## Unit 10: Distance, Area, Angles, Surface Area, & Volume

## Unit 10: Chapters 11 & 12 - Distance, Area, Angles, Surface Area, & Volume (Grade 7 Focus)

Teachers Access Only:** CSA/Resources**

### Learning Targets/Performance Indicators

** Grade 7 Geometry PI #1:** Solve problems involving area and circumference.

- Learn the formulas for area and circumference of circles
- Substitute known values into the formulas for area and circumference of circles and solve for the missing values algebraically
- Solve real world and mathematical problems involving area and circumference of circles

** Grade 7 Geometry PI #2: **Write (and solve) simple equations for an unknown in multi-step problems involving angles (supplementary, complementary, vertical, and adjacent).

- Know what supplementary, complementary, vertical, and adjacent angles are
- Create and solve algebraic equations in order to find missing values of angles given information

** Grade 7 Geometry PI #3: **Solve real world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects.

- Substitute values into formulas and solve algebraically to find missing values for the area of triangles, parallelograms, trapezoids, rectangles, and composite figures
- Substitute values into formulas and solve algebraically to find missing values for the volume of cubes and right prisms
- Substitute values into formulas and solve algebraically to find missing values for the surface area of cubes and right prisms
- Solve real-world and mathematical problems involving area, volume, and surface area

**Essential Questions:**

- What do you notice about the measures of the sides or the measures of the angles that form triangles?
- What is the differences between a vertical angle and an adjacent angle?
- How do measurements help you describe real-world objects (circumference, area)?
- How is the circumference of a circle related to its diameter?
- How are the circumference and area of a circle related?
- How do measurements help you describe real-world objects (circumference, area, volume, surface area)?
- How does the shape of a rectangular prism affect its volume and surface area?
- What is the result of a transversal cutting parallel lines?

**Prior Learning:**

In previous grades, students learned concepts of angles, measured angles, classified polygons, and constructed polygons in the coordinate plane given coordinates for the vertices. Students found the lengths of a side using the coordinates and applied these techniques to solve real-world problems.

In grade 6, students found the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. Using these methods, students discussed, developed, and justified formulas for areas of triangles and parallelograms. Work toward meeting the standard drew together grades 3–6 work with geometric measurement.

In grade 6, students found areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces they could determine. They represented three-dimensional figures using nets made up of rectangles and triangles and used the nets to find the surface area of these figures. Students applied these techniques in the context of solving real-world and mathematical problems. They reasoned about right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths.

In grade 6, students found the area of triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. They applied techniques in the context of solving real-world and mathematical problems. Students used nets to find surface area. They used cubes to find volume and compared results using formulas.

**Current Learning:**

Students gain familiarity with the relationships between angles formed by intersecting lines (supplementary, complementary, adjacent, vertical), which is part of a critical area. Solving real-world mathematical problems involving angle measures is an __additional cluster.__

Solving real-world and mathematical problems involving area is part of a critical area, an additional cluster, and an opportunity for in-depth focus. Students find the areas of various polygons, including composite figures. They must know the formulas to find the circumference and area of circles and must be able to apply these to real-world problems. Students informally derive the relationship between circumference and area of circles by decomposing a circle into wedges to form a parallelogram. In future units, they extend their knowledge of area when finding surface area and volume of cubes and right prisms.

Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross sections. They solve real-world and mathematical problems involving surface area as well as volume of three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. This is a critical area and an additional cluster for PARCC.

Meeting standard 7.G.6 is an opportunity for in-depth focus, drawing together grades 3–6 work with geometric measurement.

**Future Learning:**

Later in Pre-Algebra, students will extend their knowledge to solve problems involving volume of cylinders, cones, and spheres. In grade 8, this is a critical area and a major cluster. Students understand that two figures are similar if the second can be obtained from the first by a sequence of movements. Given two similar two-dimensional figures, they explain the sequence of movements between the two. Through investigation, students explore interior and exterior angles of triangles and angle measures created by two parallel lines cut by a transversal, and the angles of similar triangles.

All of students’ previous knowledge will be applied in high school geometry when they learn and apply geometric theorems. Students will be expected to explain volume formulas and use them to solve problems while also applying geometric concepts in modeling situations. Also, in high school geometry, they will extend their knowledge of volume of right prisms to find the volume of pyramids.

In high school, students will explain volume formulas and use them to solve problems. They will give informal arguments for the volume of a cylinder, pyramid, cone, spheres, and other solid figures. Students will use volume formulas to solve problems. These skills, along with proportional reasoning and multistep numerical problem solving, can be combined and used in flexible ways as part of modeling during high school.

In high school, students will experiment with transformations in the plane. They will understand congruence in terms of rigid motions. Students will be using concepts of similarity and congruence to prove theorems and make geometric constructions. Later in eighth grade students will use this to work with similar figures.