## Investigations

*According to Gregory (2012), an inquiry-based environment is one in which "students are given limited instruction and asked to developtheir own research questions and strategies for seeking an-*

*swers through readily available data" (p. 68). Inquiries should include scientifically-oriented questions and have students give priority to evidence, formulate explanations from evidence, connect explanations to math knowledge, and communicate/justify explanations*.

## Investigation #1

Surface Area

**Overview**

Students will be given three-dimensional figures. With only a ruler, they have to find the surface area and volume. They will be required to create a way to find the surface area of the shapes that are given to them.

**Objectives**

Students will be able to:

- demonstrate their methods they used to find the surface area of various geometric shapes
- calculate the surface area of geometric shapes
- create their own understanding of a net of a figure

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- (8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to:
- (A) find areas of regular polygons, circles, and composite figures;
- (B) find areas of sectors and arc lengths of circles using proportional reasoning;
- (C) derive, extend, and use the Pythagorean Theorem;
- (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations;
- (6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to:
- (A) describe and draw the intersection of a given plane with various three-dimensional geometric figures;
- (B) use nets to represent and construct three-dimensional geometric figures; and
- (C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems.

**Engage**

Before the day of investigation 1, students will be asked to bring a three-dimensional object from their home that resembles a geometric shape. Then they will be asked to find the surface area of their object by only using a ruler.

**Explore**

Students will be given a three-dimensional shape. With the given ruler, students will measure the dimensions to find the surface area of the cube and rectangular prism. They will be working as a group to figure out the answers. They are also required to explain their methods and reasoning for their findings.

**Explain**

As a class, we will go over the answers and have a classroom discussion to see all the different methods students used. Then the teacher will introduce the idea of a net of a figure.

**Elaboration**

Students are now given pyramids and cylinders. They are required to find the surface area with only a ruler and scissor. The idea of the lesson is for the students to figure out a way to find the surface area on their own. Once again, students will be required to write out their way of finding the answer.

**Evaluate**

At the end of the class, students will be given an exit slip (click on exit slip for worksheet).

## Investigation #2

Ratios/Scales

**Overview**

Students will go around the classroom and measure various objects (desks, chairs, chalboards, etc.) They will then map out their classroom and produce a scale relating their model of the classroom to the actual classroom.

**Objectives**

Students will be able to:

- Calculate and create rates and ratios relating their model of the sketch to their classroom
- Measure and determine the dimensions of the objects and other features of the classroom

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

- (11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to:
- (A) use and extend similarity properties and transformations to explore and justify conjectures about geometric figures;
- (B) use ratios to solve problems involving similar figures;

- (F) use conversions between measurement systems to solve problems in real-world situations.

**Engage**

Show different maps to students of their city, campus, country, etc. Have them measure certain distances between things. (Is the distance from New York City to San Francisco accurately depicted in these maps?) Introduce the idea of scales and the mathematics of proportions, rates, and ratios that goes behind all of this.

**Explore**

Ask students to first measure the dimensions of their desk using a ruler, yard stick, etc. Have them draw their desk on a paper. The dimensions don't fit. What do they have to do to make the picture of the desk fit? Have them do this with the entire classroom. They should eventually come up with a map of the classroom that includes windows, chairs. desks, etc.

**Explain**

As a class, have a gallery walk/presentation setting in which each group presents their map and scale. Have them justify their map mathematically. Discuss what difficulties students had. Clear up misconceptions.

**Elaboration**

If time permits, have students make a map of their room at home.

**Evaluate**

Students will be given an exit slip before leaving the classroom.

## Benchmark Lessons

*Benchmark lessons are needed in PBI units when introducing new ideas or techniques that students cannot learn during an investigation.*

According to Lin (2005), benchmark lessons are classroom settings that " increase student participation,promote interpersonal regard, and heighten classroom community and reflective thinking during discussions. They offer every student active learning strategies that facilitate dialogue,exchange ideas, encourage community building, and increase the physical movement in your classroom" (p. 34). They must be student-centered in order to promote learning and avoid a lecture setting in which the teacher speaks for an hour and students absent-mindedly take notes.

According to Lin (2005), benchmark lessons are classroom settings that " increase student participation,promote interpersonal regard, and heighten classroom community and reflective thinking during discussions. They offer every student active learning strategies that facilitate dialogue,exchange ideas, encourage community building, and increase the physical movement in your classroom" (p. 34). They must be student-centered in order to promote learning and avoid a lecture setting in which the teacher speaks for an hour and students absent-mindedly take notes.

## Benchmark Lesson #1

Surface Area, Area, and Volume

**Overview of the lesson**

The goal of the lesson is for the students to understand and use area surface area formulas effectively. As the professor teaches students methods used to find surface area, area, and volume, students will use a formula sheet to figure out problems presented to them on the board. At the beginning, the professor will go over the letters and what they mean using the figures. Then give some examples on how to solve for the area, surface area, and volume. Students will be up on the board to explain and show their own findings. Then on the second day, the placemat method will be used in order to promote interactive lecture in this lesson: students at each table use a different corner of a large white board. Once they have all come up with an answer, they collaborate to come up with a group answer.

**Objectives**

Students will be able to:

- communicate with other students to understand the use of area, volume, and surface area
- understand when to use which area, volume, and surface area formulas to use for each shapes
- calculate the areas, volume, and surface areas

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- (8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to:
- (A) find areas of regular polygons, circles, and composite figures;
- (B) find areas of sectors and arc lengths of circles using proportional reasoning;
- (C) derive, extend, and use the Pythagorean Theorem;
- (D) find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations;
- (F) use conversions between measurement systems to solve problems in real-world situations.

## Benchmark Lesson #2

Perpendicular and Parallel Functions

**Overview of the lesson**

The goal of the lesson is for the students to understand the properties of perpendicular and parallel lines. Students will figure out the equations that are parallel and perpendicular to the equation given. Each student will be given an equation or two and will have to come up with one equation that is parallel and one equation that is perpendicular. The think-pair-share method will then be used: students will come up with their equations then trade with another student. They will then graph the equations they were given in the trade in order to critique whether or not the lines are indeed perpendicular/parallel. Students will then come together as a class and discuss the patterns between the slopes of perpendicular and parallel lines.

**Objectives**

Students will be able to:

- determine parallel and perpendicular lines by using properties
- solve equations to find the parallel or perpendicular line to it
- find and provide equations for parallel/perpendicular lines to an equation given to them

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- (7) Dimensionality and the geometry of location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. The student is expected to:
- (A) use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures;
- (B) use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons;

## Benchmark Lesson #3

2D to 3D Transition

**Overview of the lesson**

The goal of this lesson is for students to understand how two and three dimensional figures are related. Students will discover how to make two-dimensional nets of three-dimensional figures and fold/glue them appropriately. Students will first be given printouts of different 2-D views of a three dimensional figure (top, side, front) and be asked to construct the actual figure using building blocks. Some views will be isometric and some will be orthographic. They will then be given nets and be asked to first determine what 3-D geometric figure is shown, then construct it by folding and gluing it together. Students will be using PEOE demonstrations in this lesson.

**Objectives**

Students will be able to:

- Determine what 3-D shape is represented by a particular 2-D net
- Construct 3-D figures using isometric and orthographic representations of the figure

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- (6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to:
- (A) describe and draw the intersection of a given plane with various three-dimensional geometric figures;
- (B) use nets to represent and construct three-dimensional geometric figures; and
- (C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems.

## Benchmark Lesson #4

Scales using Ratios and Rates

**Overview of the lesson**

The goal of this lesson is to get students to understand how to use rates and ratios. They will be given ratios and rates on the board to solve. They will use the placemat method to figure out the most effective methods and come up with answers. We will then go over, as a class, the effective and ineffective methods in solving rates and ratios.

**Objectives**

Students will be able to:

- Compare the ratios between similar figures and solids
- Calculate actual size of object on map using scale given (1 inch = 400 ft)

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- (F) use conversions between measurement systems to solve problems in real-world situations.
- (11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to (B) use ratios to solve problems involving similar figures;