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-1 votes
2 answers
177 views

What are your most recommended resources to teach fractions?

Please share your resources for learning and teaching fractions...sites, videos, worksheets etc. I am looking for online resource similar to this: https://worksheetgenius.com/math/fractions-add/ with ...
4 votes
2 answers
249 views

Resources for teaching decimal numbers

I am currently teaching special classes to students whose ages range from 11 to 15 and there is quite a wide spectrum in their levels of maths. The lessons are given in English and we do not have a ...
3 votes
1 answer
79 views

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

Does anyone know what is the exact program of the International Baccalaureate in math? I've been looking for the MYP and DP programs in the website of the International Baccalaureate Institute but ...
4 votes
0 answers
133 views

When can students understand the intersection of two circles?

I'm interested in learning two transitions: (1) When can students reason (intuitively, but accurately) to conclude that two circles in the plane could intersect in $0$, $1$, or $2$ points, or are ...
64 votes
17 answers
9k views

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

I have been professionally tutoring a wide range of students (from elementary school through graduate school) for many years. Most of them are from the United States. I generally focus on helping my ...
0 votes
4 answers
289 views

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?

You can safely presuppose that 13 year old (y.o.) students learned the Difference of Squares and of Cubes identities, before tackling this Difference of Powers Identity ($x^n – y^n$). The glut of ...
0 votes
3 answers
299 views

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

Michael Spivak, Calculus (4th edn 2008), p 13. I know this monograph is aimed at undergraduates (not middle schoolers), but this kind of multiple-part question resurfaces on standardized tests for 13 ...
-5 votes
1 answer
121 views

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

Does the "Middle School Mathematics domains" on page 3 of https://www.ets.org/content/dam/ets-org/pdfs/praxis/5164.pdf refer to the the following 5 topics/categories? (I) Numbers and ...
18 votes
7 answers
3k views

Do any middle-school texts indicate that irrationality requires proof?

I believe that most middle-school math curricula have at least a brief section about irrational numbers, in which students are taught (among other things) that $\sqrt{2}$ is irrational and $\pi$ is ...
5 votes
1 answer
1k views

Law of large numbers as a middle school topic?

My daughter (a biologist) is presently teaching also math at a middle school (9th grade, so about 14 years olds). Now the topic in probability seems to be the law of large numbers! More and more I ...
3 votes
5 answers
545 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 ...
10 votes
6 answers
942 views

What value is there in requiring students to declare the dimensions of an answer when it is already clear from context?

When I was in late primary and middle school (east coast US, early 1990's), we were assigned a lot of word problems of the following general form: Mary has eight self-sealing stem bolts. She sells ...
8 votes
5 answers
3k views

Exponents with Negative Base; with or without Parentheses

How can I convincingly and mathematically explain the reason behind difference between $(-1)^2$ and $-1^2$? I used to add "negation" to the order of operations, in the same row as multiplication and ...
2 votes
1 answer
514 views

Help needed to find 7th & 8th grade completed math samples

I am trying to find samples of completed homework, class work and tests for 7th and 8th grade math in the US. I can find a million blank workbooks but not copies that students have completed with ...
4 votes
6 answers
635 views

Is there an agreed upon difference between how we represent $\frac{a}{b}$ and $a \cdot \frac{1}{b}$?

When teaching addition and multiplication of fractions, I seem to recall some advice on this site that one should first treat the cases $a \cdot \frac{c}{d}$ and $a + \frac{c}{d}$ before moving on to ...
13 votes
3 answers
331 views

Middle school math games for the parking lot

I'm looking for math games that a group of students in grades 5 to 10 (ages 11 to 15, say) could play in a gym or parking lot. My school has a STEM Day each year and I get tasked with cooking up some ...
5 votes
0 answers
353 views

Word problems written in past tense, present tense, or future tense

Does anyone have extensive classroom experience regarding the best verb tense to use when writing word problems at an elementary or middle school level? I have been writing some lessons recently and I ...
3 votes
1 answer
783 views

Why is my 8th grade Algebra 1 tutoring student learning mean absolute deviation and standard deviation?

I’m tutoring an 8th grade student in Algebra 1, and he showed me that their class learned how to find standard deviation and mean absolute deviation using the following formulas: $SD=\sqrt{\...
6 votes
7 answers
1k views

Are these fraction problems different enough to warrant individual consideration?

Consider the following problems: A) You have 20 problems for your math homework this week, and you want to do 1/5 of them today. How many problems do you need to do today? B) You need to read a 20 ...
3 votes
2 answers
209 views

The value of making students copy your derivations exactly

I have a hypothesis that I have not yet implemented, and am seeking guidance before I do. I have always taught algebra (and above) in such a way that the students understand the derivations. I check ...
24 votes
9 answers
5k views

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

The assumed age of the students is 10-15 years old. What is the danger in saying that two triangles are "the same" or "equal" instead of saying that they are congruent? It seems to ...
3 votes
2 answers
672 views

Teaching words with multiple meanings in Mathematics and English

We want to teach about words that have multiple meanings, in mathematics and English for middle school children (e.g., function, root, or, volume, angle, constant). This is for students who are ...
7 votes
2 answers
489 views

When working with 12-16 year olds, how should I graph functions when the domain technically isn't $\mathbb{R}$?

Let us assume that I want to graph any of the functions below. A) A can of soda costs $\$1$. Draw a graph depicting the total cost as a function of the number of cans you buy. Comment: One cannot ...
7 votes
7 answers
4k views

What is the preferred way to denote the Pythagorean theorem equation?

I am teaching 12-16 year olds. How should I write down the Pythagorean theorem equation? Some alternatives: $a^2 + b^2 = c^2$ $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$ $\text{leg}_1^2 + \...
3 votes
2 answers
455 views

Looking for a rigorous middle school self-study math course

My son is in 5th grade (US) and since he is doing remote learning, we have been doing a lot of topics in pre-algebra just using worksheets. I'd like to start him on a formal middle school curriculum, ...
7 votes
2 answers
685 views

Logic and proofs in secondary school

Inspired by the question When do college students learn rigorous proofs?, I became curious when pupils in secondary schools learn about proofs, what kinds of proofs they are, how rigorously they are ...
1 vote
1 answer
161 views

Resources for Unit Rates

I am currently mentoring my little brother in mathematics. There is an issue with the pedagogy of unit rates. For example when given the following concept " 11.00 U.S. Dollars to 20 Planet X ...
19 votes
8 answers
6k views

Prisoner's dilemma formulation for children

I am preparing an introductory course on Game Theory for children (between 10 and 17 years old). In the course description, I want to include a prisoner's dilemma in order to catch children's ...
4 votes
2 answers
156 views

Teaching Approach at primary, middle and higher level

I would like to have a comparison or a big picture of how and why the approach for teaching math varies from primary (or pre primary) to middle to higher classes. I understand at every level one ...
1 vote
2 answers
291 views

What are the resources to learn prerequisite knowledge to latter High school and undergrad prep textbooks?

I use textbook study and am planning on studying Spivak's Calculus, Mathematics It's Content, Methods, and Meaning, How to Prove it by Velleman, etc. However, I'm worried I lack the prerequisite ...
2 votes
1 answer
194 views

Ramanujan results for middle school?

Pls I wonder what Ramanujan's results could be explained to middle school level audience, ie without using integral etc that is up to university curriculum? For example Ramanujan's infinite radicals ...
0 votes
2 answers
183 views

I find high school math very hard compared to middle school? [closed]

i hope i can get some help on how to get better at high school maths i find them very difficult compared to middle school. Whats the big difference so i can work on it ?
4 votes
5 answers
423 views

Patterns that unexpectedly fall apart at large $n$

I am constructing a learning sequence for middle grade students designed to convince them that empirical arguments (arguments by example) are not sufficient in mathematics. To motivate this, I am ...
22 votes
15 answers
7k views

Explaining why (or whether) zero and one are prime, composite or neither to younger children

There are lots of discussions out there about whether $1$ is a prime number (such as this one) and even about zero (such as this question, though note zero does generate a prime ideal in $\mathbb{Z}$ ...
-2 votes
4 answers
385 views

Inefficient methods

I see many teachers use slow methods to solve a given problem where there's another faster methods that doesn't demand much more effort. I'm not looking for mistakes like saying that $a$ is the slope ...
3 votes
3 answers
235 views

How to explain to pupils that "$\frac n{100}$ OF $a$" is equivalent to "$a$ TIMES $\frac{n}{100}$"?

How to explain to pupils that "$\frac n{100}$ OF $a$" is equivalent to "$\frac{n}{100}\times a$"? There is some difficulty in explaining that the first sentence, containing "OF" (which could suggest ...
10 votes
4 answers
5k views

Is the constant term a coefficient?

I'm a baby boomer who was taught that the constant term of a polynomial is a coefficient, being the constant factor for the x^0 term. That's not what's taught today. Current text books are vague on ...
7 votes
6 answers
244 views

Simple Number Theory Task

I am doing exercises with middle grade students and looking at their capacities to create arguments for simple number theory conjectures. I want three tasks, and I have two so far: 1. The sum of three ...
4 votes
2 answers
577 views

How to Teach Middle School Students to Read Square Roots?

This exact quote from my standard American Algebra 1 textbook states when first introducing rational square roots: $\sqrt{49} = 7$ is read "The positive square root of $49$ equals $7$." $-...
18 votes
10 answers
4k views

How to Write Steps of Solving Equations?

This is a common way to write the steps during solving equations: But in GeoGebra the steps are shown this way (the highlighted part): I'm going to use GeoGebra to teach equations. Is it OK to let ...
3 votes
1 answer
437 views

"Seeing" GCD and LCM in Word Problems

Last year, I taught GCD and LCM and then gave my students word problem relating to these concepts ("Two runners with given different speeds; when will they meet again?", "Having three kinds of flowers ...
2 votes
4 answers
349 views

Might it be helpful for students to have different symbols for subtraction (-) and negation ( _ )?

Might it be helpful for students to have two different symbols for subtraction (-) and negation ( _ )? Subtraction, after all is a binary operation (with 2 operands). Negation is a unary operation (...
6 votes
4 answers
409 views

Am I too formal teaching this middle school student?

Solve for $x$ the equation below, writing the universe set and the solution set: $$\frac{a}{b}+\frac{b}{a}-\bigg(\frac{a^2}{bx}+\frac{b^2}{ax}\bigg)=1$$ Let's choose $U=\{x\in \mathbb R;x\neq 0\}$ as ...
4 votes
1 answer
450 views

how to prove my 6th grade son knows algebra 1?

My son is taking pre-algebra in 6th grade. He mastered algebra 1 in detail more than what school can teach. How to prove to school that he mastered Algebra 1? He wants to start Algebra 2 in 7th grade. ...
0 votes
1 answer
215 views

Using Substitution in place of the balance model

How does substitution work as an alternative to the balance model in introducing solving equations? My biggest worry is that, lacking a concrete representation, is too abstract for middle schoolers. ...
8 votes
3 answers
273 views

Mathematical Task with Various Solutions

I need to come up with a mathematical task for middle school (9th grade), which involves either algebra, functions, probability or statistics (anything but geometry actually). My problem is, that the ...