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Grade 7 Overview

Ratios and Proportional Relationships

Analyze proportional relationships and use them to solve real-world and mathematical problems.

The Number System

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Expressions and Equations

Use properties of operations to generate equivalent expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Geometry

Using Random Sampling to Draw Inferences

Draw, construct and describe geometrical figures and describe the relationships between them.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Grade: 7 Unit - Geometry Time: 35 Days

In grade 7, students begin to reason about relationships among two-dimensional figures using scale and informal geometric constructions, and gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

In Unit 6, students delve further into several geometry topics they have been developing over the years. Grade 7 presents some of these topics, (e.g., angles, area, surface area, and volume) in the most challenging form students have experienced yet. Unit 6 assumes students understand the basics. The goal is to build a fluency in these difficult problems. The remaining topics, (i.e., working on constructing triangles and taking slices (or cross-sections) of three-dimensional figures) are new to students.

Composition and decomposition of shapes is used throughout geometry from Grade 6 to high school and beyond. Compositions and decompositions of regions continue to be important for solving a wide variety of area problems, including justifications of formulas and solving real world problems that involve complex shapes. Decompositions are often indicated in geometric diagrams by an auxiliary line, and using the strategy of drawing an auxiliary line to solve a problem are part of looking for and making use of structure (MP7). Recognizing the significance of an existing line in a figure is also part of looking for and making use of structure. This may involve identifying the length of an associated line segment, which in turn may rely on students’ abilities to identify relationships of line segments and angles in the figure. These abilities become more sophisticated as students gain more experience in geometry. In Grade 7, this experience includes making scale drawings of geometric figures and solving problems involving angle measure, surface area, and volume (which build on understandings
described in the Geometric Measurement Progression as well as the ability to compose and decompose figures). ||

Additional Cluster StandardsDraw, 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.
7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Solve real-life and mathematical problems in involving angle measure, area, surface area, and volume 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. ||

Additional Cluster Standards Unpacked| 7.G.2 Students draw geometric shapes with given parameters. Parameters could include parallel lines, angles, perpendicular lines, line segments, etc.

Example 1:
Draw a quadrilateral with one set of parallel sides and no right angles.

Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle or no triangle.

Example 2:
Can a triangle have more than one obtuse angle? Explain your reasoning.

Example 3:
Will three sides of any length create a triangle? Explain how you know which will work.
Possibilities to examine are:
a. 13 cm, 5 cm, and 6 cm
b. 3 cm, 3cm, and 3 cm
c. 2 cm, 7 cm, 6 cm

Solution:
“A” above will not work; “B” and “C” will work. Students recognize that the sum of the two smaller sides must be larger than the third side.

Example 4:
Is it possible to draw a triangle with a 90˚ angle and one leg that is 4 inches long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle?
(NOTE: Pythagorean Theorem is NOT expected – this is an exploration activity only)
Example 5:
Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not?

Example 6:
Draw an isosceles triangle with only one 80°angle. Is this the only possibility or can another triangle be drawn that will meet these conditions?

Through exploration, students recognize that the sum of the angles of any triangle will be 180° and the angles of any quadrilateral will sum to 360°

Other explorations would include:
• Base angles of an isosceles triangle are equal
• Angle and side length relationships between scalene, isosceles, and equilateral triangles
• Angle and side length relationships between obtuse, acute and right triangles

7.G.3 Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral (side) face. Cuts made at an angle through the right rectangular prism will produce a parallelogram;

If the pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a square cross section (red).

If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the intersection of the pyramid and the plane is a triangular cross section (red).

If the pyramid is cut with a plane (green) perpendicular to the base, but not through the top vertex, the intersection of the pyramid and the plane is a trapezoidal cross section (red).

7.G.5 Students use understandings of angles and deductive reasoning to write and solve equations

Example1: Write and solve an equation to find the measure of angle x.

Solution: Find the measure of the missing angle inside the triangle (180 – 90 – 40), or 50°.
The measure of angle x is supplementary to 50°, so subtract 50 from 180 to get a measure of 130° for x. Example 2: Find the measure of angle x.

Solution: First, find the missing angle measure of the bottom triangle (180 – 30 – 30 = 120). Since the 120 is a vertical angle to x, the measure of x is also 120°.

Example 3: Find the measure of angle b.

Note: Not drawn to scale.

Solution: Because, the 45°, 50° angles and b form are supplementary angles, the measure of angle b would be 85°. The measures of the angles of a triangle equal 180° so 75° + 85°+ a = 180°. The measure of angle a would be 20°.

7.G.6 Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects. (composite shapes) Students will not work with cylinders, as circles are not polygons. At this level, students determine the dimensions of the figures given the area or volume.

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be for all students.

Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade, students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level; however, students could create nets to aid in surface area calculations.

Students understanding of volume can be supported by focusing on the area of base times the height to calculate volume.
Students solve for missing dimensions, given the area or volume.
Students determine the surface area and volume of pyramids.

Volume of Pyramids
Students recognize the volume relationship between pyramids and prisms with the same base area and height. Since it takes 3 pyramids to fill 1 prism, the volume of a pyramid is 1/3 the volume of a prism (see figure below).

To find the volume of a pyramid, find the area of the base, multiply by the height and then divide by three.

V = BhB = Area of the Base 3 h = height of the pyramid Example 1: A triangle has an area of 6 square feet. The height is four feet. What is the length of the base?

Solution: One possible solution is to use the formula for the area of a triangle and substitute in the known values, then solve for the missing dimension. The length of the base would be 3 feet.

Example 2: The surface area of a cube is 96 in2. What is the volume of the cube?

Solution: The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in2 for each face. Because each face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying 4 x 4 x 4 or 64 in3. ||

Focus Standards for Mathematical PracticeMP.1 Make sense of problems and persevere in solving them. This mathematical practice is particularly applicable for this module, as students tackle multi-step problems that require them to tie together knowledge about their current and former topics of study (i.e., a real-life composite area question that also requires proportions and unit conversion). In many cases, students will have to make sense of new and different contexts and engage in significant struggle to solve problems.

MP.3 Construct viable arguments and critique the reasoning of others. In Topic B, students examine the conditions that determine a unique triangle, more than one triangle, or no triangle. They will have the opportunity to defend and critique the reasoning of their own arguments as well as the arguments of others. In Topic C, students will predict what a given slice through a three-dimensional figure will yield (or how to slice a three-dimensional figure for a given cross section) and must provide a basis for their predictions.

MP.5 Use appropriate tools strategically. In Topic B, students will learn how to strategically use a protractor, ruler, and compass to build triangles according to provided conditions. An example of this is when students are asked to build a triangle provided three side lengths. Proper use of the tools will help them understand the conditions by which three side lengths will determine one triangle or no triangle. Students will have opportunities to reflect on the appropriateness of a tool for a particular task.

MP.7 Look for and make use of structure. Students must examine combinations of angle facts within a given diagram in Topic A to create an equation that correctly models the angle relationships. If the unknown angle problem is a verbal problem, such as an example that asks for the measurements of three angles on a line where the values of the measurements are consecutive numbers, students will have to create an equation without a visual aid and rely on the inherent structure of the angle fact. In Topics D and E, students will find area, surface area, and volume of composite figures based on the structure of two- and three dimensional figures. ||

Skills and ConceptsPrerequisite Skills/Concepts: Students should already be able to… Geometric measurement: understand concepts of angle and measure angles.

4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Solve real-life and mathematical problems involving area, surface area, and volume.

7.G.4 Know the formulas for area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Advanced Skills/Concepts: Some students may be ready to…

Understand congruence and similarity using physical models, transparencies, or geometry software. (8.G.1-5)

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. (8.G.9)

Skills: Students will be able to …

• Use freehand, ruler, protractor and technology to draw geometric shapes with given conditions. (7.G.2)

• Construct triangles from 3 measures of angles or sides. (7.G.2)

• Given conditions, determine what and how many type(s) of triangles are possible to construct. (7.G.2)

• Describe the two-dimensional figures that result from slicing three-dimensional figures (right rectangular prisms and right rectangular pyramids). (7.G.3)

• Identify and describe supplementary, complementary, vertical, and adjacent angles. (7.G.5)

• Use understandings of supplementary, complementary, vertical and adjacent angles to write and solve equations. (7.G.5)

• Explain (verbally and in writing) the relationships between the angles formed by two intersecting lines. (7.G.5)

• Solve mathematical problems involving area, volume and surface area of two- and three dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.6)

• Solve real-world problems involving area, volume and surface area of two- and three dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (7.G.6)

Academic Vocabulary

Three-dimensional

Two-dimensional

Surface area

volume

Intersecting lines

Vertex

Complementary angles

Supplementary angles

Cross-sections

Right rectangular prism

Right rectangular pyramid

Constructions

Cube

Planar section

Compose

Decompose

Nets

Volume

Area

Polygon

Pyramid

Prism

Triangle

Angle

Right angle

Obtuse angle

Degrees

Acute angle

Angle measure

Line segment

Prism

Pyramid

Plane

Unit Resources

Prentice Hall 7th Grade Textbook - Online power point - WYSIWYG - Adams'

Statistics and Probability

Use random sampling to draw inferences about a population.

Draw informal comparative inferences about two populations.

Investigate chance processes and develop, use, and evaluate probability models.

Mathematical Practices

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Online Videos

## Grade 7 Overview

## Ratios and Proportional Relationships

## The Number System

## Expressions and Equations

## Geometry

Using Random Sampling to Draw InferencesGrade: 7Unit - GeometryTime: 35 DaysIn grade 7, students begin to reason about relationships among two-dimensional figures using scale and informal geometric constructions, and gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

In Unit 6, students delve further into several geometry topics they have been developing over the years. Grade 7 presents some of these topics, (e.g., angles, area, surface area, and volume) in the most challenging form students have experienced yet. Unit 6 assumes students understand the basics. The goal is to build a fluency in these difficult problems. The remaining topics, (i.e., working on constructing triangles and taking slices (or cross-sections) of three-dimensional figures) are new to students.

Composition and decomposition of shapes is used throughout geometry from Grade 6 to high school and beyond. Compositions and decompositions of regions continue to be important for solving a wide variety of area problems, including justifications of formulas and solving real world problems that involve complex shapes. Decompositions are often indicated in geometric diagrams by an auxiliary line, and using the strategy of drawing an auxiliary line to solve a problem are part of looking for and making use of structure (MP7). Recognizing the significance of an existing line in a figure is also part of looking for and making use of structure. This may involve identifying the length of an associated line segment, which in turn may rely on students’ abilities to identify relationships of line segments and angles in the figure. These abilities become more sophisticated as students gain more experience in geometry. In Grade 7, this experience includes making scale drawings of geometric figures and solving problems involving angle measure, surface area, and volume (which build on understandings

described in the Geometric Measurement Progression as well as the ability to compose and decompose figures). ||

Additional Cluster StandardsDraw, 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.

7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Solve real-life and mathematical problems in involving angle measure, area, surface area, and volume7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problem to write and solve simple equations for an unknown angle in a figure.

7.G.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. ||

Additional Cluster Standards Unpacked| 7.G.2 Students draw geometric shapes with given parameters. Parameters could include parallel lines, angles, perpendicular lines, line segments, etc.Example 1:

Draw a quadrilateral with one set of parallel sides and no right angles.

Students understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle or no triangle.

Example 2:

Can a triangle have more than one obtuse angle? Explain your reasoning.

Example 3:

Will three sides of any length create a triangle? Explain how you know which will work.

Possibilities to examine are:

a. 13 cm, 5 cm, and 6 cm

b. 3 cm, 3cm, and 3 cm

c. 2 cm, 7 cm, 6 cm

Solution:

“A” above will not work; “B” and “C” will work. Students recognize that the sum of the two smaller sides must be larger than the third side.

Example 4:

Is it possible to draw a triangle with a 90˚ angle and one leg that is 4 inches long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle?

(NOTE: Pythagorean Theorem is NOT expected – this is an exploration activity only)

Example 5:

Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not?

Example 6:

Draw an isosceles triangle with only one 80°angle. Is this the only possibility or can another triangle be drawn that will meet these conditions?

Through exploration, students recognize that the sum of the angles of any triangle will be 180° and the angles of any quadrilateral will sum to 360°

Other explorations would include:

• Base angles of an isosceles triangle are equal

• Angle and side length relationships between scalene, isosceles, and equilateral triangles

• Angle and side length relationships between obtuse, acute and right triangles

7.G.3 Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral (side) face. Cuts made at an angle through the right rectangular prism will produce a parallelogram;

If the pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a square cross section (red).

If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the intersection of the pyramid and the plane is a triangular cross section (red).

If the pyramid is cut with a plane (green) perpendicular to the base, but not through the top vertex, the intersection of the pyramid and the plane is a trapezoidal cross section (red).

7.G.5 Students use understandings of angles and deductive reasoning to write and solve equations

Example1:Write and solve an equation to find the measure of angle

x.Solution:Find the measure of the missing angle inside the triangle (180 – 90 – 40), or 50°.

The measure of angle

xis supplementary to 50°, so subtract 50 from 180 to get a measure of 130° forx.Example 2:Find the measure of angle

x.Solution:First, find the missing angle measure of the bottom triangle (180 – 30 – 30 = 120). Since the 120 is a vertical angle to

x, the measure ofxis also 120°.Example 3:Find the measure of angle

b.Note: Not drawn to scale.

Solution:Because, the 45°, 50° angles and

bform are supplementary angles, the measure of anglebwould be 85°. The measures of the angles of a triangle equal 180° so 75° + 85°+a= 180°. The measure of angleawould be 20°.7.G.6 Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects. (composite shapes) Students will not work with cylinders, as circles are not polygons. At this level, students determine the dimensions of the figures given the area or volume.

“Know the formula” does not mean memorization of the formula. To “know” means to have an understanding of

the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be forwhyallstudents.Surface area formulas are not the expectation with this standard. Building on work with nets in the 6th grade, students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level; however, students could create nets to aid in surface area calculations.

Students understanding of volume can be supported by focusing on the area of base times the height to calculate volume.

Students solve for missing dimensions, given the area or volume.

Students determine the surface area and volume of pyramids.

Volume ofPyramidsStudents recognize the volume relationship between pyramids and prisms with the same base area and height. Since it takes 3 pyramids to fill 1 prism, the volume of a pyramid is 1/3 the volume of a prism (see figure below).

To find the volume of a pyramid, find the area of the base, multiply by the height and then divide by three.

V =B = Area of the BaseBh3h = height of the pyramidExample 1:A triangle has an area of 6 square feet. The height is four feet. What is the length of the base?

Solution:One possible solution is to use the formula for the area of a triangle and substitute in the known values, then solve for the missing dimension. The length of the base would be 3 feet.

Example 2:The surface area of a cube is 96 in2. What is the volume of the cube?

Solution:The area of each face of the cube is equal. Dividing 96 by 6 gives an area of 16 in2 for each face. Because each face is a square, the length of the edge would be 4 in. The volume could then be found by multiplying 4 x 4 x 4 or 64 in3. ||

Focus Standards for Mathematical PracticeMP.1 Make sense of problems and persevere in solving them. This mathematical practice is particularly applicable for this module, as students tackle multi-step problems that require them to tie together knowledge about their current and former topics of study (i.e., a real-life composite area question that also requires proportions and unit conversion). In many cases, students will have to make sense of new and different contexts and engage in significant struggle to solve problems.MP.3 Construct viable arguments and critique the reasoning of others. In Topic B, students examine the conditions that determine a unique triangle, more than one triangle, or no triangle. They will have the opportunity to defend and critique the reasoning of their own arguments as well as the arguments of others. In Topic C, students will predict what a given slice through a three-dimensional figure will yield (or how to slice a three-dimensional figure for a given cross section) and must provide a basis for their predictions.MP.5 Use appropriate tools strategically. In Topic B, students will learn how to strategically use a protractor, ruler, and compass to build triangles according to provided conditions. An example of this is when students are asked to build a triangle provided three side lengths. Proper use of the tools will help them understand the conditions by which three side lengths will determine one triangle or no triangle. Students will have opportunities to reflect on the appropriateness of a tool for a particular task.MP.7 Look for and make use of structure. Students must examine combinations of angle facts within a given diagram in Topic A to create an equation that correctly models the angle relationships. If the unknown angle problem is a verbal problem, such as an example that asks for the measurements of three angles on a line where the values of the measurements are consecutive numbers, students will have to create an equation without a visual aid and rely on the inherent structure of the angle fact. In Topics D and E, students will find area, surface area, and volume of composite figures based on the structure of two- and three dimensional figures. ||Skills and ConceptsPrerequisite Skills/Concepts:Students should already be able to…Geometric measurement: understand concepts of angle and measure angles.4.MD.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Solve real-world and mathematical problems involving area, surface area, and volume.6.G.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.6.G.2Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.6.G.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Solve real-life and mathematical problems involving area, surface area, and volume.7.G.4Know the formulas for area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.Advanced Skills/Concepts:Some students may be ready to…Skills:Students will be able to …Academic VocabularyUnit ResourcesPrentice Hall 7th Grade Textbook - Online power point - WYSIWYG - Adams'## Statistics and Probability

## Mathematical Practices

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