Grade 7 Mathematics, Quarter 4, Unit 4.1

Proportional Reasoning with Scale Drawings Overview Number of instructional days:

10

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Solve problems given a scale drawing.

Model with mathematics.

•

Given a scale drawing, find lengths and areas.

•

•

Reproduce a scale drawing at a different scale.

•

Create scale drawings based on a geometric figure.

Solve problems in everyday life using the math they know according to their grade level of exposure.

•

Identify important information in a problem.

•

Look at their solutions/conclusions and decide if they are reasonable.

Use appropriate tools strategically. •

Use tools (e.g., integer models, number lines) to model and solve problems.

•

Recognize both the insight to be gained from the employment of the tools and their limitations.

•

Detect possible errors by strategically using estimation and other mathematical knowledge.

•

What is the relationship between the area of the geometric figure compared to the area of the scale representation? How are they proportional?

Essential questions •

Why are scale drawings important in the real world?

•

What steps are necessary when creating a scale drawing from a geometric figure?

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

47  

Grade 7 Mathematics, Quarter 4, Unit 4.1

Proportional Reasoning with Scale Drawings (10 days)

Written Curriculum Common Core State Standards for Mathematical Content Geometry

7.G

Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.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.

Common Core Standards for Mathematical Practice 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 early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. 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, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. 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.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

48  

Grade 7 Mathematics, Quarter 4, Unit 4.1

6

Proportional Reasoning with Scale Drawings (10 days)

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. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Clarifying the Standards Prior Learning In grade 6, students do not explicitly work with scale drawings. However, they have a lot of exposure to ratios and equivalent ratios. Students also produce drawings of polygons on a coordinate plane. Current Learning In grade 7, the students solve problems using scale drawings. They use the scale drawings to compute unknown lengths and areas. Students also are challenged with the task of creating a scale drawing from an existing scale drawing. They describe the relationships between these drawings. Future Learning In grade 8, students must describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures, using coordinates. They also explore the concept of similarity between two 2-dimensional figures.

Additional Findings According to Principles and Standards for School Mathematics, “Problems that involve constructing or interpreting scale drawings offer students opportunities to use and increase their knowledge of similarity, ratio, and proportionality. Such problems can be created from many sources, such as maps, blueprints, science, and even literature. For example, in Gulliver’s Travels, a novel by Jonathan Swift, many passages suggest problems related to scaling, similarity, and proportionality. Another interesting springboard for such problems is “One Inch Tall,” a poem by Shel Silverstein (1974) …” In connection with the poem, a teacher could pose a problem like the following: Use ratios and proportions to help you decide whether the statements in Shel Silverstein’s poems are plausible. Imagine that you are the person described in the poem, and assume that all your body parts changed in proportion to the change in your height. Choose one of the following and investigate and write a complete report of your investigation, including details of any measurements you made or calculations you performed. In the poem the author says that you could ride a worm to school. Is this statement plausible? Would it be true that you could ride a worm if you were 1 inch tall? Use the fact that common earthworms are about 5 inches long with diameters of about 1/4 inch.   Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

49  

Grade 7 Mathematics, Quarter 4, Unit 4.1

Proportional Reasoning with Scale Drawings (10 days)

In the poem the author says that you could wear a thimble on your head. Would this be true if you were only one inch tall? Use one of the thimbles in this activity box to help you decide. Another book to consider would be Two Bad Ants by Chris Van Allsburg.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

50  

Grade 7 Mathematics, Quarter 4, Unit 4.2

Length and Area in Two-Dimensional Figures Overview Number of instructional days:

10

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Solve problems with area and circumference.

Model with mathematics.

•

Determine the relationship between the circumference and the area of circles.

•

Apply proportional reasoning to plan a school event or analyze a problem in the community.

•

Find the area of select two-dimensional figures.

•

•

Determine the volume of select threedimensional figures.

•

Determine the surface area of select threedimensional figures.

Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas.

•

Mathematically analyze those relationships to draw conclusions.

Attend to precision. •

Carefully specify units of measure, and label axes to clarify the correspondence with quantities in a problem.

•

Calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context.

•

What is the difference between volume and surface area? How are they the same? How do they differ?

Essential questions •

Why is being able to determine area and volume important in the real world?

•

Why is it important to express mathematical quantities with the appropriate units?

•

•

How would one determine the surface area of a rectangular prism? What steps must be taken?

Which calculation would you use to determine the volume of a three-dimensional object?

•

•

What are the methods for finding the surface area of a rectangular prism?

What is the relationship between the area of a circle and the circumference of a circle?

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

51  

Grade 7 Mathematics, Quarter 4, Unit 4.2

Length and Area in Two-Dimensional Figures (10 days)

Written Curriculum Common Core State Standards for Mathematical Content Geometry

7.G

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

7.G.6

Solve real-world and mathematical problems involving area, volume and surface area of twoand three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Common Core Standards for Mathematical Practice 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 early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. 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, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. 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.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

52  

Grade 7 Mathematics, Quarter 4, Unit 4.2

6

Length and Area in Two-Dimensional Figures (10 days)

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. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7

Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Clarifying the Standards Prior Learning In grade 6, students’ knowledge of area is focused primarily on right triangles and quadrilaterals. They explore the volume only in terms of a right, rectangular prism. Students are introduced to the formulas for volume (V = lwh) and apply the m to real-world mathematical problems. They draw and represent threedimensional figures using nets. Current Learning In grade 7, students find the area of many two-dimensional shapes and explore more deeply the connection between the circumference of a circle and the area of a circle. Students solve real-world problems involving a variety of shapes and determine their volumes and surface areas. Future Learning In grade 8, students will be challenged to apply the concepts of congruency and similarity to twodimensional shapes. These two-dimensional figures will be dilated, rotated, translated, and reflected. Also, students will extend their knowledge of volume to cones, pyramids, and spheres.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

53  

Grade 7 Mathematics, Quarter 4, Unit 4.2

Length and Area in Two-Dimensional Figures (10 days)

Additional Findings “Whether wallpapering an odd-shaped room or painting the walls of a Gothic cathedral, you may find the area of any plane figure that is bounded by straight lines simply by dividing and conquering: Partition the figure into triangles and rectangles, find the area of each of these triangles and rectangles, and then add up these areas to get the area of the whole figure. Using this method on regular polygons, one can derive the formula A = ½ x TP, where P is the perimeter or distance around the regular polygon, and T is the perpendicular distance from the central point of the polygon to the side.” (Beyond Numeracy, Paulos, 1992, pp. 19–20). “An understanding of the concepts of perimeter, area, and volume is initiated in lower grades and extended and deepened in grades 6–8. Whenever possible, students should develop formulas and procedures meaningfully through investigation rather than memorize them. Even formulas that are difficult to justify rigorously in the middle grades, such as that for the area of the circle, should be treated in ways that help students develop an intuitive sense of their reasonableness. “One particularly accessible and rich domain for such investigation is area of parallelograms, triangles, and trapezoids. Students can develop formulas for the areas of these shapes using what they have previously learned about decomposing a shape and rearranging its component parts without overlapping does not affect the area of the shape.” (Principles and Standards for School Mathematics, NCTM, 2000, p. 244).

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

54  

Grade 7 Mathematics, Quarter 4, Unit 4.3

Plane Sections, Volume, and Surface Area of Three-Dimensional Figures Overview Number of instructional days:

10

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Slice three-dimensional figures into twodimensional components.

Attend to precision.

•

Describe the two-dimensional components that appear once the three-dimensional figure has been sliced.

•

Solve real world problems involving volume and surface area.

•

Determine the volume of select threedimensional figures.

•

Determine the surface area of select threedimensional figures.

•

Determine the differences and similarities between prisms and pyramids.

•

Carefully specify units of measure and label axes to clarify the correspondence with quantities in a problem.

•

Calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context.

Make sense of problems and persevere in solving them. •

Draw diagrams of important features and relationships, graph data, and search for regularity or trends.

•

Check answers to problems using a different method, and continually ask, “Does this make sense?”

•

What is a real-world situation in which a person would use area, surface area, angle measure, and volume?

Essential questions •

How do you slice a three-dimensional figure in order to examine its two-dimensional components?

•

How can you describe two-dimensional figures • after slicing three-dimensional figures? What do you notice about the slices of the rectangular prism compared to the slices of the rectangular pyramid?

What do right rectangular prisms and right rectangular pyramids have in common? How are they different?

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

55  

Grade 7 Mathematics, Quarter 4, Unit 4.3

Plane Sections, Volume, and Surface Area of Three-Dimensional Figures (10 days)

Written Curriculum Common Core State Standards for Mathematical Content Geometry

7.G

Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.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.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.6

Solve real-world and mathematical problems involving area, volume and surface area of twoand three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Common Core Standards for Mathematical Practice 1

Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 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 make conjectures and build a logical progression of statements to explore the truth of their conjectures. 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. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. 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.   Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

56  

Grade 7 Mathematics, Quarter 4, Unit 4.3

5

Plane Sections, Volume, and Surface Area of Three-Dimensional Figures (10 days)

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, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. 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. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Clarifying the Standards Prior Learning In grade 6, students solve a plethora of real world problems involving area, surface area, and volume. The problems are solved in such a way that the students mainly compose and decompose quadrilaterals and polygons. Current Learning In grade 7, students examine the differences/similarities of prisms and pyramids by slicing these threedimensional shapes into their two-dimensional components. They continue to solve real-world problems involving area, surface area, and volume. Future Learning In grade 8, students understand congruence and similarity using physical models as well as geometric software. They solve real-world problems involving volume of cylinders, cones, and spheres.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

57  

Grade 7 Mathematics, Quarter 4, Unit 4.3

Plane Sections, Volume, and Surface Area of Three-Dimensional Figures (10 days)

Additional Findings According to A Research Companion to Principles and Standards for School Mathematics, “At Level 3, the abstract/relational level, students can form abstract definitions, distinguish between necessary and sufficient sets of conditions for a concept, and understand and sometimes even provide logical arguments in the geometric domain. They can classify figures hierarchically by ordering their properties and can give informal arguments to justify their classifications (e.g., a square is identified as a rhombus because it can be thought of as a “rhombus with some extra properties”). They can discover properties of classes of figures by informal deduction. For example, they might deduce that in any quadrilateral, the sum of the angles must be 360 degrees. Because any quadrilateral can be decomposed into two triangles whose angles sum to 180 degrees.” (p. 153) According to Beyond Numeracy (John Allen Paulos, 1991), “There is an intriguing interplay among the notion of area, volume, and basic physics. Note that the support needed by people, animals, and general structures to stand upright is proportional to their cross sectional areas, while their weight is proportional to their volumes. For example, quadrupling the height of a structure and preserving its proportions and material makeup will result in a 64-fold (4’) increase in its weight, but only a 16-fold (42)increase in its ability to support the weight. This is why any 25-foot-tall monster ambling about the Himalayas or sunning himself on some beach in the Bermuda triangle could not possibly be proportioned as we are. This relation also put constraints on heights, proportions, and materials of trees, buildings, and bridges. Related considerations help to explain other structural features (including the surface area of lung and intestine tissues) of plants, animals, and inanimate objects (see Fractals).”

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

58  

Grade 7 Mathematics, Quarter 4, Unit 4.4

Geometric Constructions Overview Number of instructional days:

10

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Solve multistep problems involving complementary and supplementary angles.

Use appropriate tools strategically.

•

Solve multistep problems involving vertical and adjacent angles.

•

Write and solve simple equations to determine the value of an unknown angle.

•

Utilize a ruler and protractor to draw geometric shapes freehand.

• •

•

Use tools that might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.

•

Utilize geometric software packages to construct geometric shapes.

Make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

•

Determine if certain criteria/conditions form a triangle or possibly more than one triangle.

Use technological tools to explore and deepen their understanding of concepts.

Look for and make use of structure. •

Recognize the significance of an existing line in a geometric figure.

•

Use the strategy of drawing an auxiliary line for solving problems.

Essential questions •

What is the difference between complementary and supplementary angles?

•

How do you use a ruler and protractor to draw a triangle?

•

What is the difference between vertical and adjacent angles?

•

•

What steps are necessary to write and solve an equation to find a missing angle of a triangle?

What conditions must be satisfied in order for a triangle to be constructed? What conditions make it impossible for a triangle to be constructed?

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

59  

Grade 7 Mathematics, Quarter 4, Unit 4.4

Geometric Constructions (10 days)

Written Curriculum Common Core State Standards for Mathematical Content Geometry

7.G

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.5

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.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.

Common Core Standards for Mathematical Practice 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, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 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. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. 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. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

60  

Grade 7 Mathematics, Quarter 4, Unit 4.4

7

Geometric Constructions (10 days)

Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8

Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Clarifying the Standards Prior Learning In grade 6, students drew figures on a coordinate plane. Angles were not explored in great detail, students constructed vertices of polygons on a coordinate plane. Students solve real-world problems involving area of two-dimensional shapes. Current Learning In grade 7, students are introduced to many new concepts and vocabulary involving angles. Students’ ability to draw is extended beyond the coordinate plane by using a ruler and protractor and/or geometry software. Future Learning In grade 8, the geometry focus will be primarily on congruency and similarity. Students will however conjecture and solve problems regarding parallel lines cut by a transversal as well as the exterior angle theorem of triangles.

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

61  

Grade 7 Mathematics, Quarter 4, Unit 4.4

Geometric Constructions (10 days)

Additional Findings According to A Research Companion to Principles and Standards for School Mathematics, “More recent developments have led to the creation of several interactive geometry programs, such as Cabri-Geometry (Baulac, Bellemain, & Laborde, 1988) and Geometer’s Sketchpad (Jackiw, 1995). Interactive geometry software may change the nature of the relations between what Laborde (1996) calls ‘diagrams’ (physical, spatial properties) and ‘theory’ (geometrical properties). Consistent with the aforementioned discussion of diagrams, she argues that diagrams play an ambiguous role in instruction. Students believe that it is possible to abstract the properties of geometric objects from diagrams directly and thus deduce a property empirically. Interactive geometry software introduces a new type of diagram whose behaviors is controlled by theory. Students’ actions require construction of an interpretation in which visualization plays a crucial role but geometrical properties constrain such action. These characteristics produce a stronger connection between diagrams and theory because spatial invariants in the moving diagrams are almost certainly the representation of geometrical invariants. Work with students supports this hypothesis and shows that teachers must encourage students to move from performing construction tasks at the spatial-diagrammatic level to establishing links between that and the theoretical level.” (p. 159)

  Cranston  Public  Schools,  with  process  support  from  the  Charles  A.  Dana  Center  at  the  University  of  Texas  at  Austin    

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