Measuring Angles

Purpose

This unit is about estimating and measuring angles. Students begin using the ‘circular’ benchmarks of 0,1/4, 1/2, 3/4, and full turn, they then come to understand the more formal unit of measurement, the ‘degree’, and they are finally introduced to the protractor as a tool that can be used to measure an angle.

Achievement Objectives
GM3-1: Use linear scales and whole numbers of metric units for length, area, volume and capacity, weight (mass), angle, temperature, and time.
Specific Learning Outcomes
  • measure angles using degrees
Description of Mathematics

In this unit, students concept of angle is deepened and they are introduced to the measure of an angle called a ‘degree’. Angle is a fundamental quantity in geometry. It is to turning what distance is to moving in a straight line. Being able to use degrees enables us to locate our position on Earth (longitude and latitude) and navigate by using bearings. It also allows us to distinguish between triangles (by comparing their angles) and describe other polygons more accurately.

Later trigonometry is introduced in the curriculum. This first arises with respect to right-angled triangles and their use in measuring inaccessible quantities such as the height of buildings. But it moves on to wide applications relating to tides, light and sound, and in fact anything that relates to wave motion or has periodic behaviour.

Initially, in the unit the students will be required to describe an amount of turn from a particular position to another using the ‘circular’ benchmarks of 0,1/4, 1/2, 3/4, and full turn. Estimation language such as ‘just about’, ‘between’, ‘not quite’, ‘just over’, and similar terms may be used to qualify their descriptions. As the unit progresses, the students will be exposed to the more formal unit of measurement, the ‘degree’. They will be involved in constructing a link between the fractions of the circle with which they are familiar and the corresponding degree values, where 360 degrees is used as the measure of a complete turn.

Only towards the end of the unit are students introduced to the protractor as a tool that can be used to measure an angle. Estimation will continue to play an important role, however.

Required Resource Materials
  • White and black paper for angle estimator
  • Cardboard, paper, straws and split pins for angle measurer
  • Overhead projector for protractor demonstrations
  • Protractors
Activity

Getting Started

Railway crossing.Begin the unit by examining rotations in real situations. In all of these examples that are used to promote discussion, focus the students on the various ways of measuring an amount of turn about a particular point, starting at one position and finishing in another.

Focus the students on the various ways of measuring an amount of turn about a particular point, starting at one position and finishing in another. For example, the hinge of the railway crossing control arm, the horizontal position and the vertical position; the centre of the clock, when the minute’s hand is pointing at the 12 and a later time.

  1. A control arm at a railway level crossing. Perhaps ask questions like:
    What does the arm do as a train approaches?
    How would you describe its movement?
    What would happen if the mechanism controlling the arm broke and it was only able to go half-way back to where it started?
    What problems may be faced by traffic having to cross the crossing?
  2. 2. The minute’s hand on a clock.
    Ask questions like:
    How do we describe the minute’s hand when it is positioned in the following pictures (Draw on board)

 Clocks.

How long has passed when the minute’s hand goes through a quarter turn?
What fraction of a complete turn does the minute’s hand go through in 5, 10, 20, 45 minutes?

  1. Scissors, opening and closing. Perhaps ask questions like:
    How do you interpret the amount of ‘openness’?
    Can scissors open up to show a half turn?
  2. The reclining chair in the dental clinic. Have the students draw some of the different positions the chair can recline.
    How would you describe the amount of ‘recline’?
    How does this type of chair help the dental nurse and the dentist?
  3. Identify at least 4 things that turn in or about the classroom.
  4. Using a compass identify North, South, East and West and perhaps put signs up on the four walls of the classroom. Have the students stand and face to the North of the classroom. Now ask them to turn (on the spot) and face East. Ask them to draw a sketch or picture of what they have done. Get them to write a description of the amount of turn they needed to make.

    Face the students North again. Ask them to make a half-clockwise turn and describe the direction in which they are now facing. How would you draw a picture to show someone else the half turn that you have just completed?

    Starting from North each time, you might ask the students to complete a ¾ turn, a turn somewhere between a ¼ and a ½ turn, nearly a ¼ turn. These instructions must include the direction of turn, for example, clockwise and anticlockwise. Follow up with appropriate questions as well as activities that allow the students to build an image directly related to ‘amounts of turn’. For example, what would be the result of two anticlockwise quarter turns look like?

    Alternatively, have the students face the front of the classroom, and make turns ‘on the spot’ as previously indicated.
  5. Give the students a challenge:
    Starting at a point on the ground – the students might put a multi-link cube down to mark the starting point – walk out a shape that that has no more than 5 sides so that they end up where they started. They must then describe their shape to a friend and give them the instructions to enable their friend to successfully walk out the shape. Note that the students will need to pay particular attention to their instructions, which must involve the amount of turn, the direction of turn and the number of paces to walk.
  6. An example of a set of instructions might be:
    1. Begin facing the netball goal.
    2. Walk forward 10 paces then do a 1/2 turn, clockwise
    3. Walk forward 10 paces then do a 1/2 turn, clockwise
    4. Repeat instruction c. twice more.
  1. On an OHP show a reference circle, illustrated below, and ask the students to describe the amounts of turn that it might be used to measure. Cirlce
  2. Using a cut out circle, the students might make a reference guide using the benchmarks of 0, 1/4, 1/2, 3/4, and a full turn. They can also put in any other important benchmarks if they wish to. For example, 1/8 since it is a half of a quarter may also be included. This could act as a reference that they can use when estimating amounts of turn about a point. We will refer to this instrument as an angle measurer.

 Angle measurer.

Exploring

  1. Ask the students:
    How could you use the angle measurer to make a 4-sided path that begins and ends at the same place?
    The more benchmarks there are on the circle the more precise the instructions can be. Note that students should be able to specify whether the turns are to be measured clockwise or anticlockwise.
  2. Get the students to hide objects. They might mark the beginning of the turn by using a metre ruler and ‘hiding’ something at a particular point given by amount of turn (clockwise or anticlockwise) and number of paces. Instructions are given to the partner so that they can find the ‘hidden’ object.
  3. Show students an angle estimator, and ask them to make one of their own. They are not to touch or move the demonstration model. The design is shown below:
     

 

White paper
(cut along line)
Black paper
(cut along line)
Insert two circles
along cut lines

Diagram.

Diagram.

Diagram

Give students amounts of turns, and get them to use their angle estimator to sketch and check the amount of turn. Alternatively, the students could sit back to back. One student makes an amount of turn using the angle estimator and tells the amount of turn to their partner. The partner makes the turn and they then check their results.

  1. The angle estimator shows two amounts of turn. How do you think they are related? (The two together make up a full turn. This might be a good time to introduce complementary angles.)
  2. Introduce to the students to another unit for measuring an amount of turn called the degree. Start by referring to a full turn as representing a turn of 360 units (or small amounts of turn) called degrees and denoted by "° ". Ask the students to relate the benchmarks used on their angle estimator to a full turn of 360° . In particular, a 1/2 turn becomes 180° , a 1/4 turn becomes 90° , and a 3/4 turn becomes 270° . The students will need to make a new angle estimator for degrees.
  3. Introduce the word angle and ask the students to draw different pictures that show an angle (that is an amount of turn) of 45° , 100° , 180° , 30° , and 120° .
    In order to help them think about this problem, the students may need to use their angle estimator and consider some ways of refining the measurements particularly between 0° and 90° . Perhaps provide some focus on making equal divisions of 10.
  4. In pairs, get one student to draw an angle, with the direction of turn, and give it to the partner. The partner must first estimate the angle of turn and then use the angle estimator to check and refine the estimate. Both students then compare their results and defend their results if need be.
  5. Introduce students to the protractor. Demonstrate how to measure an angle.  Ask the students to write down their estimates of the different angle measurements. It is important to ensure that the angles drawn have a range of orientations. Examples are shown below:

    Angles

    Angles
     
  6. Hand out the protractors to the students explaining that they are the measuring tool used to measure angles. With the students in pairs, offer them the challenge of finding out how the protractor works. They may need to reflect on the angle estimators that they used in previous tasks.
  7. Let the students use protractors to measure accurately 1/4, 1/2 and 3/4 turns. Let them also measure accurately, 30° , 60 ° and 120° .
  8. Ask the students how they would help someone who had measured the following angle incorrectly.

 Angles

Reflecting

  1. Children draw an estimate of an angle and measure it with a protractor to see how close their estimate is.
  2. Let one member of a pair draw an angle and give it to their partner. The partner first gives a rough estimate of the angle, and then using the scale on the protractor, gives the accurate value.
  3. Give the students some straws of the same length; ask them to make triangles. Will the angles at each corner always be equal? What is the angle measure for each corner? Will the angles always be the same?

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