Pre-AP Math 7 - Quarter 2 Properties of Exponents (Chapter 13) Essential Questions: • How are the laws of integral exponents used to evaluate algebraic expressions? • What are the key differences for simplifying expressions containing bases with positive exponents and bases with negative exponents? • When is it appropriate to express numbers using scientific notation? • How can scientific notation help to make the multiplication and division of large or small numbers more efficient? • When is it appropriate to add, subtract, multiply and divide exponents when simplifying algebraic expressions? Desired Outcomes: • The student will be able to simplify expressions using the properties of powers. • The student will be able to simplify expressions containing positive, negative and zero exponents. • The student will be able to write numbers in scientific notation when given standard form. • The student will be able to write numbers in standard form when given scientific notation. • The student will be able to perform operations using numbers written in scientific notation. Evidence of Learning: 1. 2. 3. 4. 5. 6. The student will simplify expressions using the properties of powers. The student will simplify expressions containing positive, negative and zero exponents. The student will write numbers in scientific notation when given standard form. The student will write numbers in standard form when given scientific notation. The student will perform operations using numbers written in scientific notation. -1- Pre-AP Math 7 - Quarter 2 Day / Lesson Days 44-47 Lessons-13.1-13.2 Organizers from 13.6 Instructional Focus Students will: • Expand a power into a product (factored form) • Write a product as a power. • Simplify expressions containing integer exponents. • Develop a rule to simplify a product of powers. • Develop a rule to simplify a power of a power. • Develop a rule to simplify a quotient of a power. Alignment 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. Essential Lesson Components Lesson 13.1 • Pre-assess students on the lesson’s material. Students may come with much of this knowledge already. Refer to pre-test in Teacher’s Resources and Assessments for guiding questions. • Be sure to do the ‘Check for Understanding’ on page 692A of the Teacher’s Implementation Guide. This is a difficult concept for students. Lesson 13.2 • Complete all of lesson 13.2. Lesson 13.6 • Use the organizers on pages 731-733 to summarize the rules. -2- Pre-AP Math 7 - Quarter 2 Level 1 (EOL1) Simplify each of the following. • • • • • • • • (−10)5 −10 5𝑥 3 ∙ 4𝑥 7 12𝑎7 4𝑎2 • Assessment Level 2 (EOL 1) Rewrite the expression 412 with a base of 2. After rewriting the expression use the property of powers to simplify the powers. • • (𝑥 5 𝑦 3 )4 Write values for x and y that make this statement true: 5x = 5y 5 x • Simplify the expression: 9 . x State your answer in two different ways. Using mathematics, explain how you determined your answers. Level 3 (EOL 1) Explain why 43 ⋅ 42 is not165. • for 53 Choose values for a, b, c, and d so that (4𝑎 )𝑏 = 412 and 4𝑐 ∙ 4𝑑 = 412 . Explain how you determined the values a, b, c, and d. Word Wall Words: base, exponent, power, product rule of powers, power to a power rule, and quotient rule of powers. Technology: • Gizmo: Finding Factors with Area Models; Multiplying Exponential Expressions; Dividing Exponential Expressions; and Exponents and power rules. • Understanding Exponents: The Exponent Rules DI Strategies/Thinking Maps: • In problem 1, Lesson 13.2 the book uses large numbers to introduce product rule of powers. For struggling learners you may want to begin with smaller whole numbers. • A tree map can be used to organize all the rules of exponents. Resources: • • • • Good Questions for Math Teaching 5-8: pg. 141, #15 activating prior knowledge of exponents Kagan Algebra1: p. 147-150, p. 151-155 Kagan Cooperative Learning Activities for HS Mathematics: p. 28, 283, 351 Blackboard: Laws of Exponents- Coach-player; Laws of Exponents- Inside Outside Circle, and Add, Multiply…Do Not Touch; Grouping Cards- Simplifying Monomials. -3- Pre-AP Math 7 - Quarter 2 Day / Lesson Days 48-50 Lesson 13.3 Organizers from 13.6 Instructional Focus Students will: • Simplify powers that have an exponent of zero. • Simplify powers that have negative exponents. Alignment 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. Essential Lesson Components Lesson 13.3 • Complete all parts of the lesson. • Be sure to do the ‘Check for Understanding’ on page 710A of the Teacher’s Implementation Guide. Lesson 13.6 • Use the organizers on pages 734-735 to summarize the rules. −4 Rewrite 3 Level 1 (EOL 2) with a positive exponent. Assessment Level 2 (EOL 2) Simplify 35 x −2 3 5 −3 4 2 3 y x • Level 3 (EOL 2) Prove that 40 is equal to 1. • Use mathematics to prove that 3 4−1 x 3 y 2 125 x 3 −1 2 4 = 64 y 6 5 x y Explain your answer using mathematics. • -4- Choose distinct values for a,b, and c such (𝑎𝑏)𝑐 = 1. Explain how you know that your values are correct. Pre-AP Math 7 - Quarter 2 • • • Technology: • Understanding Exponents: The Exponent Rules DI Strategies/Thinking Maps • In problem 1, Lesson 13.2 the book uses large numbers to introduce product rule of powers. For struggling learners you may want to begin with smaller whole numbers. • A tree map can be used to organize all the rules of exponents. Resources: • Kagan Algebra1: p.161-166 • Kagan Cooperative Learning Activities for HS Mathematics: p. 55-58 • www.powersof10.com -5- Pre-AP Math 7 - Quarter 2 Day / Lesson Days 51-58 13.4-13.5 Organizers from 13.6 Instructional Focus Students will: • Express numbers in scientific notation. • Express numbers in standard form. • Perform operations using scientific notation. • Use rules for significant digits in computation Alignment 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. 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. 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. Essential Lesson Components Lesson 13.4 • Have students do the warm up on page 711C of the Teacher’s Implementation Guide • Complete the whole lesson. • Make sure to use the “talk the talk” on pg. 718 to pull together all the different forms of writing numbers. You could use the chart or a tree map to organize the information. Lesson 13.5 • Complete the whole lesson. Lesson 13.6 • Use the organizers on pages 736 to summarize the rules. -6- Pre-AP Math 7 - Quarter 2 • Level 1 (EOL 3,4) Express the numbers in scientific notation: 0.0000587 • 23,4000,000 • Simplify the expression and express the answer in scientific notation. (1.2 × 105 )(3.7 × 103 ) Assessment Level 2 (EOL 4,5) Jane’s hair grows at a rate of 3.3 ×10−2 cm per day. At that rate how much longer would her hair be after one year? Use mathematics to explain how you determined your answer. • The distance from the sun to the center of the Milky Way galaxy is 2.5 × 1017 km. The distance from the sun to the dwarf planet Makemake is 5.76 × 109 km. How much further is the sun from the Milky Way galaxy than the Makemake planet? Explain how you determined your answer. • • • • • • • Level 3 (EOL 4,5) The approximate diameter of Jupiter is 1.43 ×108 meters and the diameter of Pluto is approximately 227.4 ×104 meters. Which planet has the greater diameter? Justify your answer. Both M and N are expressed in scientific notation. 𝑀 = 2.6 × 105 𝑁 = ∆ × 104 Is it possible to replace ∆ with a number so that N is larger than M? Explain your reasoning. Word Wall Words: scientific notation, order of magnitude, mantissa, characteristic Materials necessary: Graphing calculator Technology: • Understanding Exponents: Scientific Notation • Middle School SMART Notebook Add-ins: Scientific Notation and Scientific Notation Calculations DI Strategies/Thinking Maps • In problem 1, Lesson 13.2 the book uses large numbers to introduce product rule of powers. For struggling learners you may want to begin with smaller whole numbers. • A tree map can be used to organize all the rules of exponents. Resources: • Kagan Cooperative Learning Activities for HS Mathematics: p. 16 -7- Pre-AP Math 7 - Quarter 2 Day / Lesson Day 59 Lesson 13.6 • Instructional Focus Students will: • Review the properties of powers. Alignment 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. 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. 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. Essential Lesson Components Using the organizer from previous sections complete problems 2 and 3 to help review all exponent rules. Level 1 (EOL 5) Simplify the expression and express the answer in scientific notation. (2.4 × 106 ) (1.7 × 103 ) Assessment Level 2 (EOL 1) The following responses were given when students were asked to evaluate 28 . Michael: 28 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16 Damon: 28 = 22 + 22 + 22 + 22 = 256 Dawn: 28 = 22 ∙ 22 ∙ 22 ∙ 22 = 64 Which student is correct? Explain why the student is correct. -8- Level 3 (EOL 1,2) Write an expression involving both positive and negative exponents that simplifies to 𝑥 8 𝑦12 . Show how to simplify your expression. Pre-AP Math 7 - Quarter 2 Circles (7th grade, chapter 12) Essential Questions: • • • • • • What is the definition of a circle? How does the radius length relate to the length of the diameter? How do the length of the radius or diameter relate to the circumference of a circle? How do you determine the circumference of a circle? How do you calculate the area of a circle? Given a composite figure, what is the area of the shaded region? Evidence of Learning: 1. 2. 3. 4. 5. The student will define a circle as a collection of points on the same plane equidistant from the same point. The student will define both radius and diameter of a circle. The student will explore the relationship between the radius and the diameters length to the circumference of a circle. The student will calculate the circumference of a circle. The student will use the area formula of a circle to calculate the area of the shaded region for a composite figure. -9- Pre-AP Math 7 - Quarter 2 Day / Lesson Day 60 Lesson 12.1 • Instructional Focus Students will: • Learn a circle is a collection of points on the same plane equidistant from the same point. • Learn the relationship between the radius length and diameter length of a circle. Alignment 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. Essential Lesson Components Problem 1 should be teacher guided. Problem 2 should be used with pairs. Level 1 Assessment Level 2 - 10 - Level 3/Transfer Construct a viable argument to prove that line segment PN is congruent to line segment ON. Using that information, justify the relationship between PN and OM. Be sure to include key vocabulary in your argument. Pre-AP Math 7 - Quarter 2 • • • • Word Wall Words: Circle, center of a circle, diameter, and radius Materials Necessary: Straight edge, ruler, and compass Technology: • See it Try it – Carnegie Learning Online Resource Center – Chapter 12 • Explore Learning Gizmos - Perimeter, Circumference and Area- Activities A and B. • Illuminations.nctm.org - http://illuminations.nctm.org/ActivityDetail.aspx?ID=116 – Circle tool – interactive • Math Warehouse- http://www.mathwarehouse.com/geometry/circle/interactive-circumference.php DI strategies/ Thinking Maps: • Problem 3 can be used for closure or group discussion. • Problem 4 can be used for higher groups. • Use a circle map to define all parts of circle with prior knowledge, then create a brace map from their circle graphs. - 11 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 61-62 Lesson 12.2 • • Instructional Focus Students will: • Measure the circumference of a circle. • Explore the relationship between the diameter and the circumference of a circle. • Write a formula for the circumference of a circle. • Use a formula to determine the circumference of a circle. Alignment 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. Essential Lesson Components Use chart from Pg. 601 #6 to record the information from 3 circles in Problem1 and 2 circles from Problem 2. This can be turned into a station activity. Answer questions from pg. 601 to help conclude activity. Problem 3 is practice with manipulating the formula. Level 1 Compute the circumference of a circle with the radius of 17 inches. Assessment Level 2 Compute the diameter of a circle with a circumference of 25 feet. Based on the cross section of this tree, what is the best estimate for the circumference of this tree? - 12 - Level 3/Transfer What is the circumference of a circle, in feet, that has a radius of 6 inches? Mark and John just bought a pizza for the night. The pizza has a circumference of 28 inches. The two friends plan to split the pizza in half and each eats half a pizza. How many inches was the cut to split the pizza? Pre-AP Math 7 - Quarter 2 • • • • • Word Wall Words: pi Materials Necessary: string, ruler, straightedge, compass Technology: • Explore Learning Gizmos – Measuring Trees – measure the height, diameter, and circumference of trees in the forest to determine the age of each tree. (This is a really good activity!) • http://www.mathgoodies.com/lessons/vol2/circumference.html DI strategies/ Thinking Maps: • Using the Gizmo from the technology section, have groups of students do different precipitation levels and compare and contrast how precipitation impacts tree growth (circumference)as a closure. • Brace Map- breaking down the circumference formula See blackboard Resources • Teaching Student Centered Math Grades 5-8: page 200 (Activity to discover Pi) - 13 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 63-65 Lesson 12.4 Addendum: Finding the area of a circle • • • Instructional Focus Students will: • Derive formula for area of a circle. • Explore the relationship between the circumference and area of a circle. • Use the area and circumference formulas to solve for unknown measurements. • Use composite figures to solve for unknown measurements. Alignment 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. Essential Lesson Components Addendum comes first (See Blackboard) Skills Practice Pgs. 811-814, word problems dealing with circumference and area. Problem 2 Student Text- work through number 1 with the class, discuss 2 with students, then use 3 and 4 as independent practice. Level 1 Assessment Level 2 Level 3/Transfer 1. Write a note to your absent seat partner explaining why the value of π ≈ 22 . 7 What does this ratio represent? 2. A circular reflecting pool is located at the entrance to Quiet Waters Park. It has a radius of 10 feet. Estimate the circumference and area of the reflecting pool. Explain how you determined your answer. - 14 - Sally is buying a trampoline that will fit in her backyard. The area of her backyard is 230 square feet. Trampolines come with a diameter of 10 ft., 15 ft., 20 ft. or 25 ft. Which trampolines could she purchase? Justify your selection. Pre-AP Math 7 - Quarter 2 • • • • • Word Wall Words: inscribed circle, composite figure Materials Necessary: straightedge, ruler, compass Technology: • Understanding Math software: Understanding Geometry: Circles DI strategies/ Thinking Maps: • Flow map- have students calculate the area of each figure used (each getting its own box), then calculate the area of the shaded region. • Bridge map- on the top would be the shape, below would be the formula used for calculating that figures area. Resources: • Teaching Student Centered Math: Activity 7.7, p. 200 • Good Questions for Math Teaching: page 141 #12-14; page 149-150 #5,7 - 15 - Pre-AP Math 7 - Quarter 2 Volume (Chapter 14) Essential Questions: • • • How are the volumes of cones, cylinders, and spheres related? What formulas and procedures can be used to determine the volume of cones, cylinders, and spheres? What happens to the volume of a shape when one or more of the dimensions are changed? Desired Outcomes: • • • The student will be able to determine a method for finding the volume of three-dimensional figures through the use of a model. The student will be able to calculate the volume of cones, cylinders, and spheres. The student will be able to use the volume of cones, cylinders, and spheres to solve real-world problems. Evidence of Learning: 1. The student will calculate the volume of cones, cylinders, and spheres. 2. The student will use the volume of cones, cylinders, and spheres to solve real-world problems. - 16 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 66-68 Lesson 14.1 Lesson 14.4 Problem #4 Project: “Popcorn Anyone?” Instructional Focus Students will: • Determine appropriate vocabulary that applies to cylinders. • Explore the volume of a cylinder using unit cubes. • Estimate the volume of a right circular cylinder. • Determine the formula for the volume of a cylinder. • Use a formula to determine the volume of a right circular cylinder. • Use appropriate units of measure when computing the volume of a right circular cylinder. Alignment 8.G.9 Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems Essential Lesson Components Lesson 14.1 – All sections should be completed. • Problems 1-2 as guided instruction. Have cylinders (cans) on which students can draw and label the parts. Students may “think, pair, share” answers as they complete lesson. • As a way to review, have students complete the warm-up from teacher’s Implementation Guide on page 747C on the day after doing Problem 2. • Problem 3: Have students put the cubes into cylinders as described so they understand the process that is being discussed. Be sure to use the formula V= Bh with students. That is often what is found on assessments. • Problem 4: Use this as a paired activity with students. • Problem 5: Make this a real problem-solving activity with students. Give half the class #1 to do and the other half #3. Require students to record their work and results on chart paper and be able to explain their results to the class. Lesson 14.4 • Problem #4 on page 788 is another excellent problem-solving activity for students. Project: “ Popcorn Anyone?” Part 1 - 17 - Pre-AP Math 7 - Quarter 2 Assessment Level 2 Find the height of a cylinder that has a volume of 608 ft3. Level 1 Find the volume. Use π = 3.14. 2 cm Level 3/Transfer Juan says that if the height of a cylinder doubles, then the volume also doubles. Sandy says the volume more than doubles. Who is correct? Explain your reasoning. 7 cm Word Wall Words: cylinder, right circular cylinder, radius of a cylinder, height of a cylinder, circumference, pi • Materials Needed: samples of cylinders (A variety of cans would be great. Including some with labels would be useful for #4 in Problem 2.), centimeter cubes • Technology: 1. Gizmo: Prisms and Cylinders—Activity A: http://www.explorelearning.com Students vary height and radius of cylinder and observe how volume is affected 2. Interactive 3D Shapes:—Volume http://www.learner.org/interactives/geometry/area_volume2.html Interactive lesson developing the formula for volume of a cylinder 3. SMART Board Add-In: 7M008 Volume of a Cylinder Interactive Lesson using unit cubes to derive the formula for volume of a cylinder 4. BrainPop: Volume of cylinders • Resources: 1. Carnegie Learning Resource: “See it, Try it” Chapter 14 Video Clip explaining the derivation of volume of right solid formula (V = Bh); specifically applies cylinder • Differentiated Instructional Strategies: Prerequisite skills that may need assistance • Order of operations • Rational Numbers (cubing) • Using a formula/Evaluating Expressions - 18 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 69-71 Lessons: 14.2 - 14.4 Project: “Popcorn Anyone?” Instructional Focus Alignment Students will: 8.G.9 Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world • Explore the volume of a cone using a and mathematical problems cylinder and birdseed/rice • Write the formula for volume of a cone • Use a formula to determine the volume of a cone • Use appropriate units of measure when calculating the volume of a cone. Essential Lesson Components Lesson 14.2 • Problems #1-2 (p.762-766) develop characteristics and conical vocabulary • Problem #3 (p.766-772) use cone and cylinder nets to determine relationship between cone and cylinder volumes • Problem #4 (p. 773) develop formula for volume of a cone Thinking Map: Double Bubble: compare/contrast cylinder and cone Lesson 14.4 • Problem #3 (p. 786-787) Project: “Popcorn Anyone?” Part 2 - 19 - Pre-AP Math 7 - Quarter 2 Level 1 Find the volume. Use π = 3.14. Assessment Level 2 A cone has a height of 6 in and a volume of V = 8 in3. What is the radius of this cone? Level 3/Transfer Rita’s Italian Ice is considering redesigning their cylinder-shaped cups in exchange for cone-shapes ones. Their cylinder-shaped cup currently costs $4.95. About how much should their new cone-shaped cups cost? (*Keep the profit margins the same!) Word Wall Words: cone, height of a cone • Materials Needed: rice/birdseed, 3D shape nets (cone, cylinder) • Technology: • Learning Math: http://www.learner.org/courses/learningmath/measurement/session8/part_b/cylinders.html Computer Lab or on SMARTBoard: Students use ratios to compare volumes of cylinders to cones • Differentiated Instructional Strategies: • Prerequisite skills that may need assistance • Order of operations • Rational Numbers (cubing) • Using a Formula/Evaluating Expressions • 14.2 Problem 3: Teacher assists as much as necessary for constructing 3D shapes and pouring • Resources: Skills Practice worksheet for Lesson 14.2 - 20 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 72-74 Lesson 14.3-14.4 Project: “Popcorn Anyone?” Instructional Focus Students will: • Explore the volume of a sphere using a hemisphere, cylinder and birdseed/rice • Write the formula for volume of a sphere • Use a formula to determine the volume of a sphere Alignment 8.G.9 Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems Essential Lesson Components Lesson 14.3 • Problem #1 (p. 776) to develop spherical vocabulary and understanding only • See Resource Section Below • As needed, Problem #2 (2-11) as guided and independent practice • Problem #2 Scavenger Hunt or Gallery Walk with problems Lesson 14.4 • Problems #1-2 (p. 784-785) Project: “Popcorn Anyone?” Part 3 Level 1 Find the volume. Use π = 3.14. Assessment Level 2 The circumference of an NBA basketball is between 29 and 30 inches. The circumference of a WNBA basketball is between 28.5 and 29 inches. Find the greatest difference between the two basketballs’ volumes. - 21 - Level 3/Transfer The volume of a The Sphere sculpture by Fritz Koenig that used to stand between the World Trade Center towers in New York City is 1766.25 ft3. Find the circumference of the sculpture. Pre-AP Math 7 - Quarter 2 • Word Wall Words: sphere, center of a sphere, radius of a sphere, diameter of a sphere, antipodes, great circle, hemisphere • Materials Needed: 3D Solids (hemisphere, cylinder), rice/birdseed • Technology: • Learning Math: http://www.learner.org/courses/learningmath/measurement/session8/part_b/cylinders.html • Computer Lab or on SMARTBoard: Students use ratios to compare volumes of cylinders to spheres • Resources: • Kagan Cooperative Learning & Geometry—Chapter 9, Lesson 5, Activity 1 (p. 379) • Students use the hemisphere and cylinder solids to determine the relationship between their volumes, and then develop the formula. a. Activity 2 (p. 380) Additional questions (exit ticket, check for understanding, etc.) • Differentiated Instructional Strategies: Prerequisite skills that may need tiering • Order of operations • Rational Numbers (cubing) • Using a Formula/Evaluating Expressions - 22 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 75-76 Project: “Popcorn Anyone?” Instructional Focus Alignment 8.G.9 Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems Students will: • Present their Popcorn projects. Essential Lesson Components Project: “Popcorn Anyone?”- Part 4 Presentations *Optional Addendum lesson if needed Resources: • There is an addendum lesson if needed for Volume of 3D Composite Figures. - 23 - Pre-AP Math 7 - Quarter 2 Essential Questions: • • • • • • What experiments demonstrate that rotations, reflections, and translations of lines and line segments are rigid? How do you write a ridged movement in transformation notation (𝐴 → 𝐴′ → 𝐴′′) ? What experiments can you use to demonstrate that rotations, reflections, and translations of angles are rigid? How can you use transformation notation to demonstrate angles are taken to angles of the same measure?(angle A → angle A' → angle A'') What experiments can you use to demonstrate that rotations, reflections, and translations of parallel lines are rigid? How can you use transformation notation to demonstrate that parallel lines are taken to parallel lines (𝐴 → 𝐴′ → 𝐴′′)? Desired Outcomes: • • • The student will be able to verify experimentally the properties of rotations, reflections, and translations. The student will be able to 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. The student will be able to describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Evidence of Learning: 1. The student will rotate, reflect and translate rigid figures. 2. The student will identify two-dimensional congruent figures after they are rotated, reflected and translated. The student will name the congruent parts. - 24 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 77-81 Lessons 7.1 -7.4 • • • • • Instructional Focus Students will: • Determine the coordinates of images following a given translation. • Translate figures on a coordinate plane. • Translate linear functions horizontally and vertically. • Use multiple representations such as tables, graphs, and equations to represent linear functions and the translations of linear functions. • Rotate given figures on a coordinate plane. • Rotate figures given the angle of rotation. • Determine vertices of reflected images. • Reflect figures over a specified axis or line. Alignment 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. Essential Lesson Components Lesson 7.1: Work through Problem 1 whole group (page 387). Lesson 7.1: Problem 3-4 students work as partners and share results. (Use share questions on page 391) Be sure students are showing their work on multiple representation charts. Lesson 7.3: The students should be using a triangle cut out as they work through the example problem on page 408. The Share Phase questions are very important for understanding. Lesson 7.3: All students should complete the hands-on rotation as described in Problem 3 (page 410). Use the Share Phase Questions 1 and 2 to lead a class discussion. - 25 - Pre-AP Math 7 - Quarter 2 Level 1 (EOL 1) Move the object 2 units to the right and 4 units up. Assessment Level 2 (EOL 1) In the diagram, the figure A is reflected in the line XY. Draw the image of A in the diagram. . - 26 - Level 3 (EOL 1,2) On a coordinate grid draw a triangle. Rotate it 90 degrees. Translate it so that it is in the 4th quadrant. Reflect it over a line y="a number" so that the square is in the 1st quadrant. Describe how you can get the shape back in its original position. Pre-AP Math 7 - Quarter 2 • • • • Word Wall Words: transformation, translation, image, pre-image, rotation, angle of rotation, point of rotation, reflection, reflection line Materials Necessary: dot paper, mirrors Technology: • Explore Learning Gizmos: Rotation, Reflections and Translations Translations Reflections Proving Triangles Congruent • Smart Add Ins: 6G010 Transformations in the Coordinate Plane • Smart add Ins: 6G009 Transformations (Could be used for review) DI Strategies: • AIMS Activities “Pieces and Patterns A Patchwork in Math and Science” Nature’s Part in Art and Math pg 40 • Discussion/ Assessment Questions: How does translating a figure affect the size, shape, and position of that figure? How does rotating a figure affect the size, shape, and position of that figure? What are the differences between a translated polygon and a reflected polygon? • • Journal/ Writing Prompt Describe what a scalene triangle looks like after being reflected over the y-axis and an equilateral triangle. Give a practical situation that represents a translation. Resources: • Teaching Student-Centered Mathematics; Van De Walle: page 209 – 211 • Hands-On Standards Grades 7-8: pages 70-77 - 27 - Pre-AP Math 7 - Quarter 2 Day / Lesson Day 82 Instructional Focus Properties of Exponents/Volume (Chapters 13 and 14) Day / Lesson Days 83-84 Lesson 8.1 – 8.2 Instructional Focus Students will: • Translate triangles in a coordinate plane. • Rotate triangles in a coordinate plane. • Reflect triangles in a coordinate plane. • Identify corresponding sides and corresponding angles of congruent triangles. • Explore the relationship between corresponding sides and angles of congruent triangles. Alignment Colorado Academic Standards Alignment 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. - 28 - Pre-AP Math 7 - Quarter 2 Essential Lesson Components • • • • • • Lesson 8.1: Have students complete the translation chart on page 432 and check for their understanding. Practice several rotations with students. Have them identify what quadrant the new image or point is in. Give students a polygon and its translation, and ask the students to describe the translation. Pair students. Have one partner choose a parallelogram, draw it on the coordinate plane, describe its original position, rotate, translate, and reflect the parallelogram several times, and list the steps. Challenge the second partner to return the parallelogram to its original location. 8.G.2 This standard is the students’ introduction to congruency. Congruent figures have the same shape and size. Translations, reflections and rotations are examples of rigid transformations. A rigid transformation is one in which the pre-image and the image both have exactly the same size and shape since the measures of the corresponding angles and corresponding line segments remain equal (are congruent). Addendum lesson Transformations (See Blackboard) Level 1 (EOL 2) If HRT ≅ MNP , complete each statement. ∠R ≅ ___________ HT ≅ __________ ∠P ≅ __________ Assessment Level 2 (EOL 2) Which point is a translation of E? A. B. C. D. Level 3 (EOL 1) The vertices of triangle XYZ are X(4,4), Y(25,9), and Z(10,5). Describe the translation used to form triangle X’Y’Z’ for X’(26,21), Y’(215, 4), and Z’(0,0). J M N L Word Wall Words: congruent line segments, congruent angles, corresponding sides, corresponding angles • Technology: • Smart Add Ins: 8G001 Translations in the Coordinate Plane • Resource: • Kagan Cooperative Learning and Geometry (Chapter 4: Lesson 5 Exploring ways to guarantee congruent triangles) • Van de Walle: Activity 7.22 Coordinate Reflections Activity 7.23 Coordinate Translations - 29 - Pre-AP Math 7 - Quarter 2 Day / Lesson Days 85-86 Lesson 8.3 • Instructional Focus Students will: • Explore the SSS Congruence Theorem. • Explore the SAS Congruence Theorem. • Use the SSS and SAS Congruence Theorems to identify congruent triangles. Alignment 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. 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. Essential Lesson Components Talk with students about the share phase questions throughout this section. . Level 1 (EOL 2) Write a congruence statement for each pair of triangles represented. a) In QRS and TUV , ∠Q ≅ ∠T , ∠S ≅ ∠U , and QS ≅ TU Assessment Level 2 (EOL 2) Determine whether the triangles shown are congruent. If so, write a congruence statement and explain why the triangles are congruent. If not explain why. b) In ABC and DEF , AC ≅ ED , ∠C ≅ ∠D , and ∠B ≅ ∠F - 30 - Level 3 (EOL 2) Draw a pair of congruent triangles. Mark the congruent parts with slashes and curves. Select SSS or SAS and write a congruence statement and explain why the triangles are congruent. Pre-AP Math 7 - Quarter 2 • Word Wall Words: SSS Congruence Theorem, Included angle, SAS Congruence Theorem, ASA Congruence Theorem, AAS Congruence Theorem • Thinking Map: • Tree Map including: Reflection, Translation, and Rotation. Dilation to be added in next chapter. • Day / Lesson Instructional Focus Alignment Day 87 Clarify and Extend Day Day / Lesson Days 88-89 Lesson 8.4 • • Instructional Focus Students will: • Explore the ASA congruence theorem. • Explore the AAS congruence theorem. • Use the ASA and AAS congruence theorems to identify congruent triangles. Alignment 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. 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. Essential Lesson Components To complete this lesson have students independently complete the graphic organizer on page 469. Then share responses as a class. Check students understanding of rigid figures. - 31 - Pre-AP Math 7 - Quarter 2 Level 1 (EOL 2) ∆ABCand ∆EDF each have one pair of sides and one pair of angles marked to show congruence. What other pair of angles needs to be marked so the two triangles are congruent by AAS? A B F C D E Assessment Level 2 (EOL 2) Level 3 (EOL 2) Sketch and label triangles XVZ and DEF. The figure shows how the Greek Mark the triangles so that they demonstrate mathematician Thales (624 B.C. – 547 congruency by the ASA theorem. B.C.) determined the distance from the shore to enemy ships during a war. He sighted the ship from point P and then duplicated the angle at ∠QPT . The angles at point Q are right angles. Explain why QT represents the distance from the shore to the ship. Shore • • Word Wall Words: ASA Congruence Theorem, AAS Congruence Theorem Technology: • http://www.nsa.gov/academia/_files/collected_learning/middle_school/geometry/exploring_transformations.pdf • http://www.nsa.gov/academia/_files/collected_learning/middle_school/geometry/creating_dream_home.pdf • http://www.nsa.gov/academia/early_opportunities/math_edu_partnership/collected_learning/middle_school/geometry.shtml http://www.ohiorc.org/orc_documents/orc/RichProblems/Discovery-Similar_Triangles.pdf - 32 -
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