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6 votes

How to motivate $x^n−y^n ≡ (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$, to 13 year olds?

Start off by observing that: $99=9 \times 11$ $999=9 \times 111$ $9999=9 \times 1111$ $99999=9 \times 11111$ The left hand side is a power of ten minus one. The second number on the right hand side is ...
Nullius in Verba's user avatar
5 votes

How to motivate $x^n−y^n ≡ (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$, to 13 year olds?

Since you're asking about teaching the general difference-of-$n$th-powers formula, I'll assume the students have already learned the difference-of-squares and difference-of-cubes formulas: $$\begin{...
Justin Skycak's user avatar
4 votes
Accepted

How to motivate $x^n−y^n ≡ (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$, to 13 year olds?

Here a slightly different proposal. Ask the students to find $A_1$, $A_2$, ... , $A_n$ such that the formula $$ x^n - y^n = (x-y)(A_1+A_2+ \dots + A_n) $$ holds. You can get that $A_1 = x^{n-1}$ such ...
pygri's user avatar
  • 156
3 votes

How to explain why we can’t factor $x^n + y^n$ for all natural numbers n, to 13 year olds?

The question is framed very oddly, so I'm not really sure what you're looking for. If the student understands that p(x) being factorable means that p(x)=0 has roots when any linear factor equals 0, ...
Sue VanHattum's user avatar
  • 20.8k
3 votes
Accepted

How to explain why we can’t factor $x^n + y^n$ for all natural numbers n, to 13 year olds?

We can use the factor theorem, unless you think it’s also too “Delphic” for your students. The factor theorem tells us that a polynomial $p(x)$ has a factor $(x-r)$ if and only if $p(r)=0$. ...
Justin Hancock's user avatar
3 votes
Accepted

Does the "Middle School Mathematics domains" refer to (I) through (V) topics?

Yes. It says all questions assess content from the above Middle School Mathematics domains, and the only thing directly above is the table. And the only text fields in the table are the content ...
Justin Skycak's user avatar
2 votes

What are your most recommended resources to teach fractions?

Update for Edit: In the updated question, the OP asks for "pizza or pie pictures." Personally (as based on research) I would advocate against circular models for introducing fraction ...
Benjamin Dickman's user avatar
2 votes
Accepted

International Baccalaureate - where to find the detail of the math programs?

As of 2024, Mathematics: Applications and Interpretation does cover Markov chains, but at the higher level only, not at the standard level. It is listed in the syllabus under subtopic AHL 4.19. ...
Justin Hancock's user avatar
2 votes

How to motivate $x^n−y^n ≡ (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$, to 13 year olds?

Would long division help? I had something like this in high school $$ \require{enclose} x - y \enclose{longdiv} {x^2 \qquad- y^2} $$ We then figured out the answer x + y. You can increase the powers ...
Daniel's user avatar
  • 229
1 vote

Is there a virtue to learning how to compute by hand?

Let me tell you the story of my father and the number 5.8: My mother had a shop in Belgium, very near to the French border. This happened in the time before the Euro currency was introduced. Normally ...
Dominique's user avatar
  • 2,119

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