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ALGORITHMS

Explore traditional and invented algorithms for addition, subtraction, multiplication, and division with practical strategies and hands-on activities. Enhance math skills using counters, ten frames, and manipulatives. Strengthen problem-solving abilities in a fun and engaging manner.

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ALGORITHMS

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  1. ALGORITHMS The procedures by which we do: 1. Addition 2. Subtraction 3. Multiplication Division Can be traditional or invented algorithms

  2. Invented vs. Traditional Algorithms • Invented are based on numbers rather than digit orientation (where the digits are) 2. Invented tend to be left to right processes 3. Invented are flexible

  3. ADDITIONPART-PART-WHOLE . Generally speaking, students begin with: ‘one more than’ and ‘two more than…’ ‘double addition facts’ (2 + 2 and 3 + 3, etc.) C. Based five and ten facts: 0 + 5 = 1 + 9 = 1 + 4 = 2 + 8 = 2 + 3 = 3 + 7 = etc.

  4. Ten Frames • Spill and fill using 10 frames: Spill ten two-coloured discs (or heads and tails with coins (CONCRETE) • Record the number of each colour (or heads and tails) • Draw results (PICTORIAL) • Translate to other manipulative (links, connect cubes…) • Record results using number sentences (SYMBOLIC)

  5. TEN FRAMES: 6 + 4 = 10

  6. TEN FRAMES: 3 + 7 = 10

  7. TEN FRAMES: 8 + 2 = 10

  8. Ten Frame War Use Aces and the 2-9 cards. Play addition war where each player must use a ‘base 10 fact’ to declare the sum. i.e., if I flip over a 4 and 8, I would say: 8 + 2 = 10 plus 2 more = 12 Natural frame 10 facts win automatically!

  9. MANIPULATIVES Counters, Connecting Cubes, Number Lines, Popsicle Sticks, Bean Sticks , Rice… EXAMPLES: A. 3 + 4 = (Students could think ‘3 + 3 + 1’) B. 9 + 6 = Students could think “If I 10 frame, I need one more to make the 9 into a 10 Therefore, the question could really be:” (9 +1) + (6 – 1) = (10) + (5) =

  10. Now, what would you think? Use three different strategies: • 7 + 9 = • Base 10 facts really 6 + 10 = ii) Double facts (7 + 1) + (9 – 1) = 8 + 8 = iii) Double facts and two more than (7 + 7 + 2 = )

  11. 7 + 9

  12. (7 – 1) + (9 + 1)

  13. Subtraction WHEN THE WHOLE AND ONE OF THE PARTS ARE KNOWN • MISSING PART: Ex.) A farmer had 9 sheep in his field but 4 of them run through a hole in the fence. How many remain in the farmer’s field? 9 – 4 = • COMPARISON MODELS: Ex.) William has 8 letters to mail. Keesha has 5 letters. How many more letters does William have?

  14. Subtraction • I have 234 stuffed animals and I gave 156 of them to my little sister. How many do I have left? • I have 145 stuffed animals and my little sister has 77 stuffed animals. How many more do I have?

  15. 10 Frame If we can help students make the connection between addition and subtraction, they can use the same principles. Family Facts: If, 6 + 4 = 10 then 10 – 6 = 4 and 10 – 4 = 6

  16. Multiplication 1. Repeated Addition: (equal groups) Ex) There were three concerts in the auditorium this week. Each was a sellout of 368 people. How many were there in total? 368 + 368 + 368 = 368 x 3 = 2. Comparison: Ex) Alexa had $8.00 in her bank account. Brenda had 3 times as much as Alexa. How much money did Brenda have?

  17. Multiplication (cont’d) 3. Array (Or Area): Ex.) A forest has 8 rows of trees with 7 trees in each row. How many trees are in the forest?

  18. Multiplication (cont’d) 4. Combinations: Given two sets of items (that usually go together) determine the number of possibilities (combinations). Ex.) Barbie has 4 pair of pants and 3 blouses. How many combinations can she wear? Ex.) A buffet serves coleslaw, Greek, Caesar, garden and bean salads. It also has lasagna, pizza, roast beef and ham for a main course. A guest at this buffet must choose one salad and one main course. How many choices does one have? USE- felt board Catalogue pictures Real clothes Grocery flyers

  19. Strategies for Multiplying x 2 = double x 4 = double, double x 5 = 10 ÷ 2 or half x 10

  20. Multiplying by 4 Tim Horton strategy! Double! Double! 23 x 4 = (remember 4 = 2 x 2) Double 23 = 46 Double 46 = 92

  21. Multiply by 5 Divide by 2 and x 10 18 x 5 = 18 / 2 = 9 (multiply by 10 or add a zero) 18 x 5 = 90

  22. Multiply by 8 Based upon 8 = 2 x 2 x 2 double, double, double 9 x 8 = double 9 = 18 double 10 = 36 double 36 = 72

  23. Based upon the strategies… The only facts that you cannot develop strategies for are: • 3 x 6 • 3 x 7 • 6 x 7

  24. One digit multipliers are a relatively easy concept for students to develop. Once the multiplier becomes two (or more) digits, things become much more difficult. • For two digit multipliers start with base ten multiplication: 20 40 30 200 x10x 20x 60x 50

  25. DIVISION 1. PARTITION DIVISION: Ex.) Ray has 24 candies and wants to share them among his 6 friends. How many will each receive? 2. MEASUREMENT DIVISION: One would use one bit of information to determine how many more times. Ex.) Wanda has 12 apples. Eric has 3 apples. How many times as many apples does Wanda have?

  26. STRATEGIES • 114 – 89 = • 56 x 8 =

  27. Strategies Traditional: Invented: 23 R4 • 142 6 142 - 60 10 82 - 60 10 22 - 183 4

  28. Invented: 142 /6 = 2/6= 0 R2 40/6 = 6 R4 100/6 = 16 R4 R = 2 + 4 + 4= 10 /6 = 1 R4 6 + 16 + 1 = 23 R4

  29. Which is the correct answer? 32 / 5 = 6 32 / 5 = 7 32 / 5 = 6.4 or 6 2/5 32 / 5 = 6 R2 It all depends upon the question asked?

  30. Flashcards http://www.aplusmath.com

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