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.
2. Not everyone’s pace will be the same, but if we walk a known distance, we can figure out the length of each person’s pace.
Q: If we pace 60ft (20yds), how could we find out the length of each pace??
= length of each pace in feet A:
Q: What assumptions are we making?
A: Paces are identical
Partners
3. Use one of the courses set up outside to count the number of paces it takes you and for your partner to travel that distance, and then find the length of each person’s pace.
Then repeat the process 4 more times. You will likely not walk exactly the same number of paces every time, so be as accurate as you be when you record the number of paces.
Find the length of each person’s pace each time (60 ft / number-of-paces).
8 INSTRUCTION ANGLE UNIT 5
Whole Group Demonstration Mathematical Concepts Unit Overview Introducing the Sample Median Materials & Preparation Mathematical Background 4. How should we deal with the different estimates of the length of your pace? Instruction Work with your partner to come up with a single value (number) to represent Formative Assessment your best estimate of the number of feet for one of your paces.
TEACHER NOTE
Let students invent their own statistic to represent their best MEASURING DISTANCE estimate before introducing them to a more conventional measure, OF TRAVEL the median.
Whole Group Conversation
The teacher selects several student inventions and asks them to describe their thinking.
Then let students know that statisticians also have several different ways of thinking about a best estimate of the number of feet in a person’s pace. One way to think about this estimate is to order the values of the measurements from least to greatest, and then to choose the value that is in the exact middle of the ordered list.
For example, when Debbie walked 60 feet, the length of her pace was 5ft, 4.75ft, 5.33ft , 5.5ft and 4.9ft. To find the median (the middle value), first order the list 4.75 ft 4.90 ft 5.00 ft 5.33 ft 5.50 ft. Then choose the middle value by finding the value that splits the list exactly in half, so that 50% of the values are less than or equal to the median and the other 50% of the values are greater than or equal to the median ( above and below). Here the middle value is 5 ft because 2 values (lower 50%) are below 5 ft and 2 values are above 5 ft (upper 50%). The median is like a hinge—50% of the values swing below and 50% swing above.
9 INSTRUCTION ANGLE UNIT 5
Individual Mathematical Concepts Unit Overview 5. Find the median for the length of your paces. Materials & Preparation
Write it down because we will need to know the length of you pace for finding Mathematical Background our way with a compass, which we will be using soon. Instruction Formative Assessment
TEACHER NOTE
Sometimes students will not order the lengths and just find the middle number as the median. For example, some students will MEASURING DISTANCE believe that 18 is the median of this collection of values (21 18 OF TRAVEL 22) because 18 is in the middle.
Other students might ask what happens if we have an even number of measurements, such as: 4.75 ft 4.90 ft 5.00 5.20 ft 5.33 ft 5.50 ft. Then the middle two values are interpolated, so that the median is 5.10 ft (midpoint between 5.00 ft and 5.20 ft). Then half the values are below (less than) 5.10 ft and half the values are above (more than) 5.10 ft. Sometimes, as in this collection of measurements, the interpolated value does not coincide with any value in the sample. This bothers some students, and it highlights the role of a statistic as representing something about a sample—not copying it exactly.
Another explanation that students find plausible is that samples tend to vary from sample-to-sample, so a median value absent in one sample’s collection of outcomes might be present in another sample of outcomes from the same process.
10 INSTRUCTION ANGLE UNIT 5
Small Group [optional]
Invent Displays Mathematical Concepts Unit Overview 6. Here is a list of the pace lengths for each person in our class. Materials & Preparation Mathematical Background Use the chart paper to make a display that will show someone else when they Instruction look at it quickly something that you noticed about our paces— some pattern Formative Assessment or something about them.
TEACHER NOTE
As students work in small groups to design displays, do not MEASURING DISTANCE insist that they conform to graphical conventions they may OF TRAVEL have learned in the past. Instead treat this as an opportunity to learn about what students have learned from their past lessons about the construction and interpretation of graphs.
Whole Group [optional]
Compare/Contrast Displays
7. Select students to talk about what particular displays show and hide about the lengths of people’s paces. Ask students to analyze how the designers of the display achieved that.
TEACHER NOTE
Be sure to highlight the mathematical foundations of the displays. For example, ordering data, grouping data into identical intervals, counting data, using the scale of measurement. If no student has invented it, introduce a dot plot and again ask students to analyze how it visualizes the class data, with an eye toward what it shows and hides about the values.
11 INSTRUCTION ANGLE UNIT 5
DETERMINING DIRECTION OF TRAVEL: Mathematical Concepts USING AN ORIENTEERING COMPASS Unit Overview Materials & Preparation Whole Group Mathematical Background Instruction 1. A compass uses the Earth’s magnetic field to locate the direction of North, Formative Assessment wherever we are on the planet.
2. If we know where North is, we can always determine a direction of travel.
Draw a circle where everone can see it and label a radius as North. Then DETERMINING DIRECTION demonstrate that due East is a 90º clockwise rotation; due South is a 180º OF TRAVEL clockwisre roation; due West is a 270º clock wise rotation.
“It’s like a protractor where North is treated as zero, and the measurement of a direction of travel is an angle—the degrees of rotation (clockwise) from North to some other direction of travel.”
12 INSTRUCTION ANGLE UNIT 5
3. Give each student an orienteering compass and help them use it to set headings of 0, 90, 180, 270, and other directions as they like. Mathematical Concepts Start out by helping students identify parts of the compass. The image below Unit Overview highlights the parts of an orienteering compass. The red arrow always points Materials & Preparation North. By rotating the housing until the red arrow aligns with the orienting Mathematical Background arrow on the base of the compass (usually indicating red-hatched lines), the Instruction Formative Assessment angle of the current direction of travel can be read from the compass at the point where it intersects the direction of travel arrow (see diagram). To set a particular heading, rotate the base of the compass until it intersects the direction of travel arrow at the desired heading (e.g. 45° or Northeast). Then rotate your body until the red arrow aligns with the orienting arrow. Sight DETERMINING DIRECTION along the direction of travel arrow to walk in the desired heading (e.g., 45°). OF TRAVEL
13 INSTRUCTION ANGLE UNIT 5
Partners
4. Students work in pairs to set their headings, first for North, South, East, Mathematical Concepts Unit Overview West, and then for other headings as they like. Materials & Preparation
Mathematical Background 5. Walk straight line distances; be sure to walk at the same heading as you pace: Instruction a. Set Heading 45º, walk 10 paces • Formative Assessment b. Set Heading 90º, walk 5 paces • c. Set Heading 180º, walk 6 paces •
d. Set Heading 225º, walk 10 paces • DETERMINING DIRECTION
OF TRAVEL
6. Walk a square using headings and paces. Write directions so that someone else could walk a square or rectangle (use the Square Walking Directions Worksheet Page 22).
Partners
7. Compare solutions for walking a square, troubleshoot any issues with using the compass or with using the compass to measure the turn angle of the square.
14 INSTRUCTION ANGLE UNIT 5
MAPPING A SPACE USING PACES Mathematical Concepts AND COMPASS HEADINGS Unit Overview Materials & Preparation Mathematical Background Partner Instruction Formative Assessment 1. Work with a partner to find the locations of some landmarks on your school’s yard, such as trees, fence, flags that were set there by your teacher, all from a starting point that you select.
From that starting point (it should be the same for all of your directions), find the heading and the paces. Be sure to convert your pace into conventional MAPPING A SPACE units, such as feet or yards. Take turns using the compass and compare your USING PACES AND estimates of distances and headings. COMPASS HEADINGS
Whole Group
2. Compare student solutions and determine the effect of different starting points on the measurement of heading and distance. (Not all students will have the same starting point, and this helps make the rationale for a common starting point—an origin—more visible. It is easier to communicate about direction and distance when there is a common starting point)
15 INSTRUCTION ANGLE UNIT 5
Mathematical Concepts Individual/Partner Unit Overview Materials & Preparation 3. Use the measurements of each landmark’s location to make a map on a Mathematical Background single piece of paper that shows all of the landmarks. You will have to Instruction decide about how to best fit everything on paper. You will need a ruler and Formative Assessment protractor.
Q: How might we show the distances between landmarks on paper?
A: A scaled unit of measure; such as 1mm = 1ft MAPPING A SPACE Q: How might we use the protractor to replicate our compass headings? USING PACES AND COMPASS HEADINGS
Partner
4. Post student solutions and solicit conjectures about what is the same and what is different about the maps constructed.
Be sure to highlight how each map had to show the starting position or origin, the use of the protractor to determine the direction of the landmark relative to the origin, the representation of North, which may depart from its orientation in the field, and the use of scale to manage the problem of representing distances on paper.
16 INSTRUCTION ANGLE UNIT 5
GENERALIZING COMPASS AND PACES: POLAR COORDINATES Mathematical Concepts Unit Overview Whole Group Materials & Preparation Mathematical Background 1. Using the compass and pacing to find our way inspired mathematicians to Instruction invent a location system that uses polar coordinates. Give all students a Formative Assessment Map of the Neighborhood Worksheet (Page 23).
“Here is an example of five landmarks represented in polar coordinates.”
Q: Where do you think the origin or starting point is? GENERALIZING COMPASS AND PACES A: Center
Q: Where do you think North is?
A: Straight up, 0º heading
Q: What is the scale for distance?
Q: What do you notice about the distance from the center to each point on the same circle?
Partner
2. Using your ruler and protractor to generate the direction of travel and the distance from the origin of each landmark. Then put a new landmark in the map and give its heading and distance from the origin.
Whole Group
3. Students share coordinates (distance, angle) for each of the five landmarks. Some of the new locations generated by students are used and checked to see if other students can use them to find the same location on the map.
REFLECTION NOTE
“All right, good work today. In your math journal, write a short note about what you learned today, and a question that you have about what we did today. Or write a question about something else you are wondering about.”
17 FORMATIVE ASSESSMENT ANGLE UNIT 5
Mathematical Concepts Formative Assessment Unit Overview Materials & Preparation Mathematical Background Instruction The formative assessment uses knowledge of a protractor and scaling a length to Formative Assessment connect angle and distance to the concept of coordinate—a way of specifying location. Although we are more familiar with Cartesian coordinates, which are essentially two rulers placed at right angles, when navigating, polar coordinates are much easier to use. This ease of use and ready connection to measuring distance and direction make polar coordinates a good way of introducing students to coordinate systems.
18 FORMATIVE ASSESSMENT ANGLE UNIT 5
Name: ______
Use the red dot below as the origin. Using a ruler and protractor, find the direction and distance from the red dot to each of the other locations (the dots marked by colors and letters). Use the scale of 1in = 2ft. Find the distance to the nearest whole number of feet and the clockwise direction from the origin for each location.
A B C D
FORMATIVE ASSESSMENT ANGLE UNIT 5
1. Buried Treasure
A pirate buried some stolen goods and left a map.
To find gold coins, start at the tree, face 0° (look North). Walk 10 paces. Make a dot and label it G to show where you would find the gold coins.
To find silver coins, start at the tree, face 75°. Walk 20 paces. Make a dot and label it S to show where you would find the silver coins.
To find diamonds, start at the tree, face 200°. Walk 35 paces. Make a dot and label it D to show where you would find the diamonds.
To find a box of cash, start at the tree, face 270°. Walk 50 paces. Make a dot and label it C to show where you would find the cash.
FORMATIVE ASSESSMENT GUIDE ANGLE UNIT 5
Item Construct Description Notes
ToM��3G A 10 ft, (3350 – 3450) Procedural B. 8 ft, (50 – 150 ) During formative assessment conversation, Protractor C 6 ft, (250 – 350) ITEM Competence D. 4 ft, (550 – 650) ask different students to how they located each landmark (A-D). Be sure to see if 1 ToML 5E Each of A-D is scored as ToML students associate due North on map with a Length-Scale Interpret markings 5E heading of 0 degree measure on the protractor. and Angle- on a standard foot if the scale is used appropriately The coordinate should specify reasonably Headings to ruler to approximate the distance. Produce accurate estimates of distance and direction of Coordinate travel. Have students enact what happens if � Connections Each is scored as ToM� 3G if only distance or only direction is specified. Polar Coordinate the degree measure is within the specified bounds.
ToM��3G G (1 radius length, 0 heading) Representational S (2 radius length, 70 +/- 10 Protractor heading-NE) Competence D (3.5 radius length, 200 +/- 10 During formative assessment conversation, ITEM degrees-SW) ask different students to demonstrate how they ToML 5E C (5 radius length, 270 -West) were thinking as they estimated the distance 2 Interpret markings and direction of each landmark. Emphasize Polar on a standard foot Each of is scored as ToML 5E if the use of cardinal direction as landmarks and Coordinates ruler the radius length is correct. 2-splits as reference points, such as 450 and 1350 Connections Each is scored as ToM��3G if Polar Coordinate the degree measure is within the specified bounds.
21 WALKING A SQUARE WITH PACES & COMPASS • ANGLE UNIT 5
NAME:______
Directions for Walking a Square with Paces and Compass
The person following these directions will not have seen you walk them, so be very specific.
To walk a square on the ground:
Start at (name of location) and at a heading of (orientation).
1.
2.
3.
4.
22 MAP OF THE NEIGHBORHOOD • ANGLE UNIT 5
NAME:______
Map of the Neighborhood
Estimate the angle (heading) and distance for each landmark.
LANDMARK ANGLE DISTANCE
23