may share attributes
(e.g., having four sides), and that the shared attributes
can define a larger category (e.g., quadrilaterals).
Recognize rhombuses, rectangles, and squares as examples
of quadrilaterals, and draw examples of quadrilaterals
that do not belong to any of these subcategories.
2.Partition shapes into parts with equal areas. Express the
area of each part as a unit fraction of the whole.
In studying shape and structure across the grades, students
progressively become able to see and reason about properties of
shape and to organize shapes into categories based on properties.
Grade 3 students reason about subcategories of shapes (for example,
rhombuses, rectangles, and others) and recognize that these shapes
may share attributes that define a larger category (for example,
quadrilaterals).
651Z | UNIT 6 | Overview
Work with Triangles
and Polygons
Lesson
1
Lesson 1 establishes a foundation for the rest of the work in this unit
by providing the vocabulary needed to discuss geometric concepts,
explaining ways to classify and name various polygons, and showing
how some figures can be composed of or decomposed into triangles.
You may want to use more than one class period for this lesson
depending on how much is review for your students.
UNIT 6
Classify Angles Students learn that a right angle forms a square
corner and they compare other angles to right angles as angles that
are larger than or smaller than a right angle. As students draw angles
that are the same as or larger than or smaller than a right angle, they
are building understanding of angle relationships that will help them
use protractors successfully in later years.
MATH BACKG ROUND
Classify Triangles Building on this understanding, students learn
to classify triangles as triangles with a right angle (right triangles),
with three angles smaller than a right angle (acute triangles), and
with an angle larger than a right angle (obtuse triangles). Students
also learn to classify triangles by the lengths of the triangle sides: 3
sides of equal length (equilateral or isosceles), 2 sides of equal length
(isosceles), and no sides of equal length (scalene). Students describe
triangles both by angles and sides but do not use the technical terms.
They understand that all closed shapes with three sides are triangles.
N
M
O
O
M
N
10.Triangle
has 11.Triangle
has 12.Triangle
has
1 angle larger than
1 right angle and
3 angles smaller
2
a right angle and
has
than a right angle
sides of
2
0
has
and has
equal length.
sides of
equal length.
sides of equal
length.
UNIT 6 | Overview | 651AA
Contents
Planning
Research & Math Background
Quadrilaterals After working with triangles, students explore
quadrilaterals by composing quadrilaterals from two congruent
triangles. They use each type of triangle based on angle measure.
They discover that two triangles usually can be put together to form
different quadrilaterals, but that in the case of two right triangles,
they can also form larger triangles. This hands-on activity helps
students develop spatial sense as they see how quadrilaterals can be
composed from triangles.
Polygons Students learn a definition of a polygon and that polygons
may be concave or convex. They see how polygons are named for the
number of sides they have and learn the names for several common
polygons: triangle, quadrilateral, pentagon, hexagon, octagon, and
decagon.
Cutting up polygons to see how they can be composed of triangles
and using triangles to form polygons provides students with another
hands-on experience that develops spatial sense. Such activities
establish a conceptual base for area that helps students understand
area formulas.
A
A
A
A
A
A
hexagon
651BB | UNIT 6 | Overview
Contents
Research & Math Background
Planning
Mathematical Practice 8
Look for and express regularity in repeated reasoning.
Students use repeated reasoning as they analyze patterns, relationships,
and calculations to generalize methods, rules, and shortcuts. As
they work to solve a problem, students maintain oversight of the
process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
Teacher Edition: Examples from Unit 6
MP.8 Use Repeated Reasoning Conclude Ask students to examine all of the rectangles
and their completed table. They should see
that as the lengths of the adjacent sides
become closer to the same length, the area
increases. Next, have students complete
Exercise 4. Ask several volunteers to share
their rectangles with the rest of the class.
Ask students to complete Exercises 5 and 6.
Then discuss their answers. They should
recognize again that the rectangle with the
least area is long and skinny. The rectangle
with the greatest area has side lengths close
to the same length.
Lesson  7
MP.8 Use Repeated Reasoning Conclude Ask volunteers to demonstrate
each of the three ways to use the blue
and green tangram pieces to find the
area of one of the purple triangles. Then
guide students to discover that the area
of a purple triangle is 2 square inches
by adding the areas of the three pieces:
1 square inch plus 2 half-square inches
equals 2 square inches.
Lesson  10
ACTIVITY 2
ACTIVITY 1
Mathematical Practice 8 is integrated into Unit 6 in the following ways:
Use Repeated Reasoning
Conclude
Student EDITION: Lesson 11 pages 341–342
Focus on Mathematical Practices
Unit 6 includes a special lesson
that involves solving real world
problems and incorporates all
eight Mathematical Practices. In
this lesson, students use what they
know about finding perimeter and
area to design a garden.
6-11
Name
Date
6-11
Class Activity
Use the dot paper below to draw a different
garden that has the same perimeter as Yoakim’s
combined garden. Beside it, draw a different
garden that has the same area as Yoakim’s garden.
Possible drawings are shown.
8 feet
B
A
3 feet
1 ft
2 feet
3 feet
6 feet
Look at the drawing of Yoakim’s garden.
It is divided into two quadrilaterals.
10 feet
7. What is the area of your garden
that has the same perimeter as
Yoakim’s garden?
© Houghton Mifflin Harcourt Publishing Company
3. Will Yoakim need more fencing to enclose the two
parts of his garden separately or to enclose the
combined garden? He will need more fencing to
8. What is the perimeter of your
garden that has the same area
as Yoakim’s garden?
enclose the two parts of his garden separately.
4. What is the area of part A?
36 square feet
What is the area of part B?
6 square feet
Possible answer: 26 feet
42 square feet
6. How does the total area of the two parts of the garden
compare to the area of the combined garden?
The area for corn is 12 square feet.
The area for beans is 25 square feet.
The area for tomatoes is 20 square feet. Possible drawings are shown.
The areas are equal.
3_MNLESE824536_U06L11.indd 341
beans
9. Use the centimeter dot paper at the
right to draw separate areas within a
garden where you would plant corn,
beans, and tomatoes.
5. What is the area of the combined garden?
UNIT 6 LESSON 11
corn
Possible answer: 40 square feet
tomatoes
24 feet
What is the perimeter of part B?
Focus on Mathematical Practices
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©PHOTO 24/Getty Images
1. What is the perimeter of part A?
2. What is the perimeter of the combined
28 feet
garden?
651X | UNIT 6 | Overview
Date
► Design a Garden
► Math and Gardening
6 feet
Name
Class Activity
341
3/27/12 3:12 AM
342
UNIT 6 LESSON 11
3_MNLESE824536_U06L11.indd 342
Focus on Mathematical Practices
28/03/12 2:42 AM
Contents
Planning
Research & Math Background
Draw Quadrilaterals Drawing quadrilaterals helps students clarify
the relationships among the different quadrilaterals. This increases
their knowledge of geometric shapes and provides support for their
discussions of attributes and properties of geometric shapes.
Before drawing the shapes on the grids in the Student Activity Book,
students describe what they know about each shape. Drawing more
than one example of each kind of shape helps students recognize that
the name of a shape simply describes the attributes of that shape.
11.Draw three different quadrilaterals that are not squares, rectangles, or
rhombuses. Drawings will vary.
Students also draw quadrilaterals to match given
descriptions. This activity requires them to take into
consideration more than one attribute at a time.
Classify Quadrilaterals Properties of shapes and
reasoning about these properties are important
in the Common Core State Standards. Among the
properties that third-graders will use to analyze and
classify quadrilaterals are:
Quadrilaterals
Parallelograms
Rectangle
Trapezoids
Rhombuses
• sides of the same length or not (all or some)
• sides parallel or not (all or some)
• angles: right angle, smaller or larger than a
right angle
Students use a category diagram to help them
classify a set of quadrilaterals that they cut out from
the Student Activity Book.
651DD | UNIT 6 | Overview
Squares
Lessons
Perimeter and Area
5
6
7
9
The concepts of perimeter and area bring together the domains of
Geometry and Measurement and Data. The Geometric measurement
standards for Grade 3 focus on area, from asking students to
recognize area as an attribute of plane figures to guiding them
to develop ways to measure area—tile with unit squares, see that
the number of unit squares in an area is the same as the product
of the side lengths, and recognize that an area model represents
the Distributive Property of Multiplication over Addition. Although
the main thrust of these lessons is area, students also investigate
perimeter and they write equations for area and perimeter.
UNIT 6
Tiling with Unit Squares Rectangular area models were used in
Unit 3 to help students understand what happens when two factors
are multiplied. They saw then that the product is the number of
unit squares that fill a rectangle. This unit begins by re-establishing
this important concept. Students construct rectangles on their
MathBoards and determine the area by tiling and counting and
by combinations of multiplying and adding. Some students may
remember that they can multiply the side lengths.
MATH BACKG ROUND
5 units
3 units
1 column is 3 sq cm.
1 row is 5 sq cm.
1 column is 3 sq cm.
1 row is 5 sq cm.
5 columns are 5 × 3 sq cm,
3 rows are 3 × 5 sq cm,
5 columns are 5 × 3 sq cm,
3 rows are 3 × 5 sq cm,
or 15 sq cm.
or 15 sq cm.
or 15 sq cm.
or 15 sq cm.
UNIT 6 | Overview | 651EE
Contents
Planning
Research & Math Background
Perimeter and Area Although students
studied perimeter as an attribute of
plane figures in Grade 2, it is important
that they understand how these two
attributes are different. Exercises that
ask students to find both measures will
help you see whether students can
differentiate perimeter and area.
Emphasize that different units are used
to describe each attribute: units of
length for perimeter and square units
for area.
In Lesson 7, students relate perimeter and area as they draw on a dot
array all the possible rectangles with a given perimeter with whole
unit side lengths. Then they find the area of the rectangles. They
observe for a given perimeter, the longest, skinniest rectangle has the
least area and the most “square-like” rectangle has the greatest area.
Students also draw all the possible rectangles with a given area
with whole unit side lengths and find the perimeters. They observe
that for a given area, the longest, skinniest rectangle has the greatest
perimeter and the most ”square-like” rectangle has the
least perimeter.
651FF | UNIT 6 | Overview
Distributive Property Two different colors of self-stick notes are used
as improvised units to construct a rectangle.
As the class members explore how to find the area of the rectangle,
they discover the Distributive Property of Multiplication over
Addition, although they do not use that term. They write an equation
that represents the area of the rectangle.
3 units +
4 units
3 units
UNIT 6
3 × 7 = (3 × 3) + (3 × 4)
MATH BACKG ROUND
Students also describe and draw rectangles that represent an
equation, such as 6 × 9 = (6 × 5) + (6 × 4).
Unknown Side Lengths When students become more familiar with
finding the area of a rectangle, they are asked to use what they
know about the inverse relationship of addition and subtraction
to find the length of a side given the length of the other side and
the perimeter and what they know about the inverse relationship
of multiplication and division to find the length of a side given the
length of the other side and the area. Be sure always to refer to the
unknown side length, and not the missing side length. The side is
not missing because it is part of the rectangle; its length is what is
not known.
Find the unknown side length in each diagram.
14.
8 cm
?
?
Area = 72 sq cm
12 cm
15.
9 cm
Perimeter = 38 cm
7 cm
UNIT 6 | Overview | 651GG
Contents
Planning
Research & Math Background
Word Problems To help them appreciate that both perimeter and
area are practical real world mathematical applications, students have
plenty of practice solving word problems about realistic situations
involving perimeter and area, including unknown side lengths.
Since there are usually several ways to solve these problems, using
Solve and Discuss helps students share ideas and learn from each.
Encourage questioning and ask students to respond with clear
explanations, using drawings to help support points they make.
Solve. Circle whether you need to find a perimeter,
an area, or an unknown side length. Draw a
diagram to represent each situation.
1.The dimensions of a rectangular picture frame
are 9 inches and 6 inches. What is the area of a picture that would fit in the frame?
Perimeter
Area
Side Length
54 square inches
2.A garden has the shape of a regular hexagon.
Each side of the garden is 5 feet long. How much
fence is needed to go around the garden?
Perimeter
Area
Side Length
30 feet
3.The length of a water slide is 9 yards. The slide
is 2 yards wide. How much of the surface of the slide must be covered with water?
Perimeter
Area
Side Length
18 square yards
4.Mr. Schmidt is installing 32 cubbies in the hallway.
He puts 8 cubbies in each row. How many rows of cubbies can he make?
Perimeter
651HH | UNIT 6 | Overview
Area
4 rows
Side Length
Lesson
Rectilinear Figures
8
Additive Nature of Area Although students have worked with
finding the area of rectangles, not all figures for which people need
to find area are perfect rectangles. Because area is additive, it is
possible to decompose or subdivide many figures into rectangles
whose area can be determined and then add the areas of those
rectangles to find the area of the figure. Such figures are sometimes
called rectilinear figures, although the word rectilinear simply means
“being made of straight lines.”
UNIT 6
Students do some investigative work with L-shaped figures on their
MathBoards to help them see that area is additive. They see that
adding the areas of two parts of a figure gives the same area as
counting all the unit squares in the figure.
MATH BACKG ROUND
To find the area of such rectilinear figures, students are asked to look
for rectangles within the figure. For some figures, students will need
to find more than two rectangles. Watch for students who may form
overlapping rectangles, as they will find too large an area for the
figure and for students who do not completely cover the figures as
they will find too small an area for the figure. You might tell students
to watch for gaps and overlaps as they work.
Decompose each figure into rectangles.
Then find the area of the figure.
7.
8.
26 square units
9.
21 square units
32 square units
10.
26 square units
UNIT 6 | Overview | 651II
Contents
Planning
Research & Math Background
Lesson
Tangrams and Area
10
Tangrams are a popular puzzle that originated in China. The object
of the puzzle is to create various figures from the set of seven shapes
that make up the original tangram square. The Math Expressions
tangrams are built on a 1-square inch grid, so that the areas of all the
shapes can be expressed in square inches. Each purple triangle has
an area of 2 square inches. Each blue shape (the square, the triangle,
and the parallelogram) has an area of 1 square inch. Each green
triangle has an area of 2_1​ ​ square inch.
Build Figures After students cut out their tangram set, they use the
shapes to build figures similar to the traditional tangram puzzles to
become familiar with the shapes.
651JJ | UNIT 6 | Overview
Use Tangram Pieces to Measure Area Once students are familiar
with the shapes and can reproduce a given figure, they use two sets
of tangram shapes and extra blue squares and green triangles to
construct figures and calculate their areas. This helps to reinforce the
concept that area is additive. Using the green triangles provides a
preview of work in the next unit on fractions.
13.
UNIT 6
What is the area of the figure?
7 square inches
MATH BACKG ROUND
Focus on Mathematical
Practices
Lesson
11
Image Credits: ©PHOTO 24/Getty Images
The Standards for Mathematical Practice are included in every lesson
of this unit. However, the last lesson in every unit focuses on all eight
Mathematical Practices. In this lesson, students apply what they have
learned about finding area and using appropriate tools to what they
have learned about decomposing shapes into rectangles to solve
problems about gardening.
UNIT 6 | Overview | 651KK
Contents
Planning
Research & Math Background
NOTES:
_________________________________________________
_________________________________________________
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_________________________________________________
Building a Math Talk Community
M AT H TA L K
Frequent opportunities for students to explain
their mathematical thinking strengthen the learning
community of your classroom. As students actively question,
listen, and express ideas, they increase their mathematical
knowledge and take on more responsibility for learning. Use
the following types of questions as you build a Math Talk
community in your classroom.
Elicit student thinking
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
• So, what is this problem about?
• Tell us what you see.
• Tell us your thinking.
Support student thinking
• What did you mean when you said _______?
• What were you thinking when you decided to _______?
• Show us on your drawing what you mean.
• Use wait time: Take your time…. We’ll wait….
_________________________________________________
Extend student thinking
_________________________________________________
• Restate: So you’re saying that _______
_________________________________________________
• Now that you have solved the problem in that way, can you
think of another way to work on this problem?
_________________________________________________
_________________________________________________
• How is your way of solving like _______’s way?
• How is your way of solving different from _________’s way?
Increase participation of other students in the conversation
_________________________________________________
• Prompt students for further participation: Would someone
like to add on?
_________________________________________________
• Ask students to restate someone else’s reasoning: Can you
repeat what _______ just said in your own words?
_________________________________________________
• Ask students to apply their own reasoning to someone
else’s reasoning:
_________________________________________________
• Do you agree or disagree, and why?
• Did anyone think of this problem in a different way?
_________________________________________________
_________________________________________________
_________________________________________________
• Does anyone have the same answer, but got it in a
different way?
• Does anyone have a different answer? Will you explain
your solution to us?
Probe specific math topics:
_________________________________________________
_________________________________________________
_________________________________________________
651LL | UNIT 6 | Overview
• What would happen if _______?
• How can we check to be sure that this is a correct answer?
• Is that true for all cases?
• What pattern do you see here?