1. Thinking problems What are the pairs of two digit numbers that have the same products when their ten’s and one’s digits are exchanged? 24 6324 63 is same as 42 3642 36 Exclude symmetrical pairs, like 23  32

2. Thinking problems Multiplication of two numbers means finding the area of a rectangle. Any two digit number can be expressed as: 10  a + b, and 10  b + a where, a and b are the unknown numbers (1, 2,…, 9)

3. Thinking problems Multiplication of two numbers means finding the area of a rectangle. 10  a b 10  b a 100  a  c 10  b  c 100  b  d 10  a  d 10  c d 10  d c 10  a  dbdbd 10  b  cacac

4. Thinking problems 10  a b 10  c d 100  a  c 10  b  c 10  a  d bdbd 10  b a 100  b  d 10  d 10  a  d 10  b  c c acac

5. Thinking problems 10  a b 10  c d 100  a  c 10  b  c 10  a  d bdbd 10  b a 100  b  d 10  d 10  a  d 10  b  c c acac Areas of both rectangles must be the same. 100  a  c + 10  b  c + 10  a  d + b  d 100  b  d + 10  a  d + 10  b  c + a  c

6. Thinking problems 10  a b 10  c d 100  a  c 10  b  c 10  a  d bdbd 10  b a 100  b  d 10  d 10  a  d 10  b  c c acac 100  a  c + 10  b  c + 10  a  d + b  d 100  b  d + 10  a  d + 10  b  c + a  c

7. Thinking problems 10  a b 10  c d 100  a  c 10  b  c 10  a  d bdbd 10  b a 100  b  d 10  d 10  a  d 10  b  c c acac 100  a  c + 10  b  c + 10  a  d + b  d 100  b  d + 10  a  d + 10  b  c + a  c

8. Thinking problems 10  a b 10  c d 100  a  c 10  b  c 10  a  d bdbd 10  b a 100  b  d 10  d 10  a  d 10  b  c c acac a b  c d b a  d c When a  c = b  d, area is the same!

9. Thinking problems 10  a b 10  c d 100  a  c 10  b  c 10  a  d bdbd 10  b a 100  b  d 10  d 10  a  d 10  b  c c acac 1 4 8 21 4 8 2 = 4 1 2 84 1 2 8

10. Thinking problems The pair of numbers: 12  63 = 21  36 12  84 = 21  48 13  62 = 31  26 13  93 = 31  39 14  82 = 41  28 23  64 = 32  46 23  96 = 32  69 24  63 = 42  36 24  84 = 42  48 26  93 = 62  39 34  86 = 43  68 36  84 = 63  48 39  62 = 93  26

11. Thinking problems What are the pairs of two digit numbers that have the same products when their ten’s and one’s digits are exchanged? Solving the problem with Algebra Any two digit number can be expressed as: 10 a + b, where a, b  (1,…,9)

12. Thinking problems Solving the problem with Algebra 10 a + b, where a, b  (1,…,9), and 10 c + d, where c, d  (1,…,9). (10 a + b)( 10 c + d) = (10 b + a)( 10 d + a) 100 ac +10( a + c )+ bd = 100 bd +10( a + c )+ ac

13. Thinking problems Solving the problem with Algebra 100 ac +10( a + c )+ bd = 100 bd +10( a + c )+ ac ac = bd

14. Thinking problems Graphical approach in solving the problem Reinforces the concept of area and multiplication of two numbers. Makes this problem accessible to students as early as 4 th grade when they would have would have mastered the topics on on area and the multiplication of two digit numbers.