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Anyone want math crosswords, take 2?


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Hey! We would love some more!

 I apologize, as it took us FOREVER to start getting around to the first ones, but my DS loved them!! 

Would something dealing with powers be possible? (maybe bases 1-10, exponents 0-3 or 0-5?) 

Also, if you know a good resource for code-breaking in different bases, let me know! I’m off to search Pinterest for binary riddles or something similar...

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On 1/23/2021 at 11:15 PM, R828 said:

Just seeing this! Yes, please! Could you please do a few with multiplication up to 12x12, multi digit addition and subtraction? 

Eeeek!! I totally flaked on this thread -- I'm so sorry. 

Here are a few: how do they look? Let me know if anything needs changing. 

Puzzle 1.pdf

Puzzle 2.pdf

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On 1/26/2021 at 9:56 PM, Shoes+Ships+SealingWax said:

Hey! We would love some more!

 I apologize, as it took us FOREVER to start getting around to the first ones, but my DS loved them!! 

Would something dealing with powers be possible? (maybe bases 1-10, exponents 0-3 or 0-5?) 

Unfortunately, I don't have exponents coded 😞 . I should code them. 

 

On 1/26/2021 at 9:56 PM, Shoes+Ships+SealingWax said:

Also, if you know a good resource for code-breaking in different bases, let me know! I’m off to search Pinterest for binary riddles or something similar...

Did you find anything? I never used anything like that! 

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13 hours ago, Not_a_Number said:

Did you find anything? I never used anything like that! 

Unfortunately, not much. I found some one-off things for Morse, Roman numerals, Mayan numbers... but that’s about it.  I think if I want any code breakers for binary, base-12, etc I’d have to make them myself & he is off to other things already so 🤷🏻‍♀️

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11 minutes ago, Shoes+Ships+SealingWax said:

Unfortunately, not much. I found some one-off things for Morse, Roman numerals, Mayan numbers... but that’s about it.  I think if I want any code breakers for binary, base-12, etc I’d have to make them myself & he is off to other things already so 🤷🏻‍♀️

Want any non-exponent crosswords? I should code up exponents, but I haven’t yet!!

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1 minute ago, Shoes+Ships+SealingWax said:

Sure; I’ll take some division crosswords. Remainders or no remainders are fine. Maybe I’ll make codes to break with the remainders 🤪

I don't have remainders coded, either! I, err, don't believe in remainders 😉 .

No fractions, right?  

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4 minutes ago, Not_a_Number said:

I don't have remainders coded, either! I, err, don't believe in remainders 😉 .

No fractions, right?  

Not yet. Maybe some addition / subtraction of fractions for later. Same- or mixed-denominators are fine so long as they’re pretty straightforward. 

Edited by Shoes+Ships+SealingWax
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7 minutes ago, Shoes+Ships+SealingWax said:

Sure; I’ll take some division crosswords. Remainders or no remainders are fine. Maybe I’ll make codes to break with the remainders 🤪

I've also been doing decoder puzzles with my classes, by the way 😄 . Here's a sample... the colors tell which kid which letter to decode, and they get the arithmetic operation that breaks the code earlier. 

Code_Feb_17.pdf

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18 hours ago, Not_a_Number said:

Eeeek!! I totally flaked on this thread -- I'm so sorry. 

Here are a few: how do they look? Let me know if anything needs changing. 

Puzzle 1.pdf 37.57 kB · 6 downloads

Puzzle 2.pdf 37.62 kB · 4 downloads

Those look great! If you get a chance we’d love some ones with just multiplication and division too. Thank you!! 

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3 hours ago, Not_a_Number said:

I don't have remainders coded, either! I, err, don't believe in remainders 😉 .

No fractions, right?  

Haha....care to explain about not believing in remainders? 😊

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1 minute ago, R828 said:

Haha....care to explain about not believing in remainders? 😊

Hahahaha, definitely, but it's a long explanation!

So, one of my main concerns with mathematical teaching is presenting consistent mental models that work for all levels of math. I think that's the set up that leads to good intuitions and confidence. 

In my opinion, kids' model of division is notoriously fragmented and disjointed. First we present a model where we "split into groups." Then we quickly jump to a model where we "count how many of a thing there are inside another thing." Then we jump to a model where we split things up... but wait, not everything can be split up! Then we jump to fractions, which are just division... except now we ARE allowed to split all the things, even if it doesn't come out even! 

This is all incredibly confusing for kids. Division isn't one single thing that makes sense. It's 5 things at once, and they don't have the chance to internalize any of the definitions properly. 

I never did remainders at all with DD8. She does know that some things don't split evenly, and if given a word problem where some things obviously don't split up (distributing people between teams or something), she won't just tell me that a team has 4.5 people on it each... but she also doesn't think of that as a pure division problem. She thinks of that as kind of adjacent to a division problem -- she uses division along the way, but the answer is not the outcome of a division. 

I probably can't explain this properly, but I know that a focus on consistent models has absolutely expedited our mathematical learning. 

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Just now, R828 said:

Not big at all, we are just starting division so only factors up to 12x12 for now! Thank you! 

So you don't want to divide numbers bigger than what by what? You don't want the outcomes to be over 12 or the thing you're dividing by to be over 12? 

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25 minutes ago, Not_a_Number said:

So you don't want to divide numbers bigger than what by what? You don't want the outcomes to be over 12 or the thing you're dividing by to be over 12? 

I mean, she knows multiplication facts upto 12x12 and we are working on understanding division as splitting into groups (so sort of as the inverse of multiplication - which we modelled as an array).  So I guess upto 144 divided by 12? And no remainders. Does that make sense? 

32 minutes ago, Not_a_Number said:

Hahahaha, definitely, but it's a long explanation!

So, one of my main concerns with mathematical teaching is presenting consistent mental models that work for all levels of math. I think that's the set up that leads to good intuitions and confidence. 

In my opinion, kids' model of division is notoriously fragmented and disjointed. First we present a model where we "split into groups." Then we quickly jump to a model where we "count how many of a thing there are inside another thing." Then we jump to a model where we split things up... but wait, not everything can be split up! Then we jump to fractions, which are just division... except now we ARE allowed to split all the things, even if it doesn't come out even! 

This is all incredibly confusing for kids. Division isn't one single thing that makes sense. It's 5 things at once, and they don't have the chance to internalize any of the definitions properly. 

I never did remainders at all with DD8. She does know that some things don't split evenly, and if given a word problem where some things obviously don't split up (distributing people between teams or something), she won't just tell me that a team has 4.5 people on it each... but she also doesn't think of that as a pure division problem. She thinks of that as kind of adjacent to a division problem -- she uses division along the way, but the answer is not the outcome of a division. 

I probably can't explain this properly, but I know that a focus on consistent models has absolutely expedited our mathematical learning. 

I understand, that makes total sense. Now that you mention it, I realize I need to be more intentional about having a consisitent model. We are modelling mutiplication and division in terms of arrays so maybe we could think of remainders as parts of arrays that are not perfect rectangles? Like the extra pieces after making a rectangle, are the remainders. Just thinking aloud here....

I just asked DD what she thinks multiplication is and she said its "taking lots of the same number, like 3 + 3 + 3"....and she said division is "separating a number into equal parts"....No mention of arrays here...lol. 

So I think her conceptual understanding is ok but could be better....

Thanks for making me think. 🙂  

 

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9 minutes ago, R828 said:

I just asked DD what she thinks multiplication is and she said its "taking lots of the same number, like 3 + 3 + 3"....and she said division is "separating a number into equal parts"....No mention of arrays here...lol. 

So I think her conceptual understanding is ok but could be better....

Those were actually the definitions we used with DD8. They are easy to work with, they correspond to real world experience, and they are very general. Most of the time you use multiplication, you aren't sorting something in an array, right? Like, if you are buying 6 dolls and each one costs 7 dollars, there are no arrays to be seen. However, if you think that 6*7 is just "six copies of 7 added together," then it's clear that to figure out the total price of the dolls, you need to multiply 6 by 7. The word problem fits the mental model. 

When I used arrays, I was very careful to constantly show a child why an array showed a multiplication. We only used arrays to explain why multiplication can be done in either order: if you arrange your multiplication in an array, it's easy to see that 6*7 is 7*6, for example.

However, most of the time we used the definition "it's a bunch of that number added together" and we worked with that definition, because it was important to me that our logic fit DD8's mental model. 

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14 minutes ago, R828 said:

....and she said division is "separating a number into equal parts"....No mention of arrays here...lol. 

I will also say that if you keep this model, without getting into remainders, it makes transitioning into fractions very painless. Because then 1/2 is just "one split into 2 equal parts" and if you already have practice splitting other things into equal parts, you can probably notice things about that quantity, like that it's less than 1, and that if you double it you get 1, and if you quadruple it you get 2, etc. 

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Well then I guess I'm doing something right?! I'll take any wins I can get lol.  😄

I think arrays fit my mental model better though....like in some weird way it's more symmetrical and beautiful. And when thinking of area, arrays make more sense than repeated addition. Plus in my mind I see repeated addition as adding stacks of blocks side by side so it does look like an array too. But that may be my weirdness! 

I do see how repeated addition may be easier to imagine and therefore better for DD. But do you really think having exposure to more than 1 mental model is bad though? Sure you're going to gravitate to one primary mental model, but in different scenarios different models fit best right? 

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20 minutes ago, Not_a_Number said:

I will also say that if you keep this model, without getting into remainders, it makes transitioning into fractions very painless. Because then 1/2 is just "one split into 2 equal parts" and if you already have practice splitting other things into equal parts, you can probably notice things about that quantity, like that it's less than 1, and that if you double it you get 1, and if you quadruple it you get 2, etc. 

We've talked about fractions quite a bit in exactly this way, even though we haven't formally done fractions yet. 🙂

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15 minutes ago, R828 said:

And when thinking of area, arrays make more sense than repeated addition.

Not really. You are adding rows of unit squares together. Or columns, if you prefer. 

 

15 minutes ago, R828 said:

But do you really think having exposure to more than 1 mental model is bad though? Sure you're going to gravitate to one primary mental model, but in different scenarios different models fit best right? 

In my opinion, what you want is a STRONG primary mental model, with an understanding that other mental models can be connected to it. But working on the initial strong model (and its logical outcomes) is very important, and I think it's something lots of people spend way too little time on. They introduce a model and think that kids are ready to move on from it, but actually what they need is to reinforce that model for it to be intuitive and easy to use. 

DD8 absolutely understands that an array shows a multiplication, but her primary multiplication model is "adding a certain number of copies of a number." That means that she'd explain why an array shows a multiplication by showing me the rows that are being added and noting that we're adding a certain number of copies of the row, and that the rows are the same. 

There were many advantages of this model. For example, the distributive property has always been very easy to her. If 7*8 means "add seven copies of 8," then it's pretty easy to see that this is the same as three 8s and another four 8s, right? And if you practice using this model to navigate, this kind of reasoning becomes easier and easier. The other thing about this model was that it was widely applicable. If she accidentally decided to add when she had more than one copy of a number, we could talk about how this was actually just the same as multiplying, and she already knew how to do that. 

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16 minutes ago, R828 said:

But do you really think having exposure to more than 1 mental model is bad though?

More on this... the important thing about being mindful about the primary model is that your explanations will make sense to your child. Like, you just checked what your child's model is, right? It's adding copies of the same number. That means that an array is one step removed from what is intuitive to her. Therefore, if you explain something with an array, she may not immediately grok it, because it's not what she thinks of when you say "and now we multiply 6 and 5." The corollary to that isn't that you can't use arrays to explain -- it's that when you do, you should remind her WHY an array is actually a multiplication and not something mysterious and random you just drew. 

So, I can tell you that when we used this model, the distributive and associative property were quite easy, but the commutative property took us a long time. She didn't visualize a multiplication as an array, so for me to explain to her why 3 copies of 5 were the same as 5 copies of 3, we'd have to draw the picture, REMIND her why that picture did actually show both 3 copies of 5 and 5 copies of 3, and then note that since it was the SAME picture, the numbers must be the same. And in the process of doing this, we both reinforced her mental model of "multiplication is just adding a certain number or copies of something" and we built on TOP of it. 

I visualize mathematics as all being built on primary models. I mindfully pick a primary model, then I spend a lot of time linking everything back to it. That way, a kid never has to wonder "but what does that symbol MEAN, anyway? Is there anything specific it refers to?", which absolutely can happen if you never establish a primary model. 

And I find that after we've spent enough time reinforcing the primary model, one can take it for granted, and one can be sure that when you say 3*5, you and your child are talking about the same thing. And at that point, you can build on that 🙂 . You can make it clear that it's also arrays, and areas, and distances traveled, and the opposite of division, and many, many things. And you're never on shaky ground 🙂 .  

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Yes, put like that I absolutely agree. Thank you for going into detail. I think my word for the year should be intentional or mindful. Being intentional in other areas of life has been on my mind, and I can see it's value in math instruction too....

Interestingly the commutative and assciative properties were pretty intuitive to DD. The distributive property has been interesting. Even before I explicitly taught it to her she would use it herself to figure out something like 17 x 5 (by adding 10x5 and 7x5). So she definitely understands it. But it's been a couple of months now and she still hasn't quite grasped that it's the same thing when she see it in the reverse form. So if she saw a (2x3) + (5x3) she would still figure them out separately and then add vs just doing 7x3. If I ask what would be an easier way or what property could we use, then she sees it but otherwise she doesn't seem to notice. It's not something I'm too concerned about, we'll just keep working on it. I just think it's interesting. 🙂

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8 hours ago, R828 said:

Yes, put like that I absolutely agree. Thank you for going into detail. I think my word for the year should be intentional or mindful. Being intentional in other areas of life has been on my mind, and I can see it's value in math instruction too....

Interestingly the commutative and assciative properties were pretty intuitive to DD. The distributive property has been interesting. Even before I explicitly taught it to her she would use it herself to figure out something like 17 x 5 (by adding 10x5 and 7x5). So she definitely understands it. But it's been a couple of months now and she still hasn't quite grasped that it's the same thing when she see it in the reverse form. So if she saw a (2x3) + (5x3) she would still figure them out separately and then add vs just doing 7x3. If I ask what would be an easier way or what property could we use, then she sees it but otherwise she doesn't seem to notice. It's not something I'm too concerned about, we'll just keep working on it. I just think it's interesting. 🙂

Yeah, I think kids have an easier time working forward than backwards! 

How would she justify the associative and commutative property, do you know? They are often things that kids have an easy time memorizing (symbolically, they are more intuitive than the distributive property) but can't necessarily explain, especially the associative property. It's actually not so simple to explain why (6*3)*5 = 6*(3*5), as it turns out! At least, DD8 had trouble. 

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On 2/28/2021 at 6:55 AM, Not_a_Number said:

Yeah, I think kids have an easier time working forward than backwards! 

How would she justify the associative and commutative property, do you know? They are often things that kids have an easy time memorizing (symbolically, they are more intuitive than the distributive property) but can't necessarily explain, especially the associative property. It's actually not so simple to explain why (6*3)*5 = 6*(3*5), as it turns out! At least, DD8 had trouble. 

I asked DD if she she could explain why the associative property works: First she gave me an example and showed that they are both the same. Then she said it’s just like the commutative property, it just works if the numbers are the same, the order doesn’t matter. 

What do you think she should have used to explain? Volume of a cuboid is how I would have. I’ll probably get out our cm cubes and demo later today. 

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Just now, R828 said:

I asked DD if she she could explain why the associative property works: First she gave me an example and showed that they are both the same. Then she said it’s just like the commutative property, it just works if the numbers are the same, the order doesn’t matter. 

Yeah, it sounds like she has it memorized but can't explain it. Believe it or not, the commutative property doesn't imply the associative property or vice versa. You can have one without the other. 

 

1 minute ago, R828 said:

What do you think she should have used to explain? Volume of a cuboid is how I would have. I’ll probably get out our cm cubes and demo later today. 

I'd ask her what she thinks 3*(2*5) means and (3*2)*5 means and listen to her answers 🙂 . Then I'll tell you what you can tell her. 

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14 hours ago, Not_a_Number said:

Yeah, it sounds like she has it memorized but can't explain it. Believe it or not, the commutative property doesn't imply the associative property or vice versa. You can have one without the other. 

 

I'd ask her what she thinks 3*(2*5) means and (3*2)*5 means and listen to her answers 🙂 . Then I'll tell you what you can tell her. 

Well she said 3*(2*5) is 3*10 and (3*2)*5 is 6*5 and they are both 30. I asked her if she could explain more or tell me why and she just looked at me like I’d lost my mind. 😂
Anyway I suggested she draw it for me and she drew 5 groups of 2 with plus signs between them and then wrote a big x3 next to it. So I said why don’t you draw them all, so then she drew 3 groups of 10 with plus signs between them. She did the same thing for (3*2)*5. 

So what do you think? 

(I also got lost on a rabbit trail reading about the commutative and associative properties and how you could have one without the other. Very interesting....and someone had a rock, paper, scissors example that I think DD will appreciate.😊)

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3 hours ago, R828 said:

Well she said 3*(2*5) is 3*10 and (3*2)*5 is 6*5 and they are both 30. I asked her if she could explain more or tell me why and she just looked at me like I’d lost my mind. 😂

Anyway I suggested she draw it for me and she drew 5 groups of 2 with plus signs between them and then wrote a big x3 next to it. So I said why don’t you draw them all, so then she drew 3 groups of 10 with plus signs between them. She did the same thing for (3*2)*5. 

So what do you think? 

So it looks like she's used to this fact but has never actually thought about it. In my experience, that's standard! 

I always prove this one by counting in 5s. We know that 3*(2*5) is 3 copies of 2*5, which means 3*2 = 6 copies of 5. Therefore, that's the same thing as (3*2)*5. I might challenge her to explain this for other examples. I would also sometimes use this fact when DD8 had trouble remembering multiplication, so it didn't feel alien to her. Like, if you're trying to figure out 6*7, you might very well remember that 3*7 is 21. But how many copies of 3*7 do we need? Well, we need three 7s, and we want six 7s, so really we're doubling the three 7s. 

I always like talking this stuff out, so the properties don't feel mysterious. Plus, it's a good skill for remembering multiplication and also for later algebraic reasoning. 

 

3 hours ago, R828 said:

(I also got lost on a rabbit trail reading about the commutative and associative properties and how you could have one without the other. Very interesting....and someone had a rock, paper, scissors example that I think DD will appreciate.😊)

For the record, one sees operations that are associative but not commutative all the time in math. It comes up with group theory and with matrices and with all sorts of stuff 🙂 . Commutative but not associative is much less common, although it's possible to come up with examples. 

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On 3/2/2021 at 7:29 AM, Not_a_Number said:

So it looks like she's used to this fact but has never actually thought about it. In my experience, that's standard! 

I always prove this one by counting in 5s. We know that 3*(2*5) is 3 copies of 2*5, which means 3*2 = 6 copies of 5. Therefore, that's the same thing as (3*2)*5. I might challenge her to explain this for other examples. I would also sometimes use this fact when DD8 had trouble remembering multiplication, so it didn't feel alien to her. Like, if you're trying to figure out 6*7, you might very well remember that 3*7 is 21. But how many copies of 3*7 do we need? Well, we need three 7s, and we want six 7s, so really we're doubling the three 7s. 

I always like talking this stuff out, so the properties don't feel mysterious. Plus, it's a good skill for remembering multiplication and also for later algebraic reasoning. 

Ok so I did point out the 3 copies of 2 copies of 5 thing while she drew everything out. And yes that is exactly how we approach remembering multiplication facts. We do really need to talk more about the how’s and why’s though....and just talk more period. I’m really inspired by your conversational style of teaching math. So thank you! 

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On 2/27/2021 at 9:33 AM, Shoes+Ships+SealingWax said:

Sure; I’ll take some division crosswords. Remainders or no remainders are fine. Maybe I’ll make codes to break with the remainders 🤪

Here are some division ones, although without remainders!! But if you want to have some specific numbers included in the puzzle to break codes with... I can do that! That's programmed in. 

Division puzzle 1.pdf

Division puzzle 2.pdf

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24 minutes ago, Not_a_Number said:

Here are some division ones, although without remainders!! But if you want to have some specific numbers included in the puzzle to break codes with... I can do that! That's programmed in. 

Division puzzle 1.pdf 37.85 kB · 2 downloads

Division puzzle 2.pdf 38.29 kB · 2 downloads

Thanks - I think he’ll really enjoy these! 

Can you do something that includes factorials?? He got a kick out of those! Or (fairly low) powers like squares & cubes? 

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7 hours ago, Shoes+Ships+SealingWax said:

Thanks - I think he’ll really enjoy these! 

Can you do something that includes factorials?? He got a kick out of those! Or (fairly low) powers like squares & cubes? 

Hahahaha, not yet, but I should!! Thinking about it, it’s easier than I thought. I’ll do it soon.

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2 minutes ago, Shoes+Ships+SealingWax said:

I’m going to single-handedly identify all the things your program cannot yet do 😝

Hey, it’s been a fun project!! And this one is easier than I’d thought... just a new option passed to the display function. Because I think an exponent or factorial would have to all be in one square, right? You can’t write it as an operation since it doesn’t fit.

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38 minutes ago, Shoes+Ships+SealingWax said:

I’m going to single-handedly identify all the things your program cannot yet do 😝

Actually, let me make this clear: would it be OK if one of the equations was something like 3^3 / 3^2 =  blank, or would you want to figure out the value of blank such that 

blank^2 = 9? 

Because I don't think I can do the letter without a symbol for exponentiation, and I don't think we have one. But it'd be easy to include an exponent or factorial in a single square. 

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17 minutes ago, Not_a_Number said:

Actually, let me make this clear: would it be OK if one of the equations was something like 3^3 / 3^2 =  blank, or would you want to figure out the value of blank such that 

blank^2 = 9? 

Because I don't think I can do the letter without a symbol for exponentiation, and I don't think we have one. But it'd be easy to include an exponent or factorial in a single square. 

The first scenario would be preferable. They can be mixed, too: 2^4 = blank = 4^2 or 4! = 6^2 - blank. 

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Just now, Shoes+Ships+SealingWax said:

The first scenario would be preferable. They can be mixed, too: 2^4 = blank = 4^2 or 4! = 6^2 - blank. 

Great! I can do those easily. And those would be good for DD8 as well.

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13 hours ago, Not_a_Number said:

Here are some division ones, although without remainders!! But if you want to have some specific numbers included in the puzzle to break codes with... I can do that! That's programmed in. 

DS found these before I could even get them off the printer & pleaded to do them in addition to his work in BA because they looked fun. I’d say that’s a win! 😅

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4 minutes ago, Shoes+Ships+SealingWax said:

DS found these before I could even get them off the printer & pleaded to do them in addition to his work in BA because they looked fun. I’d say that’s a win! 😅

Great!! I can show you what else I have. I have those honeycomb things from BA as well as “shapes puzzles” which are like variables, also magic squares.

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15 minutes ago, Not_a_Number said:

Great!! I can show you what else I have. I have those honeycomb things from BA as well as “shapes puzzles” which are like variables, also magic squares.

Sure! I’ll take whatever you want to toss this way. The kid loves math puzzles.

I should get him back into Sudoku. He liked it for a hot second, but then flitted off to other things. 

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5 minutes ago, Shoes+Ships+SealingWax said:

Sure! I’ll take whatever you want to toss this way. The kid loves math puzzles.

I should get him back into Sudoku. He liked it for a hot second, but then flitted off to other things. 

What sounds better, the magic squares or the hexagon honeycomb thing? (That can also be a "knight move" puzzle. I generalized them from the AoPS versions when I coded them.) I can also do "product" magic squares 😄 . 

Also, how big do you want the numbers? 

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6 minutes ago, Not_a_Number said:

What sounds better, the magic squares or the hexagon honeycomb thing? (That can also be a "knight move" puzzle. I generalized them from the AoPS versions when I coded them.) I can also do "product" magic squares 😄 . 

Also, how big do you want the numbers? 

Knights would be perfect, as we’re studying the Middle Ages! 🛡⚔️ 

Probably no more than double-digit for the parts. Wholes (sums, products, etc) can probably be triple-digit or quadruple-digit but he likes to complete these mentally. 

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Just now, Shoes+Ships+SealingWax said:

Knights would be perfect, as we’re studying the Middle Ages! 🛡⚔️ 

Hahahah, they are chess knights, though!! You know, the horses 😉 . Can he play chess? 

 

Just now, Shoes+Ships+SealingWax said:

Probably no more than double-digit for the parts. Wholes (sums, products, etc) can probably be triple-digit or quadruple-digit but he likes to complete these mentally. 

You mean if we do fractions? Should we have fractions? Sorry, I may be getting confused! 

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1 hour ago, Not_a_Number said:

Hahahah, they are chess knights, though!! You know, the horses 😉 . Can he play chess? 

He doesn’t play (tends to lose interest mid-game) but is familiar with them. 🙂

 

1 hour ago, Not_a_Number said:

You mean if we do fractions? Should we have fractions? Sorry, I may be getting confused! 

I only meant if you’re including multiplication, for example, let’s keep the multiplicand & multiplier to fewer than 3 digits.

2^3 = sqrt 64 = 3! + 2 🤩

258 / 6 = 5! - 77 🤔💭

27,492 x 8^7 = blank / 4,321. 🤯😭

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  • 3 weeks later...
1 minute ago, Shoes+Ships+SealingWax said:

Well, you certainly don’t owe me anything 😅 but I’ll gladly take some Knight puzzles if you’re so inclined. 

Sorry, wasn't trying to be obnoxious! Just asking what I promised you 🙂 .

Do you want any kind of special "puzzle number" as part of the puzzle? I can easily include one if you need it for codes or just fun! 

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