# Mathematical Thinking through investigation in multiplication For Primary Teacher Education programme in Hong Kong CHENG Chun Chor-Litwin The Hong Kong.

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<ul><li> Slide 1 </li> <li> Mathematical Thinking through investigation in multiplication For Primary Teacher Education programme in Hong Kong CHENG Chun Chor-Litwin The Hong Kong Institute of Education </li> <li> Slide 2 </li> <li> Introduction Multiplication is an important component in primary mathematics learning. However, most teachers taught according to the textbooks and there is little room for investigation in this topic. The set of materials introduced her wish to complement mathematical thinking for primary schools students. </li> <li> Slide 3 </li> <li> In Hong Kong, primary school teachers can go to Institutions to do an in-service release course, ranging from 5-week to a year, to refresh their teaching. This 5-week course may have different focus, such as small class teaching, children with special needs.. </li> <li> Slide 4 </li> <li> The following is the list of material that was used in the teachers course, so that teachers can try them in their classroom. The materials have been tried out in schools before they are introduced to primary teachers. In total, there are 5 topics of materials introduced in this programme. </li> <li> Slide 5 </li> <li> Topic A: Finishing the multiplication table Topic B: Filling in numbers represented by English Letters Topic C: Finding maximum / minimum product of multiplication Topic D: Using factor and common factor Topic E: Finding Equal product </li> <li> Slide 6 </li> <li> Topic A: Finishing the multiplication table Filling in table, so that multiplication established. 264 </li> <li> Slide 7 </li> <li> There are many answers find the answers and the number of answers : 6644 4 6 264264 2643388 8 3 264264 </li> <li> Slide 8 </li> <li> Question: Fill in integers so that the calculation is correct 7 </li> <li> Slide 9 </li> <li> 7969 3 3 7 237207 617181 7 7 7 427497567 There are many answers find the answers and the number of answers : </li> <li> Slide 10 </li> <li> Using the numbers 1 2 3 4 5. Fill in the boxes so that multiplication established. AB13 C 4 DE52 </li> <li> Slide 11 </li> <li> Using the numbers 1 2 3 4 5 6. Fill in the boxes so that multiplication established. 54 3 162 </li> <li> Slide 12 </li> <li> The number 1or 5 could not be placed on the unit digit (marked by a cross ) 54 3 162 </li> <li> Slide 13 </li> <li> Posing Question Using the numbers 2 3 4 5 6 7 5752 6 7 342364 </li> <li> Slide 14 </li> <li> Topic B: Filling in numbers represented by English Letters ABC208218 4 4 4 CDA832872 </li> <li> Slide 15 </li> <li> A = 4 8 2 6 0 D could be 3 or 7. Suppose D = 3, then C 3, C = 8. 2BC2D8208 4 4 4 C32832832 </li> <li> Slide 16 </li> <li> A = 4 8 2 6 0 D could be 3 or 7. Suppose D = 7, then C = 8, B = 1. 2BC2D8218 4 4 4 C72872872 </li> <li> Slide 17 </li> <li> Topic B: Fill in suitable integers ABCD2178 4 4 DCBA8172 </li> <li> Slide 18 </li> <li> A = 1 or A = 2, but A 1, and A = 2. D = 3 or D = 8, but D 3. C = 2 7. B = 1 2BCD2178 4 4 DCB28172 </li> <li> Slide 19 </li> <li> Topic B Examples used ABC41724 3 3 5ABC5172 </li> <li> Slide 20 </li> <li> ABCD1089 9 9 DCBA9801 </li> <li> Slide 21 </li> <li> Topic C: Finding maximum / minimum Using 1, 2, 3, 4 to fill in the boxes to make the greatest product 41 32 </li> <li> Slide 22 </li> <li> Topic C: Finding maximum / minimum Using 1 2 3 4 5 6 7 654321 7 </li> <li> Slide 23 </li> <li> 75321 64 </li> <li> Slide 24 </li> <li> Topic C: Finding maximum / minimum Using 1 2 3 4 5 6 7. 753 64 </li> <li> Slide 25 </li> <li> Exercise: identify the greatest product 64216531 753 742 </li> <li> Slide 26 </li> <li> 74217531 653 642 </li> <li> Slide 27 </li> <li> Topic C: Finding maximum / minimum Using 1 2 3 4. 21 3 4 </li> <li> Slide 28 </li> <li> Topic C: Finding maximum / minimum Using 1 2 3 4 5. 21 3 4 5 </li> <li> Slide 29 </li> <li> Topic C: Finding maximum / minimum Using 1 2 3 4 5 6. 32 41 5 6 </li> <li> Slide 30 </li> <li> Another type of finding greatest product From 2, 3, 4, 5, 7, add 3 number and multiply this sum with the fourth numbers, so that the product is greatest The following is a list of possibilities. 2 (3 + 5 + 7) = 30 3 (2 + 5 + 7) = 42 5 (2 + 3 + 7) = 60 7 (2 + 3 + 5) = 70 </li> <li> Slide 31 </li> <li> Topic D: Using factor / common factor Step (1), doing multiplication 815 1812 815120 1812216 144180 </li> <li> Slide 32 </li> <li> Topic D: Using factor / common factor Step (2), reverse the thinking 35 12 2120 7535 3412 73735 4 </li> <li> Slide 33 </li> <li> Topic D: Using factor / common factor Step (2), reverse the thinking 15 108 224 1448315 31515 629108 847224 1448315 </li> <li> Slide 34 </li> <li> There are 4 numbers a b c d (2 to 9). a (b + c + d) = 69 b (a + c + d)= 105 c (a + b + d) = 133 d (a + b + c) = 165 Find the values of a b c d. 69 = 3 23 105 = 5 21 = 7 15 133 = 7 19 165 = 11 15 a &lt; b &lt; c &lt; d. answer are 3 5 7 11. </li> <li> Slide 35 </li> <li> Topic E: Finding Equal product Using the numbers from 1 to 9, fill in the circle so that the product on the three sides are equal </li> <li> Slide 36 </li> <li> Topic E: Finding Equal product Using the numbers 2 3 4 6 8, fill in the circle so that the product on the three sides are equal. </li> <li> Slide 37 </li> <li> Topic E: Finding Equal product Using the numbers 2 3 4 6 8 4 2 6 3 8 X A C Y B </li> <li> Slide 38 </li> <li> Discussion (A Y B) (B C) (A X C) = T T T A B C X Y A B C.= T T T (2 3 4 6 8) A B C = T T T (2^7 3^2) A B C = T T T possible value of T = (2^4 3^1 )= 48 B C = 48 X A C Y B </li> <li> Slide 39 </li> <li> Topic E: Finding Equal product Using numbers 1 2 3 4 6 8 </li> <li> Slide 40 </li> <li> Topic E: Finding Equal product Using numbers 1 2 3 4 6 8 2 4 61 3 8 1 8 32 6 4 </li> <li> Slide 41 </li> <li> 3 2 6 4 1 9 8 </li> </ul>