Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean...

Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin

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Grade 8 Mathematics, Quarter 3, Unit 3.1

Pythagorean Theorem

Overview Number of instructional days: 15 (1 day = 45–60 minutes)

Content to be learned Mathematical practices to be integrated • Prove the Pythagorean Theorem.

• Given three side lengths, use the Pythagorean Theorem to determine if it forms a right triangle (Pythagorean Theorem converse).

• Determine unknown side lengths in right triangles using the Pythagorean Theorem (both real-world and mathematical problems) in two and three dimensions.

• Apply the theorem to find the distance between two points on the coordinate plane.

Make sense of problems and persevere in solving them.

• Check answers, and ask, “Does this make sense?”

• Draw diagrams of important features and relationships.

Construct viable arguments and critique the reasoning of others.

• Discuss with others the theorem and its converse.

• Construct arguments using prior knowledge.

• Recognize and use counterexamples, and respond to arguments of others.

Model with mathematics.

• Apply the Pythagorean Theorem to real-world problems.

• Use mathematical tools to draw conclusions about relationships.

Essential questions • How can you prove the Pythagorean Theorem?

• What does the converse of the Pythagorean Theorem prove?

• What is the difference between a leg and the hypotenuse of a right triangle?

• How do you find the length of the hypotenuse given the lengths of both legs of a right triangle?

• How do you find the length of a leg, given the length of the second leg and the hypotenuse of a right triangle?

• How could you use the Pythagorean Theorem to find the distance between two points on the coordinate plane?

• How would you apply the Pythagorean Theorem to a real-world situation?

Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean Theorem (15 days)


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Written Curriculum

Common Core State Standards for Mathematical Content

Geometry 8.G

Understand and apply the Pythagorean Theorem.

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

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



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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.

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.

Clarifying the Standards

Prior Learning

In grade 6, students found vertical and horizontal distances on the coordinate plane.

In grade 7, students focused on constructing triangles from three measures of angles or sides. They solved real-world and mathematical problems involving two- and three-dimensional objects composed of triangles.

Current Learning

In grade 8, this unit is a major cluster and a critical area. Students further their knowledge of rational and irrational numbers. They also continue to estimate the value of expressions with irrational numbers. Students explain a proof of the Pythagorean Theorem and its converse. They apply the theorem to determine missing side lengths in a right triangle in real-world and mathematical problems (with two and three dimensions). Students also use the Pythagorean Theorem to find the distance between two points on the coordinate plane.

Future Learning

In high school, students will prove theorems about lines, angles, triangles, and parallelograms. They will also construct shapes to meet certain requirements. Students will use coordinates to prove simple geometric theorems algebraically.

Additional Findings

According to Principles and Standards for School Mathematics, “Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as … the Pythagorean relationship.” (p. 397)



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According to PARCC Model Content Framework,

“By working with equations such as x2 = 2, and in geometric concepts such as Pythagorean Theorem, students enlarge their concepts of numbers beyond the system of rationals to include irrational numbers.” (p. 35)

“The Pythagorean Theorem can provide opportunities for students to construct viable arguments and critique the reasoning of others. (e.g., if a student in the class seems to be confusing the theorem with its inverse).” (p. 37)


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Rules of Exponents and Scientific Notation


Content to be learned Mathematical practices to be integrated • Understand properties of integer exponents.

• Understand the rules for division with

exponents.

• Express very large or very small quantities using scientific notation.

• Enter and interpret scientific notation on a calculator or other technology using E or EE (scientific notation), * (multiplication), and ^ (exponent) symbols.

• Perform calculations with numbers in scientific notation and standard form.

• Express solutions to calculations in scientific notation or standard form.

• Choose appropriate units for measurements of very large or very small quantities.

• Estimate how much larger or smaller numbers expressed in scientific notation are to one another.

Reason abstractly and quantitatively.

• Attend to the meaning of quantities, not just how to compute them.

• Apply numbers written in scientific notation to a real-world situation.

Use appropriate tools strategically.

• Know how to use a calculator or other technology to compute scientific notation.

• Do computations quickly on a calculator.

Look for and express regularity in repeated reasoning.

• Understand properties of integer exponents by examining patterns of exponents.

• Understand that multiplying by a power of 10 results in a repeated movement of the decimal.

Grade 8 Mathematics, Quarter 3, Unit 3.2 Rules of Exponents and Scientific Notation (15 days)


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Essential questions • How can the rules of exponents make

calculating values easier?

• How could you use patterns to explain rules of exponents?

• Why is scientific notation needed?

• How do you know when a number is properly written in scientific notation?

• How do you convert between standard form and scientific notation?

• How do you compare two numbers written in scientific notation?

Written Curriculum


Expressions and Equations 8.EE

Work with radicals and integer exponents.

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.


2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.



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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.

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.


Prior Learning

Students have denoted whole number powers of 10 with exponential notation since grade 5, and they have seen the pattern in the number of zeros when powers of 10 are multiplied.

In grade 6, students looked at whole number exponents. They also worked with operations of fractions. In grade 7, students worked with integers and fractions.

Current Learning

In grade 8, this unit is a major cluster. Students know and apply properties of integer exponents and create equivalent expressions. (Know that 32 × 3-5 = 3-3 = 1/33 = 1/27) They express very large or very small quantities using powers of 10. Students understand what scientific notation looks like through technology. They choose units of appropriate size for measurements.

Future Learning

In high school, students will extend the properties of exponents to irrational exponents. They will rewrite and evaluate expressions involving radicals and rational number exponents.



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Additional Findings

According to Principles and Standards for School Mathematics, students “develop an understanding of large numbers and recognize and approximately use exponential, scientific, and calculator notation.” (p. 393)

According to PARCC Model Content Framework, “Scientific notation presents opportunities for strategically using appropriate tools. For example, the computation (1.73 × 10-4) × (1.73 × 10-5) can be done quickly with a calculator by squaring 1.73 and then using properties of exponents to determine the exponent of the product by inspection.” (p. 37)


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Transformations on the Coordinate Plane


Content to be learned Mathematical practices to be integrated • Use transformations to show line segments are

congruent.

• Use transformations to show angles are congruent.

• Investigate side lengths and angles prior to and after the transformation to verify it is the same figure.

• Understand that figures are congruent if they can be obtained from a sequence of transformations.

• Describe the sequence of translations, reflections, and rotations to show congruency.

• Describe the effect of transformations (dilations, translations, rotations, reflections) on a figure using its coordinates.

Construct viable arguments and critique the reasoning of others.

• Use previous knowledge or results to construct arguments.

• Make conjectures and build progression of statements to explore truth.

• Use appropriate mathematical language to describe transformations.

Use appropriate tools strategically.

• Use graph paper, pencil, rulers, and geometry software for transformations.

• Use technology (computers) to visualize the results of varying assumptions.

Attend to precision.

• Communicate translations precisely to others.

• Use clear definitions in reasoning.

• State meaning of chosen symbols.

Essential questions • How do you know if two figures are

congruent?

• How does each type of transformation change a figure’s location?

• Where can you find an example of transformations and dilations in the real world?

• How do transformations affect the coordinates of the vertices of two-dimensional figures?

Grade 8 Mathematics, Quarter 3, Unit 3.3 Transformations on the Coordinate Plane (12 days)


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Written Curriculum


Geometry 8.G

Understand congruence and similarity using physical models, transparencies, or geometry software.

8.G.1 Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.


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.

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



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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.


Prior Learning

In grade 6, students worked with properties of the coordinate plane. In grade 7, they worked with corresponding lengths and the relationships between those lengths. Students used similar figures and scale to find an unknown length.

Current Learning

According to PARCC Model Content Frameworks, this is a major cluster and critical area. Students work with transformations to identify congruent figures. They verify experimentally the properties of rotations, translations, and reflections. Students describe the effect of dilations on a two-dimensional figure using coordinates.

In the next unit, students expand on the knowledge to work with similar two-dimensional figures and analyzing angles measures.

Future Learning

In high school, students will experiment with transformations in the plane. They will understand congruence in terms of ridged motions. Students will be using concepts of similarity and congruence to prove theorems and make geometric constructions.

Later in eighth grade students will use this to work with similar figures.

Additional Findings

According to Principles and Standards for School Mathematics, “Students will examine the congruence and similarity of objects using transformations.”

According to PARCC Model Content Framework, “The angle-angle criterion for triangle similarity underlies the fact that a nonvertical line in the coordinate plane has the equation y = mx + b.”



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Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean...

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