## Overview

Students will create a 3-dimensional city (blueprint) using ideas obtained from lessons on:

The driving question of this lesson is “How is math involved in the logistics of building a city?” This is relevant because students see their city on a daily basis, but fail to take into account all the mathematics and planning that went into building it. They will be able to apply previous knowledge they’ve gained from observing the city around them in designing their own city. For this unit, students are allowed to extensively collaborate with partners during the benchmarks and investigations. They will be allowed one partner for the final artifact: the miniature model of their new city. Calculators will be used to save time various times throughout the lesson. Students will be allowed computers/tablets for small amounts of research throughout certain parts of the lesson as well.

Project based learning is a way for students to gain a deeper understanding of material by actively using their previous knowledge with new ideas. Krajcik and Blumenfeld informs us that "in project based learning, students engage in real, meaningful problems that are important to them and that are similar to what scientists, mathematicians, writers, and historians do" (p. 318). "It allows them to investigate questions, propose hypotheses and explanations, discuss and challenge their ideas of others." This is why we have incorporated investigations and benchmark lessons. We allow students to cooperate with each other, creating discussions. According to Krajcik and Blumenfeld (2006) “project-based learning provides opportunities for students, teachers, and members of society to collaborate with one another to investigate questions” (p. 325). This allows students to effectively communicate and explain to one another their explanations, leading to discussions and new understandings.

- Surface area
- Parallel and perpendicular lines
- Coordinate planes
- Ratio and Rates

The driving question of this lesson is “How is math involved in the logistics of building a city?” This is relevant because students see their city on a daily basis, but fail to take into account all the mathematics and planning that went into building it. They will be able to apply previous knowledge they’ve gained from observing the city around them in designing their own city. For this unit, students are allowed to extensively collaborate with partners during the benchmarks and investigations. They will be allowed one partner for the final artifact: the miniature model of their new city. Calculators will be used to save time various times throughout the lesson. Students will be allowed computers/tablets for small amounts of research throughout certain parts of the lesson as well.

Project based learning is a way for students to gain a deeper understanding of material by actively using their previous knowledge with new ideas. Krajcik and Blumenfeld informs us that "in project based learning, students engage in real, meaningful problems that are important to them and that are similar to what scientists, mathematicians, writers, and historians do" (p. 318). "It allows them to investigate questions, propose hypotheses and explanations, discuss and challenge their ideas of others." This is why we have incorporated investigations and benchmark lessons. We allow students to cooperate with each other, creating discussions. According to Krajcik and Blumenfeld (2006) “project-based learning provides opportunities for students, teachers, and members of society to collaborate with one another to investigate questions” (p. 325). This allows students to effectively communicate and explain to one another their explanations, leading to discussions and new understandings.

## Objectives

Students will be able to:

- create a three-dimensional city by using the equation and calculation of surface area to construct buildings, including parallel and perpendicular lines based on the properties and definitions, evaluating the outcome on coordinate planes, and comparing their city’s blueprint to actual size by analyzing and calculating ratios and rates.
- define and calculate surface area

- define and recognize parallel lines, perpendicular lines, coordinate planes, ratios, and rates
- solve equations to find the parallel or perpendicular line to it

- build real-world 3-dimensional figures
- calculate and create rates and ratios relating their model of the city to a life-size city
- calculate and determine the dimensions of the buildings and other features of the city
- compare the ratios between similar figures and solids
- understand when to use area, volume, and surface area formulas to use for each shapes
- calculate the areas, volume, and surface areas
- determine what 3-D shape is represented by a particular 2-D net
- construct 3-D figures using isometric and orthographic representations of the figures
- spell all vocabulary words correctly
- listen attentively and speak clearly when presenting final artifacts to class

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

Geometry:

- (2) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to:
- (A) use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships; and
- (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.
- (5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to:
- (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties;
- (B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles;
- (C) use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and
- (D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.
- (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.
- (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; and
- (C) derive and use formulas involving length, slope, and midpoint.
- (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;
- (E) use area models to connect geometry to probability and statistics; and
- (F) use conversions between measurement systems to solve problems in real-world situations.
- (9) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to:
- (A) formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models;
- (B) formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models;
- (C) formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and
- (D) analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models.
- (10) Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to:
- (A) use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane; and
- (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;

- (19) Oral and Written Conventions/Spelling. Students spell correctly. Students are expected to spell correctly, including using various resources to determine and check correct spellings.

- (24) Listening and Speaking/Listening. Students will use comprehension skills to listen attentively to others in formal and informal settings.

- (25) Listening and Speaking/Speaking. Students speak clearly and to the point, using the conventions of language. Students will continue to apply earlier standards with greater complexity. Students are expected to give presentations using informal, formal, and technical language effectively to meet the needs of audience, purpose, and occasion, employing eye contact, speaking rate (e.g., pauses for effect), volume, enunciation, purposeful gestures, and conventions of language to communicate ideas effectively.

- (26) Listening and Speaking/Teamwork. Students work productively with others in teams. Students will continue to apply earlier standards with greater complexity. Students are expected to participate productively in teams, building on the ideas of others, contributing relevant information, developing a plan for consensus-building, and setting ground rules for decision-making.

## Alternate Conceptions/ Common Struggles to Watch For

- Many times, students have trouble deciphering between surface area and area. To avoid this, be overly clear about what makes surface area different when introducing the concept to students.
- Some students are not visual learners; they will have trouble visualizing 3-D figures using only 2-D nets. Be sure to provide students with manipulatives or 3-D pictures whenever possible.
- Students are often put off by fractions. When introducing rates and ratios try to start with whole numbers and slowly introduce fractions.
- Units of measurement must ALWAYS be specified. Get students used to writing what unit of measurement they are using in order to avoid confusion when making the final product.