Top Drawer Teachers: Facts within facts

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Top Drawer Teachers
The Australian Association of Mathematics Teachers (AAMT) Inc.
http://topdrawer.aamt.edu.au

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Top Drawer Teachers: Facts within facts

  1. 1. Facts within factsIf I know 2 × 4 = 8 what else do I know?
  2. 2. 2 × 4 = 8.Double 2 × 4 = 8gives 4 × 4 = 16.Double 4 × 4 = 16gives 8 × 4 = 32.If I know 8 × 4 = 32,I know 4 × 8 =32.2 fours are 84 fours are 168 fours are 32Facts within factsIf I know 2 × 4 = 8 what else do I know?
  3. 3. What other facts can you see in 8 × 4 ?Facts within facts
  4. 4. What can I see in 8 × 4 ?5 × 4 and 3 × 46 × 4 and 2 × 47 × 4 and 1 × 45 foursare 203 foursare 126 foursare 242 foursare 8Facts within facts
  5. 5. What can I see in 8 × 4 ?5 × 4 and 3 × 46 × 4 and 2 × 47 × 4 and 1 × 45 foursare 203 foursare 126 foursare 242 foursare 8Facts within factsThis can help workout 10 × 4 (which isdouble 5 × 4)
  6. 6. What can I see in 8 × 4 ?If I double 8 × 4 = 32I get 8 × 8 = 64What can this help meto work out?5 foursare 203 foursare 126 foursare 242 foursare 8Facts within facts
  7. 7. What else can I see?Within 8 × 4 = 32 I can see:5 × 4 and 3 × 4Facts within facts
  8. 8. What else can I see?Within 8 × 4 = 32 I can see:5 × 4 and 3 × 44 × 5 = 20 20 ÷ 5 = 44 × 3 = 12 12 ÷ 3 = 4Facts within facts
  9. 9. What else can I see?Within 8 × 4 = 32 I can see:6 × 4 and 2 × 4Facts within facts
  10. 10. What else can I see?Within 8 × 4 = 32 I can see:6 × 4 and 2 × 4Facts within facts4 × 6 = 24 24 ÷ 6 = 44 × 2 = 8 8 ÷ 2 = 4
  11. 11. What else can I see?Within 8 × 4 = 32 I can see:7 × 4 and 1 × 4Facts within facts
  12. 12. What else can I see?Within 8 × 4 = 32 I can see:7 × 4 and 1 × 4Facts within facts4 × 7 = 28 28 ÷ 7 = 44 × 1 = 4 4 ÷ 1 = 4
  13. 13. What else can I see?If I double 8 and halve 4 what do I get?Facts within facts
  14. 14. What else can I see?If I double 8 and halve 4 what do I get?Facts within facts16 × 2 = 32
  15. 15. What else can I see?If I double 8 and halve 4 what do I get?Facts within facts16 × 2 = 32Knowing this I also know:2 × 16 = 3232 ÷ 16 = 232 ÷ 2 = 16
  16. 16. × 1 2 3 4 5 6 7 8 9 101 42 83 124 4 8 12 16 20 24 28 32 405 206 247 288 32 64910 40Facts within facts
  17. 17. × 1 2 3 4 5 6 7 8 9 1012345678910Facts within facts
  18. 18. Going further: ideas and tipsIn pairs, students choose a known fact and use doubles andpartitioning to record some related facts.For example, given 3 × 4, students could double one factor(3 in this case) to get 6 × 4. They could double the otherfactor to get 3 × 8.For some facts, they could halve one and double the other.Facts within facts
  19. 19. Going further: ideas and tipsTeaching tipsDrawing the arrays on the grid provides a visual image of howthe dimensions change when one factor is doubled.It can also show ways to partition the array.Recording the product on the multiplication grid provides thelink between the visual and the symbolic representation.Facts within facts
  20. 20. Going further: ideas and tipsBuild up a class list of multiplicationfacts on the multiplication grid generatedby doubling and partitioning.Facts within facts
  21. 21. Going further: ideas and tipsStudents may also explore the use ofhalving and doubling to work out trickyfacts such as 7 lots of 8 and relateddivision facts.For example, if I know 7 fours is 28, then Ican double the fours to get 7 eights.Double the product of 28 gives 56, whichis the product of 7 and 8.Facts within facts

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