Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Sample Pages CONTENTS Algebra 1: Sequences, fractions and symbols Matching chart 1 A1 Sequences, functions and ➔ Units 2, 6 and 10 symbols (6 hours) Assumed knowledge Background Pupils will have met number sequences in Year 6. This Unit looks at number sequences in more depth and explores ways of extending number sequences without listing all the values. It will be the first time that pupils use letters to represent unknowns. They will begin to solve simple linear equations using function machines and inverse operations. Before starting this Unit, pupils should: know that 9 ⫻ 8 ⫽ 8 ⫻ 9 G be able to count on and back in both whole number multiples and simple decimals G be able to solve a problem by extracting information from tables G Main teaching objectives Pupil book sections 1.1 Sequences 1.2 Generating sequences 1.3 Investigating sequences 1.4 Function machines 1.5 More function machines Using letters to stand in for unknown numbers 1.6 Teaching objectives Generate and describe simple integer sequences Explore and predict terms in sequences generated by counting in regular steps Generate terms of a simple sequence, given a rule for finding each term from the previous term Generate terms of a simple sequence, given their positions in the sequence Generate a sequence given a rule for finding each term from its position in the sequence Generate sequences from practical contexts Begin to explore term-to-term and position-to-term relationships Suggest extensions to problems by asking ‘What if...?’ Use function machines to explore mappings and to calculate input, output and missing operations Begin to apply inverse operations where two successive operations are involved Use letter symbols to represent unknown numbers Understand algebraic conventions Use letter symbols to write expressions and construct equations Oral and mental starters Starter 27 6 7 16 23 24 Page reference Multiplication and division facts 1 Numbers in figures and in words Place value 1 Ordering positive, negative and decimal numbers Adding and subtracting positive and negative numbers Adding and subtracting pairs of numbers 1 268 253 254 260 264 265 Common difficulties G Pupils may often think that 0.9 ⫹ 0.1 ⫽ 0.10 PB Pages 2–13 Homeworks 1.1–1.6 Assessment 1B ! 2 Maths Connect 1B Tournaments (page 286) Two-step relations (page 289) Key words 1.1 Sequences Generate and describe simple integer sequences Explore and predict terms in sequences generated by counting in regular steps Links ➔ 1.2 Generating sequences Oral and mental starter 27 Introduction In this lesson, pupils learn to recognise and describe sequences. They will practise finding terms of a sequence following simple rules, and start to use the language associated with sequences correctly. Teaching activity Outline The code for a safe is 2, 4, 3, 7, 6. Can I open the safe by pressing 2, 4, 3, 6, 7? (No) Explain that a sequence is a set of numbers in a given order, but that often the order is given by a rule. Can you give me examples of other number sequences? (Answers could be: sequences following a simple rule such as square numbers; sequences following a more complex rule such as time of sunset each day; sequences following an irregular pattern, such as maximum temperature each day; random sets of numbers e.g. winning lottery numbers.) Explain that each number in a sequence is called a term. Explain that terms next to each other are called consecutive terms and are often separated by commas. If the rule for a sequence is ‘Start at 7 and count on in steps of 7’ what would the sequence look like? (7, 14, 21, 28 ...) Write the sequence on the board. What do you notice about the terms in this sequence? (They are multiples of 7.) Continue for the following three rules below. Write the first five terms of the sequence on the board each time: G Start at 3 and count on in steps of 2. (3, 5, 7, 9, 11) What do you notice about each term in the sequence? (Each term is an odd number.) G Start at 24 and count back in steps of 3. (24, 21, 18, 15, 12) What do you notice about each term in the sequence? (Each term is a multiple of 3.) G Start at 1 and count on in steps of 5. (1, 6, 11, 16, 21) What do you notice about each term in the sequence? (Each term is four less than a multiple of 5.) Is the first sequence going up or going down? (Going up) Explain that this is called an ascending sequence. Is the second sequence going up or going down? (Going down) Explain that this is called a descending sequence. Write on the board: ‘ Odd numbers between 0 and 50 ‘ Start at 1 and count on in steps of 2’ G G Ask the class if there is any difference between the two sequences generated following these rules. 4 Maths Connect 1B sequence term consecutive finite infinite Think carefully. Does either sequence contain the number 51? Why doesn’t the first sequence contain 51? (Because we have indicated where the sequence will end.) Explain that a sequence with a stated start and end is called a finite sequence. A sequence that continues indefinitely is infinite. We can show this with dots: 1, 3, 5, … 49 is finite. 1, 3, 5, 7, 9 … is infinite. Variations Teacher materials: Resource sheet 1, enlarged to make sets of cards shown below. For a more visual approach, use cards printed with patterns of dots. Show the pupils the first four cards: Tell me how to put the cards in a logical order. Arrange the cards to illustrate ascending and descending sequences, meanings of term and consecutive term etc. Plenary Questions: G G G What are examples of sequences you would find in everyday life? (Telephone numbers, house numbers, number of pupils absent each day, hours of daylight each day etc.) Does a sequence have to go up an equal amount each time? (No.) Is this sequence finite or infinite: 6, 4, 2, 0, ⫺2, ⫺4, …? Key teaching points: G G G G G G A sequence is a set of numbers in a given order. Each number in a sequence is called a term. Terms next to each other are called consecutive terms and are often separated by a comma. Sequences can be ascending or descending. Sequences can be finite or infinite. To find the next few terms of a sequence you look for a pattern; if there isn’t one you can’t find the next few terms. Exercise hints Questions 1–3 Practice Question 4–7 Problems Question 8 Investigation In Q3d pupils may find it difficult to go below zero. In Q3e, a common error is to write: 0.10, 0.11, 0.12. Remind pupils 11 tenths ⫽ 1.1. P B Exercise 1.1, page 2 Homework 1.1, page 73 Answers, page 296 Algebra 1: Sequences 5 Key words 1.2 generate term term-to-term rule Generating sequences Generate terms of a simple sequence, given a rule for finding each term from the previous term Generate terms of a simple sequence, given their positions in the sequence Links ➔ ➔ Introduction In this lesson, pupils use a starting point and a rule to generate a sequence. They learn how to generate the sequence using a calculator. 1.3 Investigating sequences 1.1 Sequences Oral and mental starter 6 Teaching activity Teacher materials: Calculator and projector (or projector and spreadsheet package) Pupil materials: Calculators Outline Discuss with pupils the meanings of sequence, term, consecutive term and rule. To generate a sequence you can use two pieces of information: a starting point and a rule. Tell pupils that you are going to generate a sequence given the following information: Starting point ⫽ 3 Rule ⫽ ⫹4 Teach pupils how to generate a sequence on a calculator: Enter 3 into the calculator and press Enter. Enter ANS ⫹4 and press enter. Press Enter repeatedly to generate the sequence 3, 7, 11, 15 … Ask the pupils to work through Q1–8. They may choose a calculator or non-calculator method for all questions, except Q5. Allow about twenty minutes, then stop the class. Tell them that you are going to look at another way of generating terms in the sequence 3, 7, 11, 15 … What were you adding to get from one term to the next? (4) Which ‘times table’ are you using in the sequence? (4) How could you get from the first term to the third term? (Add 8/ add 4 twice) How could you get from the first term to the fourth term? (Add 12/ add 4 three times ) How could you get from the second term to the fifth term? (Add 12/ add 4 three times) Can you spot a pattern that will tell you how to find any term? (It is the four times table take away 1.) Repeat this activity with: Starting Point 7 81 1 6 Maths Connect 1B Rule Add 5 Subtract 9 Add 2 Numbers generated Multiples of 5 add 2 Multiples of 9 Multiples of 2 take away 1 Variations Ask a volunteer to choose a starting point between 0 and 9 and a termto-term rule (e.g. starting point 8, ‘add 7’). Round the class, each pupil gives the next term in the sequence. The first pupil to give a value of 50 or over wins the game and starts it again. (For example, if the pupil chose the starting point of 8 and the rule of ‘add 7’, then the class would generate the sequence 8, 15, 22, 29, 36 … . The pupil who said the number 50 would win.) Question pupils about individual sequences. For example, for the sequence above: How would you work out who will call out the number 78? (Calculate how many lots of 7 you need to add and count on that many pupils) Can you work out the third term without working out the second term? (Yes) How can you do this? (By adding 2 lots of 7 to 8) Encourage pupils to plan ahead to work out who will win the game. (For example, with the sequence above, the pupil who starts should spot that the 7th person to call out a number will win, since starting at 8 and counting up in 7s you need only 6 lots of 7 before you hit 50.) You can extend this idea so that pupils give more than one rule (e.g. ‘add two, multiply by five’). Plenary Questions: G G What information do you need in order to generate a sequence? (Rule and first term, or rule and 2nd term etc.) If you know that the rule for getting from one term of sequence to the next is ‘add 2’ and your starting point is 5, how would you work out the tenth term? (3 ⫹ 10 lots of 2). Note that the first term (5) is represented as 3 ⫹ 1 lot of 2. Exercise hints Q1–5 Practice Q6–9 Problems Q10 Investigation Encourage pupils to refer to Example 1 if they struggle with Q9 and Q10. Key teaching points: G G You can find a term in a sequence, without finding all the terms in between, if you know the relationship between consecutive terms. You can generate a sequence given a first term and a rule. P B Exercise 1.2, page 4 Homework 1.2, page 73 Answers, page 296 Algebra 1: Generating sequences 7 Key words 1.3 Investigating sequences Generate a sequence given a rule for finding each term from its position in the sequence Generate sequences from practical contexts Begin to explore term-to-term and position-to-term relationships Suggest extensions to problems by asking ‘What if ...?’ Links ➔ ➔ sequence term term number rule Introduction The activity is based on diagrams and allows pupils to visualise sequences. The teaching should focus on generalising (from the patterns shown) to a rule that will enable any term in the sequence to be found. 1.4 Function machines 10.2 The general rule 1.1 Sequences 1.2 Generating sequences Oral and mental starter 7 Teaching activity Teacher materials: OHP, OHT of Resource sheet 1, sequences of patterns shown below. Outline The sequence 5, 7, 9, 11… is going up in 2s and the first term is 5. The first term is 3 plus one lot of 2, the second term is 3 plus two lots of 2 and the tenth term is 3 plus ten lots of 2. Display the pattern on the first OHT. Draw this table on the board: Pattern no. 1 2 3 4 No. of squares 5 9 13 17 5 (1 ⫻ 4) ⫹ 1 (2 ⫻ 4) ⫹ 1 (3 ⫻ 4) ⫹ 1 (4 ⫻ 4) ⫹ 1 6 10 How will this sequence continue? (You add 4 more squares each time.) How many squares will you need to make the fifth pattern? (17 ⫹ 4 ⫽ 21) How many squares will you need to make the sixth pattern? (21 ⫹ 4 ⫽ 25) Explain that you could describe this sequence by saying that the first pattern has 1 middle square and 4 more squares (1 on each of the 4 ‘arms’). The second pattern has 1 middle square and then 2 lots of 4 squares. How many squares do you think you would need to make the tenth pattern? (10 lots of 4 plus 1 ⫽ 41) How many squares would there be in the 50th pattern? (50 ⫻ 4 ⫹ 1 ⫽ 201) How many in the 100th pattern? (100 ⫻ 4 ⫹ 1 ⫽ 401) Check that the pupils recognise that this method allows us to find any term in a sequence without calculating every term. Display the second pattern on the OHT: Draw this table on the board: Pattern no. 1 2 3 4 No. of circles 4 7 10 13 (1 ⫻ 3) ⫹ 1 (2 ⫻ 3) ⫹ 1 (3 ⫻ 3) ⫹ 1 (4 ⫻ 3) ⫹ 1 8 Maths Connect 1B 5 10 15 How will this sequence continue? (You add 3 more circles each time.) How many circles will you need to make the fifth pattern? (16 ⫽ 5 lots of 3, plus 1) How many circles will you need to make the sixth pattern? (19 ⫽ 6 lots of 3, plus 1) Explain that you could describe this sequence by saying that the first pattern has 1 circle in the middle and 3 more circles (1 on each of the 3 ‘arms’). The second pattern has 1 circle in the middle and 2 lots of 3 circles. The third pattern has a circle in the middle and 3 lots of 3 circles. How many circles do you think you would need to make the tenth pattern? (10 lots of 3 plus 1 ⫽ 31) How many circles would there be in the 15th pattern? (15 lots of 3, plus 1 ⫽ 46) Variations You may prefer to omit the diagrams and focus on numerical patterns. Write the following sequence on the board: 5, 9, 13, 17, 21 … and the first two rows of Term no. 1 2 3 this table: Sequence 5 9 13 4 5 17 21 4⫹1 (4 ⫹ 4) ⫹ 1 (4 ⫹ 4 ⫹ 4) ⫹ 1 (4 ⫹ 4 ⫹ 4 ⫹ 4) ⫹ 1 (4 ⫹ 4 ⫹ 4 ⫹ 4 ⫹ 4) ⫹ 1 (1 ⫻ 4) ⫹ 1 (2 ⫻ 4) ⫹ 1 (3 ⫻ 4) ⫹ 1 (5 ⫻ 4) ⫹ 1 (4 ⫻ 4) ⫹ 1 100 Pupils will find the 6th term, by adding 4 to the 5th term and so on, but this would be a tedious way of finding the 100th term. Look for a pattern, and fill in the third and fourth row of the table as a class. Use coloured pens or chalk to match the term number with the corresponding number in the rule, if the pupils don’t see the link at first. The 100th term ⫽ 100 ⫻ 4 ⫹ 1 ⫽ 401. Repeat this for other sequences such as: 5, 10, 15, 20, 25, … 5, 8, 11, 14, 17, … Plenary Questions: G G G G G G Key teaching points: Can you think of any sequences which occur in real life? (E.g. house numbers, number of pupils absent each day, weight of an individual each year) What do you call a sequence which goes up? (Ascending) What do you call a sequence which goes down? (Descending) How can you describe a sequence? (First term and a rule) What do we call two terms that are next to one another in a sequence? (Consecutive) Are the following sequences finite or infinite: Odd numbers bigger than 20? (Infinite) Even numbers between 0 and 1000? (Finite) G You can find any term in a sequence where consecutive terms are generated by repeatedly adding the same number. To do this, you do not need to calculate all the terms in between if you know the difference between terms. P B Exercise hints Q1–4 Q5–8 Q9 Practice Problems Investigation Exercise 1.3, page 6 Homework 1.3, page 74 Answers, page 296 Algebra 1: Investigating sequences 9 Key words 1.4 Function machines Using function machines to explore mappings and to calculate input, output and missing operations Links ➔ ➔ input output function machine mapping diagram unknown Introduction 1.5 More function machines 1.3 Investigating sequences Oral and mental starter 16 Pupils will already be familiar with the idea of addition being the inverse of subtraction and multiplication being the inverse of division. In the next three lessons pupils will solve simple equations by finding missing values and represent algebraic expressions by mappings. This lesson introduces a range of methods for solving one-step function machines. Teaching activity Pupil materials: Hand-held whiteboards (optional) Outline Write the following on the board: 3 ⫹ 䊐 ⫽ 17 Write the missing number on your whiteboards (14) We can write this as a function machine: 3 ➝ ⫹䊐 ➝ 17 You need to identify the missing information in the function machine. How did you work out what to put in the box? (17 ⫺ 3 ⫽ 14) We are using an inverse operation. The inverse of addition is subtraction. The inverse of multiplication is division. Explain that a function machine enables us to find outputs for lots of different inputs. For example, write on the board: Input ➝ ⫹9 ➝ Output If the input is 3, what is the output? (12) If the input is 7, what is the output? (16) We can also work back through the machine, to find the input, if we know the output. If the output is 20, what is the input? (20 ⫺ 9 ⫽ 11) Repeat with more examples of function machines, using subtraction or multiplication. Explain that we can also use a mapping diagram. Start by writing on the board the function machine: Input ➝ ⫹4 ➝ Output When the input is 0, the output is 4. We write 0 ➝ 4 Also: 1 ➝ 5 2 ➝ 6 etc. Explain that this is called a mapping. Mappings are usually indicated by an arrow and can be shown by a mapping diagram. 10 Maths Connect 1B Adding 4 to the first number gives the second number. 0 0 1 2 3 4 5 6 1 2 3 4 5 6 Now write the following on the whiteboard: 3⫻䊐⫽4⫻3 How did you know that 䊐 stands for 4? (3 ⫻ 4 is the same as 4 ⫻ 3) Variations Play a function guessing game. Write some mappings on the board, explain that each mapping has just one operation, and ask pupils to write down the hidden rule that connects them. For example, 3 ➝ 8 (⫹5) 4 ➝ 10 (⫹6) 5 ➝ 12 (⫹7) 10 ➝ 22 (⫹12) 3 ➝ 18 (⫻6 or ⫹15) 12 ➝ 5 (⫺7) 10 ➝ 2 (⫼5 or ⫺8). Plenary Questions: G G Key teaching points: How do we find an unknown number? (By using inverse operations) Explain how you would solve the following: 䊐 ⫻ 6 ⫽ 24 (䊐 ⫽ 4) 䉭 ⫹ 5 ⫽ 11 (䉭 ⫽ 6) 䊊⫺7⫽0 (䊊 ⫽ 7) 䊐 ᎏᎏ ⫽ 6 (䊐 ⫽ 30) 5 32 ⫺ 䉭 ⫽ 12 (䉭 ⫽ 20) Exercise hints Q1–5 Q6 and 7 Q8 Practice Problems Investigation G G G You can find missing numbers using inverse operations. The inverse of multiplication is division, the inverse of addition is subtraction and vice versa. You can represent operations as function machines and in mapping diagrams. P B Exercise 1.4, page 9 Homework 1.4, page 74 Answers, page 297 Algebra 1: Function machines 11 Key words 1.5 operations inverse unknown More function machines Begin to apply inverse operations where two successive operations are involved Links ➔ ➔ Introduction 1.6 Using letters to stand in for unknown numbers 1.4 Function machines This lesson uses the methods that were introduced last lesson, and applies them to twostep function machines. Oral and mental starter 23 Teaching activity Pupil materials: Hand-held whiteboards Outline Remind pupils that if you had this function machine: and the output was 9, you could work out the input by working backwards using inverse operations. 9⫺8⫽1 Input ⫹8 ⫹8 This can be shown as: Subtraction is the inverse of addition and vice versa. Division is the inverse of multiplication and vice versa. 1 9 1 9 inverse ⫺8 Write the following on the board: Input ➝ ⫻2 ➝ ⫹5 ➝ 11 I thought of a number. I multiplied it by 2 and then added 5, and my answer was 11. What was the number? Write it on your whiteboards. (3) How did you work out the answers? (Trial and improvement, or by using inverse operations and calculating (11 ⫺ 5) ⫼ 2 Demonstrate trial and improvement for a two-step function machine: Input ➝ ⫻2 ➝ ⫹5 ➝ 11 4 ⫻ 2 ⫹ 5 ⫽ 13 2⫻2⫹5⫽9 3 ⫻ 2 ⫹ 5 ⫽ 11 (too big) (too small) So the input number was 3. 12 Maths Connect 1B Output Repeat this activity using other operations e.g. Input ➝ ⫻3 ➝ ⫹4 ➝ 10 (2) Input ➝ ⫼3 ➝ ⫹5 ➝ 7 (6) Input ➝ ⫹8 ➝ ⫺12 ➝ 6 (10) Input ➝ ⫻4 ➝ ⫼5 ➝ 20 (25) You can extend this activity by asking pupils to think of their own function machines, using more than two operations or using decimals, fractions and negative numbers. Variations Set up a spreadsheet so that column A is the input and column B is the output. Make sure you hide the formulae. A B 1 ⫽ A1 ⫹ 5 2 ⫽ A2 ⫺ 3 3 ⫽ A3 ⫻ 2 4 ⫽ A4 ⫼ 2 5 ⫽ (A5 ⫻ 2) ⫺ 4 6 ⫽ (A6 ⫺ 4) ⫻ 4 7 ⫽ (A7 ⫹ 12) ⫼ 2 8 ⫽ (A8 ⫻ 2) ⫺ 5 Ask the pupils to work out the rules being used to generate the values in column B. Encourage them to work systematically, typing in different whole number values for A1, A2 … etc. Plenary Questions: G G I choose a number, add 6 and multiply by 2 and I get 22. What number did I choose? (5) How did you work out the answer to the last question? (By trial and improvement, or by using inverse operations) Key teaching points: G G G Exercise hints Q1–2 Practice Q3–5 Problems Q6 & 7 Investigations Q6 and Q7 are extended questions that involve a large number of calculations. If you know the operations that have taken you from one number to another then by performing the inverses of these operations you can get back to your original number. Multiplication and division, addition and subtraction are inverses of one another. You should always check your answers by substituting the value back in. P B Exercise 1.5, page 10 Homework 1.5, page 75 Answers, page 297 Algebra 1: More function machines 13 Key words 1.6 Using letters to stand in for unknown numbers Use letter symbols to represent unknown numbers Recognise algebraic conventions Use letter symbols to write expressions and construct equations Links ➔ ➔ 6.1 Using algebra 10.2 The general rule 1.4 Function machines 1.5 More function machines Oral and mental starter 24 equation expression unknown algebra Introduction Pupils will be introduced to the idea of an algebraic equation in this lesson. They will practise transforming function machines and word problems into algebraic equations. The plenary draws together the learning from this and the previous two lessons. Teaching activity Pupil materials: Hand-held whiteboards (optional) Outline Explain to the class how we can use expressions. For example: I have some sweets and then I get 4 more.The expression for the number of sweets I now have is x ⫹ 4. I think of a number and multiply it by 9, The expression for this is 9x. I have some apples and give three away. The expression for the number of apples I have left is n ⫺ 3. Explain that it does not matter which letter is used for the unknown number. Revise the idea of the function machine with pupils. Input Draw this on the board: What is the input? (34) How did you work this out? (By working backwards) Explain that you can use a letter (e.g. x) to stand in for the unknown number in the function machine. ⫹73 107 Input ⫹4 10 Input ⫹8 25 Input ⫺7 22 Input ⫻2 14 x ⫼ 3 ⫽ 6, x ⫽ 6 ⫻ 3 so x ⫽ 18 (This is a good opportunity Input x to introduce the idea of writing x ⫼ 3 as ᎏᎏ) 3 ⫹3 6 Write on the board: x ⫹ 73 ⫽ 107 We call this an equation, because it includes an equals sign. What number you think x represents? (34) How did you work out the unknown number? (By working backwards using inverse operations: 107 ⫺ 73) Draw this function machine on the board: Write down the input of this function machine. (6) Write down an equation for the function machine. Use x for the unknown number. (x ⫹ 4 ⫽ 10) What is x? (6) Repeat for the other function machines below: x ⫹ 8 ⫽ 25, x ⫽ 25 ⫺ 8, x ⫽ 17 x ⫺ 7 ⫽ 22, x ⫽ 22 ⫹ 7, so x ⫽ 29 x ⫻ 2 ⫽ 14, x ⫽ 14 ⫼ 2 so x ⫽ 7 (This is a good opportunity to introduce the idea of writing x ⫻ 2 as 2x.) 14 Maths Connect 1B Variations Use puzzles to help pupils develop equations as a way of finding an unknown. For example: 1. I think of a number, add 12 and get 42. What number was I thinking of? (x ⫹ 12 ⫽ 42, x ⫽ 42 ⫺ 12, so x ⫽ 30) 2. I think of a number, subtract 7 and get 12. What number was I thinking of? (x ⫺ 7 ⫽ 12, x ⫽ 12 ⫹ 7, so x ⫽ 19) 3. I think of a number, multiply it by 5 and get 35. What number was I thinking of? (x ⫻ 5 ⫽ 35 so 5x ⫽ 35, x ⫽ 35 ⫼ 5 so x ⫽ 7) (This is a good opportunity to introduce the idea of writing x ⫹ x ⫹ x ⫹ x ⫹ x ⫽ 5 ⫻ x ⫽ 5x) 4. I think of a number, divide by 10 and get 24. What number was I x thinking of? (x ⫼ 10 ⫽ 24 so ᎏᎏ ⫽ 24, x ⫽ 24 ⫻ 10 so x ⫽ 240) 10 (This is a good opportunity to introduce the idea of writing x x ⫼ 10 as ᎏᎏ.) 10 Plenary Questions: G Key teaching points: Write on the board a function machine involving calculations that pupils will not easily be able to work out mentally, e.g. ⫻97 Input G G G ⫹343 1701 How would we find the input number? Ask pupils to suggest methods (trial and improvement, inverse operations, or transforming to an equation). Individuals could demonstrate to the class. Focus on process rather than final answer, and discuss the relative merits/disadvantages of the methods. G G G We represent an unknown by using a letter. An expression is a collection of numbers and letters, such as 2x or 3n ⫹ 2. An equation is a mathematical expression containing numbers or letters and an ⫽ sign. We can transform a function machine into an equation. We can write x ⫻ 2 as 2x. x We can write x ⫼ 2 as ᎏᎏ. 2 G w ⫻7 ⫻9 126 What is w? (2) How would the output be changed if the function machines changed places? (The output would not be changed.) G m ⫹3 ⫻8 72 What is m? (6) Would the output change if the function machines changed places? (The output would be 51 not 72.) Exercise hints Q1–8 Practice and revision Q9 Problem For Q9: tell pupils to start by finding pairs of whole numbers that sum to 72 and to multiply them together to see if they get 1100. P B Exercise 1.6, page 13 Homework 1.6, page 75 Answers, page 297 Algebra 1: Using letters to stand in for unknown numbers 15 xvi Maths Connect 1B Understand negative numbers as positions on a number line; order, add and subtract positive and negative integers in context. Recognise and use multiples, factors (divisors), common factor, highest common factor and lowest common multiple in simple cases, and primes (less than 100); use simple tests of divisibility. Recognise the first few triangular numbers, squares of numbers to at least 12 ⫻ 12 and the corresponding roots. Integers, powers and roots Understand and use decimal notation and place value; multiply and divide integers and decimals by 10, 100, 1000, and explain the effect. Compare and order decimals in different contexts; know that when comparing measurements they must be in the same units. Round positive whole numbers to the nearest 10, 100 or 1000 and decimals to the nearest whole number or one decimal place. Place value, ordering and rounding Numbers and the number system Solve word problems and investigate in a range of contexts: number; algebra; shape, space and measure; handling data; compare and evaluate solutions. Identify the necessary information to solve a problem; represent problems mathematically making correct use of symbols, words, diagrams, tables and graphs. Break a complex calculation into simpler steps, choosing and using appropriate and efficient operations, methods and resources, including ICT. Present and interpret solutions in the context of the original problem; explain and justify methods and conclusions, orally and in writing. Suggest extensions to problems by asking ‘What if …?’; begin to generalise and to understand the significance of a counter-example. Applying mathematics and solving problems Using and applying mathematics to solve problems Framework teaching objective Matching chart 2 84–85 120–121, 200–207 120–121, 206–209 86–89 100–101, 168–175 100–101, 174–177 10.1, 16.1, 16.2, 16.3, 16.4 10.1, 16.4, 16.5 104–107 102–103 22–23, 114–117 18–21, 46–47, 52–53 8–9, 202–203, 230–231 180–197 48–49, 52–53 Teacher book pages 9.2, 9.3 2.3, 9.7, 9.8 9.1 6–7, 170–171, 196–197 1.3, 16.2, 17.7 14–17, 38–39, 44–45 18–19, 96–99 152–163 Unit 15 2.1, 2.2, 4.3, 4.6 40–41, 44–45 Pupil book pages Units 1, 5, 6, 8, 10, 13, 15 & 17 4.4, 4.6 Throughout Section no. 25, 27, 28, 33 25, 29, 36 13, 16 18, 19 16, 17, 20 7, 8 Starter ref. Matching chart 2 xvii 34–39 40–43 44–45, 122–125 126–131 4.1, 4.2, 4.3 4.4, 4.5 4.6, 12.1, 12.2 12.3, 12.4, 12.5 Written methods Use standard column procedures to add and subtract whole numbers and decimals with up to two places. Multiply and divide three-digit by two-digit whole numbers; extend to multiplying and dividing decimals with one or two places by single-digit whole numbers. 24–25, 178–179 16–17, 90–93, 180–183 2.2, 9.4, 9.5, 16.7 16.8 24–25, 90–93 178–183 2.6, 9.4, 9.5, 16.6, 16.7, 16.8 2.6, 16.6 24, 30, 32, 14–25, 34–45, 122–131 Unit 2, 4, 12 20–21, 108–111, 212–215 28–29, 210–211 28–29, 108–111, 210–215 16–29, 40–53, 144–155 37 14, 24 33, 34, 35 38 26, 33 24–27, 70–71, 76–79 Starters Mental methods and rapid recall of number facts Know and use the order of operations, including brackets. Consolidate the rapid recall of number facts, including positive integer complements to 100 and multiplication facts to 10 ⫻ 10, and quickly derive associated division facts. Consolidate and extend mental methods of calculation to include decimals, fractions and percentages, accompanied where appropriate by suitable jottings; solve simple word problems mentally. Make and justify estimates and approximations of calculations. 21, 22, 23, 24 32, 33 40, 41, 43, 44 39 20, 42, 43, 44 Starter ref. 1.5, 2.1, 2.2, 2.4, See various 2.5, 2.6, 9.3, 9.4, sections 9.5, 13.2, 13.3, 16.6, 16.7, 16.8 2.4, 2.5, 6.1, 6.4, 6.5 20–21, 24–29, 106–111, 160–163, 210–215 150–155 52–53, 146–149 48–51 42–47 Teacher book pages Understand addition, subtraction, multiplication and division as they apply to whole numbers and decimals; know how to use the laws of arithmetic and inverse operations. Number operations and the relationships between them Calculations Fractions, decimals, percentages, ratio and proportion Use fraction notation to describe parts of shapes and to express a smaller whole number as a fraction of a larger one; simplify fractions by cancelling all common factors and identify equivalent fractions; convert terminating decimals to fractions, e.g. 0.23 ⫽ ᎏ120ᎏ30; use a diagram to compare two or more simple fractions. Begin to add and subtract simple fractions and those with common denominators; calculate simple fractions of quantities and measurements (whole number answers); multiply a fraction by an integer. Understand percentage as the ‘number of parts per 100’; recognise the equivalence of percentages, fractions and decimals; calculate simple percentages and use percentages to compare simple proportions. Understand the relationship between ratio and proportion; use direct proportion in simple contexts; use ratio notation, reduce a ratio to its simplest form and divide a quantity in a given ratio; solve simple problems about ratio and proportion using informal strategies. Pupil book pages Section no. Framework teaching objective xviii Maths Connect 1B 9.4, 9.5, 12.2, 12.5 16.7, 16.8 Check a result by considering whether it is of the right order of magnitude and by working the problem backwards. 2–7 4–7, 102–105, 184–187 6–7, 102–105, 184–185 8–13 106–111, 188–189 190–195 1.4, 1.5, 1.6 10.4, 10.5, 10.6, 17.3 17.4, 17.5, 17.6 Express simple functions in words, then using symbols; represent them in mappings. Generate coordinate pairs that satisfy a simple linear rule; plot graphs of simple linear functions, where y is given explicitly in terms of x, on paper and using ICT; recognise straight line graphs parallel to the x-axis or y-axis. Begin to plot and to interpret the graphs of simple linear functions arising from real-life situations. 224–229 2–9 6–9, 122–125, 218–221 8–9, 122–125, 218–219 10–15 126–131, 222–223 74–75, 156–165, 216–233 62–63, 132–139, 184–199 1.1, 1.2, 1.3 1.2, 1.3, 10.2, 10.3, 17.1, 17.2 1.3, 10.2, 10.3, 17.1 13.2, 13.3, 13.4, 17.7 6.3, Units 13 & 17 6.2, 6.5, 13.1 Starter ref. 2–15, 68–79, 118–131, 156–165, 216–233 2–5, 68–79, 118–131, 156–165, 216–233 72–73, 78–79, 158–159 160–165, 230–231 46, 47 108–111, 148–149, 154–155, 212–215 24–27, 76–79, 208–209 28–29, 114–117 Teacher book pages 2–13, 58–67, 100–111, 184–199 2–13, 58–67, 100–111, 184–199 60–61, 66–67, 132–133 134–139, 196–197 90–93, 124–125, 130–131, 180–183 20–23, 64–67, 176–177 24–25, 96–99 Pupil book pages Generate and describe simple integer sequences. Generate terms of a simple sequence, given a rule (e.g. finding a term from the previous term, finding a term given its position in the sequence). Generate sequences from practical contexts and describe the general term in simple cases. Sequences, functions and graphs Use letter symbols to represent unknown numbers or variables; know the meanings of the words term, expression and equation. Understand that algebraic operations follow the same conventions and order as arithmetic operations. Simplify linear algebraic expressions by collecting like terms; begin to multiply a single term over a bracket (integer coefficients). Construct and solve simple linear equations with integer coefficients (unknown on one side only) using an appropriate method (e.g. inverse operations). Use simple formulae from mathematics and other subjects, substitute positive integers in simple linear expressions and formulae and, in simple cases, derive a formula. Equations, formulae and identities Algebra Checking results Units 1, 6, 10, 13 & 17 Units 1, 6, 13 & 17 2.4, 2.5, 6.4, 6.5, 16.5 2.6, 9.7, 9.8 Carry out all calculations with more than one step using brackets and the memory; use the square root and sign change keys. Enter numbers and interpret the display in different contexts (decimals, percentages, money, metric measures). Calculator methods Section no. Framework teaching objective Matching chart 2 xix 7.3, 10.4, 10.5, 10.6, 14.6, 17.3, 17.4, 17.5, 17.7 14.1, 14.2, 14.3, 14.4, 14.5, 14.6, 18.3 14.1, 14.3, 14.5 Measures and mensuration Use names and abbreviations of units of measurement to measure, estimate, calculate and solve problems in everyday contexts involving length, area, mass, capacity, time and angle; convert one metric unit to another (e.g. grams to kilograms); read and interpret scales on a range of measuring instruments. Use angle measure; distinguish between and estimate the size of acute, obtuse and reflex angles. Know and use the formula for the area of a rectangle; calculate the perimeter and area of shapes made from rectangles. Calculate the surface area of cubes and cuboids. 168–169, 172–173 176–177 168–173 174–179, 240–241 70–71, 116–117 26–29 32–33, 210–211 3.1, 3.2, 5.1, 5.2, 5.3, 5.4, 7.2, 9.6, 9.7, 9.8, 18.6, Unit 8 7.2, 11.3 3.1, 3.2 3.4, 18.6 Starter ref. 62–63, 246–247 84–85, 138–139 32–35 30, 31 32–35, 56–63, 84–85, 9, 10, 11, 12, 112–117, 246–247 45 244–245 208–209 26–29, 46–53, 70–71, 94–99, 210–211 138–143, 242–245 116–121, 206–209 106–111, 150–151, , 86–87, 126–131, 188–193, 196–197 178–179, 222–231 140–141, 144–145, 148–149 140–145 146–151, 204–205 36–37, 246–247 30–31, 210–211 3.3, 18.6 82–85 Teacher book pages 112–115, 200–203 134–137, 236–239 68–71 Pupil book pages 11.1, 11.2, 18.1, 18.2 Throughout 7.1, 7.2 Section no. Use a rule and protractor to: measure and draw lines to the nearest millimetre and angles, 11.3, 11.4, 11.5, including reflex angles, to the nearest degree; construct a triangle given two sides and the included 18.4, 18.5 angle (SAS), or two angles and the included side (ASA). Explore these constructions using ICT. Use ruler and protractor to construct simple nets of 3-D shapes, e.g. cuboid, regular tetrahedron 18.5 square-based pyramid, triangular prism. Construction Use conventions and notation for 2-D coordinates in all four quadrants; find coordinates of points determined by geometric information. Coordinates Understand and use the language and notation associated with reflections, translations and rotations. Recognise and visualise the transformation and symmetry of a 2-D shape; reflection in given mirror lines, and line symmetry; rotation about a given point, and rotation symmetry; translation. Explore these transformations and symmetries using ICT. Transformations Use correctly the vocabulary, notation and labelling conventions for lines, angles and shapes. Identify parallel and perpendicular lines; know the sum of angles at a point, on a straight line and in a triangle and recognise vertically opposite angles. Begin to identify and use angle, side and symmetry properties of triangles and quadrilaterals, solve geometrical problems involving these properties, using step-by-step deduction and explaining reasoning with diagrams and text. Use 2-D representations to visualise 3-D shapes and deduce some of their properties. Geometrical reasoning: lines, angles and shapes Space, shape and measure Framework teaching objective xx Maths Connect 1B Interpreting and discussing results Use vocabulary and ideas of probability, drawing on experience. Understand and use the probability scale from 0 to 1; find and justify probabilities based on equally likely outcomes in simple contexts; identify all the possible mutually exclusive outcomes of a single event. Collect data from a simple experiment and record in a frequency table; estimate probabilities based on this data. Compare experimental and theoretical probabilities in simple contexts. Probability Interpret diagrams and graphs (including pie charts), and draw simple conclusions based on the shape of graphs and simple statistics for a single distribution. Compare two simple distributions using the range and one of the mode, median or mean. Write a short report of a statistical enquiry and illustrate with appropriate diagrams, graphs and charts, using ICT as appropriate; justify choice of what is presented. 54–57 162–163 50–51, 164–165 166–167 5.3, 15.7 15.8 52–53, 76–77, 156–157 154–155 160–161 5.5, 5.6 15.6 15.2 15.5 5.4, 8.2, 15.3 196–197 60–61, 194–195 64–67 192–193 62–63, 92–93, 186–187 184–185 190–191 92–93, 98–99, 190–191 76–77, 82–83, 160–161 96–97 56–61, 90–91, 182–183 80–81 8.4 94–95 94–95, 188–189 96–97, 188–189 Teacher book pages 46–51, 74–75, 152–153 78–79 78–79, 158–159 80–81, 158–159 Pupil book pages 8.3 8.3, 15.4 8.4, 15.4 Section no. Calculate statistics for small sets of discrete data: find the mode, median and range, and the modal 5.1, 5.2, 5.3, 8.1, class for grouped data; calculate the mean, including from a simple frequency table, using a 15.1 calculator for alarger number of items. Construct, on paper and using ICT, graphs and diagrams to represent data including: bar-line 8.2, 8.5, 15.5 graphs; frequency diagrams for grouped discrete data; use ICT to generate pie charts. Processing and representing data, using ICT as appropriate Given a problem that can be addressed by statistical methods, suggest possible answers. Decide which data would be relevant to an enquiry and possible sources. Plan how to collect and organise small sets of data; design a data collection sheet or questionnaire to use in a simple survey; construct frequency tables for discrete data, grouped where appropriate in equal class intervals. Collect small sets of data from surveys and experiments, as planned. Specifying a problem, planning and collecting data Handling data Framework teaching objective Starter ref. Maths Connect Teacher Book Transforming standards at Key Stage 3 Maths Connect Teacher Books will help you deliver interactive whole class teaching in line with the National Numeracy Strategy’s Framework. Written and developed by experienced teachers and advisors, 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. Maths Connect - everything you need to deliver effective and interactive lessons. 0 435 53503 X t 01865 888080 e [email protected] f 01865 314029 w www.heinemann.co.uk 0 435 53469 6 G958 Maths Connect 1 Blue Teacher Book