Middle School Mathematics Grade 7 Unit 6
Subject: Mathematics
Grade: 7
Timeline: 22 days
Unit 6 Title: Filling and Wrapping
Unit Overview:
This unit is a combination of the book Filling and Wrapping Investigations 2 and 5 and CMP2 Common Core Additional Investigation 4. Students will explore the volume and surface area of prisms and cylinders. They also look informally at how changing the scale of a box affects its surface area and volume. (Filling and Wrapping, pg.3). Exploring geometric topics that cover cross sections, circles, drawing triangles, and special angle relationships are also a unit focus (Common Core Additional Investigations Teacher’s Guide, pg. 27).
Unit Objectives:
At the end of this unit, all students must:
- Understand volume as a measure of filling an object and surface area as a measure of wrapping an object
- Develop strategies for finding the volumes of prisms directly and by comparison with known volumes
- Understand how changes in one or more dimensions of a rectangular prism affect the prism’s volume and surface area
- Relate their understanding of similarity and scale factors to three-dimensional figures
- Use surface area and volume to solve a variety of real-world problems
- Understand the relationships between angle measures and sides when creating unique triangles
- Describe the relationship between two-dimensional cross sections of three-dimensional figures
- Find area and circumference of a circle
- Understand complementary, vertical, and adjacent angles to write and solve simple equations for unknown angles in a figure
Focus Standards:
PA.CCSS.Math.Content.CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. (7.EE.4)
PA.CCSS.Math.Content.CC.2.3.7.A.1 Visualize and represent geometric figures and describe the relationships between them. (7.G.1, 7.EE.3, 7.EE.2)
PA.CCSS.Math.Content.CC.2.3.7.A.3 Solve real-world and mathematical problems involving angle measure, area, surface area, circumference and volume. (7.G.5, 7.G.4, 7.G.6)
Mathematical Practice Standards:
#3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
#4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
#5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
#6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
Concepts - Students will know:
- Volume and surface area of prisms
- How to use volume to solve problems
- That a variety of different three-dimensional figures can have the same volume but different surface areas
- How changes in one or more dimensions of a rectangular prism affect the prism’s volume
- How to design rectangular prisms with a given volume
- The effect on surface area of applying a scale factor to a rectangular prism
- The effect on volume of applying a scale factor to a rectangular prism
- Similarity of three-dimensional shapes
- Scale factor and its relationship to changes in one- two- and three-dimensional measures
- How to determine a unique triangle, more than one triangle, or no triangle when given three measures of their angles or sides
- How to slice three-dimensional figures to get two-dimensional cross sections
- How to use formulas for the area and circumference of a circle
- Informal derivation of the relationship between a circle’s area and its circumference
- The difference between complementary, vertical, and adjacent angles
Competencies -Students will be able to:
- Make paper prisms
- Find and compare volume and number of faces of a prism
- Find surface area of prisms
- Draw nets of a rectangular prism
- Label dimensions of prism
- Make scale models of a prism
- Compare the volume of a doubled scale model of a original prism to the actual volume of the original size prism
- Complete a table that includes dimensional measures, scale factors, surface areas, volumes of scaled up and scaled down prisms
- Describing the change in surface area and volume to different scale factors of an original prism
- Use scale models and scale factors to calculate actual dimensions and surface areas and volumes of large items
- Slice three-dimensional figures to create cross-sections that make two-dimensional shapes
- Calculate and compare the areas and ratios of areas of two different-size circles
- Calculate and compare the ratio of the radius of two different-size circles
- Calculate circumference of a circle
- Compare the circumference of a circle to its area
- Write a formula for circumference of any circle
- Draw triangles and identify the lengths of the sides
- Draw triangles and identify the angle measures
- Draw triangles given sides lengths and angle measures
- Name vertical, adjacent and complementary angles given a diagram
- Write and solve equations to find angle measures
Assessments:
Formative Assessments:
- Informal assessments on learning targets
- Common Core Additional Investigation 4 Check-Up Quiz
- Filling and Wrapping Partner Quiz
Summative Assessments:
- Common Core Unit Assessment- Filling and Wrapping
Elements of Instruction:
In Grade 6, instructional time focused on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.
Filling and Wrapping builds on the understanding that area is the number of squares that cover a two-dimensional figure and perimeter is the number of linear units that surround a two-dimensional figure, which expands on the foundation laid in the Grade 6 Unit of Covering and Surrounding by interpreting volume as the number of unit cubes that fill a three-dimensional figure and surface area as the number of squares that cover or wrap the exterior of a three-dimensional figure. The concepts of area and perimeter also lay the foundation for algorithms of volume and surface area. Enlarging, shrinking, and distorting two-dimensional figures (Stretching and Shrinking) and scaling quantities up and down using ratios and proportions (Comparing and Scaling) help students study the effects of applying scale factor to the dimensions of a prism to its volume and surface area in Filling and Wrapping (Filling and Wrapping, pg. 7).
Students compare the volumes and surface areas of a variety of prisms with regular bases and a common height. Students build prisms by folding several sheets of congruent rectangular paper into the shapes of triangular, rectangular, and hexagonal prisms. They observe that the volume of a prism increases as the number of lateral sides increases. The volume of a prism is the area of the base (the number of unit cubes in the bottom layer) of the prism multiplied by its height (the number of layers). Surface area is informally looked at as the sum of the area of the bases and lateral sides (rectangle).
Students also study the effects of changing the dimensions or the volume of a rectangular prism in the context of designing compost containers. They explore two central ideas: how to double the volume of a rectangular prism and examine how other measures change as a result, and the effects of applying scale factors to the dimensions of rectangular prisms. Students apply their knowledge of similarity and scale factors to explore the relationships between a model of a cruise ship (Filling and Wrapping, pg. 3).
As students investigate three-dimensional figures, it is recommended to have solids available to demonstrate cross sections. Providing opportunities for students to make table displays, coloring cuts, and manipulating the figures will aid in understanding.
A review of the parts of a circle will be necessary as students compare their attributes. The same is true and necessary for triangles. It will be necessary to include instruction and practice related to supplementary angles, as well. A working definition of “similar figures” will be required of students as they work comparing and contrasting figures and their measures. One purpose of this unit is to find the measures of a figure without actually measuring the angles (Common Core Additional Investigation Teacher’s Guide, pg. 27).
Major misconceptions by and struggles for students in this unit include:
- Confusing volume with surface area
- Finding only partial surface areas
- Correctly using formulas to find the area of the base of three-dimensional shapes
- Understanding that doubling volume does not mean just doubling dimensions
- Connecting scale factor to the change in size of similar three-dimensional shapes
- Accurately drawing two- and three-dimensional figures
- Confusing volume with surface area
- Confusing diameter with radius
- Properly using formulas
- Properly using protractors
- Confusing the definitions of complementary, supplementary, vertical and adjacent angles
Differentiation:
Each lesson has differentiation options for each portion of the lesson. Additional differentiation options are listed with directions and student masters in the Teacher’s Guide.
- Special Needs Handbook
- Unit Projects
- Spanish Additional Practice and Skills guide
- Strategies for English Language Learners Guide
- “Extension” homework questions
- District-created notebooks
Interdisciplinary Connections:
- Mathematical Reflections
- “Did You Know?” sections
- phschool.com and web codes
- “Connections” homework questions
- The real-world context embedded in lesson problems
Additional Resources / Games:
CMP2 and/or the Erie School District provide the following additional resources to aid students in achieving mathematical success.
- Additional Practice worksheets per investigation
- Skills Review worksheets to target key components of each investigation
- Parent letter to be sent home prior to beginning the unit to share with parents the skills, goals, and expectations of the coming unit.
- Assessment Resources workbook with extra test items (multiple choice, essay, open ended, question bank, etc)
- Investigation specific pre-generated notebooks that include tables, graphs, problem numbers, and all other items students may need to complete the investigation and all its parts. Students are provided with one per unit.
- Reflection questions at the end of each investigation to assess students’ comprehension of key concepts.
- phschool.com and web codes
- Transparencies of models, graphs, etc used within lesson(s)