Same as… Numeracy focus: Problem solving focus: Increase our understanding of what happens in multiplication specifically in relation to factors and place value. Numeracy focus: Use long multiplication or the grid method to multiply two 2-digit numbers; Explore place value and partitioning in multiplication. Problem solving focus: Explore relationships in multiples, factors and products; Work out why certain patterns arise and relate these to place value and tables facts. You will need: mini whiteboards/exercise books © Hamilton Trust

Same as… 13 x 62 32 x 69 24 x 63 Multiply 12 by 42 using grid method. Write down your answer. Reverse the digits in both numbers. Write the new multiplication. 21 x 24 Use grid method to find the product. Repeat 1 to 5, but this time using 12 x 84. Repeat for all three card multiplications. 13 x 62 Problem Solving Skills Make observations as you work out the answers you find… 32 x 69 For help with grid method click here 24 x 63 © Hamilton Trust

Ideas in the mix Same as… SAME product So, all these reversed pairs give the SAME product Interesting… but does any pair of 2-digit numbers do the same? Have you got any ideas? WHY don’t we TRY… a different pair of 2-digit numbers? Must they be even numbers? What is special about the multiples of the tens and ones? © Hamilton Trust

Same as… I think I might be onto something… Look at the multiples of the tens and the ones in each pair. I see, so in 12 x 42 the tens multiply to give 4 tens altogether… … and, in 12 x 42 the ones multiply to give 4 ones altogether… Let’s see if the same thing is true for each of the other pairs of numbers we tried? Is the multiple of the tens the same as that of the ones? Can we find a NEW pair of numbers where the reversed multiplications are the same? © Hamilton Trust

Same as… You try other numbers where the tens give a product which is the same as the product of the ones? Click here for algebra Which means that we need to start with numbers which have four single-digit factors. Problem Solving Skills Try to use algebra to generalise the calculations… Make a prediction about the answers you will find… Clicking the icon will take you to an optional page (Slide 8) where this sequence and algebraic notation is explored in detail. 9 3 x 3 9 x 1 24 6 x 4 3 x 8 © Hamilton Trust

…exploring using these pairs: Same as … Try… …exploring using these pairs: 6 x 3 = 2 x 9 6 x 2 = 3 x 4 6 x 4 = 8 x 3 2 x 8 = 4 x 4 1 x 8 = 2 x 4 6 x 1 = 2 x 3 3 x 3 = 1 x 9 Do you understand how to find a pair of numbers which have the same product when you reverse their digits? Put letters not numbers in grid method. Try using algebra! © Hamilton Trust

That’s the end of this investigation. Same as… Good job! That’s the end of this investigation. © Hamilton Trust

Algebra of reverse digit multiplications Same as… Proof Algebra of reverse digit multiplications 1 of 2 Let’s call the first number 10a + b Let’s call the second number 10c + d So the multiplication is… 10a b Reverse the digits and multiply 10c 100ac 10bc 10b a d 10ad bd 10d 100bd 10ad c 10bc ac BUT we know that the product of the tens digits is the same as the product of the ones digit, SO ac = bd Continued on next slide… © Hamilton Trust Problem Solving Term 2 Week A

Algebra of reverse digit multiplications Same as… Proof Algebra of reverse digit multiplications 2 of 2 SO our two grid additions then become 100ac + 10bc + 10ad + bd and 100bd + 10bc + 10ad + ac Since we know that ac = bd we can substitute ac for bd Therefore the two grid additions are: 100ac + 10bc + 10ad + ac and 100ac + 10bc + 10ad + ac WOW! They are IDENTICAL. BUT this only works if ac = bd The product of the tens digit = the product of the ones digit Back to investigation © Hamilton Trust Problem Solving Term 2 Week A

© Hamilton Trust Problem Solving Term 2 Week A Same as… Using grid method First draw your grid! Split the numbers into 10s and 1s. Write one number across the top. Write one number down the side. Multiply the numbers in each row by the numbers in each column. Add the four totals to get the product. 43 x 68 = ☐ 40 3 60 2400 180 8 320 24 2 4 0 0 3 2 0 1 8 0 1 2 4 2 9 2 4 Now try 34 x 86 using grid method 30 4 80 6 Back to investigation © Hamilton Trust Problem Solving Term 2 Week A