Download 2014-15 6th Grade Scope and Sequence Topic 12 and 13 Triangles

6th Grade Math ‐ Conroe ISD Third Grading Period Topic 12 and 13: Triangles and Their Properties, Area and Volume (16 days)
Mathematical Process Standards
(A) apply mathematics to problems arising in everyday life, society, and the workplace;
(B) use a problem‐solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem‐solving process and the reasonableness of the solution;
(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;
(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
(E) create and use representations to organize, record, and communicate mathematical ideas;
(F) analyze mathematical relationships to connect and communicate mathematical ideas; and
(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Unit Vocabulary
base and height of a base and height of a volume of a rectangular parallelogram
base and height of a triangle trapezoid
prism
Triangle Angle‐Sum Theorem
area of a rectangle
compose a shape
rectangular prism
angle
obtuse triangle
area of a square
area of a triangle
polygon
vertex
acute triangle
opposite
form a triangle
inequality
volume
triangle relationships
composite figure
prism
area
area of a parallelogram
trapezoid
parallelogram
decompose a shape
Math Review: During this unit, classroom Math Review should be based on the specific academic needs of each class from a variety of sources; including but not limited to benchmarks common assessments informal assessments and teacher observations
to benchmarks, common assessments, informal assessments, and teacher observations.
6.8 Expressions, equations, and relationships. The student applies mathematical process standards to use geometry to represent relationships and solve problems. The student is expected to:
TEA Supporting Readiness or Supporting
Student Expectation
Nouns Resources
Information
6.8A extend previous knowledge previous knowledge, triangles, of triangles and their properties to triangle properties, sum, angles, triangle, relationship, lengths, sides, include the sum of angles of a measures
triangle, the relationship between the lengths of sides and measures of angles in a triangle, and Verbs
determining when three lengths form a triangle
extend, include, determining, form
6th Grade Unit 13: Triangles and Their Properties
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Specificity as been provided for the Digits
relationships involving triangles. The *Digits lessons should be taught in their entirety, revised SE adds determining when including Launch, Examples, and Close and Check.
three
Vertical Alignment lengths form a triangle. Digits Teacher Guide Unit D p. 721 & p. 802
Relationships involving quadrilaterals are not
Teacher Guide
included within the Revised TEKS Topic 12 pp. 721‐801
(2012).
Topic 13 pp. 802‐848
2014‐2015
6th Grade Math ‐ Conroe ISD Readiness or Supporting
Student Expectation
Nouns area formulas, parallelograms, 6.8B model area formulas for parallelograms, trapezoids, and trapezoids, triangles, parts, shapes
triangles by decomposing and rearranging parts of these shapes
Verbs
model, decomposing, rearranging
Readiness or Supporting
TEA Supporting Information
Resources continued
Specificity has been added regarding Digits continued
the development of formulas.
Student Companion
Topic 12 pp. 261‐290
Topic 13 pp. 291‐308
Lessons 12‐1; 12‐2; 12‐3; 12‐4; 12‐5; 12‐6 and 12‐7
Lessons 13‐1; 13‐2; 13‐3 and 13‐4
Process Standards Books:
Introduction to Connections
Student Expectation
Nouns CD Activities
equations, problems, area of When the revised SE is paired with Letters I‐IX
6.8C write equations that rectangles, area of parallelograms, 6(1)(D) and 6(1)(G),the expectation Near Square Plot
represent problems related to the Angles in Polygons
area of rectangles, parallelograms, area of trapezoids, area of triangles, is that students use tables to volume of right rectangular prisms, generate equations as appropriate to Estimating Area and Volume
trapezoids, and triangles and Area of Irregular Shapes
dimensions, positive rational the problem. Specificity has been volume of right rectangular prisms Toothpick Shapes
numbers
added for formulas. The dimensions where dimensions are positive The Painted Cube
may be positive rational numbers.
rational numbers
Diagonals and Altitudes
In the revised SE, perimeter is TEA Supporting Information
Verbs
Readiness or Supporting
addressed in grade 4 and grade 5:
Algebraic reasoning 4(5)(C)
4(5)(D) 5(4)(H)
Introduction to Representations CD Activities
Picture This
TEA Supporting Missing Cubes
Student Expectation
Nouns Measuring Triangles
Information
solutions, problems, area of Specificity has been added regarding Making Triangles
6.8D determine solutions for In the Doghouse
rectangles, area of parallelograms, formulas for area and volume. problems involving the area of Box It In
area of trapezoids, area of triangles, Dimensions may be positive rational rectangles, parallelograms, Rectangular Ratio
numbers. In the revised SE, trapezoids, and triangles and Verbs
Cover It Up
measurement concepts and skills volume of right rectangular prisms Congruent Rectangles
have been focused:
where dimensions are positive determine, involving, are
Check the Volume
• Time: grades 1, 2, 3, 4
rational numbers
• Length (including perimeter): Introduction to Connections
grades 1, 2, 3, 4, 5
CD Activities
• Weight: grades 3, 4, 5
Temperature is not included within Estimating Volumes: Coal Mining
the Revised TEKS (2012) within the Measurement strand. It may be included in problems related to
everyday life.
write, represent, related, are
6th Grade Unit 13: Triangles and Their Properties
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6th Grade Math ‐ Conroe ISD 6.9 Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to represent situations. The student is expected to:
TEA Supporting Readiness or Supporting
Student Expectation
Nouns Resources continued
Information
6.9A write one‐variable, one‐step equations and inequalities to represent constraints or conditions within problems
one‐variable equations, one‐step equations, one‐variable inequalities, one‐step inequalities, constraints, conditions, problems
Verbs
write, represent
Specificity has been added regarding the type of equations which may be written in grade 6. This SE is a building block for one variable, two‐
step equations and inequalities with 7(10)(B). The revised SE extends to include inequalities. Constraints or conditions may be indicated by words such as “minimum” or “maximum.” Students will need to determine if the value in the solution is part of the solution set or Introduction to Reasoning and Proof
CD Activities
Parallelograms and Their Properties
Kites or Not?
Alike and Different
Mystery Shapes
Is It or Is It Not?
not.
6.10 Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to solve problems. The student is expected to:
TEA Supporting Readiness or Supporting
Student Expectation
Nouns
Readiness or Supporting
Student Expectation
Nouns Information
6.10A model and solve one‐
variable, one‐step equations and inequalities that represent problems, including geometric concepts
one‐variable equations, one‐step equations, one‐variable inequalities, one‐step inequalities, problems, geometric concepts
Verbs
model, solve, represent
6th Grade Unit 13: Triangles and Their Properties
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This SE is a building block for one‐
variable, two‐step equations and inequalities with 7(11)(C) and may include concepts developed in 6(8)(A)and 4(7)(E) as contexts. Geometric concepts may include complementary and supplementary angles.
2014‐2015
6th Grade Math ‐ Conroe ISD Topic 12 and 13 Instructional Information
6.8A
This standard focuses on the attributes of triangles and their relationships.
Students are focusing on three different relationships:
• Sum of the angles of a triangle
• Relationship among the lengths of sides and measures of angles in a triangle
• Determining when three lengths form a triangle.
Sum of the Angles of a Triangle
The sum of the angles of any triangle is 180°. Students should be able to find the measure of an angle of a triangle given the measures of the other two angles. They should also be able to use other information about the triangles to find missing angle measure(s) .
Example 1: Given an equilateral triangle.
Since all of the sides are congruent, this means that all of the angles are congruent too. 180 ÷ 3 = 60°.
Example 2: Given an isosceles triangle.
The two sides in red are congruent. This means that the angles that are opposite the sides are also congruent. To find the measures of the missing angles, subtract 30 from 180 to get 150.
Then 150 ÷ 2 = 75. So each base angle is 75°.
**Although this standard is in the Expressions, Equations, and Relationships strand, students in 6th grade write only one‐step equations. To write this relationship as an equation requires two steps and, thus, is beyond the reach of the 6th grade standards.
6th Grade Unit 13: Triangles and Their Properties
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6th Grade Math ‐ Conroe ISD 6.8A Continued
Relationship Between the Lengths of Sides and Measures of Angles in a Triangle
The relationship between the lengths of sides and measures of angles in a triangle is fairly intuitive. The hardest part is getting students to connect a side and an angle, especially if students haven’t had much experience with angles. In a nutshell, if the side is longest on the triangle, the angle opposite it is the largest angle in the triangle. If that angle is the smallest in the triangle, then the side opposite it is the shortest. If two sides are congruent, then the angles opposite them are also congruent (and it’s an isosceles triangle).
Determining When Three Lengths Form a Triangle
This is a great place to let students experiment with straws or pipe cleaners of different unit lengths. After their experiment, they should figure out that to make a triangle, the sum of any two lengths must be larger than the third length.
Example: Do the lengths 4, 5, and 6 make a triangle? Yes. Because 4 + 5 ˃ 6; 5 + 6 ˃ 4; 4 + 6 ˃ 5.
Example: Do the lengths 3, 5, and 8 make a triangle? No. Because 3 + 5 = 8; it isn’t larger than 8.
Example: Do the lengths 2, 3, and 7 make a triangle? No. Because 2 + 3 ˂ 7.
Since this concept would result in a one‐step equation or inequality, students should write and solve equations and inequalities that have to do with the length of sides of triangles.
6th Grade Unit 13: Triangles and Their Properties
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6th Grade Math ‐ Conroe ISD 6.8B
6.8B lays the basis for students to understand and remember the formulas for area of some two‐dimensional figures. This is a hands‐on Student Expectation (SE). Students are decomposing, which means breaking into pieces, and then rearranging the pieces to get a different figure. The relationship between the areas of the original and new figures is analyzed to find the area of the new figure. Students are familiar with area of a rectangle. This means that students should attempt to cut the shapes so that they can be rearranged into a rectangle.
Suggestion: The progression of the Pearson Digits Topic 12 works well with having students decompose and rearrange figures to discover formulas.
Parallelograms
This rectangle has the same area as the original parallelogram because all of the parts of the parallelogram were used to make the rectangle. The base of the parallelogram is the same length as the rectangle (in red), and the height of the parallelogram is the same as the height of the rectangle (in blue). Thus,
Since this formula results in a one‐step equation or inequality, students should set up and solve problems where the Area and one other measure are given. Then they have to find the 3rd measure.
6th Grade Unit 13: Triangles and Their Properties
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6th Grade Math ‐ Conroe ISD 6.8B Continued
Triangles
Triangles are often thought of as half rectangles, and the formula to find the area of a triangle is often taught by cutting a rectangle in half and finding half the area of the rectangle. However, that is NOT what this SE requires. This SE has students cutting triangles apart to figure out their area. This example will show how to cut a triangle apart to create a rectangle. The triangle should be cut so that the height is cut in half. The red dotted line is the cutting line. Then the triangle is rotated on the left side to make a rectangle. The yellow is the triangle that was cut away. The green rectangle on the right is the new one that was created. The new rectangle is half the height of the original triangle but has the same size base.
6th Grade Unit 13: Triangles and Their Properties
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6th Grade Math ‐ Conroe ISD 6.8B Continued
Trapezoid
Have students find the formula for the area of a parallelogram prior to finding a trapezoid. Cut the trapezoid in half and parallel to its bases. This creates two trapezoids. Rotate the top trapezoid around to the side of the bottom trapezoid to make a parallelogram.
Now check out the sides that are red and orange. Those are the bases of the original trapezoid. They have been transferred over to the new parallelogram.
Solving for the height or bases given the Area requires an equation or inequality that has more than one step. Students will not have to solve for the height or the bases of the trapezoid given the Area.
6th Grade Unit 13: Triangles and Their Properties
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2014‐2015
6th Grade Math ‐ Conroe ISD 6.8C & 6.8D
These two Student Expectations (SEs) have students writing equations, which are formulas, for area and volume and then doing the calculations to actually find the area or volume. The numbers involved may be whole numbers, fractions, or decimals. Each answer should be written with units.
Rather than explaining how to find area or volume, this discussion will center on teaching students to use formulas effectively. This method works from 6th grade to high school geometry, although 8th grade math and high school geometry will have one additional step.
Step 1: Read the problem and draw a diagram if one is not given. Students should put all the given dimensions on the figure. If the dimensions are in different units, check the problem to see what unit the answer should be written in. Change the dimensions to that unit.
Step 2: Identify the formula that is needed. Students should have their STAAR Reference Materials Chart available. Problems that involve volume will include filling a three‐dimensional figure. Problems that involve area will involve covering a flat surface, which is the two‐dimensional figure.
Step 3: Write the formula exactly as it appears on the chart without replacing any of the numbers. Students should leave some space between the variables in the formula.
Step 4: Rewrite the formula using the diagram to replace the variables in the formula. Write the numbers directly under the variables in the formula.
Step 5: Simplify the expression using order of operations and write the answer using units.
You can access the new STAAR reference charts at the following link:
http://www.tea.state.tx.us/student.assessment/staar/math/
6th Grade Unit 13: Triangles and Their Properties
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6th Grade Math ‐ Conroe ISD 6.9A & 6.10A
Students will need to model, solve, and write one‐variable one‐step equations as they relate to geometric concepts. This could include situations such as:
‐finding a missing dimension of a rectangle given the area
‐find a missing angle in a complementary or supplementary angle pairing
‐finding a side of square given the perimeter
‐finding the measure an angle in an equilateral triangle
**Remember that in sixth grade students only model, solve, and write one‐variable, one‐step equations.
6th Grade Unit 13: Triangles and Their Properties
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