This week's tasks involve dot patterns that can be counted using multiplication but which move away from the notion of multiplication as repeated addition.
Task 13A: Reading the diagram from left to right, the green dots are the result of each of the 3 horizontal lines cutting each of the 5 slanting lines once on their way up and again once on their way down. This makes a total of 3×5 + 3×5 intersections, or 3×5×2 = 30.
Some children might simply count the dots. If so, encourage them to confirm their result using a calculator. Counting is less likely to happen with the red dots, as they are so numerous, though some children might still partially count, leading to an expression like 30×4. Here the total can be written succinctly as 6×5×4 which equals 120.
Note that children might well write an expression like 6×5×4 with the numbers in a different order, such as 4×5×6. This could be for two reasons. First an expression like 6×5 might be read by some children as '6 lots of 5' and by others as, say, '6, 5 times'. Second, children might structure the dot pattern differently: for example, some might see the red dots as the result of 6 horizontal lines cutting 5 slanting lines and doing this 4 times, given 6×5×4; others might argue that each horizontal line produces 5 intersections 4 times, with this happening 6 times in all. giving, say, 5×4×6.
Task 13B: There are 40 green dots altogether, which can be derived from an expression like 5×2×4. Some children might argue that there are 3 dots on each edge of the pentagons, leading to an erroneous expression like 5×3×4. (There might also be children who correctly argue that each pentagon has '5×3 green dots – 5 green dots', to avoid the corner dots being counted twice.)
Similarly, there are 5×3×6 = 90 red dots (not 5×4×6 = 120).
Task 13C: There are 8 green dots and 3 times as many red dots. Children might well just count the green dots but encourage them to express both totals multiplicatively: there are 4×2 = 8 green dots and 4×2×3 = 24 red dots.
Task 13D: There are 3×3×3 green dots and 3×3×3×3 red dots, or 3³ and 3⁴. Here we get a sense of what exponential growth looks like; 1 branch produces 3 branches, then 9, 27, 81....
Task 13E: Here we gently nudge children towards the idea of a general rule. The rule is fairly simple, but not purely multiplicative:
the number of red dots is equal to 2 times the number of green dots, plus 2.
For 10 green dots there are 22 red dots; some children might conclude that for 40 green dots there will be 4 times as many red dots, which is 88 rather than 82. [Such thinking should not be dismissed as merely wrong. It is quite sophisticated in that it suggests that there is a multiplicative, or ratio, relation between the number of green dots and red dots - which is true for most of them!]
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