The Bus Stop Queue From: The Language of Functions and Graphs

‘First to 100’ Game Work with a partner. Joe Ava Total 5 10 15 6 21 8 29 39 9 48 56 66 75 85 3 88 1 89 7 96 4 100 Work with a partner. Take it in turns to choose any integer from 1 to 10. Keep a running total. Whoever chooses the number which makes the total exactly 100 is the winner. From: Problems with Patterns and Numbers.

‘First to 100’ Game Things to think about: Is there a winning strategy? What if the winning total is different, perhaps 30 or 36? What if you can only choose integers from 5 to 10? From: Problems with Patterns and Numbers.

Always, Sometimes, Never 1 On the next slide is a series of statements; decide which are always true, which are sometimes true and which are never true. Can you find an explanation to convince your teacher that you have each of the statements correctly assigned as ‘always true’, ‘sometimes true’ or ‘never true’

Always, Sometimes, Never 1 A quadrilateral which has rotational symmetry also has reflective symmetry A prime number is odd At least one of the angles in a triangle is acute. A pentagon has 4 right angles A square number is also a prime number The sum of three consecutive integers is a multiple of 3 In an isosceles triangle, the ‘odd’ side is the shortest side.

Answers (ASN 1) A prime number is odd Always Sometimes Never A prime number is odd At least one of the angles in a triangle is acute. A pentagon has 4 right angles A square number is also a prime number A quadrilateral which has rotational symmetry also has reflective symmetry The sum of three consecutive integers is a multiple of 3 In an isosceles triangle, the ‘odd’ side is the shortest side.

Fill the spaces 1 On the next slide is a Venn diagram. Find a linear function to go in each of the spaces.

Fill the spaces 1 Gradient is 3 Goes through (1,10) y intercept is -3

Fill the spaces 1 Gradient is 3 Challenges: Which regions can only have one function and which have several? Why is one region impossible to fill? Change one of the values in one of the labels so that every region is able to have a linear function in it. y intercept is -3 Goes through (1,10)

Always, Sometimes, Never 2 A shape with perpendicular lines of symmetry has rotational symmetry A square number has an even number of factors (including 1 and itself). For a multiple of 9, the sum of the digits of is also a multiple of 9. The product of two or more consecutive integers is odd. A prime number >2 is either one more or one less than a multiple of 4 The sum of three consecutive integers is a multiple of 6 For a set of three numbers, the mean value is higher than the median value.

Answers (ASN 2) Always Sometimes Never For a multiple of 9, the sum of the digits of is also a multiple of 9. The sum of three consecutive integers is a multiple of 6 A square number has an even number of factors (including 1 and itself). A prime number >2 is either one more or one less than a multiple of 4 The product of two or more consecutive integers is odd. For a set of three numbers, the mean value is higher than the median value. A shape with perpendicular lines of symmetry has rotational symmetry

Fill the spaces 2 On the next slide is a Venn diagram. Find the equation of a circle to go in each of the spaces.

Fill the Spaces 2 Touches the y-axis Goes through the origin Radius is 5

Fill the Spaces 2: Challenge Touches the y-axis Can you change one of the labels so that: There is just one space that is impossible to fill? There are two spaces that are impossible to fill? Goes through the origin Radius is 5

Teacher notes: Your Starter for 10 In this edition is a selection of short activities, inspired by the articles and resources within the M4 publication, which would work well as ‘starter’ activities to encourage discussion and thinking. One or two are reproduced from some of the original publications. The activities cover a range of content and different ones are suitable for a variety of age groups from KS3 through to post-16.

Teacher notes: The Bus Stop Queue This activity is taken from ‘The Language of Functions and Graphs’. It is the first activity in the booklet ‘Interpreting Points’. It’s often surprising how muddled pupils can become with this activity. 1 Dennis 2 Alice 3 Freda 4 Brenda 5 Errol 6 Cathy 7 Gavin One of the key things to draw out of this activity is justification or verification of pupils’ answers through referring to relative values on the chart. For example: From the chart, 1 and 4 are the same height and 2 and 5 are the same height. From the pictures, Brenda and Dennis are of equal height and Alice and Errol are of equal height.

Teacher notes: First to 100 Game This activity is taken from ‘Problems with Patterns and Numbers’. One of the objects of playing this game is for pupils to analyse the mathematical structure of situation. Making the target total lower will help them to explore this, but you might like them to grapple with the problem of target 100 initially so that they appreciate the power of ‘simplifying the problem’. However, altering the target number can have an interesting effect. Making the target number 99, for instance, ensures that Player 2 can always win simply by making the total a multiple of 11 each time. Solutions can be found in the original publication on page 79.

Teacher notes: Always, Sometimes, Never 1 These statements are accessible to the majority of pupils; decisions can be made initially by testing out some cases. The level of explanation and proof required of pupils will need to be determined by the teacher. However, it’s through answering the question ‘Why is this the case?’ that pupils are encouraged to think more deeply and develop reasoning and proof skills.

Teacher notes: Fill the spaces 1 Challenges: Which regions can only have one function and which have several? Intersection of two circles: only one possibility. Why is one region impossible to fill? Any two of these pieces of information define a line, the third piece of information does not fit with that line. Change one of the values in one of the labels so that every region is able to have a linear function in it. Actually impossible with these labels. If you have something in the centre, then you can’t have anything in the region created by two overlapping circles. Related to the second challenge. One possibility: There are several formats for each equation, this is a useful teaching point.

Teacher notes: Always, Sometimes, Never 2 These statements more challenging than those in ASN1. The level of explanation and proof required of pupils will need to be determined by the teacher. However, it’s through answering the question ‘Why is this the case?’ that pupils are encouraged to think more deeply and develop reasoning and proof skills.

Teacher notes: Fill the spaces 2 One possibility: The challenging aspect of this activity is the ‘reverse thinking’ required to ensure that a circle goes through the origin.