1 SFUSD Mathematics Core Curriculum Development Project 2014 2015 Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own understanding
Geometry G.1 Tools and Constructions 2 Number of Days Lesson Reproducibles Number of Copies Materials 1.5 Entry Task Geometric Constructions (2 pages) 1 per student Compass and straightedge (1 per student) Blank paper, markers, colored pencils 2 Lesson Series Constructing Bisectors (6 pages) 1 per student Compass, straightedge, blank paper Copying Angles and Constructing Parallel Lines (4 pages) Toolkit 1.5 Milestone Task Inscribing Polygons in Circles (3 pages) 1 per student Compass, straightedge, blank paper
Unit Overview Big Idea 3 Students learn to use tools and make formal constructions to understand basic geometric definitions. Unit Objectives Students will understand the geometric terms: circle, angle, perpendicular line, parallel line, and line segment. Students will be able to use a compass and straightedge to make formal constructions. Students will understand how formal constructions relate to definitions. Unit Description First students learn how to use basic construction tools, such as a compass and straightedge. They then use their tools to make formal geometric constructions and make connections between the constructions and the precise definitions of a circle, perpendicular, and parallel lines, etc. Finally, they use their knowledge of geometric constructions to create more complicated formal constructions such as an equilateral triangle or square. Congruence Experiment with transformations in the plane CCSS-M Content Standards G.CO.1 Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the notions of point, distance along a line, and distance around a circular arc. Make geometric constructions G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment, and constructing a line parallel to a given line through a point not on the line.
Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics 4 In elementary grades (fourth and fifth), students learn how to identify lines, angles, perpendicular, and parallel lines. In sixth grade, students draw polygons on a coordinate plane, essentially constructing polygons without using tools such as a compass and straightedge. In seventh grade, students begin to draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions, though crucially, not with a compass and straightedge. Students can distinguish between a construction of a geometric figure and a drawing of a figure. Students learn how to use tools such as folded paper, a compass and straightedge, and technology to create formal geometric constructions. Students also make explicit connections between their constructions and precise definitions of an angle, circle, perpendicular line, parallel line, and line segment (based on the undefined notions of point, line, distance along a line, and distance around a circular arc). Students will use constructions to explore the properties of transformations and use them to prove congruence and other properties throughout the geometry course.
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 central questions. 5 Entry Task: Milestone Task: What do you already know? Did you learn what was expected of you from this unit? Unlike other units, this short 5-day unit has only two tasks. 1.5 days 2 days 1.5 days Total: 5 days
6 Entry Task Geometry Constructions! Milestone Task Describing Constructions CCSS-M Standards Brief Description of Task G.CO.12 Students learn how to use geometry construction tools of compass and straightedge. G.CO.1, G.CO.12 Students identify how to use constructions to reproduce a regular hexagon and an equilateral triangle. They will then create their own directions for how to reproduce their own design. Source SFUSD Teacher Created SFUSD Teacher Created Lesson Series CCSS-M Standards Brief Description of Lessons G.CO.1, G.CO.12 Students learn how to construct circles, line segment and angle bisectors, parallel lines through a point not on the line, and how to copy angles. Students create precise definitions for circles, line segments, etc. based on their constructions. Students will also fill out a toolkit identifying each geometric figure they have formally defined. Sources SFUSD Teacher Created
Entry Task Geometry Constructions! What will students do? 7 Mathematics Objectives and Standards Students learn how to use basic geometry construction tools by making copies of circles and line segments and then creating art projects. Math Objectives: Students will learn how to use geometry construction tools. Students will define the difference between line segment and line. CCSS-M Standards Addressed: G.CO.12 Potential Misconceptions: Students may need help learning how to use the compass appropriately. They should be clear on how to construct a precise circle and arc using the compass, which will require precision regarding where to place their center point. Framing Student Experience Launch: You may want to introduce the concept of an arc to students, since the task assumes they know what it looks like. You should also think about showing them a short video to help them see how to hold their compass and use it to Video on a slide compass: https://www.youtube.com/watch?v=_vei3ji366g During: You will want to check in with students repeatedly to make sure they understand how to use their compass and straightedge, particularly when they are attempting to reconstruct the daisy drawing. Make sure students know that today is primarily about getting comfortable with the compass. They don t necessarily need to have perfectly constructed figures! Video on duplicating a line segment: http://mathopenref.com/constcopysegment.html Closure/Extension: If you have time, you could have students do a gallery walk of each other s art pieces.
8 Geometry Constructions! How will students do this? Focus Standards for Mathematical Practice: 5. Use appropriate tools strategically. Structures for Student Learning: Academic Language Support: Vocabulary: construct, compass, straightedge, point, line, line segment, circle, center, radius, arc, intersection Participation Structures (group, partners, individual, other): This task should be done within a group setting, but students will produce have individual worksheets and should produce individual pieces of art.
Lesson Series 9 Lesson Series Overview: This lesson series has students construct specific geometric figures using a compass and straightedge. After making explicit connections between precise geometric definitions and their constructions, students will practice constructing bisectors, parallel lines, and copying angles. CCSS-M Standards Addressed: G.CO.1 and G.CO.12 Time: 2 days Lesson Overview Day 1 Description of Lesson: Students will learn how to construct angle and segment bisectors using a compass and straightedge. They will also make explicit connections between the construction of a circle, bisector, and an angle and their definitions. Toward the end of the class period, you should have students start filling out their Transformations Toolkit. Constructing Bisectors Resources Constructing a Segment Bisector: http://mathopenref.com/constbisectline.html Constructing an Angle Bisector: http://mathopenref.com/constbisectangle.html Notes: You may want to run videos in the background to help students see how to make specific constructions. They will likely need help making connections between the instructions and the constructions on their paper. Use the links in the resources section if you d like video examples for students to reference. Lesson Overview Day 2 Description of Lesson: Students will continue looking at the relationship between constructions and definitions. In particular, they will learn how to copy an angle and how to construct parallel lines. Toward the end of the class period, you should have students fill out their Constructions Toolkit. Notes: As with the previous day s lesson, you may want to run videos on the specific constructions. Videos for how to copy angles and construct parallel lines are resources. Resources Copying Angles and Constructing Parallel Lines Constructions Toolkit Copy an Angle: http://mathopenref.com/constcopyangle.html Parallel Lines: http://mathopenref.com/constparallel.html
Milestone Task Describing Constructions What will students do? 10 Mathematics Objectives and Standards Students identify how to use constructions to reproduce a regular hexagon and an equilateral triangle. They will then create their own directions for how to reproduce their own design. Math Objectives: Students will master use of the compass to construct the folder. Students will construct equilateral triangles and squares using their knowledge of geometric constructions. Students will carefully describe the sequence of constructions that will reproduce a student s unique design. Framing Student Experience Launch: Students begin by producing their own lessons and ideas. Instead of recognizing their own During: Closure/Extension: CCSS-M Standards Addressed: G.CO.12 Potential Misconceptions: Students may have great difficulty reproducing the equilateral triangle and inscribed circles because they haven t worked on them before. They will also have difficulty producing the directions for how to reproduce their figure. So it s important to give them sentence starters and encourage them to take notes about what they did as they create their design.
11 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 5. Use appropriate tools strategically. Describing Constructions How will students do this? Structures for Student Learning: Academic Language Support: Vocabulary: inscribed, polygon, triangle, hexagon, square, equilateral triangle, regular hexagon Sentence frames: First, construct Place point on Adjust your compass to Place the center of your compass on Participation Structures (group, partners, individual, other): This task should be done within a group setting, but students will produce have individual worksheets and should produce individual pieces of art.