Teaching & Learning Plans Plan 7: Introduction to Angles Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on teacher input and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible and Expected Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Checking Understanding: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. Teaching & Learning Plan 7: Introduction to Angles Aims • To introduce students to the concept of an angle as a rotation and naming an angle • To help students to learn about various angle types and the measurement of angles (In degrees) • To help students to recall and/or deepen their understanding of the concepts of parallel and perpendicular lines and horizontal and vertical lines Prior Knowledge Students should understand and recall the concepts of planes, points, etc. introduced in Teaching and Learning Plan 6. They should be familiar with the circle as a shape. Learning Outcomes As a result of studying this topic, students will be able to • recognise angles in terms of rotation • classify and describe angles • describe the various angle types in terms of degrees • relate angles to shape • name an angle • identify and name parallel and perpendicular lines, vertical and horizontal lines Relationship to Junior Certificate Syllabus Sub-topics Ordinary Level 2.1 Synthetic geometry The geometrical results should be first encountered through discovery and investigation. (Refer to the appendix on geometry) Terms: Angle; Rotate; acute, right angle, obtuse, straight angle, reflex angle, ordinary angle; circle, perpendicular lines Concepts (introduced as required) Constructions: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11 Theorems: non-rigorous treatment of theorems 1 – 6. © Project Maths Development Team 2009 www.projectmaths.ie 1 Teaching & Learning Plan 7: Introduction to Angles Resources Required Geostrips and geometry sets for each student Introducing the Topic Students may well have some misconceptions about angles, which will need to be explored and overcome: • the size of an angle varies with the length of the arms • the size of an angle varies with the size of arc made with the angle vertex as centre • different orientations of an angle as a source of confusion • that a right angle only exists between vertical and horizontal lines © Project Maths Development Team 2009 www.projectmaths.ie 2 © Project Maths Development Team 2009 »» Demonstrate the concept of angle by using two (long red) Geostrips fastened together at one end. The strips will be on top of each other at first, then the top one is slowly rotated to reveal an angle. Put major emphasis on the ROTATION part of this motion. »» Start a “new words” section on the board. Insert the word “ANGLES”. Teacher’s Support and Actions • student answer/response »» Are students moving the Geostrips in a way that shows they understand the term ‘rotate’? Checking Understanding KEY: » next step »» Give each student two »» Students fasten Geostrips Geostrips: two of the same (of same colour and length) together and rotate. colour and length and an additional two strips of a different colour and length to the previous two. Student Activities: Possible and Expected Responses www.projectmaths.ie »» Today’s class is all about angles. You meet angles everywhere you go – all sorts and sizes of angles in buildings and rooms and even in natural things. Let us start by talking about how an angle is made. When we have two rays in the same place in the plane (“on top of each other”), and we rotate (or turn) one about some common point, while the other stays where it is, we get an angle, just like these two geo-strips here. As you can see when we rotate one strip, we get an angle. The angle is the amount of rotation that is done. Student Learning Tasks: Teacher Input Lesson Interaction Teaching & Learning Plan 7: Introduction to Angles 3 © Project Maths Development Team 2009 www.projectmaths.ie »» The two rays which make up the angle are called the arms of the angle. »» The key thing here is that the angle is the rotation of one ray about another ray through a common point. The angle changes as we change the amount of rotation. • student answer/response »» Are students sure what it means for something to rotate? Checking Understanding KEY: » next step »» Draw an angle on the board and indicate the rays as arms of the angle, and adds “ARMS of the ANGLE” to the “new words” list. »» Rotate the top Geostrip further to make a bigger angle, then vary the amount of rotation to emphasise the point. »» Now let us rotate the strip further; we get a bigger angle. If we rotate more, we get a bigger angle still. »» Students rotate Geostrips. »» Go to the classroom door and open it. An angle forms between the door and some wall. »» Write the word ROTATE on the board in the “new words” list. Teacher’s Support and Actions »» What about when we open the door? The door rotates around the hinges, does it not? See how the door makes an angle with the wall as I rotate it. »» Can you give examples of things that you have seen rotating? • Bottle top, steering wheel of car…your arm…your leg. • Turn…twist. »» Do you understand the meaning of “rotate”? »» Can you give me another word meaning the same? Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles 4 www.projectmaths.ie • They are the same size because they both have the same degree of rotation. »» Now have a look at these two angles on the board. They have the same degree of rotation, but the arcs are different sizes. Are these angles the same size or is one bigger than the other? © Project Maths Development Team 2009 »» Students mirror with their geostrips what the teacher has demonstrated. »» Can you see that the amount of rotation of the larger strip is the same as the amount of rotation of the longer strip? • student answer/response »» Can the students explain clearly why the two angles are the same size even though the arcs are different sizes? »» Are you satisfied that the explanations students give you, show that the length of the arms of the angle do not dictate the size of the angle? KEY: » next step »» Ask the class and give them a moment to think before asking one student. »» Draw two equal angles on the board, but insert different sized arcs on each. »» Invite students to agree or disagree, but explain they must have a valid reason for so doing. »» Do you agree or disagree with this? Teacher’s Support and Actions Checking Understanding »» Use two shorter (yellow for contrast) strips. Rotate one of these and one of the longer red strips simultaneously to show that the amount of rotation and therefore angle size, is unaffected by the length of strip used. Student Activities: Possible and Expected Responses »» Now let us see what happens • No if we change the length of the strip we use to make the angle. Let us rotate the shorter strip at the same time as the larger. Does this change the size of the angle? Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles 5 © Project Maths Development Team 2009 www.projectmaths.ie • student answer/response »» Are students actively participating in the discussion? Checking Understanding KEY: » next step »» Write the words “RIGHT ANGLE” on the board in the “new words” list. »» Brief class discussion of the meaning of the word right in the English language. • Students may say because it is…”upright”? »» Why do you think this is called a right angle? »» The word “right” has many meanings in English. »» Ask the class and give them a moment to think before asking one student. • A right angle (some may say 90°). »» Do you know what we call this angle? »» Write the word VERTEX on the board. »» Indicate on Geostrips where the vertex is. »» Move the strip to the vertical position to show a right angle. • Acute appendicitis, acute shortage etc… »» Write the words ACUTE ANGLE on the board. »» Ask the class and give them a moment to think before asking one student. »» Use the red (longer) strip to show an acute angle. Teacher’s Support and Actions »» Now let us rotate the strip further. »» Where the two arms meet is called the VERTEX of the angle. »» Have you met the word “acute” before? Where? »» Yes. Notice how sharp the point of the angle looks. “acute” means “sharp”. • Acute angle. »» Students mirror with their geostrips what the teacher has demonstrated. »» Now let us rotate the strip further. »» Do you know what we call this angle? Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles 6 • No »» Now if I were to turn the same size angle the other way, would I call it a “left angle”? www.projectmaths.ie »» A reflex angle »» Do you know what we call this angle? © Project Maths Development Team 2009 »» Students mirror with their geostrips what the teacher has demonstrated. »» Now let us rotate the strip further. »» Students may not have met this word before in other conversation? »» Notice how the point of the angle looks less sharp. “Obtuse” means “blunt”. Have you met the word “acute” before? • student answer/response »» Are any students having difficulty with the new words? KEY: » next step »» Write the words REFLEX ANGLE on the board. »» Ask the class; then select an individual student to answer. »» Move the strip to the position to show a reflex angle. »» Write the words OBTUSE ANGLE on the board. »» Ask the class; then select an individual student to answer. »» Do you know what we call this angle? • An obtuse angle. »» Move the strip to the position to show an obtuse angle. »» Ask the class to explain why. »» Turn the angle round so that the right angle is on the left as seen by the students. Teacher’s Support and Actions Checking Understanding »» Now let us rotate the strip even further. »» Students mirror with their geostrips what the teacher has demonstrated. Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles 7 © Project Maths Development Team 2009 • 360° www.projectmaths.ie »» This is a funny number to have for a full circle is it not? Well, the reason for this goes back to the ancient Babylonians of 2000 B.C. and their way of measuring the time in a day. »» How many degrees in a full circle? »» Does anyone know what unit • Degrees we use to measure angles? »» Now we must talk of measuring angles. »» Look around the classroom and identify where angles are formed. What kind of angles are they? Which is the most common angle and which is the least common angle? »» Students look for angles in the room and write a number of them into their copies. »» Students mirror with their geostrips what the teacher has demonstrated. »» Now let us rotate the strip further until we come back to where we started. »» Now we are back where we started and we have done a full revolution which means that the end of the strip has completed a full circle. Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Checking Understanding • student answer/response »» Are students actively participating? KEY: » next step »» Ask the class; then select an individual student to answer. »» Ask the class; then select an individual student to answer. »» Walk around to see what »» Has everyone completed students are writing down. the task? Some students may have difficulty. As you walk around identify and guide those students. »» Move the strip to the position where it started. Teacher’s Support and Actions Teaching & Learning Plan 7: Introduction to Angles 8 »» Move the strip to the position to show a straight angle. »» Ask the class; then select an individual student to answer. »» Students mirror with their geostrips what the teacher has demonstrated. • Perpendicular lines © Project Maths Development Team 2009 www.projectmaths.ie »» Remember you must have at least two lines to compare if you want to use the word “perpendicular”. »» Yes they are called perpendicular lines. KEY: » next step »» Write the words PERPENDICULAR LINES on the board in the “new words” list. »» Ask the class; then select an individual student to answer. • Because at this point we are half-way to a full circle and 180°= ½ of 360°. »» Why? »» There is a name for two lines which are at an angle of 90° to each other. Do you know this name? »» Move the strip to the position to show a straight angle. • Because at this point we are one quarter of the way to a full circle and 90° =1/4 of 360°. »» Why? »» Ask the class; then select an individual student to answer. »» Prepare to demonstrate various angle sizes using board and geo-strips. Move the strip to the vertical position to show a right angle. Teacher’s Support and Actions »» How many degrees in a straight angle? • 180° • 90° Student Activities: Possible and Expected Responses »» How many degrees in a right angle? »» Let us look again at the strips as we make the various angles. Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles • student answer/response Checking Understanding 9 • A vertical line • A horizontal line »» What is the name for a line which goes straight up and is perpendicular to the ground? »» What is the name for a line which is level and runs parallel to the ground? www.projectmaths.ie »» Students look for parallel »» Walk around to see what lines in the room and write students are writing down. a number of them into Some students may have difficulty. As you walk around their copies. you identify and guide those students. »» Can you see any lines parallel to each other in this room? © Project Maths Development Team 2009 • Parallel lines. »» What do you call two lines which are always the same distance apart? • student answer/response »» Do students understand these words? »» Has everyone completed the task? »» Has everyone completed the task? Checking Understanding KEY: » next step »» Write the word HORIZONTAL on the board in the “new words” list. »» Write the word VERTICAL on the board in the “new words” list. »» Ask the class; then select an individual student to answer. »» Write the words PARALLEL LINES on the board in the “new words” list. »» Ask the class; then select an individual student to answer. »» Move the strips to the position to show parallel lines. »» Walk around to see what students are writing down. Some students may have difficulty. As you walk around identify and guide those students. »» Students look for perpendicular lines in the room and write a number of them into their copies. »» Can you see any lines perpendicular to each other in this room? Teacher’s Support and Actions Student Activities: Possible and Expected Responses Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles 10 © Project Maths Development Team 2009 »» Complete Student Activity 2 and 3. www.projectmaths.ie »» Students fill out Student Activity 2 and 3. »» Students copy different ideas to their own from the board. • student answer/response »» Do students understand the words, phrases and diagrams? »» Did students have lots of different suggestions? Checking Understanding KEY: » next step »» Circulate and support students who have difficulty with the task. »» Distribute Student Activity 2 and 3. »» Take selections from each group and record them on the board/flipchart/laptop. »» Walk around to see what students are writing down. If anyone is struggling, ask questions which will give them a hint of an example. »» Distribute Student Activity 1. »» Students fill out Student Activity 1. »» Working in pairs, complete Student Activity 1. Teacher’s Support and Actions »» Demonstrate naming an angle on the board using three letters in two ways, for example <ABC or <CBA. Student Activities: Possible and Expected Responses »» Finally we want to name an angle. We name an angle by the three letters which give the two rays that make up the angle, so that the point denoting the vertex is in the middle. Student Learning Tasks: Teacher Input Teaching & Learning Plan 7: Introduction to Angles 11 © Project Maths Development Team 2009 »» Write down any question you may have. »» Write down anything you found difficult. »» Write down 3 items you learned about angles and lines today. Reflection Student Learning Tasks: Teacher Input www.projectmaths.ie 7. A line which is level and runs parallel to the ground is called a horizontal line. 6. A line which goes straight up and is parallel to the ground is called a vertical line. 5. Lines which are always the same distance apart are called parallel lines. 4. Angles are named using three letters with the vertex in the middle. 3. Two lines that meet at 90 degrees to each other are called perpendicular lines. 2. A full circle has 360 degrees. 1. There are many sizes of angles: acute, right, obtuse, straight and reflex. Student Activities: Possible and Expected Responses KEY: » next step »» Circulate and take note particularly of any questions students have and help them to answer these. Teacher’s Support and Actions Teaching & Learning Plan 7: Introduction to Angles • student answer/response Checking Understanding 12 Teaching & Learning Plan 7: Introduction to Angles Student Activity 1 Students work in pairs. 1. Find and list 5 surfaces in the classroom which are horizontal and 5 which are vertical._______________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 2. See if you can find lines or surfaces which are parallel.___________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 3. See if you can find lines or surfaces which are perpendicular to each other. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 4. Can you find perpendicular lines in the classroom where the lines are not vertical or horizontal?_ __________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 5. What type of lines are used in the pages of your copy?___________________________ ______________________________________________________________________________ ______________________________________________________________________________ 6. Why is it important to have walls vertical?______________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 7. Is it important to have floors horizontal? Why?__________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 8. Do you know what a builder uses to ensure that he builds a wall vertically? ______________________________________________________________________________ ______________________________________________________________________________ 9. What building in Italy is famous for not being vertical?___________________________ ______________________________________________________________________________ ______________________________________________________________________________ © Project Maths Development Team 2009 www.projectmaths.ie 13 Teaching & Learning Plan 7: Introduction to Angles Student Activity 2 Look at the mathematical words/phrases in the middle column below and rewrite each into the correct description box on the left and diagram box on the right. Description Mathematical words/ phrases Line going directly upwards Horizontal Line Angle greater than 90° but less than 180° Acute angle Line parallel to the ground Right Angle Angle of 180° Obtuse Angle Angle greater than 180° Straight Angle Angle of 90° Reflex Angle Angle less than 180° Ordinary Angle (more than one diagram) Lines which never meet Perpendicular Lines Lines at an angle of 90° to each other Parallel lines Angle less than 90° Vertical line © Project Maths Development Team 2009 www.projectmaths.ie Diagram 14 Teaching & Learning Plan 7: Introduction to Angles Student Activity 3 1. Name each of these angles using three letters in two ways: Angle Name one Name two 2. What is a wide-angle lens in a camera? What does it do? What type of picture does it take?_________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 3. What does it mean when we say that in football that “the goalkeeper narrowed the angle”? What angle did the goalkeeper narrow?_________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 4. Draw out the last five letters of the alphabet in capitals. What kinds of angles are formed in them?_ _____________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 5. What kinds of angle are formed by the hands of a clock? (a) At 9.00p.m?________________________________________________________________ (b) At 6.00 a.m.?_______________________________________________________________ (c) At 3.00 a.m.?_______________________________________________________________ 6. Through how many degrees will the minute hand of a clock rotate in: (a) 30 minutes?________________________________________________________________ (b) 20 minutes?________________________________________________________________ (c) 40 minutes?________________________________________________________________ (d) 45 minutes?________________________________________________________________ © Project Maths Development Team 2009 www.projectmaths.ie 15

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