# Questions tagged [secondary-education]

For questions about teaching mathematics in secondary education (in most countries approx. ages 10-18).

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### Speed math appropriate for middle-school students

There are many "rules" for speed arithmetic. List of some reference links showing speed methods or rules: https://ofpad.com/multiplication-tricks-for-mental-math/ https://ofpad.com/mental-...
1k views

### Is there a standard convention for interpreting ambiguous absolute value expressions?

Consider the expression $$|x + 2|x + 3|x + 4|.$$ One way to interpret this is that there are two products being added together: $$|x+2|x \hspace{1cm} + \hspace{1cm} 3|x+4|$$ But you could also ...
3k views

### Importance of complex numbers knowledge in real roots

Many students question the importance of complex numbers in real life. We can find many important applications of imaginary numbers in Engineering field and physics. This question is not related to ...
327 views

### Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

Back in 2018, I wrote a post about asymptotes of rational functions in which I addressed not only horizontal and slant/oblique asymptotes, but also the general case of "polynomial asymptotes.&...
1 vote
497 views

### Basic skill requirement suspension

Oregon appears to have suspended the "basic skills" requirement for graduation; see this. What will be the effect of this on the mathematical proficiency of the graduating class? Follow-up ...
380 views

### How can one lone picture prove the Triangle Inequality, $|x−y|≤|x|+|y|$, $|x|−|y|≤|x−y|$, and the Reverse Triangle Inequality?

I always showcase separate pictures of Triangle Inequality, and Reverse, to 16-years-old students in 1st class. I reshow pictures in 2nd class. I preachify Please remember these 4 inequalities. ...
1 vote
196 views

### Identifying Trigonometrical proofs

How can we identify trigonometrical proofs from geometrical proofs, do we have purely trigonometrical proof of Pythagoras theorem as claimed by two high school students ? https://www....
17k views

### To 17 year olds, how can I explain that two numbers with arbitrarily small difference are equal?

$|a – b| < ε, \forall ε > 0 \iff a = b$ resurfaces on standardized tests to 17 year old (y.o.) students, who can memorize and regurgitate the proof to earn full marks. But the glut of duplicates ...
15k views

### Why not think of derivatives as fractions?

Back in high school—back in the 1900s, as my sons say—when our calculus teacher was introducing the chain rule... $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$ ...he made a special point of ...
957 views

### Geometrical verifications for Algebraic formulae

What is the importance of using approaches related to Geometric Algebra in teaching,is it only useful when introducing Algebra to the students or can it be helpful to improve creative skills in ...
6k views

### What benefit is there to obfuscate the geometry with algebra?

Consider: In a right triangle: sin(2x + 4) = cos (46) What is the value of x? The question above is from standardized tests for a geometry course. If my goal is to have students understand ...
1 vote
67 views

### Seeking References on Deterministic and Stochastic Phenomena Suitable for High School Students

Can anyone recommend good and didactic references that delve into the dualism between deterministic and stochastic phenomena? Ideally, I'm seeking materials that provide a conceptual explanation along ...
2k views

### Explaining Sigma-Notation

I attempted to introduce the summation notation $\Sigma$ to my students. The notation was unfamiliar to the students beforehand. I worked through many examples with them, but for most of them, working ...
148 views

### Remote Teaching by Video Conferencing

I am in my early 70's and licensed to teach 8-12 math in Texas. I have an advanced degree in the same area. I used to teach in high school decades ago but have since quit because the student's ...
35 views

### Apps to make mathematics much interesting by sharing creative ideas with others

I think proofs without words is much important topic when we want to improve students interest in subject using their skills other than in mathematics. Recently I could able to find that kind of proof ...
1 vote
100 views

### highschool's mathematics journal which citable in Google Scholar

I'm a high school Mathematics teacher and I want to issue some research articles for highschool students to improve their math problem resolve skills. Is there any valuable Math journal for high ...
1 vote
263 views

### Responding to students' questions that aren't directly relevant to their exams

What would you suggest as the best way to deal with students' questions that seem irrelevant to their upcoming exams? When I was studying for my university-entrance exam, I came across a couple of ...
237 views

### Limitations of applying the factor theorem

Here are three situations in which students might try to apply the factor theorem. Proving that $x + 1$ is a factor of the polynomial $x^3 + x + 2$ can be done using the factor theorem by showing ...
124 views

### Pythagoras and Trigonometry sequencing

In teaching the high school curriculum Pythagoras is usually bundled with Trigonometry. They might be justified by way of proof of some sort. They are used to solve 2d and 3d geometry problems for ...
366 views

### What is Algebra 1 and 2 as it is in US highschool education?

I am a pre-university student who wants to help students with Algebra 1 and 2 in high school. I am curious to how the curriculum was built and what the goal of teaching both algebra 1 and 2 might be. ...
3k views

### How can we best motivate the study of polynomials to high-school students?

We all know how important and ubiquitous polynomials are in mathematics. However, when faced with a (not so much in love with the subject) 14-year-old asking us why they should care about these things,...
1 vote
175 views

### Whole numbers as sets vs abstracted properties of sets

I recently landed on a book written for elementary school teachers which introduced the concept of whole numbers in the following manner: We have a set $\{\alpha, \beta, \gamma\}$. There are other ...
929 views

### Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?

One common way to introduce Euler's number $\mathrm e$ is $$\mathrm e = \lim_{n\to \infty} \left(1+\frac{1}{n}\right)^n,$$ where the right-hand expression has an "interest rate interpretation&... 2k views

### Can this be a better way of defining subsets?

I remember my high school days where subsets were defined in the following manner: Given two sets A and B, if every element of B is an element of A, then B is called a subset of A. A common ...
305 views

### Recommendations for secondary student interested in maths

As a student attending a grammar school in the UK ,I have been fortunate to have access to various opportunities to showcase my mathematical abilities. These include participating in maths challenges ...
110 views

### Partitioning objects in combinatorics

When you come to explain dividing given n number of objects into k number of groups, is it good to describe the cases involved using an example to cover as many cases as possible in order to give ...
115 views

### Infinite descent method

We have plenty of examples in mathematical induction for advanced level mathematics students. Can we introduce infinite descent method as extremely opposite approach to mathematical induction and is ...
4k views

### Why do we teach linear algebra in precalculus classes?

When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't ...
1k views

### Are Error-Analysis Lessons Effective?

I recently came across a thought-provoking video where Simon Sinek explains that the human brain struggles to process negative statements. In the video, Sinek states that skiers should not spend their ...
272 views

### Differentiation in integer solutions

What would you suggest as examples to demonstrate as applications of differentiation in finding integer solutions of an equation for advanced level students? Here you have one example which I have ...
1 vote
99 views

### simpson paradox in classroom: reports?

he Simpson's Paradox is a statistical phenomenon in which a trend or relationship observed within a dataset disappears or reverses when the dataset is divided into smaller groups. It occurs when a ...
88 views

### What books were used to teach the old Scholarship level exams in the UK?

The scholarship level looks like it could have some interesting questions: https://en.wikipedia.org/wiki/Scholarship_level Any ideas on what books or resources were used to teach this level?
1 vote
156 views

### Questions to test highest level of competency

In mathematics we ask so many types of questions to check the student's knowledge of the subject. More oftenly we ask to define terms, state a formula or application of theorems. What would you ...
159 views

### Multiple proofs for the same problem

One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics ...
679 views

### Mathematical induction without simplifying equations or inequalities

We discuss lot of questions related to mathematical expressions consisting equations or inequalities in mathematical induction. What are the examples where we can apply mathematical induction as the ...
168 views

### Geometrical approaches in algebra

Usually we describe proofs in algebra by algebraic means, I think it may be useful to introduce geometrical approaches to those proofs to improve creativity skills of students, what are the examples ...
3k views

### When writing log, do you indicate the base, even when 10?

I’ve been working with many students on logarithms and have noted that log has a base of 10 unless specified. Further, I commented that putting a 10 as a subscript to log is redundant, or at least not ...
136 views

### Online platforms for teachers to discuss matters related to mathematics

As we all agree mathematics educators community is doing a great service as an international platform regarding teachers issues related to mathematics education. Not all but only personally motivated ...
1k views

### Difference between the Cambridge IGCSE 0580 and 0607 mathematics courses

I am a high school mathematics teacher, in our school students take the Cambridge IGCSE 0580 exam. After IGCSE our school offers the IB Diploma programme and I am thinking about proposing the ...
725 views

### Definite integrals with equal limits

As a property of definite integrals, we teach that definite integrals are zero if the lower and upper limits are the same (Wolfram mathworld says this too). Is this valid in general? In the case of ...
779 views

### Composite functions

How would you describe the existence of a composite function $f(g(x))$in terms of range of $g$ and domain of $f$ . Does range of $g$ need to be subset of domain of $f$ or is it sufficient if the two ...
206 views

### How to formalize high-school (Euclidean) geometry?

I have unsuccessfully attempted several times over the years to formalize high-school (Euclidean) geometry, or even a working subset of it. Think very simple, diagramless geometry. The usual two-...
5k views

### Teaching math too soon in middle school and high school

I'm a retired university math prof and I now have a retirement job teaching at a small private high school. This is my 4th year. This school teaches Algebra 1 to 8th graders. Geometry to 9th ...
505 views

### Why do we explicitly state the equality of two things when we know they're equal

Recently my brother in high school and I were talking about some math when he said If we know two things are the same i.e. equal why do we need to state that they're the same? What he was trying to ...
215 views

### Which are the most used Greek letters in math textbooks?

I am looking for a list of the most frequent Greek letters used in high school and college textbooks or some other corpora. I've realized my students don't know Greek letters and I would like to teach ...
229 views

### How to introduce the use of Greek letters in high school?

I am looking for any hints or experience reports or materials/potential difficulties about how to introduce the use of Greek letters in high school Math/Physics.
1k views

### How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38,\$ and that \$\ 3.4 \times 10^{-6} \times 2.1 \times 10^{-5} = 3.4\times 2.1 ...
88 views

### Suggestion for IB program Analysis and Approaches SL book?

What is the most suitable book for the IB program Analysis and Approaches SL for a student with significant weaknesses? I had suggested the book from HAESE Mathematics yet he finds it particularly ...