Geometry and Measurement
Unit Length and Description:
Unit 5 is designed to engage students in Geometry.† This unit has students drawing, constructing, and describing geometrical figures and describing the relationships between them. The constructions the students do in this unit will focus on triangles and provide students the opportunity to formalize their understanding of what conditions define a unique triangle, more than one triangle, or no triangle.† Students also solve a variety of real-life and mathematical problems involving angle measure, area, surface area, and volume of two- and three-dimensional objects. They revisit unknown angles and continue to use proportional reasoning as they revisit and analyze scale drawings.† This unit is the first time the students have engaged in circles since 4th grade where they gained a conceptual understanding of circles and their properties with an emphasis on angle measures.† Students will apply their conceptual understanding of area and perimeter to circles, and will be introduced to pi for the first time.† Students use the formulas for the area and circumference of a circle to solve problems involving circles. Additionally, the focus of this unit is on irregular figures which distinguish this unit from similar units on area, volume, and surface area in previous courses. All two- and three-dimensional objects used in this unit should be a composition of triangles, quadrilaterals, polygons, cubes, and right prisms. This will require students to be able to recognize and decompose the objects into pieces they can work with. To support their work with irregular two- and three-dimensional objects, students will slice three dimensional figures to describe the resulting two-dimensional figure.
7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.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.A.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.
7.G.B.4 Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
o Standards Clarification:† Irrational numbers are not introduced until the 8th grade. An approximate value of pi can be discovered by allowing the students to derive experimentally the formulas for area and circumference. The approximations that are discovered should be used for pi and the actual value of pi as an irrational number will be discussed in the 8th grade.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an unknown angle in a figure.
o Standards Clarification:† Students will be expected to write and solve equations for an unknown angle in a figure. This provides an opportunity to reinforce studentsí fluency in solving equations; also, this unit provides an opportunity to model real-world situations involving angles with equations and inequalities. The standards do not explicitly address at any grade level the measure of a straight angle. It may need to be discovered by applying the studentsí understanding of right angles and angle addition.
7.G.B.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.
Focus Standards of Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
∑ Scale drawings and scale models are used to represent objects that are too large or too small to be drawn or built at actual size.
∑ The scale gives the ratio that compares the measurements of the drawing or model to the measurements of the real object.
∑ The measurements on a drawing or model are proportional to the measurements on the actual objects.
∑ Scale drawings can be applied to problem solving situations involving geometric figures.
∑ Geometrical figures can be used to reproduce a drawing at a different scale.
∑ Writing and solving real-life and mathematical problems involving simple equations for an unknown angle in a figure helps students as they engage in higher geometry concepts.
∑ Reason about relationships among two-dimensional figures, which leads to gaining familiarity with the relationships between angles formed by intersecting lines.
∑ Solve problems involving the area and circumference of a circle and surface area of three-dimensional objects.
∑ Solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.
∑ Geometry and spatial sense offer ways to interpret and reflect on our physical environment.
∑ Analyzing geometric relationships develops reasoning and justification skills.
∑ How does geometry help us describe real-world objects?
∑ How can I use geometric figures to reproduce a drawing at a different scale?
∑ How can I use and relate facts about special pairs of angles to write and solve simple equations involving unknown angles?
∑ What is the total number of degrees in supplementary and complementary angles?
∑ What is the relationship between vertical and adjacent angles?
∑ How can geometry be used to solve problems about real-world situations, spatial relationships, and logical reasoning?
∑ How do geometric models describe spatial relationships?
∑ How are geometric shapes and object classified?
∑ Where do you see shapes in the world around you?
∑ How are angles related to the sides that create them?
∑ What characteristics are necessary for shapes to be defined?
∑ How do you describe triangles?
∑ What two-dimensional figures result from slicing prisms, pyramids, cubes, cylinders, and cones?