I am unsure if you welcome suggestions, but I had a couple of ideas which I thought I'd share.
- Implementation of a levels based system, say from Level A to Level Z. Each level will introduce only a few new facts. Around 50% of questions will be new facts, while the remaining 50% of questions will be from previous facts. Each fact can have a "basic mastery level", and the higher the mastery level, the less the chance of it appearing. Thus, students will be examined on facts which they don't know until they master it. This is based off well known and much researched programs like Rocket Maths (commercial - https://www.rocketmath.com/wp-content/uploads/2019/04/problems-set-L-partial-page.png) and Spaceship Maths (free - https://www.dadsworksheets.com/worksheets/multiplication-spaceship-math.html)
I've drawn up sample levels, as below:
Multiplication:
Level 01: 1 x anything up to 12
Level 02: 0 x anything up to 12
Level 03: 2x2 2x3
Level 04: 2x4 2x5
Level 05: 2x6 2x7
Level 06: 2x8 2x9
Level 07: 3x3 3x4
Level 08: 3x5 3x9
Level 09: 3x6 4x9
Level 10: 3x7 5x9
Level 11: 3x8 6x9
Level 12: 7x9 8x9
Level 13: 4x4 4x5
Level 14: 4x6 4x7
Level 15: 5x5 4x8
Level 16: 5x6 5x7
Level 17: 6x6 5x8
Level 18: 6x7 7x7
Level 19: 6x8 7x8
Level 20: 8x8 9x9
Level 21: 10x10 11x10 12x10
Level 22: 11 x anything up to 9
Level 23: 11x11 12x2 12x3
Level 24: 12x4 12x5 12x6
Level 25: 12x7 12x8 12x9
Level 26: 12x10 12x11 12x12
Addition levels:
Level 1: 0 + (all numbers), 1 + 2 , 1 + 3
Level 2: 1 + 4 , 1 + 5
Level 3: 1 + 6 , 1 + 7
Level 4: 1 + 8 , 1 + 9
Level 5: 2 + 3 , 2 + 4
Level 6: 2 + 5 , 2 + 6
Level 7: 2 + 7 , 2 + 8
Level 8: 2 + 9 , 3 + 3
Level 9: 3 + 9 , 3 + 4
Level 10: 4 + 9 , 3 + 5
Level 11: 5 + 9 , 3 + 6
Level 12: 6 + 9 , 3 + 7
Level 13: 7 + 9 , 3 + 8
Level 14: 8 + 9 , 9 + 9
Level 15: 4 + 4 , 4 + 5
Level 16: 4 + 6 , 4 + 7
Level 17: 4 + 8 , 5 + 5
Level 18: 5 + 6 , 5 + 7
Level 19: 5 + 8 , 6 + 6
Level 20: 6 + 7 , 6 + 8
Level 21: 7 + 8 , 8 + 8
Level 22: 10 + (all numbers)
Level 23: 11 + (all numbers)
Level 24: All addition facts, tested equally. To pass level, perhaps 40 questions must be done in 60 seconds?
Level 25: All addition facts, tested equally, but no facts involving adding 0, 1 or 2.
Level 26: All addition facts, tested equally. To pass level, perhaps 50 questions in 60 seconds?
- My second suggestion is an implementation of a section between multiplication and division sections. In this section, students will essentially be doing multiplication, but they'll be answering questions like:
5 x ? = 35
In essence, this helps them associated 5, 7 and 35 together, making division far simpler.
This is implemented in https://mathwithoutborders.com/learn-em-forwards-and-backwards/ and I've read comments from many teachers that it works quite well.
Please let me know what you think.
edit: edited addition levels for clarity. Note, the levels are merely suggestions, and can certainly be modified. The core idea is to have levels which allow for incremental but consistent improvement over time. We want students to feel sense of growth and achievement, and be able to hopefully TRACK this improvement and visibly see it. Success is incredibly important as a motivator, and certainly begets more success.
I am unsure if you welcome suggestions, but I had a couple of ideas which I thought I'd share.
I've drawn up sample levels, as below:
Multiplication:
Level 01: 1 x anything up to 12
Level 02: 0 x anything up to 12
Level 03: 2x2 2x3
Level 04: 2x4 2x5
Level 05: 2x6 2x7
Level 06: 2x8 2x9
Level 07: 3x3 3x4
Level 08: 3x5 3x9
Level 09: 3x6 4x9
Level 10: 3x7 5x9
Level 11: 3x8 6x9
Level 12: 7x9 8x9
Level 13: 4x4 4x5
Level 14: 4x6 4x7
Level 15: 5x5 4x8
Level 16: 5x6 5x7
Level 17: 6x6 5x8
Level 18: 6x7 7x7
Level 19: 6x8 7x8
Level 20: 8x8 9x9
Level 21: 10x10 11x10 12x10
Level 22: 11 x anything up to 9
Level 23: 11x11 12x2 12x3
Level 24: 12x4 12x5 12x6
Level 25: 12x7 12x8 12x9
Level 26: 12x10 12x11 12x12
Addition levels:
Level 1: 0 + (all numbers), 1 + 2 , 1 + 3
Level 2: 1 + 4 , 1 + 5
Level 3: 1 + 6 , 1 + 7
Level 4: 1 + 8 , 1 + 9
Level 5: 2 + 3 , 2 + 4
Level 6: 2 + 5 , 2 + 6
Level 7: 2 + 7 , 2 + 8
Level 8: 2 + 9 , 3 + 3
Level 9: 3 + 9 , 3 + 4
Level 10: 4 + 9 , 3 + 5
Level 11: 5 + 9 , 3 + 6
Level 12: 6 + 9 , 3 + 7
Level 13: 7 + 9 , 3 + 8
Level 14: 8 + 9 , 9 + 9
Level 15: 4 + 4 , 4 + 5
Level 16: 4 + 6 , 4 + 7
Level 17: 4 + 8 , 5 + 5
Level 18: 5 + 6 , 5 + 7
Level 19: 5 + 8 , 6 + 6
Level 20: 6 + 7 , 6 + 8
Level 21: 7 + 8 , 8 + 8
Level 22: 10 + (all numbers)
Level 23: 11 + (all numbers)
Level 24: All addition facts, tested equally. To pass level, perhaps 40 questions must be done in 60 seconds?
Level 25: All addition facts, tested equally, but no facts involving adding 0, 1 or 2.
Level 26: All addition facts, tested equally. To pass level, perhaps 50 questions in 60 seconds?
5 x ? = 35
In essence, this helps them associated 5, 7 and 35 together, making division far simpler.
This is implemented in https://mathwithoutborders.com/learn-em-forwards-and-backwards/ and I've read comments from many teachers that it works quite well.
Please let me know what you think.
edit: edited addition levels for clarity. Note, the levels are merely suggestions, and can certainly be modified. The core idea is to have levels which allow for incremental but consistent improvement over time. We want students to feel sense of growth and achievement, and be able to hopefully TRACK this improvement and visibly see it. Success is incredibly important as a motivator, and certainly begets more success.