ANGLE UNIT FINDING OUR WAY: MAPPING AND POLAR COORDINATES • A walking pace, a unit of length measure, is the distance traveled from toe to toe each time your right foot strikes the ground. It is approximately 5 feet in most adults. • An orienteering compass is a tool to determine direction of travel relative to magnetic North. • A heading is a measure of the direction of travel, represented by the angle of travel relative to North measured in degrees. A heading of 0° indicates North, 90° East, 180° South, and 270° West. Contents • The radius of a circle is a constant distance from the center of the circle to the circle’s edge. Mathematical Concepts 1 Unit Overview 2 • A polar coordinate locates a place in space as a distance traveled at a constant heading from an origin. The origin is the center of a set of Materials & Preparation 3 concentric circles, with locations on the circumference specified by heading Mathematical Background 4 and the length of each radius. Each polar coordinate specifies length Instruction 8 (distance) and angle (heading) from the origin. Formative Assessment 18 • The median is a measure of the central tendency of a collection of Unit 5 Worksheets 22 ranked values (e.g., arranged from least value to greatest value). It is the point in the collection where 50% of the values are of greater magnitude and 50% are of lesser magnitude. ©2019, Richard Lehrer, all rights reserved. UNIT OVERVIEW ANGLE UNIT 5 Mathematical Concepts Unit Overview Unit Overview Materials & Preparation Students determine the approximate length of their walking pace by pacing Mathematical Background Instruction a 20-yard (60ft) length at least 5 times. They are introduced to the median Formative Assessment as a way to estimate the length of their pace in light of measurement variability. Students learn to use an orienteering compass to set their heading, and then use paces and the compass to write directions to create familiar paths, such as squares and rectangles. Paces are converted to feet to facilitate comparisons among directions. Using distance and direction, students make a map that shows the location of a set of objects designated by flags, given their unique starting points. During whole-class conversation, students’ maps are compared. The teacher highlights the virtue of a common origin, the need to re-represent feet or yards with a scale, such as 1-inch = 5ft, and the convention of orienting maps so that “up” is North. Students are introduced to polar coordinates as a solution to generating a map with a common origin and agreed-upon conventions for representing distance and direction on the map. Students use their maps to solve problems posed by their teacher to challenge way finding in a large- scale space, such as a school yard. 2 MATERIALS & PREPARATION ANGLE UNIT 5 Read Mathematical Concepts • This Unit Unit Overview Start by reading the unit to learn the content and become familiar with the Materials & Preparation activities. Pay special attention to questions that reveal student thinking and Mathematical Background that provoke student thinking. Read the TEACHER NOTES to help guide Instruction productive conversation about important ideas and procedures. Formative Assessment • Mathematical Background Reread the mathematical background carefully to help you think about the important mathematical ideas within the unit, especially the idea of a coordinate. • Lenth and Angle Measurement Construct Maps Read the construct maps to help you recognize the mathematical elements in student thinking, especially unit iteration for length and the use of degrees as angle measures. Gather ☐ Circular protractors, one for each student ☐ Orienteering compasses, one for each student or pair of students ☐ 8.5in × 11in paper ☐ A tape measure and cones to set up a few 20-yard straight paths ☐ Rulers marked in either inches or centimeters Prepare Prepare a few 20-yard straight paths so that students can determine the length of their paces. From an origin of your choice in a large scale space (e.g, a playground), set flags at locations of your choice. It would be most helpful if the flags were at substantially different headings, such as 120, 180 and 270. Determine the coordinates of each flag in terms of distance (your paces) and headings. 3 MATHEMATICAL BACKGROUND ANGLE UNIT 5 Mathematical Background Mathematical Concepts Straight When is a line straight? Often, when we measure a distance between Unit Overview Materials & Preparation two points, we imagine a line. A traditional way of thinking about a straight line is Mathematical Background as the shortest distance between two points. But it is more consistent with bodily Instruction Formative Assessment experience to consider a line to be straight when it is formed by moving without any change in direction. For instance, walking at a constant heading while towing a piece of chalk on a flat surface ideally creates a straight line. Pace to Measure Straight Distance Traveled Although we usually measure distances with inches, feet, and yards, a more handy way of estimating distance traveled when walking is a pace. A pace is the distance from the toe of the foot to the toe of the same foot when it next strikes the ground, as illustrated below. For most adults, it is approximately 5 ft. What counts as a pace is two steps, so the count of paces can be the number of times you strike with your left foot or the number of times you strike with your right foot. People count paces as they travel: “And 1 (right foot strike), And 2 (right foot strike), And 3 (right foot strike).” 4 MATHEMATICAL BACKGROUND ANGLE UNIT 5 Repeated Measure All measures of an attribute are approximate, even those Mathematical Concepts Unit Overview that are conducted with the finest of instruments and with Materials & Preparation the greatest care. In the preceding units, we encountered Mathematical Background Instruction these errors of measure but did not characterize them. But Formative Assessment the presence of error means that repeated measures of an attribute are necessary to obtain a good estimate of its true measure. So, scientists always conduct repeated measurements of the same attribute to obtain an estimate of its true measure and the precision of the measurements (how alike the measurements tend to be). Statistic A statistic measures a characteristic of a collection of measures. Many characteristics can be measured, including estimating the central tendency of the collection of values (i.e., what value best describes the middle or “average” value?) and estimating the tendency of the values to agree or to cluster and clump together (i.e., the variability of the values. More clumping=less variability). Angle An angle is a directed rotation from a heading or the length of arc relative to circumference between two intersecting line segments. In previous units, we investigated measures of angles in degrees, and we employ this degree unit of angle measure in this unit. 5 MATHEMATICAL BACKGROUND ANGLE UNIT 5 Compass Heading A compass uses a magnetized needle to show the Mathematical Concepts Unit Overview constant orientation of magnetic North. The direction of Materials & Preparation North is represented as a heading of 0, and all angles from Mathematical Background Instruction North are represented by headings (clockwise angles) Formative Assessment relative to this absolute or fixed heading. For example, due East is represented by a heading of 90°, due South by a heading of 180°, and due West by a heading of 270°. In the diagram below, the direction of travel is approximately 45º 6 MATHEMATICAL BACKGROUND ANGLE UNIT 5 Polar Coordinate A polar coordinate represents a location in space as a Mathematical Concepts Unit Overview distance from an origin and the direction of travel relative to North. By convention, Materials & Preparation 0° (North) is represented as a vertical axis, so that different headings are Mathematical Background Instruction represented by clockwise rotation from 0°. The radii of concentric circles centered Formative Assessment at the origin represent distances at different headings. The effect is like a bicycle’s wheel’s hub (origin) and spokes (headings), with the rim of wheel as one of the possible circles. Some cities, such as Washington DC, are configured with this model in mind. Maps Maps represent locations in space for purposes of navigation. Polar coordinates are one way of representing location, so an orienteering compass and walking can be used to make maps and to read maps. Locations are represented by polar coordinates specifying distance and angles relative to North from a common origin. This kind of map preserves the angles between locations in the world and changes all lengths by the same proportion or scale factor. This is a similarity transformation. 7 INSTRUCTION ANGLE UNIT 5 MEASURING DISTANCE OF TRAVEL Mathematical Concepts Unit Overview Whole Group Demonstration Materials & Preparation Mathematical Background 1. When we walk, we could carry a tape measure or yardstick to measure Instruction the distance we have traveled, or we could travel with a wheel with a Formative Assessment counter that indicates the number of complete revolutions of the wheel. But an easier way to estimate how far we have walked is to count our paces. A pace is counted like this: We start by standing with our feet about shoulder width apart. We’ll mark the position of our toes. Step with your left foot as you MEASURING DISTANCE normally would, like this, and say “And.” Then step with your right foot as OF TRAVEL you normally would, counting 1. I’ll mark where the toe of my right foot lands, like this (mark). The distance between the two toe marks that we have made is called a pace. The next pace is walked the same way, but the count changes, “2.” Enact for students, and have each students walk and count 2 paces.