Making modular arithmetic interesting for school kids
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This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:
$$colorredmathbf2cdot 9 = 1colorredmathbf8$$
$$colorredmathbf8cdot 9 = 7colorredmathbf2$$
$$colorbluemathbf3cdot 9 = 2colorbluemathbf7$$
$$colorbluemathbf7cdot 9 = 6colorbluemathbf3$$
$$colorgreenmathbf4cdot 9 = 3colorgreenmathbf6$$
$$colorgreenmathbf6cdot 9 = 5colorgreenmathbf4$$
which may come as kind of a miracle when first discovering it.
In mathematical terms
$$boxedacdot (10-1) equiv b mod 10 Leftrightarrow bcdot (10-1) equiv a mod 10 \
acdot (10-1) equiv b mod 10 Leftrightarrow a + b = 10 equiv 0 mod 10$$
This holds not only for $10$ but for every $p in mathbbN$, i.e. in every "number system":
$$boxedacdot (p-1) equiv b mod p Leftrightarrow bcdot (p-1) equiv a mod p \
acdot (p-1) equiv b mod p Leftrightarrow a + b = p equiv 0 mod p$$
and is responsible for the fact that the graphical multiplication tables of $mathbbZ/pmathbbZ$ always looks the same for $p-1$:
I wonder if there are attempts (in educational research and literature) to make use of the simple observability of the pattern above to explain to (clever) school kids that the observed regularity is not by pure coincidence, why it is so, and what it does "mean".
elementary-number-theory reference-request modular-arithmetic education
|
show 5 more comments
up vote
11
down vote
favorite
This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:
$$colorredmathbf2cdot 9 = 1colorredmathbf8$$
$$colorredmathbf8cdot 9 = 7colorredmathbf2$$
$$colorbluemathbf3cdot 9 = 2colorbluemathbf7$$
$$colorbluemathbf7cdot 9 = 6colorbluemathbf3$$
$$colorgreenmathbf4cdot 9 = 3colorgreenmathbf6$$
$$colorgreenmathbf6cdot 9 = 5colorgreenmathbf4$$
which may come as kind of a miracle when first discovering it.
In mathematical terms
$$boxedacdot (10-1) equiv b mod 10 Leftrightarrow bcdot (10-1) equiv a mod 10 \
acdot (10-1) equiv b mod 10 Leftrightarrow a + b = 10 equiv 0 mod 10$$
This holds not only for $10$ but for every $p in mathbbN$, i.e. in every "number system":
$$boxedacdot (p-1) equiv b mod p Leftrightarrow bcdot (p-1) equiv a mod p \
acdot (p-1) equiv b mod p Leftrightarrow a + b = p equiv 0 mod p$$
and is responsible for the fact that the graphical multiplication tables of $mathbbZ/pmathbbZ$ always looks the same for $p-1$:
I wonder if there are attempts (in educational research and literature) to make use of the simple observability of the pattern above to explain to (clever) school kids that the observed regularity is not by pure coincidence, why it is so, and what it does "mean".
elementary-number-theory reference-request modular-arithmetic education
1
There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area
– Yuriy S
15 hours ago
What do you mean by "math clubs"?
– Hans Stricker
15 hours ago
Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English
– Yuriy S
15 hours ago
I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries?
– Hans Stricker
15 hours ago
4
Maybe this question is better suited for the Mathematics Educators SE site?
– YiFan
15 hours ago
|
show 5 more comments
up vote
11
down vote
favorite
up vote
11
down vote
favorite
This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:
$$colorredmathbf2cdot 9 = 1colorredmathbf8$$
$$colorredmathbf8cdot 9 = 7colorredmathbf2$$
$$colorbluemathbf3cdot 9 = 2colorbluemathbf7$$
$$colorbluemathbf7cdot 9 = 6colorbluemathbf3$$
$$colorgreenmathbf4cdot 9 = 3colorgreenmathbf6$$
$$colorgreenmathbf6cdot 9 = 5colorgreenmathbf4$$
which may come as kind of a miracle when first discovering it.
In mathematical terms
$$boxedacdot (10-1) equiv b mod 10 Leftrightarrow bcdot (10-1) equiv a mod 10 \
acdot (10-1) equiv b mod 10 Leftrightarrow a + b = 10 equiv 0 mod 10$$
This holds not only for $10$ but for every $p in mathbbN$, i.e. in every "number system":
$$boxedacdot (p-1) equiv b mod p Leftrightarrow bcdot (p-1) equiv a mod p \
acdot (p-1) equiv b mod p Leftrightarrow a + b = p equiv 0 mod p$$
and is responsible for the fact that the graphical multiplication tables of $mathbbZ/pmathbbZ$ always looks the same for $p-1$:
I wonder if there are attempts (in educational research and literature) to make use of the simple observability of the pattern above to explain to (clever) school kids that the observed regularity is not by pure coincidence, why it is so, and what it does "mean".
elementary-number-theory reference-request modular-arithmetic education
This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:
$$colorredmathbf2cdot 9 = 1colorredmathbf8$$
$$colorredmathbf8cdot 9 = 7colorredmathbf2$$
$$colorbluemathbf3cdot 9 = 2colorbluemathbf7$$
$$colorbluemathbf7cdot 9 = 6colorbluemathbf3$$
$$colorgreenmathbf4cdot 9 = 3colorgreenmathbf6$$
$$colorgreenmathbf6cdot 9 = 5colorgreenmathbf4$$
which may come as kind of a miracle when first discovering it.
In mathematical terms
$$boxedacdot (10-1) equiv b mod 10 Leftrightarrow bcdot (10-1) equiv a mod 10 \
acdot (10-1) equiv b mod 10 Leftrightarrow a + b = 10 equiv 0 mod 10$$
This holds not only for $10$ but for every $p in mathbbN$, i.e. in every "number system":
$$boxedacdot (p-1) equiv b mod p Leftrightarrow bcdot (p-1) equiv a mod p \
acdot (p-1) equiv b mod p Leftrightarrow a + b = p equiv 0 mod p$$
and is responsible for the fact that the graphical multiplication tables of $mathbbZ/pmathbbZ$ always looks the same for $p-1$:
I wonder if there are attempts (in educational research and literature) to make use of the simple observability of the pattern above to explain to (clever) school kids that the observed regularity is not by pure coincidence, why it is so, and what it does "mean".
elementary-number-theory reference-request modular-arithmetic education
elementary-number-theory reference-request modular-arithmetic education
edited 13 hours ago
asked 15 hours ago
Hans Stricker
5,50433884
5,50433884
1
There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area
– Yuriy S
15 hours ago
What do you mean by "math clubs"?
– Hans Stricker
15 hours ago
Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English
– Yuriy S
15 hours ago
I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries?
– Hans Stricker
15 hours ago
4
Maybe this question is better suited for the Mathematics Educators SE site?
– YiFan
15 hours ago
|
show 5 more comments
1
There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area
– Yuriy S
15 hours ago
What do you mean by "math clubs"?
– Hans Stricker
15 hours ago
Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English
– Yuriy S
15 hours ago
I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries?
– Hans Stricker
15 hours ago
4
Maybe this question is better suited for the Mathematics Educators SE site?
– YiFan
15 hours ago
1
1
There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area
– Yuriy S
15 hours ago
There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area
– Yuriy S
15 hours ago
What do you mean by "math clubs"?
– Hans Stricker
15 hours ago
What do you mean by "math clubs"?
– Hans Stricker
15 hours ago
Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English
– Yuriy S
15 hours ago
Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English
– Yuriy S
15 hours ago
I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries?
– Hans Stricker
15 hours ago
I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries?
– Hans Stricker
15 hours ago
4
4
Maybe this question is better suited for the Mathematics Educators SE site?
– YiFan
15 hours ago
Maybe this question is better suited for the Mathematics Educators SE site?
– YiFan
15 hours ago
|
show 5 more comments
2 Answers
2
active
oldest
votes
up vote
4
down vote
After having the class talk about the quote
And yet patterns exist, and we slowly discover them. Seasons,
migrations, moons: the template is there. Consciously or
unconsciously, most people accept certain components of cycle theory.
We seek and see patterns in things. It is the way our minds work,
presumably for the purpose of survival.
--- Nick Paumgarten
the teacher should then gently explain to their students that to survive in his class, they MUST memorize the 1-digit multiplication tables; with $bar n = 0, 1, 2, 3,4,5,6,7,8$,
$tag 1 (m,n) mapsto m times n quad text for m,n in bar n$
She can then discuss an interesting pattern:
If one of the digits in $(m,n)$, say $n$, is a nine, and $m$ is non-zero then
$tag 2 m times n = string(m - 1) text string(k), text where k text is chosen so that m - 1 + k = 9$
Notice how the concept of string concatenation from computer languages is being introduced!
Example 1: $7 times 9$: Since $6 + 3 = 9$, ANSWER: $63$.
Example 2: $9 times 9$: Since $8 + 1 = 9$, ANSWER: $81$.
Example 3: $1 times 9$: Since $0 + 9 = 9$, ANSWER: $09$ (but tell them to drop the leading $0$).
The teacher can also explain why (2) works. Whey you multiply a digit $n$ by $9$ it has to be under that nice round number $n times 10$, and the pattern describes by how much.
Of course the students who don't care about understanding patterns can keep working the flash cards.
Interestingly, this cycle repeats. Students who can grasp patterns really don't have to memorize a bunch of stuff to do mathematics. The majority are always trying memorize all the formulas, when, if in fact, they UNDERSTAND just a couple of things, they are much better off.
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
1
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
add a comment |
up vote
1
down vote
Not sure about this pattern. But the computation (addition and multiplication) using the clock modulo 12 or modulo 24 is something that kids are capable of to understand. I'd start with say a clock modulo 4 to explain addition (say $2+3=1$) and multiplication (say $2cdot 3 = 2$).
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
After having the class talk about the quote
And yet patterns exist, and we slowly discover them. Seasons,
migrations, moons: the template is there. Consciously or
unconsciously, most people accept certain components of cycle theory.
We seek and see patterns in things. It is the way our minds work,
presumably for the purpose of survival.
--- Nick Paumgarten
the teacher should then gently explain to their students that to survive in his class, they MUST memorize the 1-digit multiplication tables; with $bar n = 0, 1, 2, 3,4,5,6,7,8$,
$tag 1 (m,n) mapsto m times n quad text for m,n in bar n$
She can then discuss an interesting pattern:
If one of the digits in $(m,n)$, say $n$, is a nine, and $m$ is non-zero then
$tag 2 m times n = string(m - 1) text string(k), text where k text is chosen so that m - 1 + k = 9$
Notice how the concept of string concatenation from computer languages is being introduced!
Example 1: $7 times 9$: Since $6 + 3 = 9$, ANSWER: $63$.
Example 2: $9 times 9$: Since $8 + 1 = 9$, ANSWER: $81$.
Example 3: $1 times 9$: Since $0 + 9 = 9$, ANSWER: $09$ (but tell them to drop the leading $0$).
The teacher can also explain why (2) works. Whey you multiply a digit $n$ by $9$ it has to be under that nice round number $n times 10$, and the pattern describes by how much.
Of course the students who don't care about understanding patterns can keep working the flash cards.
Interestingly, this cycle repeats. Students who can grasp patterns really don't have to memorize a bunch of stuff to do mathematics. The majority are always trying memorize all the formulas, when, if in fact, they UNDERSTAND just a couple of things, they are much better off.
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
1
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
add a comment |
up vote
4
down vote
After having the class talk about the quote
And yet patterns exist, and we slowly discover them. Seasons,
migrations, moons: the template is there. Consciously or
unconsciously, most people accept certain components of cycle theory.
We seek and see patterns in things. It is the way our minds work,
presumably for the purpose of survival.
--- Nick Paumgarten
the teacher should then gently explain to their students that to survive in his class, they MUST memorize the 1-digit multiplication tables; with $bar n = 0, 1, 2, 3,4,5,6,7,8$,
$tag 1 (m,n) mapsto m times n quad text for m,n in bar n$
She can then discuss an interesting pattern:
If one of the digits in $(m,n)$, say $n$, is a nine, and $m$ is non-zero then
$tag 2 m times n = string(m - 1) text string(k), text where k text is chosen so that m - 1 + k = 9$
Notice how the concept of string concatenation from computer languages is being introduced!
Example 1: $7 times 9$: Since $6 + 3 = 9$, ANSWER: $63$.
Example 2: $9 times 9$: Since $8 + 1 = 9$, ANSWER: $81$.
Example 3: $1 times 9$: Since $0 + 9 = 9$, ANSWER: $09$ (but tell them to drop the leading $0$).
The teacher can also explain why (2) works. Whey you multiply a digit $n$ by $9$ it has to be under that nice round number $n times 10$, and the pattern describes by how much.
Of course the students who don't care about understanding patterns can keep working the flash cards.
Interestingly, this cycle repeats. Students who can grasp patterns really don't have to memorize a bunch of stuff to do mathematics. The majority are always trying memorize all the formulas, when, if in fact, they UNDERSTAND just a couple of things, they are much better off.
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
1
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
add a comment |
up vote
4
down vote
up vote
4
down vote
After having the class talk about the quote
And yet patterns exist, and we slowly discover them. Seasons,
migrations, moons: the template is there. Consciously or
unconsciously, most people accept certain components of cycle theory.
We seek and see patterns in things. It is the way our minds work,
presumably for the purpose of survival.
--- Nick Paumgarten
the teacher should then gently explain to their students that to survive in his class, they MUST memorize the 1-digit multiplication tables; with $bar n = 0, 1, 2, 3,4,5,6,7,8$,
$tag 1 (m,n) mapsto m times n quad text for m,n in bar n$
She can then discuss an interesting pattern:
If one of the digits in $(m,n)$, say $n$, is a nine, and $m$ is non-zero then
$tag 2 m times n = string(m - 1) text string(k), text where k text is chosen so that m - 1 + k = 9$
Notice how the concept of string concatenation from computer languages is being introduced!
Example 1: $7 times 9$: Since $6 + 3 = 9$, ANSWER: $63$.
Example 2: $9 times 9$: Since $8 + 1 = 9$, ANSWER: $81$.
Example 3: $1 times 9$: Since $0 + 9 = 9$, ANSWER: $09$ (but tell them to drop the leading $0$).
The teacher can also explain why (2) works. Whey you multiply a digit $n$ by $9$ it has to be under that nice round number $n times 10$, and the pattern describes by how much.
Of course the students who don't care about understanding patterns can keep working the flash cards.
Interestingly, this cycle repeats. Students who can grasp patterns really don't have to memorize a bunch of stuff to do mathematics. The majority are always trying memorize all the formulas, when, if in fact, they UNDERSTAND just a couple of things, they are much better off.
After having the class talk about the quote
And yet patterns exist, and we slowly discover them. Seasons,
migrations, moons: the template is there. Consciously or
unconsciously, most people accept certain components of cycle theory.
We seek and see patterns in things. It is the way our minds work,
presumably for the purpose of survival.
--- Nick Paumgarten
the teacher should then gently explain to their students that to survive in his class, they MUST memorize the 1-digit multiplication tables; with $bar n = 0, 1, 2, 3,4,5,6,7,8$,
$tag 1 (m,n) mapsto m times n quad text for m,n in bar n$
She can then discuss an interesting pattern:
If one of the digits in $(m,n)$, say $n$, is a nine, and $m$ is non-zero then
$tag 2 m times n = string(m - 1) text string(k), text where k text is chosen so that m - 1 + k = 9$
Notice how the concept of string concatenation from computer languages is being introduced!
Example 1: $7 times 9$: Since $6 + 3 = 9$, ANSWER: $63$.
Example 2: $9 times 9$: Since $8 + 1 = 9$, ANSWER: $81$.
Example 3: $1 times 9$: Since $0 + 9 = 9$, ANSWER: $09$ (but tell them to drop the leading $0$).
The teacher can also explain why (2) works. Whey you multiply a digit $n$ by $9$ it has to be under that nice round number $n times 10$, and the pattern describes by how much.
Of course the students who don't care about understanding patterns can keep working the flash cards.
Interestingly, this cycle repeats. Students who can grasp patterns really don't have to memorize a bunch of stuff to do mathematics. The majority are always trying memorize all the formulas, when, if in fact, they UNDERSTAND just a couple of things, they are much better off.
edited 12 hours ago
answered 12 hours ago
CopyPasteIt
3,2981525
3,2981525
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
1
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
add a comment |
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
1
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
(2) is quite hard stuff - to read, to understand, and to apply. Why not just memorizing the 1-digit multiplication table for $n=9$, too?
– Hans Stricker
12 hours ago
1
1
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
Actually first the the students memorize the table, but then they can learn to understand the pattern that makes it easier to recall.
– CopyPasteIt
12 hours ago
add a comment |
up vote
1
down vote
Not sure about this pattern. But the computation (addition and multiplication) using the clock modulo 12 or modulo 24 is something that kids are capable of to understand. I'd start with say a clock modulo 4 to explain addition (say $2+3=1$) and multiplication (say $2cdot 3 = 2$).
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
add a comment |
up vote
1
down vote
Not sure about this pattern. But the computation (addition and multiplication) using the clock modulo 12 or modulo 24 is something that kids are capable of to understand. I'd start with say a clock modulo 4 to explain addition (say $2+3=1$) and multiplication (say $2cdot 3 = 2$).
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
add a comment |
up vote
1
down vote
up vote
1
down vote
Not sure about this pattern. But the computation (addition and multiplication) using the clock modulo 12 or modulo 24 is something that kids are capable of to understand. I'd start with say a clock modulo 4 to explain addition (say $2+3=1$) and multiplication (say $2cdot 3 = 2$).
Not sure about this pattern. But the computation (addition and multiplication) using the clock modulo 12 or modulo 24 is something that kids are capable of to understand. I'd start with say a clock modulo 4 to explain addition (say $2+3=1$) and multiplication (say $2cdot 3 = 2$).
answered 14 hours ago
Wuestenfux
2,2161410
2,2161410
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
add a comment |
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Right, that's the standard way to do it - and it surely is a good one. This is how you would introduce modular arithmetic from scratch.
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Alternatively one could ask: Where do you find modularity in the decimal system? (Of course when you add two decimal numbers digit-wise. But you would not mention "modularity" as a teacher - just teach the technique and use (mental) addition tables with entries like $7+8 = 15$ to be used like "write $5$, carry $1$".)
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
Conceptually it's two tables: $7+8operatornamemod10 = 5$ and $lfloor(7+8)/10rfloor=1$ the students do learn in one.
– Hans Stricker
14 hours ago
add a comment |
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1
There's plenty of advanced math clubs for middle to high school students all over the world. This is sure to be a popular topic there. Not that I'm familiar with the area
– Yuriy S
15 hours ago
What do you mean by "math clubs"?
– Hans Stricker
15 hours ago
Organized meetings of students in their free times with a tutor or tutors. I'm not sure how is it called in English
– Yuriy S
15 hours ago
I'm not sure if there are plenty such clubs e.g. in Germany. Where do you live, possibly Russia? What about other countries?
– Hans Stricker
15 hours ago
4
Maybe this question is better suited for the Mathematics Educators SE site?
– YiFan
15 hours ago