SFUSD Mathematics Core Curriculum Development Project

Transcription

1 1 SFUSD Mathematics Core Curriculum Development Project Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own understanding

2 Grade Constructions and Angles 2 Number of Days Lesson Reproducibles Number of Copies Materials 1 Entry Task Introduction to Geometric Shapes (2 pages) 1 per student Pattern Blocks 4 Lesson Series 1 CPM CCC (3 pages) 8.3.1A Resource Page 8.3.1B Resource Page CPM CCC (2 pages) Connected Mathematics 1.4 (4 pages) Engage NY Exit Slip Complement & Supplement Connected Mathematics ACE Practice (6 pages) Engage NY Exit Slip Unknown Angles 1 Apprentice Task Connected Math p. 90 #1: House Problem / p. 38 #69: Angle Map (2 pages) 3 Lesson Series 2 Connected Math Problem 3.1 (2 pages) ACE Practice Problems p. 76, 77 (2 pages) CPM CCC (2 pages) CPM CCC (3 pages) Design Challenge II (2 pages) 2 Expert Task SERP Triangles to Order Handout #1 SERP Triangles to Order Handout #2 SERP Triangles to Order Handout #3 SERP Triangles to Order Handout #4 2 Lesson Series 3 Connected Mathematics 3.3 (2 pages) CPM CCC2 Problems 8-81, 8-82b, 8-84 CPM CCC2 Problem Milestone Task Angles, Angles, and More Angles / Engage NY problems (2 pages) 1 per student 1 per student 1 per student 1 per student 1 per student Protractors Rulers 1 per student 1 per student 1 per student Protractors Protractors Rulers Angle Rulers Angles Cards - (reproduce one set per group on card stock) Brass Fasteners (at least 3/student) Strips with holes (use attendance strip side rip-outs; at least 3/student) Protractors/Angle Rulers Rulers Scissors Rulers Protractors Poster Paper Brass Fasteners (at least 4/student) Attendance strips with holes (at least 4/student)

3 Unit Overview Big Idea 3 Two-dimensional objects can be described, classified, and analyzed by their attributes. These objects can be oriented in an infinite number of ways. Some attributes of objects are measurable and can be quantified using unit amounts. Unit Objectives Students will be able to construct shapes, particularly triangles, with given measures of angles and/or sides. Students will be able to determine if three given measures produce a unique triangle, more than one triangle, or no triangle. Students will be able to apply facts about supplementary, complementary, vertical, and adjacent angles to solve for an unknown angle in a figure. Unit Description In Lesson Series 1, students become familiar with protractor use and how to draw, construct, and describe angles with ruler and protractors. They will learn about supplementary, complementary, vertical, and adjacent angles facts and the use of these to find a missing angle. In Lesson Series 2, students focus on conditions that determine a unique triangle, more than one triangle, or no triangle. The last lesson series has students practice drawing quadrilaterals with given sides and review what they ve learned. Geometry CCSS-M Content Standards Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

4 Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics 4 Students have built their work with geometric shapes by reasoning about relationships among shapes. Students have worked on constructions by drawing polygons in the coordinate plane, including drawing points, lines, line segments, rays, and perpendicular and parallel lines. Students are familiar with identifying these in twodimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size as well as recognize right triangles as a category, and identify right triangles. Students also understand that attributes belonging to a category of two-dimensional figures belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. Students are able to classify two-dimensional figures in a hierarchy based on properties. Students are first introduced to protractors and angle relationships in seventh grade. Students gain familiarity with relationships between angles formed by intersecting lines. Students work extensively with a ruler and protractor to construct geometric shapes, mainly triangles, with given conditions. They explore the conditions that determine a unique triangle, more than one triangle, or no triangle and develop an intuitive understanding of the structure of a triangle. Students focus on congruence and similarity. Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. In High School, students formalize their understanding of congruence in terms of rigid motions, and build on rigid motions as a familiar starting point for development of concept of geometric proof.

5 Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit? 5 1 Day 4 Days 1 Day 3 Days 2 Days 2 Days 1 Day Total: 14 days

6 6 Entry Task Intro to Geometric Shapes Apprentice Task Connected Mathematics: Shapes and Designs Expert Task Triangles to Order Milestone Task Pentagon Problem - MARS Task and Problem Solving from Engage NY CCSS-M Standards 7.G.5 7.G.5 7.G.2 7.G.2 7.G.5 Brief Description of Task Students use pattern blocks mainly to practice describing self-created designs to be replicated by a partner, and secondly as an introduction to angles and practice estimating angles. Students look for, classify, and measure angles in a picture of a house. They also apply angle rules on a map. Students explore building a triangle given certain conditions and work to generalize when unique triangles and cannot be created. Students find angle measures in geometric and real-life objects using protractors and knowledge of angle relationships. Source SFUSD Teacher Created Connected Mathematics 3 Shapes and Designs p. 38 and p. 90 SERP Poster Problem Adapted from MARS Task Lesson Series 1 Lesson Series 2 Lesson Series 3 CCSS-M Standards 7.G.5 7.G.2 7.G.2, 7.G.5 Brief Description of Lessons Students create and measure angles with a protractor. They learn about angle relationships and use this to find missing angles. Students explore triangles with given measures of sides and/or angles to determine if they create one, more than one, or no triangle. Students continue to explore angles in geometric shapes and have time to put into practice concepts they have learned. Sources CPM Core Connections Course 2 Chapter 8, Section 8.3.1, Section Connected Mathematics 3 Shapes and Designs, Saskatoon Public School, CPM Core Connections Course 2 Chapter 8, Section 8.3.3, Section Connected Mathematics 3 Shapes and Designs Illuminations.nctm.org, CPM Core Connections Course 2 Chapter 8, Section 8.3.3, Connected Mathematics 3 Shapes and Designs, MARS task

7 Entry Task Introduction to Geometric Shapes What will students do? 7 Mathematics Objectives and Standards Math Objectives: Students will be familiar with working with pattern blocks to create patterns and develop their ability to describe their pattern specifications. Students will draw shapes and patterns while learning their names as an introduction to geometric shapes. Students will identify angles and estimate their degree of measurements. CCSS-M Standards Addressed: 7.G.5 Potential Misconceptions: Framing Student Experience Launch: Set up pattern blocks for each group of students. Have them freely explore while guiding them to notice the types of shapes, kinds of angles in each shape, and any other shape descriptions. Have them talk to their elbow partner about what they know about polygons, angles, and their relationships. In pairs and/or as a whole group, do the first worksheet, Introduction to Geometric Shapes except the column in which you measure the angles with a protractor. Use the opportunity to introduce/remind students about vocabulary and concepts such as Obtuse Angle etc. During: Students will make their own drawing or pattern from geometric shapes. Give students individual work time to create and trace their creation on the worksheet. Next, have students practice describing their patterns. Teacher may start off modeling an example of your own to walk them through the exercise. Finally, students will write down in detail their exact pattern on a scratch piece of paper to exchange with a partner who will try to re-create this pattern. After the exercise, teachers can lead a discussion about how important it was to be very detailed in their description. They will leave the last two boxes ( write an improved description for your creation... ) to complete when they revisit their work. Closure/Extension: Students should estimate the angles in their creation to prepare for Lesson Series 1. Note: Students need to keep this sheet for Day 4 in Lesson Series 1.

8 8 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 4. Use appropriate tools strategically. Structures for Student Learning: Academic Language Support: Introduction to Geometric Shapes How will students do this? Vocabulary: geometric shapes (hexagon, triangle, rhombus, square, trapezoid), patterns, angles, acute angle, obtuse angle, right angle Sentence frames: The is above the. There are 2 to the right of the. The and touch at one vertex. Differentiation Strategies: Students can take pictures of their creations rather than tracing it. Hand out a sheet for direction descriptions to recreate shapes with a partner. Participation Structures (group, partners, individual, other): Partners, group

9 Lesson Series #1 9 Lesson Series Overview: Students are introduced to angles and protractor use. They explore angle properties and learn how to apply them. CCSS-M Standards Addressed: 7.G.5 Time: 4 days Lesson Overview Day 1 Description of Lesson: Students begin with showing what it means to do a 360 and various other angles. Students practice a few, then estimate on paper an angle. They create their own angle ruler first, then compare it to a protractor. Lesson Overview Day 2 Description of Lesson: Students begin to practice measuring angles with a protractor. They classify angles as right angle, acute angle, and obtuse angles. Complementary and supplementary angles are introduced. There is an optional Exit Slip from EngageNY to assess student s progress. Resources CPM CCC2 Lesson Resource Pages 8.3.1A, 8.3.1B Optional: Computer for short video clip find a video of an ice dancer doing spins Resources CPM CCC2 Lesson 8.3.2: problems 8-66 through 8-68 Connected Mathematics 3 Shapes and Design Problem 1.4 A E Exit Slip: Complementary and Supplementary Angles Notes: CPM lesson is split into two days. Only complete Problems 8-66 through Then use Connected Mathematics 3 Shapes and Design for more practice. Lesson Overview Day 3 Description of Lesson: Students learn about angle pair properties to write and solve simple equations. There are problems in Connected Mathematics for extra practice with estimating and measuring angles and another exit slip available from EngageNY. Notes: Complete the rest of from Problems 8-69 through Attempt additional challenge for students who finish early. Resources CPM CCC2 Lesson 8.3.2: 8-69 through 8-72 Connected Mathematics 3 Shapes and Designs ACE problems p : 5, 6, 9, 10, 12 28, Exit Slip: Solving for Unknown Angles Using Equations

10 10 Lesson Overview Day 4 Description of Lesson: Students take their entry task and now find all the measurements using a protractor. Direct students to apply angle pair properties to find angles so they are not measuring all the angles. Have students label a few complementary and supplementary angles, acute, obtuse and any right angles, as well as adjacent and vertical angles. Students may begin to realize the sum of the angles in quadrilaterals and triangles. They will explore the sum of angles in a triangle in Lesson Series 1, and quadrilaterals in Lesson Series 3. Lastly, have students complete the last two boxes to improve their description using the new angle words and vocabulary. Resources Students need to have their copy of their Entry Task. Saskatoon Public School Card Game Copy Angles Card game - reproduce one set per group on card stock Students can then play a complementary and supplementary card game. Modify the game based on student needs. There are three different versions included in the directions. Notes: Not all the angles may appear. Complementary and vertical angles may be hard to find or mislabeled. Complementary angles are formed by an equilateral triangle and the long skinny rhombus. Be sure to correct misunderstandings.

11 Apprentice Task Connected Mathematics Extension: House Problem/Angle Map 11 What will students do? Mathematics Objectives and Standards Math Objectives: Students will demonstrate geometric knowledge and reasoning of triangle work through a pentagon problem. Students will communicate their thinking by working in partners. CCSS-M Standards Addressed: 7.G.5 Potential Misconceptions: The lines drawn are not Amelia Earhart s path. Students should draw a path to the two islands indicated by a small arrow to the dot. Framing Student Experience Launch: Ask students to turn and talk to their elbow partners about characteristics of angles. Some observations that may come up include: The sum of the measures of supplementary angles is 180 o The sum of the measures of complementary angles is 90 o Students may bring up that the sum of the measures of all angles in triangles add up to 180 o, but this is not explored until the next lesson series. During: Connected Mathematics p. 90 #1 (House Problem) & p. 38 #69 (Angle Map) Note: Have students complete p. 90 independently first, then p. 38 in partners for Problem 69: Part b should not be a graded item. Closure/Extension: Have students share their thinking with another pair of partners.

12 12 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 3. Use appropriate tools strategically. Structures for Student Learning: Academic Language Support: Vocabulary: angle, measure, obtuse, isosceles Sentence frames: Connected Mathematics Extension How will students do this? Differentiation Strategies: Use angle rulers for friendly version of protractors. Provide a vocabulary list of shapes if students don t remember the names. Participation Structures (group, partners, individual, other): Partners

13 Lesson Series #2 13 Lesson Series Overview: Students will use strips, rulers, protractors, and technology to draw geometric shapes with given conditions. CCSS-M Standards Addressed: 7.G.2 Time: 3 days Lesson Overview Day 1 Description of Lesson: Students use paper strips to build different triangles. They discover the conditions that make or do not make a triangle. This lesson only focuses on three given sides. As a wrap-up/challenge, pose the problem: A triangle has a perimeter of 20 units. What might its dimensions be? Resources Connected Mathematics 3 Shapes and Designs 3.1 and ACE problems p. 76: 1 4 NOTE: Collect enough attendance strip sides for 4 per student. They will be reusing these strips the rest of the unit. Notes: If there is extra time, discuss whether the three sides can ever create more than one triangle. A great conversation about congruent triangles may arise, but informal conversation should focus mainly on how triangles can be flipped, rotated, and put on top of each other to match the other exactly. Lesson Overview Day 2 Resources Description of Lesson: Give students conditions for triangles and allow students to individually draw their own triangle. Students also discover that the sum of the three angles in a triangle equals 180. The lesson then leads into students constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. CPM CCC2 Lesson CPM CCC2 Lesson Notes: This lesson jumps around to only focus on triangles. Only do Problems 8-80 and 8-83 from 8.3.3, and then jump to Lesson

14 14 Lesson Overview Day 3 Description of Lesson: Allow students to finish the lesson from Day 2. If there is any extra time, use any of the resource material for students to deepen their understanding and gain more practice with angles and triangles. Group students to focus on their struggles or have stations for students to practice all. Notes: Resources Illuminations website (exploring when three sides will create a triangle: Website to explore and create SSS, SAS, AAS, ASA, SSA triangles: Connected Mathematics 3 p. 77 Problems 6 9 (Continued from Day 1) Connected Mathematics 3 Section 3.2 Design Challenge II

15 Expert Task Triangles to Order - SERP Poster Problem What will students do? 15 Mathematics Objectives and Standards Math Objectives: Students will build triangles to particular specifications (specs). The specs can include side lengths, angles, or a combination of both. Students go on to generalize, and develop an understanding about when they can determine a triangle from partial information and when they cannot. CCSS-M Standards Addressed: 7.G.2, 7.G.5 Potential Misconceptions: Framing Student Experience Launch: Students continue their study with triangles. Explain why triangles are important. For example, triangles are at the center of geometry. If you understand triangles really understand them you ll be in great shape. One reason is that you can make things out of triangles. In fact, most realistic computer graphics, from games to movies, are made up of triangles. During: Day 1: Introduce students to group work on working with an open-ended problem. Students plan and produce a mathematical poster to communicate their work. Day 2: Lead a whole group discussion making connections with posters. G uide students to understand from a more basic to a more general approach. Closure/Extension: Have class do a gallery walk (described in the Math Teaching Toolkit) to observe work and generate discussions among each other.

16 16 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 3. Use appropriate tools strategically. Structures for Student Learning: Academic Language Support: Vocabulary: specifications (specs), conditions Sentence frames: Differentiation Strategies: Participation Structures (group, partners, individual, other): Partners Triangles to Order - SERP Poster Problem How will students do this?

17 Lesson Series #3 17 Lesson Series Overview: Students build quadrilaterals based on given conditions. There is also a day to review any material before the Milestone Task. CCSS-M Standards Addressed: 7.G.2, 7.G.5 Time: 2 days Lesson Overview Day 1 Description of Lesson: Students use rulers, protractors, and technology, or to draw geometric shapes with given conditions. The focus this day is on quadrilaterals and given sides in Connected Mathematics, then with angles in CPM. Resources Connected Mathematics 3 Shapes and Designs Section 3.3 CPM CCC2 Lesson (Teacher notes can be found in Lesson Series 2) Resource Page (projected) Notes: Complete Connected Mathematics 3.3. Then move into CPM Lesson problems 8-81, 8-82(b) pertaining to quadrilaterals, and Problem If there is no technology, use the Resource Page. Lesson Overview Day 2 Description of Lesson: Facilitate a math talk lesson on one of the problems in Closure Problem or have them work in pairs to find each of the angles. Resources CPM CCC2 Closure Problem A or B If possible, send students out around campus or outdoors to take pictures of a few angles or geometric shapes they see. Back in the classroom, students estimate and measure the angles and share with the class. If using an ipad is an option, students can use the Skitch app to mark up their picture and share.

18 Milestone Task Pentagon Task and Problem Solving from EngageNY What will students do? 18 Mathematics Objectives and Standards Math Objectives: Students will extend their knowledge of two-dimensional geometry in problem solving. Students will test their understanding of shapes and designs using a given set of conditions. CCSS-M Standards Addressed: 7.G.2, 7.G.5 Potential Misconceptions: Students do not know where to draw the angles for the first two problems or assume the Leaning Tower of Pisa is 90 degrees. Framing Student Experience Launch: Go over concepts and skills covered: 1. Angles and measurements 2. Appropriate use of tools for given task 3. Communicate thinking During: Pentagon Task: adapted from MARS and p. 5 problems 1 3 student version Closure/Extension: Exit Ticket/Slip: Describe at least two skills that you used to solve for today s work. Where in life would you see these skills applied? Give two examples.

19 19 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. Structures for Student Learning: Academic Language Support: Vocabulary: geometric shapes, angles, polygons Sentence frames: Differentiation Strategies: Use angle rulers instead of protractors. Pentagon Task and Problem Solving from Engage NY How will students do this? Participation Structures (group, partners, individual, other): Students will work independently on this in order to show mastery of concepts and skills.