• Identify corresponding, alternate and co-interior relationships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
•
7.1 Classifying angles and lines
7.3 Angles and parallel lines
7.4 Triangles
7.5 Quadrilaterals
7.6 Polygons
are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04) Student Book pages 295–301
• Identify corresponding, alternate and co-interior relationships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Classify triangles, quadrilaterals and other polygons according to their side and angle properties; identify and reason about relationships (VC2M7SP02)
• Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Classify triangles, quadrilaterals and other polygons according to their side and angle properties; identify and reason about relationships (VC2M7SP02)
• Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Classify triangles, quadrilaterals and other polygons according to their side and angle properties; identify and reason about relationships (VC2M7SP02)
7.5
7.3
7.4
7.5
• Identify corresponding, alternate and co-interior relationships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Classify triangles, quadrilaterals and other polygons according to their side and angle properties; identify and reason about relationships (VC2M7SP02)
Curriculum links
Measurement
Iden�fy corresponding, alternate and co-interior rela�onships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Constructing a pair of parallel lines and a pair of perpendicular lines using their properties, a pair of compasses and a ruler and set squares, or using dynamic geometry software
• Using dynamic geometry software to identify relationships between alternate, corresponding and co-interior angles for a pair of parallel lines cut by a transversal
• Using dynamic geometry software to demonstrate how angles and their properties are involved in the design and construction of scissor lifts, folding umbrellas, toolboxes and cherry pickers
• Using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°
Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Using concrete materials to demonstrate that the sum of the interior angles of a triangle is 180°; for example, using paper triangles and tearing to demonstrate that the interior angles when combined form 180°
• Using decomposition and the angle sum of a triangle to generalise the interior angle sum of an ����-sided polygon, as 180(����− 2) = 180����− 360 Space
Classify triangles, quadrilaterals and other polygons according to their side and angle proper�es; iden�fy and reason about rela�onships (VC2M7SP02)
• Using strips of paper with parallel sides to make triangles and quadrilaterals, and contrasting the rigidity of triangles with the flexibility of quadrilaterals
• Constructing triangles with 3 given side lengths and discussing the question ‘Can any 3 lengths be used to form the sides of a triangle?’
• Identifying and communicating about side and angle properties of scalene, isosceles, equilateral, right-angled, acute and obtuse triangles using geometric conventions
• Describing, comparing and contrasting squares, rectangles, rhombuses, parallelograms, kites and trapeziums, explaining the relationships between these shapes
Suppor�ng resources (available online)
• Interactive skillsheets:
o The interactive skillsheets both teach and assess a discrete mathematical skill. The ‘learn this skill’ section introduces the skill, with opportunities for students to fill in gaps in examples and check their comprehension in situ. The ‘practise this skill’ section contains a variety of different questions to truly interrogate student understanding of the skill.
• Investigations:
o The investigations explore the concepts covered in the exercises in more detail, in a real-life context. Each investigation has a clearly stated focus, with questions leading to a logical conclusion and extension material supplied in a ‘going further’ section.
• Worksheets:
o The worksheets offer additional practice, with a clearly stated focus and questions in both ‘show your understanding’ and ‘now try this’ sections.
• Online assessments:
o Pre-tests
- The pre-tests assess the listed prerequisite skills (found in each module opening spread), with 5 questions assessing each skill. The results of the pre-tests are reported against each skill at three levels: ‘has this skill’, ‘needs to practise’ and ‘needs to learn’.
o Quick quizzes
- The quick quizzes are multiple-choice quizzes with 9 questions that assess the key concepts from each lesson in the series.
o Checkpoint quizzes
- The checkpoint quizzes consist of 15 questions covering content from each module up to that point. All the questions in the checkpoint quizzes are distinct from the questions in the checkpoint quiz in the student book.
o Checklists
- An online version of the success criteria checklist at the end of each module which students can use to rate their confidence for each skill. Their overall confidence level is displayed in the assessment report to provide insights into student confidence levels.
o Review quizzes
- The review quizzes consist of 20 questions covering content from across each module. All the questions in the review quizzes are distinct from the questions in the review section in each module.
• Quizlet
o The Quizlet resource provides an opportunity for students to practice their mathematical literacy skills, by matching definitions to key terms.
• Class tests
o The class tests are quarantined (teacher-only) resources that cover content from across each module, split into multiple choice, short answer and analysis questions. Two class tests (A and B) are provided for each module, with test B being a mirror of test A (e.g. the same questions with numbers, etc. changed where possible).
Things to know before teaching Module 7
Diagnos�c pre-test and interac�ve skillsheets
A diagnostic pre-test has been developed to determine whether students have the prerequisite skills and background knowledge to complete this module. The pre-test assesses their understanding of the following prerequisite skills:
• Comparing angles less than 180º
• Classifying shapes
• Substitution
• Solving equations using inverse operations
DRAFT
For each skill, students receive one of the following results: ‘needs to learn’, ‘needs to practise’ or ‘has this skill’. Students assigned either ‘needs to learn’ or ‘needs to practise’ a particular skill are deemed to require further support before they begin work on the new content in the module. These students are assigned an interactive
skillsheet which includes relevant theory, sample questions and practice questions. Students may work through these interactive skillsheets as many times as required until they are confident with the skill.
Following the pre-test, students are also assigned one of three levels of understanding based on their overall result: Above level (A), On level (O) and Below level (B). These levels are used to assign suggested questions and extra resources to each student as part of a Learning Pathway Report (LPR). You can access these levels, student skill results and the related LPRs in the Skills Report for this module online.
It is recommended that students complete the diagnostic pre-test and any assigned interactive skillsheets before starting work on the module.
Lesson 7.1: Classifying angles and lines
Pages 286–294
Total time: 50 minutes
Learning inten�ons
By the end of this lesson, students will be able to:
• identify, measure and classify angles and lines.
Success criteria
By the end of this lesson, students will be able to:
• identify and name intervals, rays, lines, parallel lines and perpendicular lines
• measure and draw angles using a protractor
• identify and classify angles.
Curriculum links
Measurement
Iden�fy corresponding, alternate and co-interior rela�onships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Constructing a pair of parallel lines and a pair of perpendicular lines using their properties, a pair of compasses and a ruler and set squares, or using dynamic geometry software
• Using dynamic geometry software to identify relationships between alternate, corresponding and co-interior angles for a pair of parallel lines cut by a transversal
• Using dynamic geometry software to demonstrate how angles and their properties are involved in the design and construction of scissor lifts, folding umbrellas, toolboxes and cherry pickers
• Using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°
Things to know before you start teaching this lesson
DRAFT
Poten�al difficul�es
• Students may become confused when angles are shown in different orientations. Where angles are in different orientations, encourage students to look at the arc for a clear indication of which angle they are working with. Students should be encouraged to physically rotate their book if it makes it easier for them to ‘see’ the angle.
• Students may find it difficult to remember the different types of angles. They could make posters for a classroom display. Students could also be encouraged to think of ways to remember different types of angles; for example, an acute angle is the smallest (it is cute).
• Students can find it difficult to place their protractor correctly. An activity where students draw a range of acute angles and measure them, then draw obtuse angles and measure them, not only enables students to practise using their protractor, but also reinforces the difference between these angle types.
• Some students may find it difficult to measure reflex angles. There are two strategies that can be used to assist them.
1. Students can be given a 360° protractor to use for these questions.
2. Students can measure the acute or obtuse angle and then write a number sentence where they subtract this angle from 360° to obtain the size of the reflex angle. Differen�a�on Support
• Students experiencing difficulties identifying angles by their size could be encouraged to use concrete models to create a range of different angles. As they physically move the arms of their model, they will become more aware of the sizes of angles.
• Some students may find it difficult to draw an accurate representation of an angle as they are unsure of where to place the protractor once they have ruled a line. These students will need to be stepped through the process. Q8 can be used as the basis for this, but students who are struggling may require additional practice at this skill.
At level
• Remind students that the letter on the vertex of the angle must be in the middle of the angle name.
• You may need to undertake some explicit teaching so that students understand how to identify the major and minor intervals on a scale. It may be beneficial to review simple number lines and their scales.
• Ensure that all students understand how to use a protractor correctly. In particular, they need to understand when each scale is appropriate; that is, always start from 0°. Students should be encouraged to estimate the size of the angle before measuring, so they can assess the reasonableness of their measurement.
Extension
DRAFT
• Q18 explores angles made by pool balls as they rebound on the sides of the pool table. Refer students to the diagram and ensure they understand the related angles. For interested students, they can be challenged to verify this with a golf ball rebounded against a wall. Students can place tape on the ground, using angles to map out the predicted path of the rebounding ball.
Inter-year links
• Support: Understanding angles
• Year 8: 7.1 Angles
Classroom ac�vi�es
• Introduce the definitions in this lesson by showing students an image from daily life that contains a combination of different angles and lines such as a logo or a road sign. An example is provided below. Guide students to look at the red lines and consider if the two lines would ever touch if they continued in both directions. Compare the red lines with the grey lines, how are they different? Introduce the term perpendicular and discuss what it means to say that objects are at right angles to each other.
• Read through the theory/worked examples and start the understanding and fluency questions.
• Students will require access to a straight edge or ruler and a pair of compasses to complete some questions. An angle is formed anywhere that two lines meet. Introduce the terms an arm and a vertex. Different-sized angles have different names. From smallest to largest they are acute, right, obtuse, straight, reflex and revolution. Give an example of each type of angle and arrange the angles in order. Show students how to measure each angle with a protractor and label the size of each angle in the example.
• Student book
• Ruler
• Compasses
• Protractor
Problem solving and reasoning 20 minutes
• After students have had some time to work, go through one question related to each worked example, with students talking through each step and suggesting what to write
Extension
• More capable students could move on to the rest of the exercise without further instruction.
• Model one of the problem solving and reasoning questions with student input about the required steps.
Extension
Available online
• Investigation: Minigolf challenge
• Investigation: Constructing parallel and perpendicular lines Addi�onal ac�vi�es
• Students who complete their assigned problem solving and reasoning questions could attempt the challenge questions, or the investigations titled ‘Mini-golf challenge’ and ‘Constructing parallel and perpendicular lines’.
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from the lesson if they are unable to complete all set questions.
• Revise the lesson by doing any of the following:
o Complete the Classifying angles interactive skillsheet
o Complete the Classifying angles, Classifying lines and Measuring angles worksheets
o Watch the worked example videos
DRAFT
o Watch the related key content video
o Complete the 7.1 Quick quiz
• Create a one-page summary sheet for the module by adding relevant notes for each lesson after each class. Students could use this summary as their notes for a test or other assessment. Encourage students to write in their own words and to include examples which help them to understand key skills in the module.
Available online
• Interactive skillsheet: Classifying angles
• Worksheet: Classifying angles, Classifying lines, and Measuring angles
• Worked example videos: 7.1A, 7.1B, 7.1C and 7.1D
• Key content video: Classifying angles and lines
• 7.1 Quick quiz
Homework
Optional activity: Investigation 30 minutes
Optional activity: Drawing and estimating angles 5–10 minutes
• Investigation ‘Mini-golf challenge’ to allow for greater student understanding of this lesson before moving on to 7.2 Angles at a point.
• Investigation ‘Constructing parallel and perpendicular lines’ to allow for greater student understanding of this lesson before moving on to 7.2 Angles at a point.
• You may like to ask students to draw an acute or obtuse angle. Underneath the drawing, the students are to write step-by-step instructions on how to estimate and then measure the angle. They should swap with a classmate and follow the instructions. Students can then discuss their results and discuss ways of improving their instructions. A good way to do this is to draw a range of different-sized acute and obtuse angles on the whiteboard and ask students to write their estimate of the size of each angle into their workbooks. Measure the angles using the whiteboard protractor. Compare student estimates with your measurements. Discuss how recognising whether the angle is acute or obtuse helps with estimation.
Lesson 7.2: Angles at a point
Pages 295–301
Total time: 50 minutes
Available online
• Investigation: Minigolf challenge
• Investigation: Constructing parallel and perpendicular lines
• Whiteboard
• Whiteboard protractor
Learning inten�ons
By the end of this lesson, students will be able to:
• identify and find angles at a point.
Success criteria
By the end of this lesson, students will be able to:
• identify complementary, supplementary and vertically opposite angles
• find unknown angles using complementary and supplementary relationships
• find unknown angles at a point.
Curriculum links
Measurement
Iden�fy corresponding, alternate and co-interior rela�onships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Constructing a pair of parallel lines and a pair of perpendicular lines using their properties, a pair of compasses and a ruler and set squares, or using dynamic geometry software
• Using dynamic geometry software to identify relationships between alternate, corresponding and co-interior angles for a pair of parallel lines cut by a transversal
• Using dynamic geometry software to demonstrate how angles and their properties are involved in the design and construction of scissor lifts, folding umbrellas, toolboxes and cherry pickers
• Using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°
DRAFT
Things to know before you start teaching this lesson
Poten�al difficul�es
• Students may find it difficult to remember the different types of angles. They could make posters for a classroom display. Students could also be encouraged to think of ways to remember different types of angles; for example, students can remember the difference between complementary and supplementary angles: C is before S in the alphabet, so C is smaller.
• When the diagram contains three angles, two given and one unknown, students may have difficulty determining which angles are to be used in the calculation. Suggest to students that they first look to see whether all of the angles make a straight line or a right angle. It may be helpful for students to write out an equation, for example, ���� + 31° + 29° = 90° becomes ���� + 60° = 90°.
Differen�a�on
Support
• In Q1, students find the size of an unknown angle in a diagram by using complementary or supplementary angles. Encourage students to first look at the overall angle given in the diagram. Is it a right angle or a straight angle? Complementary or supplementary angle?
• You may need to undertake some explicit teaching so that students understand how to find the missing angle in a number sentence by inspection.
At level
• For Q9, it may be beneficial to demonstrate to students how to ‘break up’ the diagrams into two parts; they can physically do this by placing their hand over the section not required.
• For Q10, encourage students to draw clear diagrams of each of the situations to correctly identify each of the angles. Diagrams will be best drawn from a bird’s eye view.
• Q10d involves using a fraction to determine the angle travelled. Students can convert the angle into degrees. Remind students of their work on multiplying fractions. What does the ‘of’ mean in 2 3 of 360°?
• In Q14, students will explore the angles made by the aerial rotations of skateboarding and snowboarding tricks. Ensure students understand that rotations greater than 360° indicate more than a full turn around. Students may like to demonstrate this by turning around on the spot or using a skateboard shaped object. Suggest to students they break up rotations greater than 360° into 360° and 180° components. This will help them with Q14.
• You may like to draw a diagram on the board, perhaps using three straight lines to give three pairs of vertically opposite angles. Label the size of two of the angles and ask students to identify the size of the remaining four angles.
• You may introduce some questions where students need to use algebra to solve for the unknown angles. For example: George has sliced a pizza using straight line cuts through the centre of the pizza. The slices are not the same size. George notices that two adjacent slices are complementary. If one of the slices has an angle of 2���� and the other an angle of 3���� , what are the sizes of the two angles? (36° and 54°)
Inter-year links
• Support: Understanding angles
• Year 8: 7.1 Angles
Classroom ac�vi�es
Starter activity
5–10 minutes
Understanding and fluency
20 minutes
• Draw two intersecting straight lines and ask students to measure all the angles formed with a protractor. Introduce the idea of supplementary and vertically opposite angles around a point. Guide students to use the measurements to discover that angles around a point add to 360° and the equivalence of vertically opposite angles.
• Read through the theory/worked examples and start the understanding and fluency questions.
• After students have had some time to work, go through one question related to each worked example, with students talking through each step and suggesting what to write.
Extension
• More capable students could move on to the rest of the exercise without further instruction.
• Protractor
Student book
Problem solving and reasoning
20 minutes
• Model one of the problem solving and reasoning questions with student input about the required steps.
Extension
Available online
• Investigation: Sharing the poles
• Students who complete their assigned problem solving and reasoning questions could attempt the challenge questions, or the investigation titled ‘Sharing the poles’.
Addi�onal ac�vi�es
Homework 20–30 minutes
Optional activity: Investigation 30 minutes
Optional activity: Angles on a clock 10 minutes
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from the lesson if they are unable to complete all set questions.
• Revise the lesson by doing any of the following:
o Complete the Angles at a point interactive skillsheet
o Complete the Angles at a point worksheet
o Watch the worked example videos
o Watch the related key content video
o Complete the 7.2 Quick quiz
• Create a one-page summary sheet for the module by adding relevant notes for each lesson after each class. Students could use this summary as their notes for a test or other assessment. Encourage students to write in their own words and to include examples which help them to understand key skills in the module.
• Investigation ‘Sharing the poles’ to allow for greater student understanding of this lesson before moving on to 7.3 Angles and parallel lines.
• Construct adjacent angles, complementary angles and supplementary angles by drawing lines on an image of a clock without hands. Ask students to identify the complementary angles, supplementary angles and vertically opposite angles given the following angles:
• angle formed by connecting 12, the centre and 1
• angle formed by connecting 1, the centre and 3.
Available online
• Interactive skillsheet: Angles at a point
• Worksheet: Angles at a point
• Worked example videos: 7.2A, 7.2B, 7.2C and 7.2D
• Key content video: Angles at a point
• 7.2 Quick quiz
Available online
• Investigation: Sharing the poles
• Image of a clock without hands
• Pencil
• Ruler
Optional activity: Revolution cards 15 minutes
• Guide students to consider if they can find the complementary angle of an obtuse angle or the supplementary angle of a reflex angle.
• Play an angle-matching card game using knowledge of angles around a point to complete a revolution. Prepare 24 cards and make four identical sets with 20°, 30°, 40°, 50°, 80° and 110° written on each card. Designate one player as the ‘drawer’. This player needs a protractor, a pencil and a sheet of paper. Each player (including the drawer) gets three cards. The remaining cards are placed face down in a pile at the centre of the table. The ‘drawer ’ draws a single line with a point at one end as one arm and the vertex of an angle. Players take turns playing angle cards, which add together around a point. As each angle card is played, the angle drawer adds the angle to the drawing. If an angle card makes the total:
o 90°, the player scores 2 points
o 180°, the player scores 3 points
o 360°, the player scores 5 points
• Custom made cards
• Protractor
• Pencil
• Piece of paper
• When all the angles make a revolution (or no player can make exactly 360°), start again with a new drawing.
• Players pick up new cards as they progress, so they always have three cards in their hand. The winner is the player with the most points once all 24 cards have been played.
Lesson 7.3: Angles and parallel lines
Pages 302–309
Total time: 50 minutes
DRAFT
Learning inten�ons
By the end of this lesson, students will be able to:
• identify and find angles on parallel lines
• determine if two lines are parallel.
Success criteria
By the end of this lesson, students will be able to:
• identify alternate, corresponding and co-interior angles
• find unknown angles by identifying alternate, corresponding and co-interior angles
• determine if two lines are parallel using alternate, corresponding and co-interior angles.
Curriculum links
Measurement
Iden�fy corresponding, alternate and co-interior rela�onships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Constructing a pair of parallel lines and a pair of perpendicular lines using their properties, a pair of compasses and a ruler and set squares, or using dynamic geometry software
• Using dynamic geometry software to identify relationships between alternate, corresponding and co-interior angles for a pair of parallel lines cut by a transversal
• Using dynamic geometry software to demonstrate how angles and their properties are involved in the design and construction of scissor lifts, folding umbrellas, toolboxes and cherry pickers
• Using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°
Things to know before you start teaching this lesson
Poten�al difficul�es
• Some students will find it difficult to identify which angle type they are working with. Encourage these students to rely on their knowledge of straight-line angles and supplementary angles to find missing angle values. It may be beneficial for students to create a poster indicating all angle relationships created when a pair of parallel lines is crossed by a transversal.
Differen�a�on
Support
• Q2 asks students to identify whether the angle pairs in Q1 have the same value or are supplementary. It may be beneficial to remind students that supplementary angles add to 180°.
• Ensure that students can recognise the difference between vertically opposite, alternate, corresponding and co-interior angles.
At level
• In Q4, students are asked to list all pairs of alternate, co-interior and corresponding angles shown on a diagram. Challenge students to use their knowledge of vertically opposite angles and angles around a point to justify or prove how we know these angles are equal.
• Q10 requires students to find the value of an angle represented by a pronumeral where the question is set in a real-life context. This question can be used to discuss angles of elevation and depression with more able students.
• For Q11, it may be helpful for students to draw a diagram highlighting the two pairs of parallel lines from the railway track diagram.
Extension
• Eleni makes a frame for a painting with four sides. She wants to check that opposite sides are parallel by measuring the angles at the corners and seeing if they are right angles. How many corners does she need to check to be sure that opposite sides are parallel? (3 corners)
Inter-year links
• Support: Understanding angles
• Year 8: 7.1 Angles
Classroom ac�vi�es
Starter activity
5–10 minutes
Understanding and fluency
20 minutes
• Draw a pair of parallel lines and then rule another line (the transversal) across the parallel lines on the board as shown.
• Label all the angles formed and use a protractor to measure all the angles, a–h.
• Write down all the measurements and ask students if they can find all equivalent angles.
• Introduce the idea of alternate angles, corresponding angles and co-interior angles.
• Ask students to create their own graph by drawing the transversal line from a different direction or changing the distance between the two parallel lines.
• Ask them to measure the angles formed and show them the following rules are always true:
o alternate angles are equal in size
o corresponding angles are equal in size
o co-interior angles are supplementary.
• Read through the theory/worked examples and start the understanding and fluency questions.
• To find angles in a diagram where the angle represented by the pronumeral is not part of a given angle pair can be difficult for some students. A good strategy is for students to find the value of all angles in the diagram, using their knowledge of angles formed by parallel lines and a transversal. Once all angles have a value, finding the value of the pronumeral is easier.
Step through a sample of the questions from Q8 on the board, demonstrating how all the angles can be labelled with a value. Students can then attempt the remaining questions on their own.
• Ruler
• Whiteboard
• Protractor
Student book
Problem solving and reasoning 20 minutes
• After students have had some time to work, go through one question related to each worked example, with students talking through each step and suggesting what to write.
Extension
• More capable students could move on to the rest of the exercise without further instruction.
• Model one of the problem solving and reasoning questions with student input about the required steps.
Extension
Available online
• Investigation: The streets of New York Addi�onal
Homework
20–30 minutes
• Students who complete their assigned problem solving and reasoning questions could attempt the challenge questions or the investigation titled ‘The streets of New York’.
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from the lesson if they are unable to complete all set questions.
• Revise the lesson by doing any of the following:
o Complete the Angles and parallel lines interactive skillsheet
o Complete the Angles and parallel lines worksheet
o Watch the worked example videos
o Watch the related key content video
o Share the Desmos activity ‘Transversals and parallel lines’
DRAFT
o Complete the 7.3 Quick quiz
• Create a one-page summary sheet for the module by adding relevant notes for each lesson after each class. Students could use this summary as their notes for a test or other assessment. Encourage students to write in their own words and to include examples which help them to understand key skills in the module.
Available online
• Interactive skillsheet: Angles and parallel lines
• Worksheet: Angles and parallel lines
• Worked example videos: 7.3A, 7.3B, 7.3C and 7.3D
• Key content video: Angles and parallel lines
• Desmos activity: Transversals and parallel lines
• 7.3 Quick quiz
Revision activity 20–30 minutes
Optional activity: Investigation 30 minutes
Optional activity: Angles memory game 5–10 minutes
• Complete the Checkpoint questions on pages 310–311 on the student book. These are designed to consolidate student knowledge and understanding of the first half of the module before moving on to the remaining lessons.
• Complete the online Checkpoint quiz. The questions in this quiz are distinct from the questions in the student book. All questions are auto marked and provide immediate feedback to students who may resit the quiz as many times as desired.
• Investigation ‘The streets of New York’ to allow for greater student understanding of this lesson before moving on to 7.4 Triangles.
• Create a deck of cards with the following definitions and corresponding graphs to play a game of memory. This game will test students’ knowledge of all angle facts covered to this stage. The cards can be laminated and retained.
Acute angle Vertically opposite angles
Right angle Corresponding angles
Obtuse angle Alternate angles
Straight angle Co-interior angles
Reflex angle Supplementary angles
Revolution Complementary angles
Student book
Available online
• Checkpoint quiz
Available online
• Investigation: The streets of New York
• Custom made cards
Lesson 7.4: Triangles
Pages 312–317
Total time: 50 minutes
Learning inten�ons
By the end of this lesson, students will be able to:
• identify and classify triangles
• find unknown angles in triangles.
Success criteria
By the end of this lesson, students will be able to:
• identify and classify triangles using side lengths and angle sizes
• find unknown angles using the sum of angles in a triangle.
Curriculum links
Measurement
Iden�fy corresponding, alternate and co-interior rela�onships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons (VC2M7M04)
• Constructing a pair of parallel lines and a pair of perpendicular lines using their properties, a pair of compasses and a ruler and set squares, or using dynamic geometry software
• Using dynamic geometry software to identify relationships between alternate, corresponding and co-interior angles for a pair of parallel lines cut by a transversal
• Using dynamic geometry software to demonstrate how angles and their properties are involved in the design and construction of scissor lifts, folding umbrellas, toolboxes and cherry pickers
• Using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°
Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Using concrete materials to demonstrate that the sum of the interior angles of a triangle is 180°; for example, using paper triangles and tearing to demonstrate that the interior angles when combined form 180°
• Using decomposition and the angle sum of a triangle to generalise the interior angle sum of an ����-sided polygon, as 180(����− 2) = 180����− 360
Things to know before you start teaching this lesson
DRAFT
• Some students may mistakenly confuse the value of the sum of all internal angles in a triangle as 360° instead of 180°.
• Remind students triangles could be classified into equilateral, isosceles, scalene based on their side lengths, or into right-angled-triangle, obtuse triangle, acute triangle based on their interior angles. The same triangle could be classified into two different groups depending on what the question is asking for. For example, a right-angled-triangle could also be an isosceles triangle.
Differen�a�on
Support
• Students need a sound knowledge of angle types and their features. They should be able to recall features of the following angles: acute, right, obtuse, straight, reflex and revolution. It is helpful for students to have a detailed description and diagram of these angle types to continually refer to when completing this lesson.
• It may be helpful for students to include diagrams in their notes of several standard types of triangles and their angle relationships. For example, ensure students understand that each of the three angles in an equilateral triangle is 60°.
At level
• When finding the size of an angle in a triangle, encourage students to first identify the type of triangle. For example, in Q4b, students will need to identify the triangle as equilateral so they can assume angles a, b, c are equal.
• Remind students of their work in algebra. One way of approaching Q4 is to use angle properties to find and solve an equation. This can be an opportunity to discuss the interconnection between the different topics and techniques studied in mathematics.
Extension
• Students can explore the relationship between the exterior angle of a triangle and the two interior angles that are opposite the exterior angle.
Inter-year links
• Year 8: 7.2 Triangles
Classroom ac�vi�es
Starter activity 5–10 minutes
• Ask students to use a ruler to draw a triangle and cut it out. Students can discover that the angle sum of a triangle is 180° by accurately cutting out a copy of the triangle, tearing off the corners, and placing the angles side by side. The three angles should form a straight angle.
• Draw an equilateral triangle, an isosceles triangle, a scalene triangle, a right-angled-triangle, an acute triangle and an obtuse triangle on the board. Measure and mark the length of each side and angle of each triangle. Ensure students understand side length and angle labelling • Scissors • Paper
Pen
Ruler • Whiteboard
Understanding and fluency
Problem solving and reasoning
20 minutes
conventions. They will need this knowledge to correctly identify equilateral, isosceles, scalene, right-angled, acute and obtuse triangles.
• Read through the theory/worked examples and start the understanding and fluency questions.
• After students have had some time to work, go through one question related to each worked example, with students talking through each step and suggesting what to write.
Extension
20 minutes
Addi�onal ac�vi�es
Student book
Homework 20–30 minutes
• More capable students could move on to the rest of the exercise without further instruction.
• Model one of the problem solving and reasoning questions with student input about the required steps.
Extension
• Students who complete their assigned problem solving and reasoning questions could attempt the challenge questions or either of the investigations titled ‘Mosaic triangles’ and ‘Constructing triangles’.
Available online
• Investigation: Mosaic triangles
• Investigation: Constructing triangles
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from the lesson if they are unable to complete all set questions.
• Revise the lesson by doing any of the following:
DRAFT
o Complete the Triangle properties interactive skillsheet
o Complete the Classifying triangles worksheet
o Watch the worked example videos
o Watch the related key content video
o Complete the 7.4 Quick quiz
• Create a one-page summary sheet for the module by adding relevant notes for each lesson after each class. Students could use this summary as their notes for a test or other assessment.
Available online
• Interactive skillsheet: Triangle properties
• Worksheet: Classifying triangles
• Worked example videos: 7.4A and 7.4B
• Key content video: Classifying triangles
• 7.4 Quick quiz
Optional activity: Investigation 30 minutes
Optional activity: Triangles with technology 15 minutes
Optional activity: Features of a triangle 10 minutes
Encourage students to write in their own words and to include examples which help them to understand key skills in the module.
• Investigations ‘Mosaic triangles’ or ‘Constructing triangles’ to allow for greater student understanding of this lesson before moving on to 7.5 Quadrilaterals.
Pages 318–325
Total time: 50 minutes
• Use a geometry software, such as the Desmos geometry tool or GeoGebra, to explore equilateral, isosceles and scalene triangles. Have students identify each type of triangle and predict what will happen when a point is clicked and dragged.
• Provide students with a selection of different triangles. Students can measure the lengths of the triangles and classify them based on their side lengths. Students can use their protractors to measure the angles of each triangle and look at the relationship between the length of the sides and the interior angles of each triangle.
Available online
• Investigation: Mosaic triangles
• Investigation: Constructing triangles
• Access to geometry software such as Desmos or GeoGebra
• Different sized and shaped triangles
Lesson 7.5: Quadrilaterals
Learning inten�ons
By the end of this lesson, students will be able to:
• identify and classify quadrilaterals
• find unknown angles in quadrilaterals.
Success criteria
By the end of this lesson, students will be able to:
• identify and classify squares, rectangles, rhombuses, parallelograms, kites and trapeziums
• find unknown angles using the sum of angles in a quadrilateral.
Curriculum links
Measurement
Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
• Using concrete materials to demonstrate that the sum of the interior angles of a triangle is 180°; for example, using paper triangles and tearing to demonstrate that the interior angles when combined form 180°
• Using decomposition and the angle sum of a triangle to generalise the interior angle sum of an ����-sided polygon, as 180(����− 2) = 180����− 360
Space
Classify triangles, quadrilaterals and other polygons according to their side and angle proper�es; iden�fy and reason about rela�onships (VC2M7SP02)
• Using strips of paper with parallel sides to make triangles and quadrilaterals, and contrasting the rigidity of triangles with the flexibility of quadrilaterals
• Constructing triangles with 3 given side lengths and discussing the question ‘Can any 3 lengths be used to form the sides of a triangle?’
• Identifying and communicating about side and angle properties of scalene, isosceles, equilateral, right-angled, acute and obtuse triangles using geometric conventions
• Describing, comparing and contrasting squares, rectangles, rhombuses, parallelograms, kites and trapeziums, explaining the relationships between these shapes
Things to know before you start teaching this lesson
Poten�al difficul�es
• When classifying special quadrilaterals, ensure that all students understand the different features that can be used to classify quadrilaterals. They need to check if the shape provided satisfies all the properties given. For example, for a shape to be classified as a kite, students need to check all the following conditions:
o two pairs of sides that are equal in length
o one pair of opposite angles that are equal in size
o no parallel sides.
• If students only check the first condition, then students could mistakenly classify a parallelogram as a kite.
Differen�a�on
Support
• Students need to be able to recognise conventions for indicating equal side lengths, parallel sides, perpendicular sides and right angles.
• Q9 is an open question asking students to identify the types of quadrilaterals they see in an image. Suggest students first count the squares, then the rectangles, then the parallelograms etc.
At level
• Q8 is a more challenging question that relies on students’ understanding of basic algebra to find the value of the pronumeral. Prompt students to use the information contained in the diagram to write an equation.
• Q18 explores the angles contained in a rectangle divided by a diagonal line. This question leads students to see that the total sum of angles in a quadrilateral is the sum of the angles in the two triangles formed inside when it is divided. This rule can be generalised to all polygons. Students can tear off the ‘corners’ of a quadrilateral in Q18 and place the angles together to form a revolution. This confirms that the angle sum is 360°.
Extension
• For advanced students, Q14 could be an opportunity to introduce the concept of one class of object being a subset of a more general class of objects.
Inter-year links
• Year 8: 9.6 Quadrilaterals
Classroom ac�vi�es
Starter activity
5–10 minutes
Understanding and fluency
20 minutes
• Introduce the definition and explore the features of different quadrilaterals with the following table. Draw a parallelogram, a rectangle, a square, a rhombus, a trapezium and a kite in the first row. Ask students to investigate one shape at a time and write yes or no in the following rows. Encourage students to check their answers with a classmate and discuss any differences.
• Read through the theory/worked examples and start the understanding and fluency questions.
• Pen and paper
Student book
Problem solving and reasoning 20 minutes
• After students have had some time to work, go through one question related to each worked example, with students talking through each step and suggesting what to write.
Extension
• More capable students could move on to the rest of the exercise without further instruction.
• Model one of the problem solving and reasoning questions with student input about the required steps.
Extension
• Students who complete their assigned problem solving and reasoning questions could attempt the challenge questions or the investigation titled ‘Quadrilaterals everywhere’.
Available online
• Investigation: Quadrilaterals everywhere
Homework
20–30 minutes
Optional activity: Investigation 30 minutes
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from the lesson if they are unable to complete all set questions.
• Revise the lesson by doing any of the following:
o Complete the Quadrilateral properties interactive skillsheet
o Watch the worked example videos
o Watch the related key content video
o Complete the 7.5 Quick quiz
• Create a one-page summary sheet for the module by adding relevant notes for each lesson after each class. Students could use this summary as their notes for a test or other assessment. Encourage students to write in their own words and to include examples which help them to understand key skills in the module.
• Investigation ‘Quadrilaterals everywhere’ to allow for greater student understanding of this lesson before moving on to 7.6 Polygons.
• To support students to recognise that the angle sum of any quadrilateral is 360°:
o Provide examples of both regular and irregular quadrilaterals for students to examine.
o Some students may benefit from a practical activity where the angles inside different quadrilaterals are measured using their protractor.
o Students can then find the sum of the interior angles for the quadrilaterals and discover the angle sum property.
• Print and cut out a range of shapes, ask students to classify them as quadrilaterals and other shapes. Students further classify them as irregular quadrilaterals or special quadrilaterals introduced in this lesson. You may need to undertake some explicit teaching so that students understand that all quadrilaterals are 2D shapes with four straight sides. It is important to ensure that students recognise that not all quadrilateral shapes are regular and that irregular quadrilaterals also have four straight sides.
• Investigation: Quadrilaterals everywhere
• Different types of quadrilaterals
• Protractor
• Different types of quadrilaterals
Pages
Total
Lesson 7.6: Polygons
Success criteria
By the end of this lesson, students will be able to:
• classify, describe and name polygons.
Success criteria
By the end of this lesson, students will be able to:
• classify polygons as regular or irregular
• classify polygons as convex or concave
• describe and name polygons
Curriculum links
Measurement
• Using geometric reasoning of angle properties to generalise the angle relationships of parallel lines and transversals, and related properties, such as the size of an exterior angle of a triangle is equal to the sum of the sizes of opposite and non-adjacent interior angles, and the sum of the sizes of interior angles in a triangle in the plane is equal to the size of 2 right angles or 180°
Demonstrate that the interior angle sum of a triangle in the plane is 180° and apply this to determine the interior angle sum of other shapes and the size of unknown angles (VC2M7M05)
DRAFT
• Using concrete materials to demonstrate that the sum of the interior angles of a triangle is 180°; for example, using paper triangles and tearing to demonstrate that the interior angles when combined form 180°
• Using decomposition and the angle sum of a triangle to generalise the interior angle sum of an ����-sided polygon, as 180(����− 2) = 180����− 360
Classify triangles, quadrilaterals and other polygons according to their side and angle proper�es; iden�fy and reason about rela�onships (VC2M7SP02)
• Using strips of paper with parallel sides to make triangles and quadrilaterals, and contrasting the rigidity of triangles with the flexibility of quadrilaterals
• Constructing triangles with 3 given side lengths and discussing the question ‘Can any 3 lengths be used to form the sides of a triangle?’
• Identifying and communicating about side and angle properties of scalene, isosceles, equilateral, right-angled, acute and obtuse triangles using geometric conventions
• Describing, comparing and contrasting squares, rectangles, rhombuses, parallelograms, kites and trapeziums, explaining the relationships between these shapes
Things to know before you start teaching this lesson
• Students may experience difficulty using the flowchart on page 327 to classify shapes. It may be beneficial for students to collect a group of related objects and create a flowchart that separates the items into specific groups. This is a valuable exercise as students will gain a deeper understanding of how flowcharts work.
Differen�a�on
Support
• Ensure that all students understand the definitions of regular, irregular, convex and concave polygons. They should be able to come up some examples for each type of 2D shape.
• Students should be aware of the prefixes used to describe the number of sides of a polygon, from three to ten. Any new terms should be added to their glossaries.
At level
DRAFT
• Q5 asks students to name various polygons. They should start by deciding whether the polygon is regular or irregular, and convex or concave. Students should then count the number of sides to identify the correct prefix to use when naming the polygon.
• Q7 guides students through the process of calculating the number of degrees in a decagon by dividing the decagon into triangles. Ensure students understand the correct means of dividing the decagon into triangles and that no lines can overlap. How do we know that the sum of all the angles in the triangles is equal to the sum of the angles in the decagon? Can students explain this geometrically?
Poten�al difficul�es
• For Q10, students should recognise that the internal angles of a regular polygon are equal.
• For Q11, students will first need to use the method from Q7 to determine the sum of the angles in the various polygons.
Inter-year links
• Support: Polygons
Classroom ac�vi�es
Starter activity
5–10 minutes
• Introduce the concept of the polygon being a closed shape with all straight sides. In Greek, “poly-gonia” means many-angles. Are triangles and quadrilaterals, like hexagons and decagons, “many-angles” shapes? Prompt students to recognise that the polygon name for ‘trigon” means literally in Greek “tri-angle” and hence refers to the same shape. Draw examples of different polygons on a whiteboard or show examples to the class and have them count the number of sides and internal angles.
• Whiteboard
• Whiteboard markers
• Polygon examples
Understanding and fluency
20 minutes
Problem solving and reasoning
20 minutes
• Read through the theory/worked examples and start the understanding and fluency questions.
• After students have had some time to work, go through one question related to each worked example, with students talking through each step and suggesting what to write.
Extension
• More capable students could move on to the rest of the exercise without further instruction.
• Model one of the problem solving and reasoning questions with student input about the required steps.
Extension
• Students who complete their assigned problem solving and reasoning questions could attempt the challenge questions, or the investigation titled “Constructing regular polygons”.
Student book
Student book
Available online Investigation: Constructing regular polygons
Addi�onal ac�vi�es
Homework
20–30 minutes
Optional activity: Investigation 30 minutes
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from the lesson if they are unable to complete all set questions.
• Revise the lesson by doing any of the following:
o Complete the Polygons interactive skillsheet
o Watch the worked example videos
o Watch the related key content video
o Complete the 7.6 Quick quiz
Optional activity: Sum of interior angles 10 minutes
Available online
• Interactive skillsheet: Polygons
• Worked example videos: 7.6A and 7.6B
• Key content video: Polygons
• 7.6 Quick quiz
• Create a one-page summary sheet for the module by adding relevant notes for each lesson after each class. Students could use this summary as their notes for a test or other assessment. Encourage students to write in their own words and to include examples which help them to understand key skills in the module.
• Investigation “Constructing regular polygons” to allow for greater student understanding of this lesson before moving on to 7.7 Review.
DRAFT
• Remind students that the angle sum of a triangle is 180° and the angle sum of a quadrilateral is 360°. Start with a pentagon. Have students trace a pentagon from their Mathomat, or other geometry template. Alternatively, students can construct a pentagon within a circle, or online using geometry software like Desmos or GeoGebra.
• Divide the pentagon into triangles.
• You may like to number each triangle (1, 2 and 3) and label the three angles in each triangle (����, ����, ���� ,…, ℎ, ���� ). This can be used to show the link between the angles in the three triangles and the angles of the pentagon. Working clockwise, the angles to be added for the angle sum
Available online
• Investigation: Constructing regular polygons
• Access to online Geometry software like Desmos or GeoGebra, or to a Mathomat or other geometry template
• Paper
• Pens
Lesson 7.7: Review
Pages 331–337
Total time: 50 minutes
Review quiz 30 minutes
Homework 20–30 minutes
of the pentagon are
and
Show that this is equivalent to finding the sum of the three angles in each of the three triangles. That is, 3 × 180° = 540°.
• Students would benefit from repeating this exercise for a hexagon, heptagon, octagon, nonagon and decagon. They should be guided to observe the pattern where: angle sum = (number of sides – 2) × 180°.
• Ask students to predict the angle sum of larger polygons using any patterns that they can see. This information can be recorded into a table in their workbooks.
DRAFT
• The auto-marked 20 question review quiz covers key concepts from across the module.
• Complete unfinished questions from class. It is advised to set students a time limit (e.g. 30 minutes) after which they can stop working on the questions. Encourage students to ensure they have completed a variety of questions from each lesson if they are unable to complete all set questions.
Module 7 review quiz
Class test
50 minutes
Additional activity: Key terms revision
Additional activity: Rhombus rings
Additional activity: Ambiguous stacks
5–10 minutes
• Complete the checklist. Students can complete the relevant interactive skillsheets if they need further learning and practice of the skills.
• Set the class test with alternate students receiving class tests A and B.
20–30 minutes
20–30 minutes
• Review key terms introduced in this module as a whole class using Quizlet flashcards. Encourage students to define each term in their own language and give an example for each term where possible.
• Divide students in groups of 2 to 4 to attempt the low-floor, high-ceiling question using skills covered in this module. Encourage students to explore different approaches to solve the question.
• Divide students in groups of 2 to 4 to attempt the low-floor, high-ceiling question using skills covered in this module. Encourage students to explore different approaches to solve the question.
Available online
• Module 7 class test A
• Module 7 class test B
Available online
• Module 7 Quizlet
Student book AMT Explorations: page 486 Question 2
