Te Kete Ipurangi
Communities
Schools

### Te Kete Ipurangi user options:

Level Two > Geometry and Measurement

# Angles

Keywords:
Purpose:

This unit uses one of the digital learning objects, Angles, to support students as they investigate measuring and drawing angles using other angles as units of measurement. It is suitable for students working at level 2 of the curriculum because they estimate and measure the size of other angles using other angles and not with compasses and protractors. This unit includes background information on teaching about angles, a sequence that can be used by the teacher when working with a group of students on the learning object, and ideas for independent student work.

Achievement Objectives:

Specific Learning Outcomes:

estimate and measure angles using other angles

Description of mathematics:

Angle can be seen as and thought of in at least three ways. These are as:

• the spread between two rays
• the corner of a 2-dimensional figure
• an amount of turning

The final one of these underpins the others and leads on naturally to the definition of degree and the ability to measure angles with a standard unit. This leads students on to being able to apply their knowledge of angle in a variety of situations.

We see angle as developing over the following progression:
Level 1: quarter and half turns as angles
Level 2: quarter and half turns in either a clockwise or anti-clockwise direction and angle as an amount of turning
Level 3: sharp (acute) angles, blunt (obtuse) angles and right angles degrees applied to simple angles – 90°, 180°, 360°, 45°, 30°, 60°
Level 4: degrees applied to all acute angles, degrees applied to all angles, angles applied in simple practical situations
Level 5: angles applied in more complex practical situations

The concept of angle is something that we see students developing gradually over several years. As their concept matures, they will be able to apply it in a range of situations including giving instructions for directions and finding heights.

Outside school and university, angle is something that is used regularly by surveyors and engineers both as an immediate practical tool and as a means to solve mathematics that arises from practical situations. So angle is important in many applications in the ‘real’ world as well as an ‘abstract’ tool. This all means that angles have a fundamental role to play in mathematics and its application.

### Prior to using the Angles Learning Objects

The unit Clockwise helps students to explore the idea that turns can be clockwise and anticlockwise in their direction. This would also be a useful unit to do prior to this Angles unit.

Activity:

### Working with the learning object with students (Measure)

1. Show students the learning object and explain that it provides a tool for measuring angles.
2. Choose the Measure button from the front page of the learning object.
3. Choose Measuring Angle.

4. Discuss the first screen with students and clarify with them what the problem is asking.
5. Discuss ways to solve the question. For example, start by estimating how many red angles would be in half the circle. Ask for a volunteer to discuss a way to solve the question and enter an answer.
6. After a correct answer has been entered wait to click the Make an Angle to Measure button.

7. Discuss with the students ways to solve the question. Ask for a volunteer to enter a number.
8. Watch the red angle move around the blue angle and count the number of turns.
9. Correct the answer if necessary.
10. By clicking the New Angle button you can measure more angles using the tool.

### Working with the object with students (Draw)

1. Choose the Draw button from the front page of the learning object.
2. The first screen of the draw problem is the same as the first screen of the Measuring Angles. Again discuss with students ways to solve the problem and ask a volunteer to enter an answer.
3. Click on Draw an Angle.

4. Ask for a volunteer to watch the blue angle and click the stop button when it is the correct size.
5. Discuss ways to solve the problem. For example, counting out loud the number of red angles as the blue angle grows, 1, 2, 3, 4, 5, 6.
6. The learning object provides feedback to help if the answer is incorrect and you can click Draw Angle Again to try again.

### Students working independently with the learning object

This learning object generates problems for the student so once they are familiar with how it works you could allow individual students or pairs of students to work with the learning object independently. The learning object rewards correct answers with balloons and students could be challenged to collect a bunch of balloons.

### Students working independently without the learning object

Independent activities that develop the same concepts as the learning object include:

• Matching angles the same size but in different orientations.
• Ordering angles of different sizes from smallest to biggest.
• Drawing angles 2, 3, 4 times bigger than given angles.

## Angles, Parallel Lines, and Polygons

In this unit, students will learn to investigate polygons using an instrument of their own construction. They should be able to prove the general formula for the number of degrees in any polygon (including a triangle). Finally they will investigate an interesting locus problem related to polygons.

## Simple Angles

identify and construct right, acute and obtuse angles

begin to appreciate the degree, unit of measurement of angle

know the degree value of angles that are simple fractions of a whole turn

know that the angle at a point is 360°

## Measuring Angles

This unit is about estimating and measuring angles. Initially, 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.

## Direction and Position

This unit of work is based on the Māori medium unit Te Ahunga me te Taunga, and investigates the main compass directions, degree measures of direction and the use of a grid system to define locations on a map.

## Street Maps

In this unit students use street maps as the context to learn about grid references, and for giving and following instructions involving directions and distances. In this unit students use street maps as the context to learn about grid references, and for giving and following instructions involving directions and distances.