Middle School Mathematics Grade 8 Unit 1
Subject: MathematicsGrade: 8
Timeline: 27 days
Unit 1 Title: Looking for Pythagoras
In Looking For Pythagoras, students explore two main ideas: The Pythagorean Theorem and square roots. They also review and make connections among the concepts of area, distance and irrational numbers.
At the end of this unit, students should have mastered the following skills and abilities:
- Relate the area of a square to the side length.
- Estimate the values of square roots of whole numbers.
- Locate irrational numbers on a number line.
- Develop strategies for finding the distance between two points on a coordinate grid.
- Understand and apply the Pythagorean Theorem.
- Use the Pythagorean Theorem to solve everyday problems.
- Distinguish between rational and irrational numbers.
- Estimate irrational numbers by comparing them to rational numbers.
PA.CCSS.Math.Content.CC.2.1.6.E.4 Apply and extend previous understandings of numbers to the system of rational numbers. (6.NS.C.8)
PA.CCSS.Math.Content.CC.2.3.7.A.1. Visualize and represent geometric figures and describe the relationships between them. (7.G.A.2)
PA.CCSS.Math.Content.CC.2.3.7.A.3. Solve real-world and mathematical problems involving angles measure, area, surface area, circumference, and volume.(7.G.B.6)
PA.CCSS.Math.Content.CC.2.3.8.A.2. Understand and apply the Pythagorean Theorem to solve problems. (8.G.B.6, 8.G.B.7, 8.G.B.8)
PA.CCSS.Math.Content.CC.2.2.8.B.1. Apply concepts of radicals and integer exponents to generate equivalent expressions. (8.EE.A.2)
PA.CCSS.Math.Content.CC.2.2.8.B.2. Understand connections between proportional relationships, lines, and linear equations.. (8.EE.B.6)
PA.CCSS. Math.Content.CC.2.1.8.E.1. Distinguish between rational and irrational numbers using their properties. (8.NS.1)
PA.CCSS.Math.Content.CC.2.1.8.E.4. Estimate irrational numbers by comparing them to rational numbers. (8.NS.A.2)
Mathematical Practice Standards:
#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 consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. 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.
#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.
#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.
#7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.
#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. 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.
Concepts - Students will know:
- The coordinate system by exploring distances on a coordinate grid.
- Properties of quadrilaterals by drawing shapes on a coordinate grid.
- Similar triangles have similar slopes.
- How to find the area of a square using dot paper.
- How to find the side length of a square by taking the square root of its area.
- The Pythagorean Theorem is the area of square one added to square two which equals the area of square three.
- The Pythagorean Theorem can be written as a2 + b2 = c2
- The Pythagorean Theorem can be used to find the distance between two points on a grid.
- How to use the Pythagorean Theorem to differentiate between a right and non-right triangle.
- How to estimate the square roots of irrational numbers by using the Wheel of Theodorus.
- Irrational numbers can be estimated by comparing the number to rational numbers.
- How to distinguish between rational and irrational numbers using their properties.
Competencies -Students will be able to:
- Review the coordinate system by exploring distances on a coordinate grid.
- Review properties of quadrilaterals by drawing shapes on a coordinate grid.
- Draw similar triangles and determine their slopes.
- Find the area of squares using dot paper.
- Find how to find the side length of a square by using the square root of a known area.
- Collect data about the area of three squares and look for patterns between their areas.
- Work with a puzzle to prove at a2 + b2 = c2 in a right triangle.
- Apply the Pythagorean Theorem to find the distance between two points on a grid.
- Use the Pythagorean Theorem to differentiate between right and non-right triangles.
- Estimate the square roots of irrational numbers by using the wheel of Theodorus.
- Estimate irrational numbers using rational numbers.
- Show decimal expansion that repeats and convert a decimal expansion that repeats into a rational number.
- Informal assessments on learning targets
- Check-up #1
- Partner Quiz
- Common Core unit assessment- Looking For Pythagoras
Elements of Instruction:
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Each lesson and/or unit offers a wide variety of ways to differentiate for all levels of learners. These include:
- Special Needs Handbook
- Unit Projects
- Spanish Additional Practice and Skills guide
- Strategies for English Language Learners Guide
- “Extension” homework questions
- District-created notebooks
- 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)