PAGE 1: Overview OVERVIEW ======== An angle is made when two lines meet at a point. In this sub-strand you will find angles made by lines on a chart, spot angles on objects such as cubes and cuboids, and see how the amount of a turn is the same thing as the size of an angle. You will also look at how people use angles every day. ## Strand 3.0: Geometry Sub-strand 3.2: Angles Suggested lessons: 5 ## What this sub-strand covers • Angles at a point • Relating turns to angles ## Source • KICD Grade 4 Mathematics Curriculum Design (Revised 2024) — Strand 3.0 Geometry, Sub-strand 3.2 Angles ## Learning outcomes By the end of the sub-strand, the learner should be able to: a) identify an angle at a point in lines b) identify angles from the objects in the environment c) relate a turn to angles in real-life situations d) appreciate use of angles in real-life situations ## Key inquiry questions 1. How can we use angles in daily life? PAGE 2: How we will learn ## Learning experiences • discuss and recognise angles that are made by 2 lines that meet at point using a chart with a different line • team u[ with peers to explore and identify angles from objects such as cubes and cuboids in the environment • make clockwise quarter and half turns, and relate the turns to angles • play digital games and learn more about angles ## Core competencies • Communication and collaboration: learner collaboratively works with others and recognises angles that are made by 2 lines that meet at a point using a chart with a different line. ## Values • Responsibility: learner observes safety precautions while identifying angles from objects such as cubes, cuboids in the environment. ## Pertinent and contemporary issues • Environmental Education: learner explores and identifies angles from objects such as cubes, and cuboids in the environment. ## Links to other learning areas • The learner is able to relate angles from objects such as cubes and cuboids in the environment to perspective in Creative Arts. LESSON 1: ANGLES AT A POINT =========================== ## What you will learn • Explain what an angle is. • Identify the angles made where lines meet at a point. • Recognise a right angle and tell whether an angle is smaller or bigger than a right angle. Curriculum outcomes: a) identify an angle at a point in lines ## Introduction Open a book a little, then wider, then wider still. The two covers meet along the spine, and the gap between them grows. That opening between two lines that meet is an angle. PAGE 3: Lesson 1: Angles at a point — What is an angle? ## Notes ### What is an angle? When two straight lines meet at a point, they make an angle. The point where the lines meet is called the vertex. The two lines are called the arms of the angle. The angle is the amount of opening, or turn, between the two arms. The size of an angle does not depend on how long the arms are. If you draw the same two arms longer, the opening between them stays the same, so the angle is the same. ### Counting angles at a point - Two lines that meet at their ends, like the letter L or V, make 1 angle. - A line that meets the middle of another line, like the letter T, makes 2 angles, one on each side. - Two lines that cross each other, like the letter X or a plus sign +, make 4 angles at the point where they cross. ### The right angle The corner of a book, the corner of a door and the corner of this page are all the same kind of angle. It is called a right angle, or a square corner. Right angles are often marked with a small square drawn in the corner. You can make a right angle tester: fold a piece of paper in half, then fold it in half again so the folded edges lie on top of each other. The corner you get is a right angle. PAGE 4: Lesson 1: Angles at a point — Example 2 ### Comparing angles Put the corner of your tester on the vertex of an angle with one edge along one arm. - If the other arm is inside the tester, the angle is smaller than a right angle. - If the other arm lies exactly along the other edge, it is a right angle. - If the other arm is outside the tester, the angle is bigger than a right angle. When two lines cross to make a plus sign, all 4 angles are right angles. [DIAGRAM] TYPE: svg ALT: Angles are made where lines meet at a point. PAYLOAD: CAPTION: Angles are made where lines meet at a point. [/DIAGRAM] ## Key words • angle — the amount of opening or turn between two lines that meet at a point • vertex — the point where the two arms of an angle meet • arms — the two lines that make an angle • right angle — a square corner, like the corner of a book ## Worked examples ### Example 1 How many angles are made at the point where the two lines of the letter X cross? Working: 1. The two lines cross and make four openings around the point: top, bottom, left and right. Answer: 4 ## Worked examples ### Example 2 How many angles does the letter T make where its lines meet? Working: 1. The upright line meets the middle of the top line. 2. There is one angle on its left and one on its right. Answer: 2 ## Worked examples ### Example 3 Three straight lines cross at one point. How many angles are there at the point? Working: 1. Each line crossing through the point adds 2 angles, one on each side. 2. 3 lines × 2 = 6 Answer: 6 PAGE 5: Lesson 1: Angles at a point — Answers to practice ## Activity: Angles on a line chart You will need: Chart with different lines meeting at points, Plain paper for a right angle tester, Pencil and ruler 1. In your group, look at the chart. Point to each place where 2 lines meet at a point. 2. Count the angles made at each point and write the number next to it. 3. Fold a piece of paper twice to make a right angle tester. 4. Use the tester to sort each angle on the chart: smaller than a right angle, a right angle, or bigger than a right angle. 5. Draw your own letters L, T, V and X with a ruler and mark the angles in each. ## Practice 1. What is the name of the point where the two arms of an angle meet? 2. How many angles are made where the lines of a plus sign (+) cross? 3. The letter V is drawn with longer arms. Does the angle get bigger? 4. Four straight lines cross at one point. How many angles are there at the point? ## Answers to practice 1. The vertex 2. 4 3. No. The size of an angle depends on the opening between the arms, not on how long the arms are. 4. 8 ## Summary • An angle is made where two lines meet at a point called the vertex. • Two crossing lines make 4 angles. • A right angle is a square corner. PAGE 6: Lesson 2: Angles on objects in the environment LESSON 2: ANGLES ON OBJECTS IN THE ENVIRONMENT ============================================== ## What you will learn • Identify angles on cubes, cuboids and other objects. • Count the right angles on the faces of a cube or cuboid. Curriculum outcomes: b) identify angles from the objects in the environment; d) appreciate use of angles in real-life situations ## Introduction A box of chalk, a brick, a match box and a dice all have corners. Every corner is a place where edges meet, and wherever edges meet there are angles. ## Notes Angles are all around us. You can find them wherever two edges or lines meet. ### Cubes and cuboids A cube is a solid shape like a dice. All its faces are squares of the same size. A cuboid is a solid shape like a box of chalk, a brick or a match box. Its faces are rectangles, and some may be squares. Cubes and cuboids have: - 6 flat faces - 12 edges, where two faces meet - 8 corners, where three edges meet PAGE 7: Lesson 2: Angles on objects in the environment — Key words ### Angles on the faces Each face of a cube or cuboid has 4 corners, and each corner of the face is a right angle. Check this by putting your right angle tester into the corner of a box. Since there are 6 faces and each face has 4 right angles, there are 6 × 4 = 24 right angles on the faces of a cube or cuboid. ### Angles in the classroom and at home - Corners of a door, a window, a desk top, the chalkboard and a book are right angles. - The two hands of a clock make an angle at the centre of the clock. - The legs of a folding ladder or an open pair of scissors make angles that can be smaller than a right angle. - The roof of many houses meets at the top to make an angle. ### Why angles matter Builders check that walls meet at right angles so that the house stands strong and doors and windows fit. Carpenters make the corners of tables and cupboards square. When a door is opened wider, the angle between the door and the wall becomes bigger. ### Safety When looking for angles, do not touch sharp corners of metal sheets or broken glass, and ask an adult before climbing to look at a roof. ## Key words • cube — a solid with 6 equal square faces, like a dice • cuboid — a solid with 6 rectangular faces, like a box of chalk • edge — a line where two faces of a solid meet • face — a flat surface of a solid ## Worked examples ### Example 1 How many right angles are there on all the faces of a match box? Working: 1. A match box is a cuboid with 6 faces. 2. Each face has 4 right angles. 3. 6 × 4 = 24 Answer: 24 PAGE 8: Lesson 2: Angles on objects in the environment — Angle hunt with cubes and cuboids ## Worked examples ### Example 2 How many right angles are there on the top face of a desk? Working: 1. The top of a desk is a rectangle. 2. A rectangle has 4 corners, all right angles. Answer: 4 ## Worked examples ### Example 3 A class has 3 windows. Each window is a rectangle. How many right angles do the corners of the windows make altogether? Working: 1. Each window has 4 right angles. 2. 3 × 4 = 12 Answer: 12 ## Activity: Angle hunt with cubes and cuboids You will need: Empty boxes (cuboids), A dice or cube, Right angle tester, Exercise book 1. In your group, hold a box and point to its faces, edges and corners. Count each. 2. Use your right angle tester to check the corners of each face. 3. Count the right angles on all the faces and record the number. 4. Walk safely round the classroom and list 5 other objects that have right angles. 5. Share your list with another group. ## Practice 1. How many faces does a cube have? 2. How many right angles are on 2 faces of a cube? 3. Name two objects in the classroom that have right angles. 4. Why do builders make walls meet at right angles? ## Answers to practice 1. 6 2. 8 3. For example a book, the chalkboard, a desk top, a door or a window. 4. So that the building is strong and doors, windows and roofs fit well. ## Summary • Cubes and cuboids have 6 faces, 12 edges and 8 corners. • Every face of a cube or cuboid has 4 right angles. • Angles are found wherever edges or lines meet. PAGE 9: Lesson 3: Relating turns to angles — Quarter turn and right angle LESSON 3: RELATING TURNS TO ANGLES ================================== ## What you will learn • Explain that an angle is an amount of turn. • Relate quarter, half and full turns to right angles. Curriculum outcomes: c) relate a turn to angles in real-life situations; d) appreciate use of angles in real-life situations ## Introduction Stand facing the chalkboard, then make a quarter turn clockwise to face the side wall. The line you were looking along and the new line you are looking along make a square corner at your feet. A turn makes an angle. ## Notes In Position and Direction you made quarter, half and full turns. Every turn makes an angle between where you started facing and where you end up facing. ### Quarter turn and right angle A quarter turn makes a right angle. When the minute hand of a clock turns from 12 to 3, the hand at the start and the hand at the end make a right angle at the centre of the clock. At 3 o'clock the hour hand points to 3 and the minute hand points to 12. The two hands make a right angle. At 9 o'clock they also make a right angle. PAGE 10: Lesson 3: Relating turns to angles — Turns in daily life ### Half turn and a straight line A half turn is made of two quarter turns, so it is the same as 2 right angles. The start and end make a straight line. At 6 o'clock the two hands of a clock point in opposite directions and make a straight line. ### Full turn A full turn is made of four quarter turns, so it is the same as 4 right angles. You end facing where you started. ### Turns in daily life - A door that is opened until it makes a square corner with the wall has turned a quarter turn. - When you turn a key in a lock, a tap or a steering wheel, you are making an angle. - A football player who turns to face the opposite goal has made a half turn, which is 2 right angles. - Road signs show arrows that bend at an angle to tell drivers which way to turn. The bigger the turn, the bigger the angle. A turn smaller than a quarter turn makes an angle smaller than a right angle. PAGE 11: Lesson 3: Relating turns to angles — Example 3 ## Key words • amount of turn — how far something turns; this is the size of an angle • straight line — what a half turn makes; it is equal to 2 right angles ## Worked examples ### Example 1 How many right angles are there in a half turn? Working: 1. A half turn is 2 quarter turns. 2. Each quarter turn is 1 right angle. Answer: 2 ## Worked examples ### Example 2 How many right angles are there in a full turn? Working: 1. A full turn is 4 quarter turns. 2. Each quarter turn is 1 right angle. Answer: 4 ## Worked examples ### Example 3 The minute hand turns clockwise from 12 to 9. How many right angles has it turned through? Working: 1. 12 to 3 is one right angle, 3 to 6 is another, 6 to 9 is another. 2. Each right angle is 3 numbers on the clock: 9 ÷ 3 = 3 Answer: 3 right angles ## Activity: Make turns, make angles You will need: Open space, Two long sticks or rulers, Paper clock with two hands 1. One learner stands on a point and holds a stick pointing forward. A partner places another stick on the ground along the first direction. 2. The learner makes a clockwise quarter turn and points the stick again. Place the second stick on the ground. Check the angle between the two sticks with a right angle tester. 3. Repeat for a clockwise half turn and see that the sticks make a straight line. 4. On the paper clock, set the hands to 3 o'clock, 6 o'clock and 9 o'clock. Say whether the hands make a right angle or a straight line. PAGE 12: Lesson 3: Relating turns to angles — Summary ## Practice 1. What angle is made by a quarter turn? 2. At what time between 1 o'clock and 11 o'clock do the two hands of a clock make a straight line? 3. A learner makes 3 quarter turns. How many right angles has she turned through? 4. Give one example from daily life where a turn makes an angle. ## Answers to practice 1. A right angle 2. 6 o'clock 3. 3 4. For example opening a door, turning a key, turning a steering wheel or turning to face another direction. ## Summary • An angle is an amount of turn. • A quarter turn makes a right angle. • A half turn is 2 right angles and a full turn is 4 right angles. LESSON 4: QUARTER TURNS, HALF TURNS AND ANGLES AT A POINT ========================================================= ## What you will learn • Make a clockwise quarter turn and a clockwise half turn and name the angle. • Find how many right angles fit around a point. Curriculum outcomes: a) identify an angle at a point in lines; c) relate a turn to angles in real-life situations ## Introduction Put two sticks on the ground so that they meet at your toes. Turn on the spot until you face along the second stick. The gap between the sticks did not grow because the sticks got longer. It grew because you turned. PAGE 13: Lesson 4: Quarter turns, half turns and angles at a point — A clockwise quarter turn ## Notes The size of an angle is the amount of turn from one arm to the other. You can make that turn and then look back at the angle it left behind. ### A clockwise quarter turn Face the front of the class and lay a stick on the ground pointing forward. Turn clockwise until you face the right-hand wall, and lay a second stick that way. The sticks meet at a square corner. That angle is one right angle. The minute hand of a clock makes the same turn from 12 to 3. At 3 o'clock the two hands also make one right angle. ### A clockwise half turn Keep turning clockwise until you face the back of the room. You have made two quarter turns, so the angle is 2 right angles. The two sticks now lie on one straight line. At 6 o'clock the clock hands point opposite ways and lie on a straight line. Two angles that sit next to each other and make a straight line add up to 2 right angles. If one of them is a right angle, the other one is a right angle as well, because 2 − 1 = 1. ### A three-quarter turn One more quarter turn, from the back of the room to the left-hand wall, makes 3 right angles altogether. The minute hand does this when it moves clockwise from 12 to 9. The other way from 12 to 9 is only one quarter turn, so the same two directions can have a small angle between them one way and a large angle the other way. ### Turns that are not exact quarters Open a door a little. The turn is less than a quarter turn, so the angle between the door and the frame is smaller than a right angle. Open the door past the square corner but not as far as a straight line. The angle is now bigger than a right angle. A longer door does not make a bigger angle. Only a bigger turn does. PAGE 14: Lesson 4: Quarter turns, half turns and angles at a point — Right angles around a point ### Right angles around a point A full turn is 4 right angles. Stand at a point and turn all the way round: the angles you pass, with no gaps and no overlaps, add to 4 right angles. Two lines that cross like a plus sign make exactly those 4 right angles. On a clock they are the turns from 12 to 3, 3 to 6, 6 to 9, and 9 back to 12. If one angle at a point is a right angle, the rest of the full turn is 4 − 1 = 3 right angles. [DIAGRAM] TYPE: svg ALT: A quarter turn is 1 right angle, a half turn is a straight line of 2 right angles, and a full turn around a point is 4 right angles. PAYLOAD: CAPTION: A quarter turn is 1 right angle, a half turn is a straight line of 2 right angles, and a full turn around a point is 4 right angles. [/DIAGRAM] PAGE 15: Lesson 4: Quarter turns, half turns and angles at a point — Example 3 ## Key words • angles at a point — the angles that fit all the way round a point and add up to 4 right angles • straight line — an angle of 2 right angles, made by a half turn ## Worked examples ### Example 1 The minute hand moves clockwise from 12 to 9. How many right angles has it turned through? Working: 1. From 12 to 3 is 1 right angle, from 3 to 6 is another, and from 6 to 9 is another. 2. Each right angle is 3 numbers on the clock: 9 ÷ 3 = 3. Answer: 3 right angles ## Worked examples ### Example 2 One angle on a straight line is a right angle. How many right angles is the angle beside it? Working: 1. A straight line is 2 right angles. 2. 2 − 1 = 1, so the angle beside it is also 1 right angle. Answer: 1 right angle ## Worked examples ### Example 3 A learner makes 3 half turns clockwise. How many right angles is that, and how many full turns does it contain? Working: 1. A half turn is 2 right angles. 2. 3 × 2 = 6 right angles. 3. A full turn is 4 right angles, and 6 = 4 + 2, so this is 1 full turn and 1 half turn. Answer: 6 right angles, which is 1 full turn and 1 half turn PAGE 16: Lesson 4: Quarter turns, half turns and angles at a point — Turn and mark the angle ## Activity: Turn and mark the angle You will need: Two long sticks or rulers, Chalk, A right angle tester, A paper clock 1. In pairs, mark a point. One learner faces a wall and the partner lays a stick along that direction. 2. The learner makes a clockwise quarter turn. Lay the second stick and test the angle. It should match the right angle tester. 3. Return to the start and make a clockwise half turn. Check that the sticks make a straight line, which is 2 right angles. 4. On the paper clock, move the minute hand from 12 to 3, to 6 and to 9. Say how many right angles each move is. 5. Find one angle on the clock chart or on a drawing of crossing lines, and say whether it is smaller than, equal to, or bigger than a right angle. ## Practice 1. How many right angles are there in a three-quarter turn? 2. How many half turns make 6 right angles? 3. Angles around a point add up to 4 right angles. One of them is 1 right angle. How many right angles are in the rest of the turn? 4. The minute hand turns clockwise from 6 to 12. How many right angles is that? ## Answers to practice 1. 3 2. 3 3. 3 4. 2 ## Summary • A clockwise quarter turn makes 1 right angle. A half turn makes a straight line of 2 right angles. • A three-quarter turn is 3 right angles. • The angles around a point add up to 4 right angles. PAGE 17: Lesson 5: Using angles in daily life LESSON 5: USING ANGLES IN DAILY LIFE ==================================== ## What you will learn • Find angles on toys and other objects and say how they compare with a right angle. • Give examples of how people use angles in daily life. Curriculum outcomes: b) identify angles from the objects in the environment; d) appreciate use of angles in real-life situations ## Introduction A toy car made from a small box still has square corners, and its wheels turn about their centres. The same angles you found on a clock and on a chalk box are on the toys you make and on the doors you open. ## Notes An angle is useful when we need a corner to fit, a door to open, or a turn to be clear. ### Angles you can test The front cover of a book has 4 right angles. A window frame and the corner of a set square do too. Press a folded-paper right angle tester into the corner. If the edges match, the angle is a right angle. A dice is a cube and a chalk box is a cuboid. Each face is a square or a rectangle, so each face has 4 right angles. A toy car or a doll's bed made from a box has those same face angles. Count faces and corners before you decorate the toy, so you know which corners should be square. PAGE 18: Lesson 5: Using angles in daily life — Care, and learning more A wheel turns about its centre. A quarter turn of the wheel is 1 right angle, a half turn is 2 right angles, and a full turn is 4 right angles. The wheel looks the same after a full turn, but it has still turned. ### Angles that open and close Scissors, a door and a folding chair have angles that change. Closed scissors make a very small angle between the blades. Open them wider and the angle grows. A door that has turned a quarter away from the frame makes a right angle with the wall. Open it further and the angle becomes bigger than a right angle. A doll's arm or a toy crane can be set to a new angle in the same way. These objects show that an angle is an amount of turn, not a fixed mark on the page. ### Where people use angles A builder checks that a wall meets the floor in a right angle so the wall stands upright and the door can close. A carpenter checks the corners of a table or a cupboard before fixing them. Two paths that cross make angles at the crossing. If they cross like a plus sign, the four angles are right angles. A bent arrow on a sign tells a driver or a cyclist which way to turn. ### Care, and learning more Point to sharp metal corners, broken glass and thorns, but do not touch them. When you make a toy car or a doll from card, fold on a table, keep scissors closed while you walk, and ask the teacher before you cut. With the teacher, a digital game can ask you to spot right angles or to turn a character by a quarter or a half turn. The angles on the screen are the same ones you made with sticks. PAGE 19: Lesson 5: Using angles in daily life — Example 3 ## Key words • right angle tester — a square corner, often a twice-folded paper, used to check an angle • crossing — the point where two paths or two lines meet and make angles ## Worked examples ### Example 1 Each of 5 exercise books has 4 right angles on its front cover. How many right angles are there on the 5 covers? Working: 1. Each front cover is a rectangle with 4 right angles. 2. 5 × 4 = 20. Answer: 20 ## Worked examples ### Example 2 A wheel makes 2 full turns. How many right angles has it turned through? Working: 1. One full turn is 4 right angles. 2. 2 × 4 = 8. Answer: 8 ## Worked examples ### Example 3 Two paths cross like a plus sign. How many right angles do they make? If 6 such crossings are built, how many right angles is that? Working: 1. A plus-sign crossing makes 4 right angles. 2. 6 × 4 = 24. Answer: 4 at one crossing, and 24 at 6 crossings PAGE 20: Lesson 5: Using angles in daily life — Angles on toys and in the room ## Activity: Angles on toys and in the room You will need: Boxes, card and bottle tops for a toy car or doll, Right angle tester, Scissors used only with the teacher, Exercise book 1. In a group, make a simple toy car or doll from a box. Point to the right angles on its faces and check them with the tester. 2. Turn one wheel or a paper circle through a quarter turn and through a half turn. Say how many right angles each turn is. 3. Open a door or a pair of scissors a little, then more. Say whether each angle is smaller than, equal to, or bigger than a right angle. 4. List two jobs or daily actions that need right angles, such as fitting a door or making a table. 5. If a digital device is available, play a game that asks you to find angles. Share one new angle you noticed. ## Practice 1. A wheel turns through 8 right angles. How many full turns is that? 2. Three window frames each have 4 right angles. How many right angles is that? 3. A door is opened by a quarter turn from the closed position. What angle does it make with the frame? 4. Name one worker who checks right angles, and say why. ## Answers to practice 1. 2 2. 12 3. A right angle 4. For example a builder, so that walls stand upright and doors fit, or a carpenter, so that table corners are square. ## Summary • Faces of cubes, cuboids, books and windows have right angles you can test. • Doors, scissors and wheels show angles as turns. • Builders and carpenters use right angles so that parts fit. PAGE 21: Revision quiz REVISION QUIZ ============= Answer all the questions, then mark yourself on the Answers tab. 1. What is the point where two arms of an angle meet called? A. Edge B. Vertex C. Face D. Arm 2. How many angles are made where two straight lines cross? A. 1 B. 2 C. 3 D. 4 3. How many angles does the letter L make? A. 1 B. 2 C. 3 D. 4 4. Which of these corners is a right angle? A. The corner of a book B. The tip of a sharpened pencil C. The edge of a ball D. The point of a thorn 5. How many edges does a cuboid have? A. 6 B. 8 C. 12 D. 24 6. How many right angles are there on all the faces of a cube? A. 8 B. 12 C. 24 D. 6 7. Which turn makes a right angle? A. A quarter turn B. A half turn C. A full turn D. No turn 8. How many right angles make a full turn? 9. At 3 o'clock, what angle do the hands of a clock make? A. A straight line B. A right angle C. No angle D. A full turn 10. Why should a builder make the corners of a door frame right angles? 11. How many right angles are there in a three-quarter turn? A. 1 B. 2 C. 3 D. 4 12. Angles around a point add up to how many right angles? A. 2 B. 4 C. 6 D. 8 PAGE 22: Quiz answers ANSWERS ======= 1. B. Vertex The vertex is the meeting point of the arms. 2. D. 4 Crossing lines make 4 angles round the point. 3. A. 1 The two lines meet at their ends, making one angle. 4. A. The corner of a book A book corner is a square corner. 5. C. 12 A cuboid has 12 edges. 6. C. 24 6 faces × 4 right angles each = 24. 7. A. A quarter turn A quarter turn is the same as one right angle. 8. 4 A full turn is 4 quarter turns, each a right angle. 9. B. A right angle One hand points to 12 and the other to 3, a quarter turn apart. 10. So that the door fits well and closes properly. Square corners let doors, windows and walls fit together. 11. C. 3 Each quarter turn is 1 right angle, so three of them make 3 right angles. 12. B. 4 A full turn around a point is 4 right angles.