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NYCDOE Magnet Program District 25 & 28 JHS185 Magnet School Grade 8 Essential Question: How does transformational geometry allow us to analyze and predict population changes? Suggested Time Frame: 5 weeks Magnet Theme: Social Justice Stage 1- Desired Results Standards-Based Learning Goals: 8.G.1 Identify pairs of vertical angles as congruent 8.G.2 Identify pairs of supplementary and complementary angles 8.G.3 Calculate the missing angle in a supplementary or complementary pair 8.G.4 Determine angle pair relationships when given two parallel lines cut by a transversal 8.G.5 Calculate the missing angle measurements when given two parallel lines cut by a transversal 8.G.6 Calculate the missing angle measurement when given two intersecting lines and an angle. 8.G.7 Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations.) 8.G.8. Draw the image of a figure under rotations of 90 and 180 degrees 8.G.9. Draw the image of a figure under a reflection over a given line 8.G.10. Draw the image of a figure under a translation 8.G.11 Draw the image of a figure under a dilation 8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation Concepts Big Ideas for this Unit: Magnet School Theme: Change/continuity Social Justice Balance/Relationships Relevant/Connected Big Idea: Interdependence/Connections Angle relationships are crucial in how we plan and design Patterns cities. Regions in population density maps can be represented as geometric figures o Translation, Reflection , Rotation can be related to movements in the population (e.g. relocation) o Dilation can be related to population growth or shrinkage Enduring Understandings: Students will understand that: Changes in life and the world can be represented as translations, rotations, reflections or dilations Reflections of geometric forms can be considered as creating balance Transformations of a geometric figure mimic population transformation (relocation, gentrification) Changes in demographics can be studied as transformations Content (nouns) Students will know… Angle relationships o Vertical angles o Interior angles o Exterior angles o Alternate interior angles o Alternate exterior angles o Corresponding angles o Complementary angles o Supplementary angles o Adjacent angles Parallel lines Perpendicular lines Transversal Overarching Essential Question(s): How does transformational geometry allow us to analyze, predict, and prepare for population changes? Content and Skills Skills (verbs) Students will be able to… Identify parallel lines and distinguish them from intersecting lines Identify a transversal Identify and label pairs of angles in angle relationships Distinguish and explain the difference between one angle relationship and the other. Determine which pairs of angles formed by parallel lines and a transversal are congruent and which are supplementary Determine the measure of an angle when its complement or supplement is given. Construct an equation to determine the measure of angles in angle relationships when the measures are given as Congruent Transformations Rotations Reflections Translations Dilations Properties preserved Properties not preserved algebraic expressions. Describe transformations in the plane Graph and identify transformations in the plane using proper notation Draw the image of a figure under rotations of 90° clockwise and counterclockwise using proper notation Draw the image of a figure under rotations of 180° using proper notation Draw the image of a figure under translations Draw the image of a figure under reflections of x-axis, yaxis, and line x=y. Draw the image of a figure under dilations. Draw the image of a figure under rotation, reflection, translations, or dilations after being rotated, reflected, translated or dilated. Identify the properties preserved and not preserved under a reflection, rotation, translations, and dilation. Stage 2- Summative Assessment Evidence If students understand, know and are able to do the items in Stage 1, they should be able to show their understanding by completing an authentic task found in the world beyond the classroom. G- (goal) The students will create a land-use proposal for the fictional town of Greenville. They will use transformational and analytical geometry to create a proposal that deals with emergency evacuation situations and population growth or shrinkage. R- (role) The students are urban planners. They are working in groups that are competing to have their proposal approved by the Greenville Office of City Planning. A- (audience) The audience is the Greenville Office of City Planning. The proposals from each group of urban planners will be reviewed and voted on. S- (situation) Greenville is a newly founded town. The town has a small population, but expects population growth in the coming years. The town government wants to have a development plan in place to deal with population growth. They want a map that shows the town’s boundaries after doubling in size. Greenville also wants a map that shows what the town could do if they had to evacuate and relocate to another area. This map should show the main roads, which form a set of parallel lines and transversal. The urban planners will decide where the roads go. Then, they will use tools to measure only one angle. They will use analytical geometry skills to determine the angle measurement formed by the intersecting roads. The urban planners must write a letter to accompany their proposal. The letter should describe how they came up with the population growth map, relocation maps, and how they decided to put in the roads. P- (purpose and product) Each map below must include coordinates of all vertices. Coordinates must also be included for each transformation. The transformation that is applied in each step must be identified and explained (i.e. what is the transformation and how do figures move or change under the transformation) Population growth map (dilation) o For scale factor of 2. Population shrinkage map (dilation) o For scale factor of 0.5. Hurricane relocation maps o Relocation (translation with rule of x+3,y-4) o Relocation and change (translation with same rule, and rotation of 90-degrees clockwise) o Relocation and change (translation with same rule, and rotation of 90-degrees counterclockwise) o Relocation and change (translation with same rule, and rotation of 180-degrees) o Relocation and change (translation with same rule, and reflection across x-axis) o Relocation and change (translation with same rule, and reflection across y-axis) Town description letter for relocated town (roads and angle relationships) o Urban planners place North, South, and Main Street on the relocated-town map, indicating the angles formed by the roads. o In a letter, they describe their vision for the relocated town. Accompanying each map with a graph drawn on a graph paper. S- (standards for performance) Transform each map and label them correctly. Correctly identify and describe the different transformations. Draw the map with roads, and correctly determine and label the angle measurements. Correctly identify the angle relationships. Culminating Project You are urban planners. You are working in groups that competing to have your proposal approved the Greenville Office of City Planning. You will use transformational and analytical geometry to create a proposal that deals with emergency evacuation situations, population growth or shrinkage, and various economic developments. Greenville is a newly founded town. The town has a small population, but expects population growth in the coming years. The town government wants to have a development plan in place to deal with population growth. They want a map that shows the town’s boundaries, if the town were to grow double, triple, etc. in size. At its present size, they are also requiring a map that shows what the town could do if they had to evacuate and relocate to another area. This is because the town is currently located in an area frequently struck by hurricanes. The town government wants a relocation plan that show where the town would be located if they had to evacuate. Also to be included in the plan is a proposal for how the town could be changed. The council requires a few details in the map. North Avenue and South Avenue are two roads in Greenville. They are parallel to each other. Main Street runs diagonal to North and Main Avenues. The proposal map must include measurements for the angles formed by these roads. The urban planners will use this information in a descriptive letter that will accompany their proposal. The proposals from each group of urban planners will be reviewed by the Review Board of the Greenville Office of City planning, as well as the Greenville City Council. Each map below must include coordinates of all vertices. Coordinates must also be included for each transformation. The transformation that is applied in each step must be identified and explained (i.e. what is the transformation and how do figures move or change under the transformation) Population growth map (dilation) o For scale factor of 2. Population shrinkage map (dilation) o For scale factor of 0.5. Hurricane relocation maps o Relocation (translation with rule of x+3,y-4) o Relocation and change (translation with same rule, and rotation of 90-degrees clockwise) o Relocation and change (translation with same rule, and rotation of 90-degrees counterclockwise) o Relocation and change (translation with same rule, and rotation of 180-degrees) o Relocation and change (translation with same rule, and reflection across x-axis) o Relocation and change (translation with same rule, and reflection across y-axis) Town description letter for relocated town (roads and angle relationships) o Urban planners place North, South, and Main Street on the relocated-town map, indicating the angles formed by the roads. o In a letter, they describe their vision for the relocated town. Each map with a graph drawn on a graph paper and tape them on to a presentation cardboard. Rubric for Culminating Project Unit Essential Question: 4 3 2 Complete Good solid response with response with Explanation Explanation detailed clear unclear. explanation. explanation. Use of Visuals Mechanics 1 Misses key points. Correct maps or sketch with Incorrect maps Correct maps No maps or appropriate with incorrect or sketch. sketch. labels. labeling No math errors. No major math May be some Major math errors or serious math errors or serious flaws errors or flaws serious flaws in reasoning. in reasoning. in reasoning. Shows Shows Shows complete lack Demonstrated complete Shows some substantial of Knowledge understanding. understanding. understanding. understanding. Does not meet Hardly meets Goes beyond Meets the Requirements the requirements. requirements. requirements requirements. . Mini-Unit Title Line and angle relationships Translation in a coordinate plane Big Ideas of the miniunit/ Concept Statement Interdepende nce/Connect ions Fairness Change/cont inuity Balance/relat ionships Pattern Knowledge Important content to know about mini-unit (nouns) Vertical angles Interior angles Exterior angles Alternate interior angles Alternate exterior angles Correspondin g angles Complementa ry angles Supplementary angles Adjacent angles Parallel lines Perpendicular lines Transversal Congruent Transformatio ns Translations Dilations Properties preserved Properties not preserved Skills What should students be able to do? (verbs) Identify parallel lines and distinguish them from intersecting lines Identify a transversal Identify and label pairs of angles in angle relationships Connection to Magnet Theme Possible Topical Essential/ Focus Questions Mini-Unit Assessment How can you describe your vision for a relocated town (roads and angle relationships)? If measure of angle D is 53° and angle D and angle E are complementary, what is measure of angle E? Given one angle, find the other angle if they are complement to each other. Angles PQR and STU are supplementary. If measure angle PQR = x-15 and measure angle STU = x-65, find the measure of each angle. Given one angle, find the other angle if they are supplement to each other. Distinguish and explain the difference between one angle relationship and the other. Determine which pairs of angles formed by parallel lines and a transversal are congruent and which are supplementary Identify all congruent angles in vertical angles. Determine the measure of an angle when its complement or supplement is given. Identify all congruent angles when two parallel lines cut by a transversal line Construct an equation to determine the measure of angles in angle relationships when the measures are given as algebraic expressions. Describe transformations in the plane Graph and identify transformations in the plane using proper notation Draw the image of a figure under translations Identify the properties preserved and not preserved under a reflection, rotation, How does transformation al geometry allow us to analyze and predict population changes? The vertices of triangle MNP are M(4,-2), N(0,2), and P(5,2). Graph the triangle and the image of triangle MNP after a translation 3 units left and 6 units up. Given sets of coordinates, students will translate them according to the given rules (some units to the left or right and some units up or down). Benchmarks, Scaffolding Towards Culminating Project Draw maps roads (maps) and identify angle relationships for relocated town Draw hurricane relocation maps in which translation is involved translations, and dilation. Reflection in a coordinate plane Change/cont inuity Balance/relat ionships Rotation in a coordinate plane Change/cont inuity Balance/relat ionships Dilation in a coordinate plane Change/cont inuity Balance/relat ionships Transformatio ns Reflections Properties preserved Properties not preserved Draw the image of a figure under reflections of x-axis, y-axis, and line x=y. Identify the properties preserved and not preserved under a reflection, rotation, translations, and dilation. Transformatio ns Rotations Properties preserved Properties not preserved Draw the image of a figure under rotations of 90° clockwise and counterclockwise using proper notation Transformatio ns Dilations Properties preserved Properties not preserved Draw the image of a figure under dilations. Draw the image of a figure under rotation, reflection, translations, or dilations after being rotated, reflected, translated or dilated. Draw the image of a figure under rotations of 180° using proper notation Identify the properties preserved and not preserved under a reflection, rotation, translations, and dilation. Identify the properties preserved and not preserved under a reflection, rotation, translations, and dilation. How does transformation al geometry allow us to analyze and predict population changes? The vertices of a figure are A(-2,3), B(0,5), C(3,1), and D(3,3). Graph the figure and the image of the figure after a reflection over the y-axis Given sets of coordinates, students will reflect them according to the given rules (across xaxis, y-axis, or line of y=x). Draw hurricane relocation maps in which reflection across x-axis and y-axis are involved How does transformation al geometry allow us to analyze and predict population changes? A figure has vertices A(-4,5), B(-2,4), C(1,2), D(-3,1), and E(5,3). Graph the figure and the image of the figure after a rotation of 180° Given sets of coordinates, students will rotate them 90° clockwise, 90° counterclockwise, or 180°. How does transformation al geometry allow us to analyze and predict population changes? A trapezoid has vertices A(-1,2), B(0,3), C(4,1), and D(2,-1). Graph the figure and the image of the figure after a dilation of a scale factor of 2. Given sets of coordinates, students will dilate them with a given scale factor). Draw hurricane relocation maps in which rotation of 90° counterclock wise, 90° clockwise and 180° are involved Draw population growth and population shrinkage maps WHERE is the student going and what is expected HOOK with needed skills to experience and explore Opportunity to REVISE and RETHINK their understanding Allow students to EVALUATE work and implications TAILOR work to student needs Be ORGANIZED to maximize engagement Session1 Session2 Session3 Session4 Session5 Content Focus: Adjacent & vertical angles. Content Focus: Identify complementary & supplementary angles. Content Focus: Calculate the missing angle in a supplementary or complementary pair. Content Focus: Continue Calculate the missing angle in a supplementary or complementary pair. Content Focus: Determine line and angle relationships in which two parallel lines cut by a transversal Hook: In parallelogram SRQT, angles R and Q are supplementary. If measure of angle R = (3x+22)° and measure of angle Q = (5x2)°, what is the measure of angle Q? Hook: The measure of the supplement of an angle is 15° less than four times the measure of the complement. Find the measure of the angle. Hook: Two streets are parallel to each other and cut by a railroad track diagonally. Identify angles which are congruent. Daily Assessment: Glencoe workbook p. 201203. Daily Assessment: Pre-Algebra p. 495-496 #69, 22-25 Hook: Beth used a pizza cutter to mate 2 intersection cuts and divide a pizza into 4 slices. She expressed the angle of one slice as (4x + 20)° and the one opposite as (3x + 35)°. What was the measure of each angle? Daily Assessment: Glencoe workbook p. 195197 Hook: If a-short-hand points to 1, a-long-hand points to 7, and a-third-hand points to 9on a clock, what kind of angles do they formed? What are their measures? Daily Assessment: Given several sets of right angles and straight angles that are each formed by two angles, students are going to determine the missing angles. Weekly Assessment: Multiple Choice and Extended Response questions testing the sessions’ content. What have the students produced that scaffolds towards the units culminating assessment? The students have drawn a relocated-town map, indicating the angles formed by the roads. Daily Assessment: Have students draw two parallel lines and a transversal. Ask them to name all of the congruent angles. WHERE is the student going and what is expected HOOK with needed skills to experience and explore Opportunity to REVISE and RETHINK their understanding Allow students to EVALUATE work and implications TAILOR work to student needs Be ORGANIZED to maximize engagement Session6 Session7 Session8 Session9 Session10 Content Focus: Calculate missing angles when given two parallel lines cut by a transversal. Content Focus: Calculate missing angles (algebraically) when given two parallel lines cut by a transversal. Content Focus: Continue calculate missing angles (algebraically) when given two parallel lines cut by a transversal. Content Focus: Describe and identify transformations Content Focus: Translation Hook: A road crosses two parallel railroad tracks. Given an angle, find the missing angles. Daily Assessment: Have students draw two parallel lines and a transversal. Give them one of the angle measures and have them fill in the measures of the other angles. Hook: To measure the angle between a sloped ceiling and a wall, a carpenter uses a plumb line (a string with a weight attached). Given one angle algebraically, determine the relationship and find the missing angles. Hook: A road crosses two parallel railroad tracks. Given two angles as expressions, find the missing angles. Daily Assessment: Pre-Algebra p. 496 #29-33 Hook: Michael saw snowflakes falling down and he wondered what they have to do with transformations? Daily Assessment: Glencoe workbook p. 221222 Daily Assessment: Line m and l are parallel lines and t is a transversal. Find the value of x if measure of angle 2 = 2x +3 and measure of angle 4 = 4x – 7. (Draw the figure and place angle 2 and 4 so that they form vertical angle relationships). Weekly Assessment: Multiple Choice and Extended Response questions testing the sessions’ content. What have the students produced that scaffolds towards the units culminating assessment? The students have drawn a relocated map with translation with rule of x+3, y-4. Hook: Translation can be related to movements in the population such as relocation due to hurricane Daily Assessment: Glencoe p. 238-240. WHERE is the student going and what is expected HOOK with needed skills to experience and explore Opportunity to REVISE and RETHINK their understanding Allow students to EVALUATE work and implications TAILOR work to student needs Be ORGANIZED to maximize engagement Session11 Session12 Session13 Session14 Session15 Content Focus: Reflection. Content Focus: Rotation 90° clockwise Content Focus: Rotation 180°. Content Focus: Dilation Hook: Which transformation exists when you look into a mirror? Hook: Rotation can be related to movements in the blades of a ceiling fan, a dartboard or a ferris wheel Content Focus: Rotation 90° counterclockwise Hook: Given a-downward-facing picture, students have to rotate it so that it faces upward. Daily Assessment: Pre-Algebra Lesson 10-3 #14 or Glencoe workbook p. 231233. Daily Assessment: Given a set of coordinates of a figure, students will draw and write the coordinates of its image after a rotation of 90° clockwise. Hook: Population growth (dilation with a scale factor greater than 1) Population shrinkage (dilation with a scale factor smaller than 1) Hook: Given a landscape orientation picture, students have to rotate it into a portrait orientation so that they can upload it into an online profile. Daily Assessment: Given a set of coordinates of a figure, students will draw and write the coordinates of its image after a rotation of 90° counterclockwise. Daily Assessment: Given a set of coordinates of a figure, students will draw and write the coordinates of its image after a rotation of 180°. Weekly Assessment: Multiple Choice and Extended Response questions testing the sessions’ content. What have the students produced that scaffolds towards the units culminating assessment? The students have drawn almost all of the required relocated maps. Daily Assessment: Given a set of coordinates of a figure and scale factors, students will draw and write the coordinates of their images. WHERE is the student going and what is expected HOOK with needed skills to experience and explore Opportunity to REVISE and RETHINK their understanding Allow students to EVALUATE work and implications TAILOR work to student needs Be ORGANIZED to maximize engagement Session16 Session17 Session18 Session19 Session20 Content Focus: The properties preserved and not preserved Content Focus: Content Focus: Content Focus: Content Focus: Hook: Hook: Hook: Hook: Daily Assessment: Daily Assessment: Daily Assessment: Hook: For the relocation proposal to be useful, students have to understand the properties preserved and not preserved under each transformation Daily Assessment: Daily Assessment: Glencoe workbook p. 248250. Weekly Assessment: Multiple Choice and Extended Response questions testing the whole unit’s content. What have the students produced that scaffolds towards the units culminating assessment? The students are completing their project. Unit Resources Books: Pre-Algebra, Glencoe (McGraw Hill), 2005 Glencoe – New York Review Series (Debra L. Harley & Erica J. Shatz) Websites: Teacher Materials: Other: