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Grade 7 Mathematics, Quarter 3, Unit 3.2
Constructing Triangles
Overview
Number of instruction days:
3–5
Content to Be Learned
Mathematical Practices to Be Integrated
Draw and construct possible triangles when
given three measures of their angles.
3 Construct viable arguments and critique the
reasoning of others.
Draw and construct possible triangles when
given three measures of their sides.
Construct viable arguments to determine
whether the given constraints form a triangle.
Recognize when given conditions determine
unique triangles, more than one triangle, or no
triangle.
Critique the reasoning of others.
Understand that the sum of two side lengths
must be greater than the third side length to
make a triangle.
Justify conclusions and communicate them to
others.
Analyze situations by breaking them into cases.
5 Use appropriate tools strategically.
Demonstrate familiarity with tools (i.e. rulers,
protractors) appropriate for grade level.
Make sound decisions about when these tools
might be helpful, recognize the insight to be
gained from them and their limitations.
Use technological tools to explore and deepen
understanding.
7 Look for and make use of structure.
Look closely to discern a pattern or structure.
Recognize the significance of an existing line
in a geometric figure.
Step back to gain an overview and shift
perspective.
Essential Questions
How can you tell whether three line segments
will form a triangle?
What conditions determine a unique triangle,
more than one triangle, or no triangle?
If it is possible to build one triangle, is it
possible to build a different triangle with the
same three segments? Explain.
How many different triangles can be drawn
given the measurement of the three angles?
Providence Public Schools
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Grade 7 Mathematics, Quarter 3, Unit 3.2
Version 4
Constructing Triangles (3–5 days)
Standards
Common Core State Standards for Mathematical Content
Geometry
7.G
Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given
conditions. Focus on constructing triangles from three measures of angles or sides, noticing
when the conditions determine a unique triangle, more than one triangle, or no triangle.
Common Core State Standards for Mathematical Practice
3
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them
into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them
to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient
students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or
reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they
make sense, and ask useful questions to clarify or improve the arguments.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their
limitations. For example, mathematically proficient high school students analyze graphs of functions and
solutions generated using a graphing calculator. They detect possible errors by strategically using
estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are able to
identify relevant external mathematical resources, such as digital content located on a website, and use
them to pose or solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may
sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8
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Constructing Triangles (3–5 days)
Grade 7 Mathematics, Quarter 3, Unit 3.2
Version 4
equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the
significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line
for solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed of several
objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
Students drew polygons using the coordinates for the vertices in sixth grade. They applied these
techniques in the context of solving real-world and mathematical problems. The work toward meeting
standard 7.G.2 draws together the work with geometric measurements that students did in grades 3-6.
Current Learning
Students construct triangles with given constraints. They focus on constructing triangles from measures of
angles of sides, leading toward finding unique triangles, more than one triangle, or none at all.
Future Learning
In high school geometry, students will make formal geometric constructions with a variety of tools and
methods. Students will construct equilateral triangles inscribed in a circle.
Additional Findings
Students enter the middle grades with informal knowledge of two-dimensional shapes. They have had
experience in drawing lines, angles, triangles, and other polygons. “They should develop an
understanding of different angle relationships and be proficient in measuring angles. Toward this end,
they should learn to use a protractor to measure angles directly . . . so middle-grades students also need
help with mechanics of using a protractor.”(Principles and Standards for School Mathematics, p. 233)
Principles and Standards for School Mathematics also asserts that all students in grades 6–8 should “draw
geometric objects with specified properties, such as side lengths and angle measures [and be able to]
precisely describe, classify, and understand relationships among types of two- and three-dimensional
objects using their defining properties.” (p. 397)
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
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Grade 7 Mathematics, Quarter 3, Unit 3.2
Version 4
Constructing Triangles (3–5 days)
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your students’
attainment of the mathematics within the unit.
Draw triangles given three measures of their sides.
Draw triangles given three measures of their angles.
Show that the sum of two side lengths of a triangle must be greater than the third side of the same
triangle.
Recognize when given conditions determine unique triangles, more than one triangle, or no triangle.
Instruction
Learning Objectives
Students will be able to:
Build triangles given three side lengths and decide whether any three lengths will make a triangle.
Build triangles given three angle measures and decide whether any three angle measures will make a
triangle.
Determine whether given conditions make a unique triangle, more than one triangle, or no triangle.
Demonstrate understanding of constructing triangles.
Resources
Connected Mathematics 2, Pearson/Prentice Hall, 2008: Shapes and Designs
Problem 4.1: Building Triangles, Student Book (pages 70-71)
Teacher’s Guide
Implementing and Teaching Guide
Teaching Transparencies
Assessment Resource Book
Additional Practice and Skills Workbook
Strategies for English Language Learners
Special Needs Handbook
Parent Guide
Prentice Hall Teacher Station Software
Exam View Software
www.phschool.com (Students can enter web-codes)
Connected Mathematics 2, Pearson/Prentice Hall, 2011: Common Core Additional Investigations
Grade 7
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Constructing Triangles (3–5 days)
Grade 7 Mathematics, Quarter 3, Unit 3.2
Version 4
CC Investigation 4: Geometry Topics; Problem 4.3
https://www.pearsonsuccessnet.com/snpapp/login/login.jsp
Investigations are located in the “Worksheets” tab
Common Core Investigations Teacher’s Guide
Implementing a Common Core Curriculum
Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010); available with the Algebra
resources
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.
Materials
Angle rulers or protractors, polystrips, fasteners, 3 number cubes per group, larger poster paper for
student work (optional), blank transparencies for student work (optional), transparency markers
Key Vocabulary
There is no new vocabulary for this unit.
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary from previous grades or units: triangle, protractor.
Living word walls assist all students in developing content language. Word walls should be visible to all
students, focus on the current unit’s vocabulary, both new and reinforced, and have pictures, examples,
and/or diagrams to accompany the definitions.
Teachers should review the “Mathematics of the Unit” found on page 3 of all CMP2 teacher editions. For
planning considerations read through the teacher edition for suggestions about scaffolding techniques,
using additional examples, and differentiated instructional guidelines as suggested by the CMP2 resource.
For Problem 4.1, in Shapes and Designs, you will need to borrow a set of polystrips from a 6th-grade
teacher. If you don’t have access to polystrips, you can use spaghetti, and students can measure, mark,
and break the spaghetti into appropriately sized pieces. When working the problems in 4.1, students often
conjecture that if you add the two smaller side lengths of the triangle and the sum is larger than the third
side length, then the three sides will always form a triangle. Discuss this conjecture along with the
Triangle Inequality Theorem.
For the CC Investigation 4: Geometry Topics, in Problem 4.3 students will need to recall that the sum of
the measures of the angles in any triangle is 180 . Students will use protractors or angle rulers to measure
angles and construct triangles. Some students may not remember how to use these tools, so you may want
to use a “Do Now” to review this skill with them.
Incorporate the Essential Questions as part of the daily lesson. Options include using them as a “do now”
to activate prior knowledge of the previous day’s lesson, using them as an exit ticket by having students
respond to it and post it, or hand it in as they exit the classroom, or using them as other formative
assessments. Essential questions should be included in the unit assessment.
CMP2 has online resources that may be helpful in planning for all units of study. Visit
www.phschools.com and sign on to SuccessNet. You will find the Common Core Additional
Investigations and Common Core Investigations Teacher’s Guide under the “worksheet” tab.
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Grade 7 Mathematics, Quarter 3, Unit 3.2
Version 4
Constructing Triangles (3–5 days)
Notes
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