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2 R Teacher Book Transforming standards at Key Stage 3 Maths Connect Teacher Books will help you deliver interactive whole class teaching in line with the Framework. Written and developed by experienced teachers and advisers, Maths Connect Teacher Books offer you: A practical and realistic route through the Framework and Sample medium-term plans for Mathematics. ● Practical ideas for whole class teaching based on real Framework practice. ● Complete lesson plans that include starters, plenaries and teaching ideas. ● Key words, teaching objectives and common difficulties highlighted for each lesson. ● Links showing where you can find relevant pupil resources, homeworks and assessments. ● Links between concepts and skills to help you build confidence and understanding. ● t 01865 888080 e [email protected] f 01865 314029 w www.heinemann.co.uk 0 435 53658 3 J262 Maths Connect - everything you need to deliver effective and interactive lessons. ● TEACHER BOOK Sample Pages Contents of Maths Connect 2R: Teacher Book Page 2 Algebra 1, Number 1: Integers, powers and roots Pages 3-11 2 R Teacher Book contents Maths Connect 2R follows the objectives from the teaching programme for Year 8 as suggested in the extension tier of the Sample medium-term plans. It is written specifically for Year 8 extension groups. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 N1/A1 SSM1 HD1 N2 A2 SSM2 A3 N3 SSM3 A4 HD2 N4 A5 N5 SSM4 HD3 Integers, powers and roots Angles and shapes Probability Fractions, decimals and percentages Equations and formulae Measures Diagrams and graphs Place value and calculations Transformations Solving equations Analysing statistics Calculations Equations and graphs Ratio and proportion and solving problems Construction Collecting, displaying and analysing data Features Thinking Maths activities adapted from the King’s College CAME team – proven to build pupils’ thinking skills and improve performance across Key Stage 3. 2 Teacher Book 2R Each unit features an overview page that summarises objectives, outlines assumed 18/5/04 4:23 pm Page 2 knowledge and common difficulties and links to other components of the course. A_Teachers.qxd 1 N/A1 Integers, powers and � Units 5, 8 and 12 roots (6 hours) Background Assumed knowledge In Number 1 pupils add, subtract, multiply and divide positive and negative integers, using the rules of signs and the sign change key on the calculator. They extend their knowledge of squares and square roots to include negative square roots and cubes and cube roots. Index notation is introduced with powers of 2 and 3 and beyond, leading to simple index rules for multiplying and dividing. Pupils apply their calculator skills and knowledge of square roots to estimate square roots by the method of trial and improvement. Algebra 1 (Lessons 1.5 and 1.6) deals with sequences, including finding the general term of linear sequences. Lesson 1.6 considers non-linear sequences, finding the general term where possible, or using the pattern of differences to generate the next few terms in the sequence. Before starting this Unit, pupils should: � be able to add and subtract positive and negative integers using a number line � understand the concept of squares and positive square roots � be able to find the general term of a linear sequence Main teaching objectives Pupil book sections 1.1 Calculations with integers 1.2 Powers and roots 1.3 Indices 1.4 Square roots 1.5 Sequences 1.6 Special sequences Teaching objectives Add and subtract integers Multiply and divide integers Use the sign change key to a calculator Use index notation for small integer powers Know cubes of 1, 2, 3, 4, 5 and 10 and the corresponding roots Use a calculator to find squares and cubes Know that 100 � 102, 1000 � 103, 1 million � 106, 1 billion � 109 Use index notation for integer powers Use simple instances of the index laws Use ICT to estimate square roots Use trial and improvement to find approximate solutions to equations Use linear expressions to describe the nth term of an arithmetic sequence justifying its form by referring to the activity or practical context from which it was generated Introduce the vocabulary T(n) for the general term Know that an arithmetic sequence is generated by starting with a number a and adding a constant number d to the previous term Continue familiar sequences (square numbers, powers of 10, 2, etc.) Generate sequences by multiplying or dividing by a constant factor Oral and mental starters Starter 1 8 9 12 2 10 Page 245 249 250 252 245 250 Common difficulties � Pupils may confuse operation signs of addition and subtraction with signs indicating that an integer is positive or negative. For this reason, integers are recorded like this: �6, �3. PB Pages 2–13 Homeworks 1.1–1.6 Assessment 2R ! Thinking Maths 2 Maths Connect 2R Sample page from Maths Connect 2R: Teacher Book 3 Teacher Book 2R Follows the structure of the Sample medium-term plans, featuring a suggested 18/5/04 4:23 pm Page 4 starter, main teaching activity and plenary for each lesson. A_Teachers.qxd Key words 1.1 positive negative integer product Calculations with integers Add and subtract integers Multiply and divide integers Use the sign change key on a calculator Links � 1.3 Indices Oral and mental starter 1 Introduction In Year 7, pupils added and subtracted positive and negative integers using a number line and met the rules of signs for multiplication and division. In this lesson they revise these skills. They convert subtractions into addition of the inverse. For multiplications and divisions, they calculate the value for positive integers and find the sign using the rules. Pupils also use the sign change key on a calculator. Teaching activity Teacher materials: OHP, an OHP calculator, or a calculator on a whiteboard, OHT of Resource sheet 1 (�10 to �10 number lines), OHT of Resource sheet 2 (multiplication grid) Pupil materials: Calculators Outline Explain to the pupils that this lesson focuses on calculating with integers. Remind them that integers are positive and negative whole numbers, including zero. Write ‘�4 � �5 �’ on the board. Emphasise that we use raised signs for positive and negative integers, to avoid confusion with the operations of addition and subtraction. Show the �10 to �10 number line from Resource sheet 1. Remind pupils that we can use a number line to add integers. The method is (a) start at the first number on the line, then (b) count right if the second number is �ve, and count left if the second number is �ve. Demonstrate by starting at �4, and counting five places left to land on �1. Next, revise subtraction of integers. Write ‘�4 � �5 �’ on the board and invite answers. Remind pupils that subtracting an integer is equivalent to adding its inverse. Choose a pupil to write the converted calculation, ‘�4 � �5 �’. Confirm that the answer is �9. Next consider multiplication of integers. Show the multiplication table from Resource sheet 2 and start by doing the multiplications of positive integers in the bottom right hand corner, writing in the products. Complete the table together, by following the sequence pattern for each row/column. Use the results in the multiplication table to confirm the rules of signs for multiplication. When multiplying two integers: if they both have the same sign, the product is positive; if they have different signs, the product is negative. Write ‘3 � 2 � 6’ on the board. What division facts can we write from this multiplication fact? Record the two possibilities: (6 � 3 � 2 and 6 � 2 � 3). � � 3 � 3 � 2 �1 � 9 � 6 � � 2 � 6 � 4 �2 � 1 0 � 3 0 � 2 0 � 0 3 0 0 � 1 �2 � 3 � 3 1 0 � 0 0 0 � 1 � 3 � 2 �1 0 � 6 � 9 � 2 �4 � 6 1 � 2 0 � 3 0 � 1 �2 � 3 � 2 � 6 � 4 � 2 0 � 2 � 4 � 6 � 3 � 9 � 6 � 3 0 � 3 � 6 � 9 4 Maths Connect 2R 4 Sample page from Maths Connect 2R: Teacher Book Teacher Book 2R For each lesson in the Pupil Book, there is a corresponding double page spread in 18/5/04 4:23 pm Page 5 the Teacher’s book for ease of use. A_Teachers.qxd Choose other multiplication facts from the multiplication table, and ask pupils for the related division facts, for example �2 � �3 � �6, leads to � 6 � �3 � �2, and �6 � �2 � �3 Confirm that the rules of signs for division exactly match those for multiplication – the answer is positive for like signs, and negative for unlike signs. Write calculations on the board, for example, �6 � �2, �7 � �4, �35 � � 5, �12/�3, �5 � �1, … Choose pupils to explain how they calculated the answers. Demonstrate how to use the sign change key on a calculator, and how to use it to perform these calculations. 6 � 7 � � 2 � � � � 4 � � � Variations Teacher materials: Coin, two sets of 1–9 integer cards from Resource sheet 3 Revise the methods for calculating with integers, as in the Outline. To generate the integers to use in calculations, pick cards to give integer values and flip the coin for the sign: heads positive, tails negative. As an extension to the activity in the Outline, discuss the result of multiplying together three integers, for example, �5 � �2 � �4. Plenary � � � � The answer to an addition of two integers is � 5. What could the addition be? (Examples could be: �10 � �15, �3 � �2, �5 � �10 etc.) The result of multiplying two integers is � 10. What could the two integers be? What is (�4)2, (�2)3? (Answers are 16, �8) What is �5 � �3 � �2? (Answer is �30) Key teaching points: Integers are the positive and negative whole numbers, together with zero. •We can add integers by counting on a number line. Start at the position of the first number, then count left to add a negative number and count right to add a positive number. •To subtract one integer from another, convert the subtraction into an addition of the inverse, then use the number line. •When multiplying two integers, multiply the whole number parts, and then use the rule of signs to decide if the result is positive or negative. The same method applies to division. •Rules of signs for multiplication and division: •If both signs are the same, then the result is positive. •If both signs are different, then the result is negative. •You can use the sign change key on a calculator to do calculations with negative numbers. � Exercise hints Q1–3 Practice Q4–7 Problems Q8 Activity Q9 Investigation Photocopies of Resource sheet 1, number lines may be useful for the exercise. P B Exercise 1.1, page 3 Homework 1.1 � Answers, page 276 Number 1, Algebra 1: Calculations with integers 5 Sample page from Maths Connect 2R: Teacher Book 5 Teacher Book 2R Features clear teaching objectives and learning 18/5/04 4:23 pm Page 6 outcomes to set the scene for each lesson. A_Teachers.qxd Key words 1.2 square index power square root cube cube root Powers and roots Use index notation for small integer powers Know cubes of 1, 2, 3, 4, 5 and 10 and the corresponding roots Use a calculator to find squares and cubes Know that 100 � 102, 1000 � 103, 1 million � 106, 1 billion � 109 Links � 1.3 Indices, 1.4 Square roots, 5.1 Index notation � Oral and mental starter 8 Introduction This lesson revises squares and square roots from Year 7 and extends to cubes and cube roots, and how to find these using a calculator. Index notation is extended to powers of 2 and 3, and beyond for powers of 10, including one million and one billion. Teaching activity Teacher materials: OHP calculator or a calculator on a whiteboard Pupil materials: Calculators Outline Revise square numbers and square roots. Invite pupils to state some square numbers. (1, 4, 9, 16, …) Write ‘62 � 6 � 6 � 36’. Explain that the raised 2 is called the index or power of 2. It indicates that we multiply the 6 by itself 2 times. Clarify different ways of reading this statement: ‘Six squared is thirty-six’ and ‘Six to the power of two is thirty-six’. � � 6’. Read this statement together: ‘The square root of thirtyWrite ‘�36 six is six’. Use a variety of examples to clarify that finding a square root is the inverse of finding a square. What is (�6)2? (36) So what is another �? (�6) Remind pupils that every square root has a positive solution to �36 and a negative solution, but by convention we take the positive value, unless told otherwise. Challenge pupils to calculate the squares of more complex numbers, 2 2 4� e.g. (0.2)2. You might wish to show that (0.2)2 � �1�0 � �1�0 � � 100 � 0.04. Demonstrate how to use a calculator to find squares, e.g. 7 and square roots 4 9 . x2 � Introduce the concept of cube numbers and cube roots. Write ‘23 � 2 � 2 � 2 � 8’. Explain that the index is 3, and we read it as ‘Two cubed is eight’ and ‘Two to the power of three is eight’. 2 Demonstrate the relationship between two squared and two cubed, by drawing a 2 � 2 square and a 2 � 2 � 2 cube. Challenge pupils to state the cubes of 1 (1), 2 (8), 3 (27), 4 (64), 5 (125) and 10 (1000). Explain that the inverse of finding a cube is finding a cube root. Clarify that the cube root of 8 is 2, for example. Challenge pupils to find the cubes of more complex numbers, for example (�2)3, (0.1)3, … 2 2 2 2 (�2)3 � �8 (0.1)3 � 0.1 � 0.1 � 0.1 � 0.001 6 Maths Connect 2R 6 Sample page from Maths Connect 2R: Teacher Book Teacher Book 2R Questions and main teaching points provide support for the plenary. 4:23 pm Page 7 A_Teachers.qxd 18/5/04 Demonstrate how to use a calculator to find cubes and cube roots, e.g. 2 7 3 � Invite pupils to complete this table of base 10. 101 � 10 102 � 10 � 10 � 100 103 � 10 � 10 � 10 � 104 � 105 � 106 � 107 � 108 � 109 � 1010 � Choose pupils to read each number, confirming that 106 is one million and 109 is one billion. Variations Introduce the lesson by drawing a 2 � 2 square. How do we calculate the area? (2 � 2) How else can we write this? (22) Explain that we call this index notation. The index, or power, is the small raised 2. How do we read 22? (2 squared or 2 to the power 2) Next draw a 2 � 2 � 2 cube. How do we calculate the volume? (2 � 2 � 2) How do we write this in index notation? (23) How do we read 23? (2 to the power 3 or two cubed) Plenary � � � � Which number is both a square number and a cube number? (64) What is (�3)2, (�5)3? (9, �125) What is (0.1)2, (0.2)3? (0.01, 0.008) How many thousands make a million? a billion? (1000, 1 000 000) Exercise hints Q1–4 Q5–7 Q8 Practice Problems Investigation Key teaching points: 72 means 7 � 7, and is read as ‘seven squared’ and ‘seven to the power of two’. •The inverse of finding the square of a number is finding the square root of a number. •43 means 4 � 4 � 4, and is read as ‘four cubed’ and ‘four to the power of three’. •The inverse of finding the cube of a number is finding the cube root of a number. •The first few cube numbers are 1, 8, 27, 64, 125, … •106 � 1 million, 109 � 1 billion. •We can use a calculator to find squares, square roots, cubes and cube roots. � P B Exercise 1.2, page 5 Homework 1.2 � Answers, page 276 Number 1, Algebra 1: Powers and roots 7 Sample page from Maths Connect 2R: Teacher Book 7 Teacher Book 2R A_Teachers.qxd 18/5/04 4:23 pm Page 8 Key words 1.3 index power indices Indices Use index notation for integer powers Use simple instances of the index laws Links � � Provides links to other relevant sections to help enrich your teaching. Introduction 1.4 Square roots, 5.1 Index notation, 12.3 Brackets and powers 1.2 Powers and roots Oral and mental starter 9 In the previous lesson pupils used index notation to write squares and cubes and powers of 10. In this lesson they extend index notation to higher powers, leading to simple index rules for multiplying and dividing. Teaching activity Outline Explain that this lesson is about multiplying and dividing quantities written in index notation. Write ‘21 � 2’ on the board. Write ‘22 � 4’ underneath. Choose pupils to extend the list up to the tenth power of 2, by doubling. 21 � 2 22 � 4 23 � 8 24 � 16 25 � 32 26 � 64 27 � 128 28 � 256 29 � 512 210 � 1012 Explain that we can find many simple patterns when multiplying and dividing the powers of 2. Write ‘23 � 24 �’ on the board. Invite pupils to use the table to help perform the calculation. (8 � 16 � 128) Is this answer also in the table? (Yes, 27) Repeat for some similar calculations, using a loop, to demonstrate the patterns. 23 � 24 � 27 22 � 23 � 25 8 � 16 � 128 4 � 8 � 32 Challenge pupils to describe the patterns relating to the powers of 2, for example ‘when multiplying two powers of 2, the result is also a power of 2, and the power is the sum of the powers’. Consolidate this idea by writing out the multiplication in full: ‘23 � 24 � 2 � 2 � 2 � 2 � 2 � 2 � 2 � 27’ Extend to division. Write ‘26 � 24 �’ on the board. Record the answer using a loop as before. Repeat for ‘27 � 22 �’. Search for the pattern: ‘when dividing two powers of 2, the result is also a power of 2, and the power is the difference of the powers. 26 � 24 � 22 64 � 16 � 4 8 Maths Connect 2R 8 Sample page from Maths Connect 2R: Teacher Book Teacher Book 2R A_Teachers.qxd 18/5/04 4:23 pm Page 9 Consolidate this by writing out the division in full, and cancelling. 2�2�2 �1 � 2 �1 � 2 �1 � 2 �1 26 � 24 � ��� � 22 �1 � 2 2 �1 � 2 �1 � 2 �1 Will these patterns appear in powers of other numbers? Draw a table of powers of 3 together, up to 36 � 729. Use examples to demonstrate that 34 � 32 � 36, 35 � 32 � 33. Challenge pupils to express the rules more generally, using algebra. na � nb � na�b na � nb � na�b Consolidate the rules by extending to examples using powers of 10. Finally, demonstrate that for the rules to apply, the numbers must have the same base. In other words, they must be powers of the same number. Use some examples to demonstrate this: 23 � 102, 34 � 23. Variations After working through the lesson Outline above, extend to developing the meaning of the power zero, by using the laws to consider 34 � 30 � 34, and 26 � 20 � 26. Consolidate this by writing the calculations in full, i.e. 3 � 3 � 3 � 3 � ? � 3 � 3 � 3 � 3. Clarify that any number to the power zero is one. n0 � 1. Variation sections provide an alternative way of covering the same teaching objectives, giving choice in terms of resources used and approach. Plenary � � � � � What is 102 � 104? (106) How many thousands make one million? Demonstrate using powers. What is (23)2? (26) What is 24 � 24? (1) What is 50? 80? n0? (1) Key teaching points: � � � � Exercise hints Q1–4 Q5–7 Q8 Q9 The plural of ‘index’ is ‘indices’. When multiplying two powers of a number, the result is also a power of that number, given by the sum of the powers. When dividing two powers of a number, the result is also a power of that number, given by the difference of the powers. The rules can be expressed algebraically: na � nb � na�b, na � nb � na�b Practice Problems Activity Investigation P B Exercise 1.3, page 7 Homework 1.3 � Answers, page 276 Number 1, Algebra 1: Indices 9 Sample page from Maths Connect 2R: Teacher Book 9 Teacher Book 2R A_Teachers.qxd 18/5/04 4:23 pm Page 10 Key words 1.4 trial and improvement estimate square root Square roots Use ICT to estimate square roots Use trial and improvement methods to find approximate solutions to equations Links � � 8.8 More square roots, 8.9 Cube roots 1.2 Powers and roots Oral and mental starter 12 Introduction In Lesson 1.2, pupils used the square root key on a calculator to find square roots. In this lesson, pupils are introduced to trial and improvement methods for finding a square root, using a table of square roots to help with an initial estimate. They use calculators to find the squares of estimated numbers, and compare results with a targeted square root. Teaching activity Teacher materials: OHP calculator, or a calculator on a whiteboard Pupil materials: Calculators Outline Explain to the pupils that in this lesson they will be using a method called trial and improvement to find square roots. With pupils’ help, construct a table of squares and square roots on the board. Number � �’ on the board. Use the table to confirm that the Write ‘�21 � � 5’, root lies between the numbers 4 and 5. Write ‘4 � �21 clarifying the notation and explaining that 4 is the lower bound and 5 is the upper bound. Suggest that we make an estimate, for example 4.5. 1 2 3 4 5 6 7 8 9 10 Square 1 4 9 16 25 36 49 64 81 100 Number 10 20 30 40 50 60 70 80 90 100 Square 100 400 900 1600 2500 3600 4900 6400 8100 10 000 Ask pupils to find (4.5)2 using their calculators. (20.25) Too small. Write ‘4.5 � �21 � � 5’. Next try (4.6)2. (21.166) Too large. Write ‘4.5 � �21 � � 4.6’. Invite suggestions, for example try (4.58)2, giving 20.9764. Too small. � � 4.6’. Write ‘4.58 � �21 Invite suggestions, for example try (4.59)2, giving 21.0681. Too large. Features detailed guidance for interactive teaching of concepts and skills – perfect for the less experienced teacher. Write ‘4.58 � �21 � � 4.59’. Ask pupils to suggest a number between 4.58 and 4.59. (For example 4.582) If necessary, demonstrate this on a 10-point number line, with the ends labelled 4.58 and 4.59. Find (4.582)2 using a calculator. (20.994 724) Too small. Write ‘4.582 � �21 � � 4.59’. Invite suggestions for the next number to try, for example 4.583. Try (4.583)2 using their calculators. (21.003 889) Too large. � � 4.583’. Write ‘4.582 � �21 10 Maths Connect 2R 10 Sample page from Maths Connect 2R: Teacher Book Teacher Book 2R A_Teachers.qxd 18/5/04 4:23 pm Page 11 Invite suggestions of a number between 4.582 and 4.583. (For example 4.5825) If necessary, demonstrate this on a 10-point number line, with the ends labelled 4.582 and 4.583. The pupils try (4.5825)2 using their calculators. (20.999 306) Confirm that we are very close to finding the square root of 21. Ask pupils to use � � 4.582 575 6. Discuss the closeness of this to 4.5825. the square root key to find �21 Challenge pupils to round both solutions to the nearest two decimal places (4.58), demonstrating that these are equal. � � 4.583. Ask pupils to Look back at the results written on the board to 4.582 � �21 round the upper and lower bounds to 2 decimal places. (Both 4.58) Explain that if we had been asked to find the square root to 2 decimal places we could have stopped � � 4.583. So, �21 � � 4.58 to 2 decimal places. the calculation here, as 4.582 � �21 Explain that this method is known as ‘trial and improvement’, because you try a number, look at the result, and use this to improve the try. Repeat the whole process to find the square roots of other numbers, for example, �75 �, �4200 �. Remind pupils that �4200 � can be expressed as �42 �00 � 1� or �42 � � 10. Variations Explain the method of trial and improvement, as in the Outline. Write a square root �. Stress that pupils may not use the square root key on the board, for example �3100 of their calculators. Pupils work in pairs, and are allowed four iterations of the trial and improvement method. Stress the importance of making a good first estimate. Pupils then make an estimate in the form of an 8-digit number, e.g. 55.552 660. Each pair writes their answer on a whiteboard. Finally show the result obtained, using the square root key on the board calculator. (55.677 643) Eliminate the results of pupils’ estimates, leaving the closest three as winners. Repeat for a different number. Plenary � � � What are upper and lower bounds for �6.7 �, �350 �, �8600 �,… ? Try finding �27 � using the square root key (5.196 152 4), then try squaring the answer (26.999 999). Discuss these results. Use the trial and improvement method to find the cube root of 40. Key teaching points: � � � � Exercise hints Q1–3 Q4–7 Q8–9 Practice Problems Investigations For square roots of numbers up to 100, it is possible to state two consecutive whole numbers as the upper and lower bound of the root. For square roots of numbers up to 10 000, it is possible to state two consecutive multiples of 10 as the upper and lower bounds of the root. By making an estimate of the square root of a number, and then squaring this estimate, it is possible, by comparing, to make an improved estimate. The method of estimating a solution, comparing it with the target solution, and then making an improved estimate is called the method of ‘trial and improvement’. P B Includes answers to all the Pupil Book exercises, with a helpful reference on each double page. Exercise 1.4, page 9 Homework 1.4 � Answers, page 276 Number 1, Algebra 1: Square roots 11 Sample page from Maths Connect 2R: Teacher Book 11