Stars and Crosses Numeracy focus: Problem solving focus: Students apply an awareness of the number system through 100-square position & algebra. Numeracy focus: Calculate the total of 5 or more two-digit numbers. Problem solving focus: Recognise and describe patterns: orally and in writing; Generalise relationships between numbers; Use algebra to express a generalisation. You will need: mini whiteboards/ exercise books © Hamilton Trust

Stars and Crosses 1. Ring a number on the 1-100 grid that is not in a row or column on the edge of the square. 2. Ring the number below, the number above, the number to the left and the number to the right to form a cross. 14 23 24 25 + 34 _2_ 120 3. Find the total of the five numbers and make a record of this along with the central number of the cross. © Hamilton Trust

Stars and Crosses e.g. 4. Repeat elsewhere on the grid… Problem Solving Skills Can you see a relationship between the central number and the sum of the five numbers in the cross? Encourage the students to work through plenty of specific examples – it will help with their generalisation. © Hamilton Trust

So, what’s going on with the numbers to produce this relationship? Stars and Crosses Ideas in the mix So, what’s going on with the numbers to produce this relationship? Can you write down the relationship between the central number and the totals using a clear, concise sentence? Can you write an algebraic expression for finding the total of the five numbers in any cross laid out like this? Encourage students to discuss ideas aloud. Clicking the ‘f(x)’ icon will take you to an optional page (Slide 11) where the sequences and algebraic notation are explored more explicitly. Click to explore the algebra of this sequence… © Hamilton Trust

Stars and Crosses 5. Now investigate numbers in a bigger cross: maybe nine, thirteen or even more numbers! You may want to use a calculator to speed up the additions if your cross is really big… Problem Solving Skills Can you predict the total for each size of cross? Problem Solving Skills Can you write a formula for finding the total of ANY cross that will fit on the grid? Ask students to give reasons for their predictions. © Hamilton Trust

Stars and Crosses …this was not a 1-100 grid, but a 1-81 grid, arranged in nine rows of 9? Clicking the ‘f(x)’ icon will take you to an optional page (Slide 12) where the sequences and algebraic notation are explored more explicitly. © Hamilton Trust

Stars and Crosses …the cross is arranged on the diagonals? Clicking the ‘f(x)’ icon will take you to an optional page (Slide 13) where the sequences and algebraic notation are explored more explicitly. © Hamilton Trust

Stars and Crosses …you make up your own shape…? © Hamilton Trust Clicking the ‘f(x)’ icon will take you to an optional page (Slide 10) where the sequences and algebraic notation are explored more explicitly. © Hamilton Trust

Show that… Stars and Crosses = 6n - 66 …the total of any 6 numbers in this arrangement (where the bottom left number is n) is always: = 6n - 66 Clicking the ‘f(x)’ icon will take you to an optional page (Slide 14) where the sequences and algebraic notation are explored more explicitly. © Hamilton Trust

That’s the end of this investigation. Stars and Crosses Good job! That’s the end of this investigation. © Hamilton Trust

Now, what is the total of these 5 amounts? Stars and Crosses Back to investigation Algebraic notation If we describe the central number as n, how can we write each number in the cross in terms of n? Now, what is the total of these 5 amounts? total = n + n+10 + n-10 + n+1 + n-1 total = 5n + 10 – 10 + 1 - 1 total = 5n © Hamilton Trust

Stars and Crosses Algebraic notation Back to investigation Algebraic notation If we describe the central number as n, how can we write each number in the cross in terms of n? n-18 n-9 n-2 n-1 n n+1 n+2 n+9 n+18 total = n + n+9 + n+18 + n+1 + n+2 + n-9 + n-18 + n-1 + n-2 total = 9n + 9 + 18 + 1 + 2 - 9 - 18 - 1 - 2 total = 9n + 30 - 30 total = 9n © Hamilton Trust

Stars and Crosses Algebraic notation Back to investigation Algebraic notation If we describe the central number as n, how can we write each number in the cross in terms of n? total = n + n+9 + n+11 + n+18 + n+22 + n-9 + n-11 + n-18 + n-22 total = 9n + 60 - 60 total = 9n © Hamilton Trust

Stars and Crosses total = n + n-1 + n-10 + n-12 + n-21 + n-22 Back to investigation I wonder why this total is not an exact multiple of n, like the other patterns explored…? total = n + n-1 + n-10 + n-12 + n-21 + n-22 total = 6n - 66 © Hamilton Trust