Download Ready Florida MAFS Instruction TRB grade 5

5
Mathematics Teacher Resource Book
2015
Florida MAFS
Table of Contents
Ready® Florida MAFS Program Overview
A6
Supporting the Implementation of the Florida MAFS
A7
Answering the Demands of the MAFS with ReadyA8
The Standards for Mathematical Practice
A9
Depth of Knowledge Level 3 Items in Ready Florida MAFSA10
Cognitive Rigor Matrix
A11
Using Ready Florida MAFSA12
Teaching with Ready Florida MAFS Instruction
A14
Content Emphasis in the MAFS
A16
Connecting with the Ready® Teacher Toolbox
A18
Using i-Ready® Diagnostic with Ready Florida MAFS
A20
Using Ready Practice and Problem SolvingA22
Features of Ready Florida MAFS Instruction
A24
Supporting Research
A40
Correlation Charts
Florida MAFS Coverage by Ready Instruction
Interim Assessment Correlations
A44
A48
Lesson Plans (with Answers)
MAFS Emphasis
Unit 1: Number and Operations in Base Ten
Lesson 1
Understand Place Value
1
5
M
13
M
21
M
31
M
41
M
49
M
MAFS Focus - 5.NBT.1.1 Embedded SMPs - 1–8
Lesson 2
Understand Powers of Ten
MAFS Focus - 5.NBT.1.2 Embedded SMPs - 2–8
Lesson 3
Read and Write Decimals
MAFS Focus - 5.NBT.1.3a Embedded SMPs - 2, 4–7
Lesson 4
Compare and Round Decimals
MAFS Focus - 5.NBT.1.3b, 5.NBT.1.4 Embedded SMPs - 1, 2, 4–7
Lesson 5
Multiply Whole Numbers
MAFS Focus - 5.NBT.2.5 Embedded SMPs - 1–8
Lesson 6
Divide Whole Numbers
MAFS Focus - 5.NBT.2.6 Embedded SMPs - 1–5, 7
M = Lessons that have a major emphasis in the MAFS
S/A = Lessons that have supporting/additional emphasis in the MAFS
MAFS Emphasis
Unit 1: Number and Operations in Base Ten (continued)
Lesson 7
Add and Subtract Decimals
57
M
67
M
77
M
MAFS Focus - 5.NBT.2.7 Embedded SMPs - 2–7
Lesson 8
Multiply Decimals
MAFS Focus - 5.NBT.2.7 Embedded SMPs - 1–5, 7
Lesson 9
Divide Decimals
MAFS Focus - 5.NBT.2.7 Embedded SMPs - 1–5, 7
Unit 1 Interim Assessment
89
Unit 2: Number and Operations—Fractions
92
Lesson 10 Add and Subtract Fractions
96
M
106
M
114
M
122
M
130
M
140
M
148
M
158
M
166
M
MAFS Focus - 5.NF.1.1 Embedded SMPs - 1, 2, 4, 7
Lesson 11 Add and Subtract Fractions in Word Problems
MAFS Focus - 5.NF.1.2 Embedded SMPs - 1–8
Lesson 12 Fractions as Division
MAFS Focus - 5.NF.2.3 Embedded SMPs - 1–5, 7
Lesson 13 Understand Products of Fractions
MAFS Focus - 5.NF.2.4a Embedded SMPs - 1–8
Lesson 14 Multiply Fractions Using an Area Model
MAFS Focus - 5.NF.2.4b Embedded SMPs - 1–8
Lesson 15 Understand Multiplication as Scaling
MAFS Focus - 5.NF.2.5a, 5.NF.2.5b Embedded SMPs - 1, 2, 4, 6, 7
Lesson 16 Multiply Fractions in Word Problems
MAFS Focus - 5.NF.2.6 Embedded SMPs - 1–8
Lesson 17 Understand Division with Unit Fractions
MAFS Focus - 5.NF.2.7a, 5.NF.2.7b Embedded SMPs - 1–8
Lesson 18 Divide Unit Fractions in Word Problems
MAFS Focus - 5.NF.2.7c Embedded SMPs - 1–8
Unit 2 Interim Assessment
M = Lessons that have a major emphasis in the MAFS
S/A = Lessons that have supporting/additional emphasis in the MAFS
177
MAFS Emphasis
Unit 3: Operations and Algebraic Thinking
Lesson 19 Evaluate and Write Expressions
180
183
S/A
193
S/A
MAFS Focus - 5.OA.1.1, 5.OA.1.2 Embedded SMPs - 1, 2, 5, 7, 8
Lesson 20 Analyze Patterns and Relationships
MAFS Focus - 5.OA.2.3 Embedded SMPs - 1, 2, 7, 8
Unit 3 Interim Assessment
Unit 4: Measurement and Data
Lesson 21 Convert Measurement Units
203
206
208
S/A
218
S/A
228
S/A
238
M
246
M
254
M
262
M
MAFS Focus - 5.MD.1.1 Embedded SMPs - 1, 2, 5–7
Lesson 22 Solve Word Problems Involving Conversions
MAFS Focus - 5.MD.1.1 Embedded SMPs - 1, 2, 5–7
Lesson 23 Make Line Plots and Interpret Data
MAFS Focus - 5.MD.2.2 Embedded SMPs - 1, 2, 4–7
Lesson 24 Understand Volume
MAFS Focus - 5.MD.3.3a, 5.MD.3.3b Embedded SMPs - 2, 4–7
Lesson 25 Find Volume Using Unit Cubes
MAFS Focus - 5.MD.3.4 Embedded SMPs - 2, 4–7
Lesson 26 Find Volume Using Formulas
MAFS Focus - 5.MD.3.5a, 5.MD.3.5b Embedded SMPs - 1–8
Lesson 27 Find Volume of Composite Figures
MAFS Focus - 5.MD.3.5c Embedded SMPs - 1–8
Unit 4 Interim Assessment
Unit 5: Geometry
Lesson 28 Understand the Coordinate Plane
271
274
277
S/A
285
S/A
295
S/A
303
S/A
MAFS Focus - 5.G.1.1 Embedded SMPs - 4, 6, 7
Lesson 29 Graph Points in the Coordinate Plane
MAFS Focus - 5.G.1.2 Embedded SMPs - 1, 2, 4–7
Lesson 30 Classify Two-Dimensional Figures
MAFS Focus - 5.G.2.4 Embedded SMPs - 2, 3, 5–7
Lesson 31 Understand Properties of Two-Dimensional Figures
MAFS Focus - 5.G.2.3 Embedded SMPs - 2, 6, 7
Unit 5 Interim Assessment
M = Lessons that have a major emphasis in the MAFS
S/A = Lessons that have supporting/additional emphasis in the MAFS
311
Develop Skills and Strategies
Lesson 30
(Student Book pages 268–275)
Classify Two-Dimensional Figures
Lesson Objectives
The Learning Progression
•Classify two-dimensional figures in a hierarchy
based on properties.
This standard emphasizes that two-dimensional figures
can be classified on many different levels based on
their properties.
•Use flow charts, Venn diagrams, and tree diagrams
to show the hierarchical relationship of twodimensional figures.
Prerequisite Skills
•Identify two-dimensional figures.
•Recognize parallel and perpendicular lines.
•Recognize right, acute, and obtuse angles.
Vocabulary
hierarchy: a ranking of categories based on properties
This standard could be used alongside 5.G.3 to
demonstrate that while the figures in a certain category
of two-dimensional figure have the same attributes,
these figures can often be classified more specifically
based on those attributes they do not share. Students
should understand that two figures could be called the
same thing based on one level of classification but be
called two different things at the next level of
classification.
There are two definitions for trapezoids that are used
in the United States. This lesson defines a trapezoid as
a quadrilateral with at least one pair of parallel sides.
In Grade 6, students will apply what they know about
two-dimensional figures to help them find the area of
polygons and graph polygons on the coordinate plane.
Teacher Toolbox
Teacher-Toolbox.com
Prerequisite
Skills
Ready Lessons
Tools for Instruction
✓
5.G.2.4
✓
✓✓
Interactive Tutorials
MAFS Focus
5.G.2.4 Classify two-dimensional figures in a hierarchy based on properties.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 3, 5, 6, 7 (see page A9 for full text)
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Part 1: Introduction
Lesson 30
At a Glance
Students identify the properties of a rectangle, square,
and parallelogram. Then students determine the most
specific name for three polygons shown.
Develop skills and strategies
Lesson 30
Arrange the polygons below so that a polygon can also be called by the name of
the polygon before it. Order them from left to right.
A
•Have students read the problem at the top of the page.
B
C
explore it
use the math you already know to solve the problem.
•Review the meaning of parallel lines. [Lines that
never intersect and always remain the same distance
apart.] Draw examples of intersecting, parallel, and
perpendicular lines on the board. Ask students to
describe the lines.
What are the properties of polygon A?
2 pairs of parallel and congruent sides,
4 right angles
Circle all the names that polygon A can be called.
quadrilateral
parallelogram
rectangle
square
What are the properties of polygon B? 2 pairs of parallel and congruent sides
Circle all the names that polygon B can be called.
quadrilateral
•Work through Explore It as a class. As you work
through the first question, have students justify why
polygon A can be called a quadrilateral [a closed
figure with 4 sides], a parallelogram [opposite sides
are parallel], and a rectangle [4 right angles].
•Ask student pairs or groups to explain their answers
for the remaining questions.
5.g.2.4
in this lesson, you will classify polygons by their properties such as sides and
angles. take a look at this problem.
•Tell students that this page models using properties
of polygons to determine the most specific name for
a polygon.
•Ask students what properties all three polygons have
in common. [4 sides, parallel sides] Ask what
polygon this describes. [parallelogram]
MaFs
classify two-Dimensional Figures
Step By Step
•Have students use the same reasoning to justify why
polygon B can be called a quadrilateral and a
parallelogram and to justify why polygon C can be
called a quadrilateral, a parallelogram, a rectangle,
and a square.
Part 1: introduction
parallelogram
rectangle
square
What are the properties of polygon C? 4 congruent sides, 4 right angles,
parallel opposite sides
Circle all the names that polygon C can be called.
quadrilateral
parallelogram
rectangle
square
Write the most specific name for each polygon shown above.
rectangle
parallelogram
A:
B:
C:
square
How can you order the polygons so that each can also be called by the name of the
ones before it? parallelogram, rectangle, square
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L30: Classify Two-Dimensional Figures
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Mathematical Discourse
•Which is the most specific description of a breakfast
menu item: a fried egg, an egg, or a fried egg over
easy? Describe how this compares to determining the
most specific name for polygons A, B, and C.
Responses should indicate an understanding
that the description that provides the most
information is the most specific description.
•Why might it make more sense to call polygon C
a square when you could also call it a rectangle,
a parallelogram, or a quadrilateral?
Calling polygon C a square gives the most
information possible about the figure.
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L30: Classify Two-Dimensional Figures
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Part 1: Introduction
Lesson 30
At a Glance
Students use a Venn diagram and a flow chart to show
the hierarchy of the polygons from the previous page.
Step By Step
•You may wish to review Venn diagrams with your
class before discussing this page.
Part 1: introduction
Find out More
When you order categories of shapes by their properties, you put them in a hierarchy.
A hierarchy starts with the most general group. Then it shows how more specific groups
are related.
A Venn diagram can show categories and subcategories. This Venn diagram shows that
parallelograms have all the properties that quadrilaterals have and some new ones.
Rectangles have all the properties that parallelograms have and some new ones.
Squares have all the properties that rectangles have and some new ones.
Quadrilaterals
•Read Find Out More as a class.
•Have students look at the Venn diagram. Work from
the outermost figure to the innermost. Guide
students to understand that a parallelogram is a
quadrilateral because it has all of the properties of a
quadrilateral plus some additional properties. Ask
students to name those additional properties.
[opposite sides are parallel, opposite sides are
congruent]
•Similarly, a rectangle is a parallelogram because it
has all of the properties of a parallelogram plus some
more. Ask students to name those additional
properties. [four right angles]
•Finally, have students explain why a square is a
rectangle. [A square has all of the properties of a
rectangle plus all sides are congruent.]
Lesson 30
Parallelograms
Rectangles
Squares
A flow chart also can show the hierarchy of the categories and subcategories of shapes.
The most general category is at the left, while the most specific is at the right. This flow
chart shows the three quadrilaterals from the most general to the most specific.
Quadrilaterals
Parallelograms
Rectangles
Squares
reflect
1 How are the flow chart and the Venn diagram alike? How are they different?
Possible answer: they are alike because they order quadrilaterals first
and squares last. they are different because the venn diagram has a larger
area for the largest category, and the flow chart shows each category in
same-sized boxes.
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269
•Point out that the flow chart shows the same
relationships.
Concept Extension
Explore ordering three-dimensional figures.
Point out that just as two-dimensional figures
can be ordered from general to most specific, so can
three-dimensional figures. Show students an example
of a generic prism [faces are parallelograms],
a rectangular prism, and a cube. Point out that the
faces of a generic prism are parallelograms, the faces
of a rectangular prism are rectangles, and the faces of
a cube are squares. Ask students to use what they
learned about parallelograms, rectangles, and squares
to order the three-dimensional figures from least to
most specific.
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Real-World Connection
Encourage students to think about everyday places
or situations where people might see or talk about a
hierarchy.
Example: A hierarchy can be applied to answer the
question “Where do you live?” For example, you
could answer based on your continent, country,
state, city, town, neighborhood, or street name. All
answers would be accurate, but the name of your
street would be the most specific answer for where
you live.
297
Part 2: Modeled Instruction
Lesson 30
At a Glance
Part 2: Modeled instruction
Students use a table to organize the properties of
triangles. Then students use a tree diagram to order the
triangles in a hierarchy.
Lesson 30
read the problem below. then explore different ways to understand ordering
shapes in a hierarchy.
Order the following triangles from the most general to the most specific:
scalene triangle, isosceles triangle, and equilateral triangle. Use a tree diagram
to order them.
Step By Step
•Read the problem at the top of the page as a class.
Model it
you can understand the problem by organizing the properties of the triangles in a
table before ordering them in a tree diagram.
•Read the first Model It. Be sure that students
understand that a triangle can be called isosceles if it
has 2 or 3 congruent sides. Point out that you could
also say that an isosceles triangle has at least
2 congruent sides.
triangle
Isosceles
Scalene
Equilateral
side Properties
2 or 3 congruent sides
no congruent sides
3 congruent sides
Model it
you can represent the problem by starting a tree diagram of the triangles.
A tree diagram can show the hierarchy of the categories and subcategories of shapes.
The most general category, triangles, is the top box of the tree. The subcategories of
triangles, the different kinds of triangles to be ordered, are the branches.
•Read the second Model It. Ask students to explain
why Triangle is the most general category. [All of the
other categories are specific types of triangles.]
Triangles
Category
Subcategories
SMP Tip: Students use clear and precise language
to list the properties of isosceles, scalene, and
equilateral triangles (SMP 6).
270
ELL Support
Point out that a tree diagram gets its name because
its shape resembles a tree with many branches.
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Mathematical Discourse
•How does ordering the triangles compare to ordering
the quadrilaterals on the previous page?
Responses may vary. Possible answer: You
could use a list of properties when ordering the
quadrilaterals, but the list of properties for the
triangles is too involved.
•How is a tree diagram similar to a Venn diagram?
Responses may vary but should indicate an
understanding that both show a hierarchy of
items. Both would show the relationships
between isosceles, scalene, and equilateral
triangles.
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Part 2: Guided Instruction
Lesson 30
At a Glance
Students revisit the problem on page 270 to complete a
tree diagram to understand the relationships among
isosceles, scalene, and equilateral triangles.
Step By Step
•Read Connect It as a class. Be sure to point out that
the questions refer to the problem on page 270.
•Guide students to understand that all triangles can
be classified as either scalene or isosceles because
scalene triangles have no congruent sides and
isosceles triangles have at least 2 congruent sides.
Since “at least 2” means 2 or more, equilateral
triangles are also isosceles triangles.
SMP Tip: Students model the hierarchical order
of triangles using a tree diagram (SMP 4). Point
out that they could also use a flow chart or
Venn diagram to model the order.
•Ask students how they would show that all
equilateral triangles are isosceles triangles using a
Venn diagram. [The equilateral box (or circle) should
be nested inside the isosceles box.]
Concept Extension
Explore Venn diagrams.
Point out that the Venn diagram for quadrilaterals,
parallelograms, rectangles, and squares from the
Introduction included space outside of the
parallelogram category because there are figures that
are quadrilaterals but not parallelograms. Ask
students to name or draw such a shape. [kite]
Explain that when drawing a Venn diagram for
triangles, scalene and isosceles triangles make up
the whole category. There are no triangles that do
not fit in one of these two categories.
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Part 2: guided instruction
Lesson 30
connect it
now you will solve the problem from the previous page using the tree diagram
and the shared properties of the triangles from the table.
2 Why is “Triangles” in the top row of the tree diagram?
Possible answer: that’s the biggest category.
3 Write “Scalene” and “Isosceles” in the
second row of the tree diagram.
Triangles
Why can their categories NOT overlap?
a triangle cannot have congruent
sides and be classified as scalene.
Scalene
Isosceles
4 Write “Equilateral” beneath “Isosceles.”
Why can all equilateral triangles be
classified as isosceles?
Equilateral
they have at least 2 congruent sides.
5 How can you use a tree diagram to order shapes? Possible answer: Place the most
general category of shape at top and more specific categories of shapes beneath.
try it
use what you learned about ordering shapes in a hierarchy to solve this problem.
6 Complete the Venn diagram below to show the hierarchy of isosceles, scalene, and
equilateral triangles.
Triangles
scalene
Isosceles
equilateral
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Try It Solutions
6Solution: See possible Venn diagram above; Students
may draw a Venn diagram with an outside category
of triangles, two non-overlapping subcategories of
scalene and isosceles, and a category of equilateral
that is nested inside isosceles.
ERROR ALERT: Students who draw equilateral such
that it overlaps both scalene and isosceles may not
understand what overlapping categories of a Venn
diagram represent.
299
Part 3: Guided Practice
Lesson 30
Part 3: guided Practice
Lesson 30
study the model below. then solve problems 7–9.
Lesson 30
8 Create a Venn diagram to show the ranking of the polygons
Student Model
Categories that can never
overlap, like hexagons
and quadrilaterals, are
shown as separate circles.
Part 3: guided Practice
Create a Venn diagram to show the hierarchy of quadrilaterals,
polygons, trapezoids, and hexagons.
Look at how you could show your work using a Venn diagram.
described in the chart.
shape
Trapezoid
Isosceles trapezoid
Parallelogram
“At least 2” means
2 or more.
Description
quadrilateral with at least 1 pair of parallel sides
trapezoid with at least 2 congruent sides
quadrilateral with 2 pairs of parallel sides
Possible venn diagram:
Polygons
Trapezoids
Quadrilaterals
Hexagons
Isosceles Trapezoids
Trapezoids
Pair/share
Recreate the hierarchy
with a tree diagram.
9 Look at the flow chart below.
Which type of triangle is
the most specific?
7 Look at the tree diagram below. Write a statement about the
Quadrilaterals
relationship between acute triangles and equilateral triangles.
Plane Figures
Triangles
Polygons
Pentagons
Hexagons
Acute
Pair/share
Write a statement about
the relationship
between acute triangles
and obtuse triangles.
Right
Obtuse
Pair/share
Draw one example of a
polygon in each
separate category of
your Venn diagram.
Parallelograms
The flow chart is like a
tree diagram. But the
arrows show that the
ranking moves from left
to right instead of top to
bottom.
Which statement is true? Circle the letter of the correct answer.
Equilateral
Solution: Possible answer: all equilateral triangles are
acute triangles.
a
A plane figure will always be a polygon.
b
All polygons are plane figures.
c
A polygon is always a quadrilateral, a pentagon, or a hexagon.
D A hexagon is not a plane figure.
Brad chose c as the correct answer. How did he get that answer?
Possible answer: brad read the chart backward.
Pair/share
Does Brad’s answer
make sense?
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L30: Classify Two-Dimensional Figures
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At a Glance
Solutions
Students use tables, Venn diagrams, tree diagrams, and
flow charts to classify plane figures in a hierarchy.
Ex A Venn diagram illustrating the hierarchy is shown.
Students may begin by creating a table or list of
properties.
Step By Step
•Ask students to solve the problems individually and
label categories and subcategories in their diagrams.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
7Solution: Students may say that all equilateral
triangles are acute triangles. They may also say that
some acute triangles are equilateral triangles.
(DOK 2)
8Solution: See possible student work above;
Students use the descriptions in the table to create
a Venn diagram. (DOK 2)
9Solution: B; “Polygons” is a subcategory of
“Plane Figures,” which means that all polygons
are plane figures.
Explain to students why the other two answer
choices are not correct:
A is not correct because circles are plane figures
but are not polygons.
D is not correct because hexagons are plane figures.
(DOK 3)
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Part 4: MAFS Practice
Lesson 30
Part 4: MaFs Practice
Lesson 30
Part 4: MaFs Practice
3
Solve the problems.
Lesson 30
The word “isosceles” can be used to describe any polygon with at least 2 congruent sides.
Look at the following flow chart.
1
Look at the shape below.
Isosceles
Trapezoids
Quadrilaterals
Which is a correct classification for this shape from LEAST specific to MOST specific?
2
A
polygon, quadrilateral, rectangle
B
quadrilateral, parallelogram, square
C
polygon, quadrilateral, rhombus
D
quadrilateral, rectangle, square
Parallelograms
Rectangles
Squares
Part A
Draw an example of an isosceles trapezoid.
Possible isosceles trapezoid:
Classify the triangles as scalene, isosceles, or obtuse. Draw the triangles in the correct box of
the table. If a triangle fits more than one classification, draw it in all the boxes that apply.
If none of these classifications apply, leave it outside the boxes.
Part B
Explain how isosceles trapezoids relate to parallelograms.
Possible answer: all parallelograms are isosceles trapezoids. trapezoids have at least
2 congruent sides and at least 2 parallel sides.
SCALENE
ISOSCELES
OBTUSE
OBTUSE
Part C
Can you use the term isosceles to describe a rectangle? Explain your reasoning.
Possible answer: yes. a rectangle has at least two congruent sides.
self check Go back and see what you can check off on the Self Check on page 251.
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L30: Classify Two-Dimensional Figures
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At a Glance
Students classify plane figures in a hierarchy based on
properties to answer questions that might appear on a
mathematics test.
Solutions
1Solution: A; The most general name for the shape
shown is polygon. The shape has 4 sides, so it is
also a quadrilateral. The shape has 2 pairs of
congruent sides and 4 right angles, so it is also a
rectangle. (DOK 2)
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3Part A Solution: Students draw a trapezoid that has
two sides of the same length; See possible drawing
above.
Part B Solution: See possible student explanation
above.
Part C Solution: Yes; See possible student
explanation above. (DOK 2)
2Solution: See above for solution; A scalene triangle
has 3 sides of different lengths. An isosceles
triangle has at least 2 congruent sides. An obtuse
triangle has an angle greater than 90°. (DOK 2)
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Differentiated Instruction
Lesson 30
Assessment and Remediation
•Ask students to draw a Venn diagram to classify the following shapes in order from least specific to most
specific: parallelogram, polygon, rhombus, quadrilateral [polygon, quadrilateral, parallelogram, rhombus].
Remind students that a rhombus is a parallelogram with four congruent sides.
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Ask students to draw a Venn diagram to classify
the following shapes in order from least specific to most specific: parallelogram, square, rhombus,
quadrilateral [quadrilateral, parallelogram, rhombus, square]
•If a student is still having difficulty, use Ready Instruction, Level 4, Lesson 32.
If the error is . . .
Students may . . .
To remediate . . .
any order other
than the correct
one
not be able to identify
all of the properties of
the given shapes
Have students make a table with the names of the shapes as the
headings of four columns. Work with students to list the properties of
each of the shapes so that students can see, for example, that a
rhombus has all the properties that parallelograms have plus one more.
Hands-On Activity
Challenge Activity
Build quadrilaterals that fit the given
conditions.
Name the figure in different ways.
Materials: geoboards and geobands
Challenge students to provide as many different
names as they can for figures that you draw.
Have students make a shape that fits conditions you
supply. Ask students to name the shape they made.
Label important features such as congruent sides and
right angles in your drawings.
•four sides, opposite sides are parallel, no right
angles [parallelogram (or rhombus)]
Include figures such as an equilateral triangle, a
parallelogram, a square, a rectangle, and a rhombus.
Have students justify why each of the names they use
applies to the figure.
•four sides, opposite sides are parallel, four right
angles [rectangle (or square)]
•four sides, opposite sides are parallel, four right
angles, all sides are congruent [square]
If time permits, provide conditions for students to
build different triangles and have students build and
name the triangles you have described.
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Focus on Math Concepts
Lesson 31
(Student Book pages 276–281)
Understand Properties of Two-Dimensional Figures
Lesson Objectives
The Learning Progression
•Understand that attributes belonging to a category of
two-dimensional figures also belong to all
subcategories of that category.
This standard emphasizes that two-dimensional figures
are categorized and subcategorized based on their
attributes, and that a figure in one subcategory shares
category-defining attributes with figures from the other
subcategories with its category.
•Use Venn diagrams, flow charts, and tree diagrams
to model how properties are shared by categories
of polygons.
Prerequisite Skills
•Identify two-dimensional figures.
•Recognize parallel and perpendicular lines.
•Recognize right, acute, and obtuse angles.
Students should be familiar with examples of various
categories of two-dimensional figures and the
attributes shared by figures of any subcategories. For
example, figures categorized as triangles include
scalene, isosceles, equilateral, right, obtuse, and acute.
While triangles in these subcategories are different,
they all share the category attribute of three sides.
Vocabulary
Teacher Toolbox
attribute: a characteristic of a polygon (e.g., the number
of sides)
subcategory: a category that is completely nested
inside another category; it shares all the same
properties as the category it is nested inside of.
Teacher-Toolbox.com
Prerequisite
Skills
Ready Lessons
Tools for Instruction
Interactive Tutorials
✓
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5.G.2.3
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MAFS Focus
5.G.2.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 6, 7 (see page A9 for full text)
L31: Understand Properties of Two-Dimensional Figures
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303
Part 1: Introduction
Lesson 31
At a Glance
Students explore how to describe polygons. Students
learn that a polygon can be categorized in more than
one way.
Focus on Math concepts
Lesson 31
MaFs
Understand Properties of Two-Dimensional Figures
5.g.2.3
how do we group polygons into categories?
Step By Step
Polygons are grouped into categories by their attributes, such as the number of sides or
angles, the side lengths, and the angle measures. All polygons in the same category
share certain attributes. Some attributes of polygons are described in the table below.
•Introduce the Question at the top of the page.
•Review the meanings of congruent [same size and
same shape], parallel [lines that never intersect and
stay the same distance apart], and perpendicular
[lines that intersect at a 90 degree angle].
•Review the marks that indicate congruent sides and
right angles.
•Review (or introduce) the meaning of a regular
polygon [all sides and angles are congruent]. Ask
students to name a regular quadrilateral [square].
Polygon Attribute
Description
Scalene
No congruent sides
Example
Isosceles
At least 2 congruent sides
Equilateral
All congruent sides
Regular
All congruent sides
Irregular
At least 1 side and 1 interior angle
are not congruent to the others
Right
At least 1 pair of perpendicular sides
Parallel sides
At least 1 pair of opposite sides, if
extended forever, will never intersect
think Can a polygon be categorized in more than 1 way?
Think about how a quadrilateral is defined. It is a polygon with
4 sides. So any shape with 4 straight sides can be called both a
polygon and a quadrilateral. If the quadrilateral has two pairs
of parallel sides, then it can also be called a parallelogram.
•Read the Think question with students.
•Point out that since an isosceles triangle has at least
two congruent sides and an equilateral triangle has
three congruent sides, an equilateral triangle can
also be classified as an isosceles triangle.
Part 1: introduction
shade a polygon
above that can be
named both a
quadrilateral and
parallelogram.
Every parallelogram is a quadrilateral because every
parallelogram has 4 sides. But not all quadrilaterals are
parallelograms because not all quadrilaterals have two pairs
of parallel sides.
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Mathematical Discourse
ELL Support
It is important that students understand the phrase
“at least” when discussing attributes of polygons.
Use the phrase “at least” in sentences to help
students better understand the meaning of the
phrase. For example, to say that a cell phone costs at
least $75 means it costs $75 or more. If you say that
you need to study for at least two hours tonight, you
mean two hours or more.
Have students tell you what “at least two congruent
sides” means. [two or more congruent sides] Ask, If
a triangle has three congruent sides, can you say that it
has at least two congruent sides? [yes]
304
•Is it more useful to categorize triangles by their side
lengths or by their angle measures? Explain your
reasoning.
Responses should indicate an understanding
that both categories are useful depending on
the context of the situation.
•In how many different ways can a polygon can
categorized? Explain your reasoning.
Responses may vary but should indicate an
understanding that it depends on the polygon.
For example, a quadrilateral can be categorized
two ways, as a polygon and a quadrilateral. A
parallelogram can be categorized three ways, as
a polygon, quadrilateral, and parallelogram.
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Part 1: Introduction
Lesson 31
At a Glance
Students use a Venn diagram to show the relationship
between isosceles, equilateral, right, and obtuse
triangles.
Part 1: introduction
think How can we show the relationships among polygons with a diagram?
A Venn diagram is a useful tool for organizing categories of
polygons that share characteristics.
Step By Step
Triangles
•Read the Think question with students.
•Tell students that categories that do not overlap do
not share any properties. Ask students why an
equilateral triangle can never also be an obtuse
triangle. [Obtuse triangles have an angle that
measures greater than 90 degrees, but all the angles
of an equilateral triangle measure 60 degrees.]
•Point out that categories that overlap can share
properties. For example, you can have a triangle that
is both right and isosceles.
•A category that is nested completely inside another
category shares all the properties of the category it is
nested inside of.
Lesson 31
The Venn diagram shows
me that right triangles
never overlap with
equilateral triangles.
Obtuse
Isosceles
Equilateral
Acute
Right
Notice that Right partly overlaps Isosceles. A right triangle can share all the properties of
an isosceles triangle. Also notice that Right does not overlap Obtuse. A right triangle will
never share all the properties of an obtuse triangle.
The category for equilateral triangles is nested completely inside the category for
isosceles triangles to show that it shares all of the same properties and is a subcategory
of isosceles triangles.
reflect
1 What do you learn from the diagram showing that the space for obtuse triangles
partly overlaps the space for isosceles triangles?
Possible answer: an obtuse triangle can also be an isosceles triangle.
•Have students read and reply to the Reflect directive.
L31: Understand Properties of Two-Dimensional Figures
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Visual Model
Work with students to draw an example inside each
category and overlapping categories in the Venn
diagram. Include markings to indicate right angles
and congruent sides as appropriate.
Be sure to include an acute scalene triangle in the
space labeled “triangles” but outside of any other
categories.
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277
Mathematical Discourse
•Can you think of another relationship that could be
modeled with a Venn diagram?
Responses may vary. Possible answers:
whole numbers, prime numbers, and
composite numbers; integers, whole numbers,
and natural numbers
•How could you explain to a friend why a Venn
diagram is a useful tool?
Responses should indicate an understanding
that a Venn diagram visually shows categories
and subcategories to easily determine what
characteristics items do and do not share.
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Part 2: Guided Instruction
Lesson 31
At a Glance
Part 2: guided instruction
Students complete a Venn diagram and table of
properties for quadrilaterals.
explore it
a venn diagram can help you understand what properties are shared by
categories of polygons.
Step By Step
2 The Venn diagram shows categories of quadrilaterals with different properties. Write
the name of each category that fits the description.
•Tell students that they will have time to work
individually on the Explore It problems on this page
and then share their responses in groups. You may
choose to work through the first problem together
as a class.
A. 4 sides
Quadrilaterals
B. At least 1 pair of parallel sides
C. 2 pairs of parallel sides
D. 4 congruent sides
Rhombuses
•Note that a trapezoid is defined as a quadrilateral
with at least one pair of parallel sides.
•Take note of students who are still having difficulty
and wait to see if their understanding progresses as
they work in their groups during the next part of the
lesson.
STUDENT MISCONCEPTION ALERT: When
completing the table, students may notice that
categories A through D repeat the properties of the
preceding category. Students may mistakenly extend
this pattern to category E. Point out that category E
is not nested inside category D, so it does not share
all the properties of category D. Emphasize that it is
nested inside categories A, B, and C, so it does share
all the properties of those categories.
Lesson 31
Trapezoids
Parallelograms
F. Squares
E. 4 right angles
Rectangles
3 Use the Venn diagram to fill in the table below.
278
category
Properties
name
A
4 sides
quadrilateral
B
4 sides, at least 1 pair of parallel sides
trapezoid
C
4 sides, 2 pairs of parallel sides
parallelogram
D
4 sides, 2 pairs of parallel sides, 4 congruent sides
rhombus
E
4 sides, 2 pairs of parallel and congruent sides,
4 right angles
rectangle
F
4 sides, 2 pairs of parallel sides,
4 congruent sides, 4 right angles
square
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Mathematical Discourse
•What pattern do you see in the chart?
Responses should indicate an understanding
that a nested category repeats all the properties
of the category in which it is nested.
•Explain why a square is also a rectangle, a rhombus,
a parallelogram, a trapezoid, and a quadrilateral.
Responses may vary but should indicate an
understanding that a square shares the
properties of each of the other figures.
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Part 2: Guided Instruction
Lesson 31
At a Glance
Students use a Venn diagram to understand how
properties are shared by categories of quadrilaterals.
Then students use a flow chart to answer questions
about the properties of quadrilaterals.
Part 2: guided instruction
Lesson 31
talk about it
use the venn diagram to help you understand how properties are shared by
categories of quadrilaterals.
yes
4 Is every property of parallelograms also a property of all rectangles?
Is every property of rectangles also a property of all parallelograms?
Step By Step
•Organize students into pairs or groups. You may
choose to work through the Talk About It problems
together as a class.
•Walk around to each group, listen to, and join in on
discussions at different points. Use the Mathematical
Discourse questions to help support or extend
students’ thinking.
•Ask students to draw examples of figures in each
category. Discuss whether or not the figure drawn
could also belong to another category.
no
Explain what the Venn diagram shows about the relationship between rectangles
and parallelograms. Possible answer: all rectangles are parallelograms, but
only some parallelograms are rectangles.
classify each inference statement as true or false. if false, explain.
5 The opposite angles of any parallelogram are congruent. Therefore, the opposite
angles of any rhombus are congruent. true
6 The diagonals of any square are congruent. Therefore, the diagonals of any rhombus
are congruent. false; Possible explanation: not all rhombuses are squares.
try it another Way
the flow chart below shows another way to think about how quadrilaterals are
categorized and ranked.
Rectangles
Quadrilaterals
Trapezoids
Squares
Parallelograms
Rhombuses
use the flow chart to describe the statements as true or false:
7 The diagonals of a rectangle are congruent. Therefore, the diagonals of any
SMP Tip: Students use reasoning skills to classify
statements about properties of quadrilaterals as true
or false (SMP 1). Have students share their
reasoning with a partner.
parallelogram must be congruent.
true
L31: Understand Properties of Two-Dimensional Figures
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•Direct the group’s attention to Try It Another Way.
Have a volunteer from each group come to the board
to explain the group’s solutions to problems 7 and 8.
false
8 A rhombus has 2 lines of symmetry. Therefore, a square has 2 lines of symmetry.
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279
Mathematical Discourse
•Luca says a four-sided plane figure with two pairs of
parallel sides is a quadrilateral. Federico says it is a
parallelogram. Who is correct? Explain your
reasoning.
Responses should indicate an understanding
that both are correct, but parallelogram is a
more precise name.
•How could you explain that not all rectangles are
squares?
Responses should include the fact that not all
rectangles have four congruent sides.
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Part 3: Guided Practice
Lesson 31
At a Glance
Part 3: guided Practice
Students demonstrate their understanding of
categorizing polygons.
Lesson 31
connect it
talk through these problems as a class. then write your answers below.
Step By Step
9 categorize: Polygons are either convex or concave. A convex polygon has all
interior angles less than 180°. A triangle is an example. A concave polygon has at least
1 interior angle greater than 180°. The quadrilateral below is an example.
•Discuss each Connect It problem as a class using the
discussion points outlined below.
210°
Categorize concave polygons, convex polygons, triangles, quadrilaterals, and
rectangles in a Venn diagram. Draw an example of each polygon in the diagram.
Possible venn diagram.
Categorize:
Polygons
•You may choose to have students work in pairs to
encourage sharing ideas.
Concave
Convex
Triangles
Quadrilaterals
SMP Tip: Students draw a Venn diagram to model
Rectangles
the relationship between properties of concave
polygons, convex polygons, quadrilaterals, and
rectangles (SMP 4). Point out that they could also
draw a flow chart or tree diagram to model the
relationships.
•Have students compare their Venn diagram with a
partner’s. Ask a volunteer to explain any differences
between their diagram and their partner’s diagram.
•Ask students how they decided which were the most
general categories. [Triangles, quadrilaterals, and
rectangles are all types of polygons; polygons are the
most general category.]
•Ask students to explain why triangles can never
be concave polygons. [Each of the angles of a triangle
measures less than 180 degrees. A concave polygon
has at least one interior angle greater than
180 degrees.]
Explain:
10 explain: Nadriette said that a rectangle can never be called a trapezoid. Explain what
is incorrect with Nadriette’s statement. Possible answer: since a rectangle has at
least 1 pair of parallel sides, it can be called a trapezoid.
11 create: Describe the properties of a shape that is both a rectangle and a rhombus.
Draw an example.
a parallelogram with 4 congruent sides and 4 right angles: a square
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Create:
•This discussion gives students an opportunity to
describe the properties of and draw a shape given
certain criteria.
•Ask students to describe what methods they can use
to answer the question. [List and compare the
properties of a rectangle and the properties of a
rhombus, draw a Venn diagram, flow chart, or
tree diagram.]
•The second problem focuses on the ability to explain,
in words, how two shapes are related.
•Ask students to describe how rectangles are related
to trapezoids based on the Venn diagram and the
flow chart on the previous pages. [All parallelograms
are trapezoids.] Based on those diagrams, what other
figures could be classified as trapezoids? [rectangles,
rhombuses, and squares]
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Part 4: Performance Task
Lesson 31
At a Glance
Part 4: Performance task
Students create a tree diagram showing the relationship
between types of triangles. Then students use the
diagram to write statements about types of triangles
that are always true and sometimes true.
Lesson 31
Put it together
12 Use what you just learned about classifying polygons to complete this task.
a Create a tree diagram to show the following types of triangles: acute, obtuse,
right, isosceles, and equilateral. Use information in the table to help you.
triangle
Acute
Right
Obtuse
Scalene
Isosceles
Equilateral
Step By Step
•Direct students to complete the Put It Together task
on their own.
angle Properties
all acute angles
2 acute angles and 1 angle exactly 90°
2 acute angles and 1 obtuse angle
acute, right, or obtuse
acute, right, or obtuse
all acute angles
Possible tree diagram:
•Explain that the categories in each row of the tree
diagram can be placed in any order as long as they
are connected to the subcategories correctly.
Triangles
•As students work on their own, walk around to
assess their progress and understanding, to answer
their questions, and to give additional support,
if needed.
Acute
Right
Obtuse
Equilateral
Isosceles
Scalene
b Write a statement that is always true about the relationship between obtuse
triangles and equilateral triangles.
Possible answer: an equilateral triangle can never be classified as an
obtuse triangle.
•If time permits, have students share their tree
diagrams with a partner.
c Write a statement that is sometimes true about the relationship between acute
triangles and isosceles triangles.
Possible answer: an isosceles triangle can be an acute triangle.
Scoring Rubrics
See student facsimile page for possible student answers.
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A
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281
Points Expectations
2
1
0
The tree diagram indicates a student’s
understanding of the relationship between
the properties of triangles. The tree diagram
correctly connects categories and
subcategories.
B
An effort was made to accomplish the task.
The tree diagram demonstrates some
understanding of the relationship between
the properties of triangles, but the student’s
diagram is missing categories, misplaces
categories or subcategories, or incorrectly
connects categories and subcategories.
There is no tree diagram or the diagram
shows little or no understanding of the
relationship between the properties of
triangles.
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C
Points Expectations
2
The student wrote a statement about the
relationship between obtuse triangles and
equilateral triangles that is always true.
1
An effort was made to accomplish the task,
but the statement the student wrote is not
always true.
0
There is no response or the response shows
no evidence of understanding the
relationship between obtuse and
equilateral triangles.
Points Expectations
2
The student wrote a statement about the
relationship between acute triangles and
isosceles triangles that is sometimes true.
1
An effort was made to accomplish the task,
but the statement the student wrote is
not true.
0
There is no response or the response shows
no evidence of understanding the
relationship between acute and isosceles
triangles.
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Differentiated Instruction
Lesson 31
Intervention Activity
On-Level Activity
Model and categorize triangles.
Create a model of a Venn diagram.
Materials: geoboards, strips of paper
Materials: string, paper, scissors
Group students into pairs. Tell students to write each
of the following categories on its own strip of paper,
and include a description of each: triangle, right
triangle, acute triangle, obtuse triangle, scalene
triangle, isosceles triangle, and equilateral triangle.
Have one student display a triangle on the geoboard.
Have the other student choose as many strips of
paper as possible that describe the triangle shown.
Then have students switch roles.
Distribute string and scissors to each student. Have
students draw and cut out a quadrilateral, trapezoid,
parallelogram, rhombus, rectangle, and square. Tell
students to label and write the properties of each
figure on their cutouts. Then have students use string
to make a Venn diagram for the figures, using their
figures to label each section. Have students draw and
cut out other quadrilaterals, write all the categories
that describe them on the cutouts, and place them
appropriately in their Venn diagram.
If time allows, repeat for quadrilaterals,
parallelograms, trapezoids, rectangles, rhombuses,
and squares.
Challenge Activity
Justify classifications of polygons.
Ask students to answer each of the following questions and to provide an explanation for each answer.
•Are all rectangles squares? [No, students may draw a rectangle that does not have 4 congruent sides.]
•Are all squares rectangles? [Yes, students may list all of the properties of a rectangle and note that a square
has all of these properties.]
•Is every quadrilateral a parallelogram or a trapezoid? [No, students may draw a quadrilateral that does not
have at least one pair of parallel sides.]
•Are all parallelograms rectangles? [No. Not all parallelograms have 90 degree angles.]
•Is there a quadrilateral that can be classified as a quadrilateral, trapezoid, parallelogram, rectangle, and
rhombus? [Yes. A square shares all of the properties of these figures.]
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