Download Topic 16 - Milwaukee Public Schools

Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16
Standards for Mathematical Practice
Beyond the Critical Area(s): Properties of Geometric Figures
Major Work
70% of time
4.OA.A.1-2-3
4.NBT.A.1-2-3
4.NBT.B.4-5-6
4.NF.A.1-2
4.NF.B.3-4
4.NF.C.6-7
FOCUS for Grade 4
Supporting Work
20% of time
4.OA.B.4
4.MD.A.1-2-3
4.MD.B.4
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Additional Work
10% of time
4.OA.C.5
4.MD.C.5-6-7
4.G.A.1-2-3
Standards in bold are specifically targeted within instructional materials.
Required fluency: 4.NBT.B.4
Domains:
Geometry
Measurement and Data
Clusters:
Clusters outlined in bold should drive the learning for this period of instruction.
4.G.A Draw and identify lines and angles, and classify shapes by properties of 4.MD.C Geometric measurement: understand concepts of angle and measure
their lines and angles.
angles.
Standards:
4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines. Identify these in two-dimensional
figures.
4.G.A.2 Classify two-dimensional figures based on the presence or absence of
parallel or perpendicular lines, or the presence or absence of angles of a
specified size. Recognize right triangles as a category, and identify right
triangles.
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure as a line
across the figure such that the figure can be folded along the line into
matching parts. Identify line-symmetric figures and draw lines of symmetry.
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Property of MPS
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever
two rays share a common endpoint, and understand concepts of angle
measurement:
a. An angle is measured with reference to a circle with its center at the
common endpoint of the rays, by considering the fraction of the circular
arc between the points where the two rays intersect the circle. An angle
that turns through 1/360 of a circle is called a “one-degree angle,” and
can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle
measure of n degrees.
4.MD.C.6 Measure angles in whole-number degrees using a protractor.
Sketch angles of specified measure.
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16
Foundational Learning
3.G.A.1
4.G.A.1
Key Student Understandings


Future Learning
5.G.B
Assessments
Students will understand properties and characteristics of (2-dimensional) geometric figures, including
perpendicular and parallel lines, classification of angles, and lines of symmetry.
Students will understand that geometric figures can be analyzed and classified based on their
properties, such as having parallel sides, particular angle measures, and/or symmetry.

Formative Assessment Strategies

Evidence for Standards-Based Grading
Common Misconceptions/Challenges
4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
 Students believe a wide angle with short sides may seem smaller than a narrow angle with long sides. Students can compare two angles by tracing one
and placing it over the other. Students will then realize that the length of the sides does not determine whether one angle is larger or smaller than
another angle.
 Students sometimes find more lines of symmetry than actually exist. Simple or convenient definitions that lines of symmetry “chop” shapes in half do not
necessarily imply that these lines must also create one half the exact mirror image of the other (e.g., in the rectangle below, the horizontal and vertical
dotted lines are lines of symmetry, but the diagonal lines are not). Teachers should encourage the use of cut-out shapes that students can actually fold to
“test” possible lines of symmetry, as shown with the parallelograms (below right).
4.MD.C Geometric measurement: understand concepts of angle and measure angles.
 Students are confused as to which number to use when determining the measuring an angle with a protractor because most protractors have a double
set of numbers. Students should have multiple experiences estimating and comparing angles to the benchmark 90°/right angle. They should explain their
reasoning by deciding first if the angle appears to be an angle that is less than the measure of a right angle (90°) or greater than the measure of a right
angle (90°). If the angle appears to be less than 90°, it is an acute angle and its measure ranges from 0° to 89°. If the angle appears to be an angle that is
greater than 90°, it is an obtuse angle and its measure ranges from 91° to 179°. Ask questions about the appearance of the angle to help students in
deciding which set of degree measurements to use.
 Students may mistakenly interpret a “right angle” as an angle that opens to the right, relying on their understanding of “right” as a directional word. Take
time to discuss how “right” is a word with multiple meanings, including its use mathematically to describe angles that are 90o.
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16
Instructional Practices
Domain: 4.G
Cluster: 4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional
figures.

This standard asks students to draw two-dimensional geometric objects and to also identify them in 2-D figures. This is the first time that students are
exposed to rays, angles, and perpendicular and parallel lines. Examples of points, line segments, lines, angles, parallelism, and perpendicularity can be
seen daily. Students may not easily identify lines and rays because they are more abstract.
o Examples:
 Draw two different types of quadrilaterals that have two pairs of parallel sides.
 Is it possible to have an acute right triangle? Justify your reasoning using pictures and words.
 Draw and list the properties of a parallelogram. Draw and list the properties of a rectangle. How are
your drawings and lists alike? How are they different? Be ready to share your thinking with the class.
 How many acute, obtuse and right angles are in the shape pictured at right?

Students should be able to use side length to classify triangles as equilateral, equiangular, isosceles, or scalene; and can
use angle size to classify them as acute, right, or obtuse. They then learn to cross-classify, for example, naming a shape
as a right isosceles triangle. Thus, students develop explicit awareness of and vocabulary for many concepts they have
been developing, including points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Such mathematical terms are useful in communicating geometric ideas, but more important is that
constructing examples of these concepts, such as drawing angles and triangles that are acute, obtuse, and right, help
students form richer concept images connected to verbal definitions. That is, students have more complete and
accurate mental images and associated vocabulary for geometric ideas (e.g., they understand that angles can be larger
than 90 and their concept images for angles include many images of such obtuse angles).

Students also learn to apply these concepts in varied contexts. For example, they learn to represent angles that occur in various contexts as two rays,
explicitly including the reference line, e.g., a horizontal or vertical line when considering slope or a “line of sight” in turn contexts. They understand the
size of the angle as a rotation of a ray on the reference line to a line depicting slope or as the “line of sight” in computer environments.

Analyzing the shapes in order to construct them requires students to explicitly formulate their ideas about the shapes. For instance, what series of
commands would produce a square? How many degrees are the angles? What is the measure of the resulting angle? What would be the commands for
creating an equilateral triangle? How many degrees are the angles? What is the measure of the resulting angle? Such experiences help students connect
what are often initially isolated ideas about the concept of angle. (Progressions for the CCSSM, Geometry, CCSS Writing Team, June 2012, page 14)
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16
4.G.A.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a
specified size. Recognize right triangles as a category, and identify right triangles.

Two-dimensional figures may be classified using different characteristics, such as parallel or perpendicular lines or by angle measurement.

Parallel or Perpendicular Lines: Students should become familiar with the concept of parallel and perpendicular lines. Two lines are parallel if they never
intersect and are always equidistant. Two lines are perpendicular if they intersect in a way that forms right angles (90º). Take time to discuss examples
and nonexamples with students, allowing them to develop their understanding of line concepts. Lines that intersect but aren’t parallel are simply called
intersecting lines.
o
o
See below: Lines AB and CD are parallel; regardless of how far the lines continue, they will never intersect.
See below: Lines FG and CD are perpendicular; right angles are formed at the point where they intersect. This is also true for lines FG and AB.
Students may use transparencies with lines to arrange two lines in different ways to determine if 2 lines might intersect at one point or may never
intersect. Further investigations may be initiated using geometry software. These types of explorations may lead to a discussion on angles.
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16

This standard calls for students to sort objects based on parallelism, perpendicularity and angle types.
o Example:
- Do you agree with the label for each category of the Venn diagram below? Describe why some shapes fall in the overlapping sections of the
diagram.
- Which figure in the Venn diagram below is in the wrong place, explain how do you know?
o
Example: Draw and name a figure that has two parallel sides and exactly 2 right angles.
o
Example: For each of the following, sketch an example if it is possible. If it is impossible, say so, and explain why or show a counter example.
 a parallelogram with exactly one right angle
 an isosceles right triangle
 a rectangle that is not a parallelogram (impossible)
 every square is a quadrilateral
 every trapezoid is a parallelogram
o
Example:
Identify which of these shapes have perpendicular (or parallel) sides and justify your selection.
A possible justification that students might give is: “The square has perpendicular lines because the sides meet at a corner, forming right angles.”
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16
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Angle Measurement: This expectation is closely connected to 4.MD.5, 4.MD.6, and 4.G.1. Students’ experiences with drawing and identifying right, acute,
and obtuse angles support them in classifying two-dimensional figures based on specified angle measurements. They use the benchmark angles of 90°,
180°, and 360° to approximate the measurement of angles.

Right triangles can be a category for classification. A right triangle has one right angle. There are different types of right triangles. An isosceles right
triangle has two or more congruent sides and a scalene right triangle has no congruent sides.

The notion of congruence (“same size and same shape”) may be part of classroom conversation but the concepts of congruence and similarity do not
appear until middle school.

In the U.S., the term “trapezoid” may have two different meanings; research identifies these as inclusive and exclusive definitions. (Progressions for the
CCSSM: Geometry, The Common Core Standards Writing Team, June 2012.)
o
The inclusive definition states: A trapezoid is a quadrilateral with at least one pair of parallel sides.
o
The exclusive definition states: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Progressions for the CCSSM, K-6 Geometry, CCSS Writing Team, June 2012, p 15
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16
4.G.A.3 Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching
parts. Identify line-symmetric figures and draw lines of symmetry.

Students need experiences with figures which are symmetrical and non-symmetrical. When exploring
this concept, figures should include both regular and non-regular polygons (see right), as well as other
geometric figures (see below). Folding cut-out figures will help students determine whether a figure
has one or more lines of symmetry.

Students understand that a figure could have one, two or more, or an infinite number of lines of
symmetry (circle); or a figure may not be symmetrical at all. Encourage students to make and then try
to prove conjectures that relate lines of symmetry to other geometric properties, i.e., number of sides,
number of angles, regular vs irregular polygons.
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Mathematics 2016-17—Grade 4
Weeks 35-36—May
enVisionmath2.0—Topic 16

This standard only includes line symmetry, not rotational symmetry.
o Example: For each figure below, draw all of the lines of symmetry. What pattern do you notice? How many lines of symmetry do you think there
would be for regular polygons with 9 and 11 sides. Sketch each figure and check your predictions.
Differentiation
4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
 Encourage the use of concrete, as well as pictorial models of geometric figures while exploring concepts.
 Use different colors to draw or trace over shapes when decomposing them into parts (e.g., use red to identify all acute
angles, use green to identify all right angles).
 Model and encourage the use of precise language when identifying and classifying shapes—have students rephrase
their own thinking, and allow other students to repeat a peer’s thinking to develop students’ mathematical language.
4.MD.C Geometric measurement: understand concepts of angle and measure angles.
 Encourage students to use the benchmark angle of 90o when measuring and drawing angles. Push students to explain
how they know the angle they measured/drew is less than/greater than a right angle.
Literacy Connections
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Academic Vocabulary Terms
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Vocabulary Strategies
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Literacy Strategies
The Common Core Approach to Differentiating Instruction (engageny How to Implement a Story of Units, p. 14-20)
Linked document includes scaffolds for English Language Learners, Students with Disabilities, Below Level Students, and
Above Level Students.
Resources
enVisionmath2.0
Developing Fluency
Multiplication Fact Thinking Strategies
Topic 16 Pacing Guide
Grade 4 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
Revised 2/2017
Property of MPS
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