Today I gave Rocket Boy this problem from the big book of puzzles.
On a six-sided dice, the two opposite sides are marked in such a way that the sum of their numbers (or dots) adds up to 7. Given a picture of a dice with three sides visible, figure out what should appear on the sides hidden from view. No problem there.
And then we got to the final question – what is the sum of the numbers (dots) on all the sides of this dice?
Here’s what happened:
Rocket Boy (after adding all the numbers): 21
I: Yes, I believe you’re right. I overheard you adding all the numbers together. Is there a different way of getting to the answer?
RB: I don’t understand
I: You know how you’ve been solving a lot of multiplication problems lately (using Beast Academy workbooks)? Is there a way to apply your knowledge here?
RB: Multiplication… ok… how can I get 21 if multiplication… Oh, 3×7 is 21!
I: Yes, that’s true. 3×7=21. But looking at this dice, what exactly is 3 here and what is 7?
RB: Wait, can you say it again? I don’t understand.
Note to self: He’s getting quite good at recognizing these moments of not understanding and asking for clarification instead of just hiding them and pushing forward without really getting what’s being asked.
I: Sure, glad you asked. 3×7=21. You already established (by adding) that there is a total of 21 dots on this dice. So 21 is dots, right? So now we just need to understand what 3 refers to and what 7 refers to.
RB: Ok… well, 7 is what the dots on opposite sides add up to. That’s 7.
I: I agree. How about 3?
RB (confused and cautious): Well, it’s the number of sides.
I: Number of sides? Can you be more precise?
RB: Number of sides on the dice.
I: Ok, but isn’t there 6 sides on this dice?
RB: Yes… ok… it’s 3 sides that are visible when we look at the dice.
I (turning the dice): ok, yes, I can see 3 sides. And on them I see 1, 2 and 4 dots.
RB: See, that’s 7 dots total.
I (turning the dice): but now I see 3 different sides – 3, 5, and 6 dots. That’s not 7.
RB: I’m confused.
I: ok, maybe we should try to record what you already know. You tell me what you know about this dice and I’ll write it down.
Note: RB has severe graphomotor delays. Writing something down legibly takes too much effort and takes the focus away from the math of it. So sometimes I am his “recorder”.
RB (pointing to a picture in the book): I know that this side has 3 dots. And that side has 2 dots. And…
I: hold on a second, writing “this” and “that” is too long. Can we call “this” side – side A. And “that” side – side B. And the “other” side – side C?
RB: good idea, let’s do it.
I: What do you want to call the sides we can’t see?
RB: sides D, E and F
I: Ok, now, we know that each of these sides is opposite to one of the visible sides. Is there a way to code this information in their names?
RB: well, let’s name them “negative A”, “negative B”, “negative C”
We agree on notations.
RB: Side A is 4. So side -A is 3. Side B is 6, so side -B is 1. Side C is 5, so side -C is 2.
I (writing it all down): Great. What else?
RB: The total number of dots on all sides is 21.
I: How should I write it down?
I: What else do you know?
RB: That sum of opposite sides is 7. So A+”-A”=7, B+”-B”=7, C+”-C”=7 … Ah, ok, so it’s three times seven. I see! 3 pairs!!!