James Tanton
@jamestanton
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An Aussie fellow promoting uplifting joyful genuine math thinking and doing for students & teachers alike. Thrilled: reaching millions!
Phoenix, AZ
Joined December 2009
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Little known "fact" about base ten arithmetic: It is okay to cancel 3s, 6s, and 9s from fractions.
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Classic: A circular pen has a fence whose height varies continuously. Does there exist two points 180-degrees "apart" on the fence such that a bar placed at those points is perfectly horizontal?
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Every triangle has a line through it that simultaneously divides its area and its perimeter each in half. (Why?) For an isosceles triangle that special line is its line of symmetry. Where is that special line for a 3-4-5 right triangle?
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Yes ... this is how the multiplication table should appear: drawn to scale!
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At what value between 0 and 1 does a horizontal line at that height produce two regions of equal area as shown on the graph of y = x^2?
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A wordless self-replication puzzle. (Though someone will no doubt point out to me that I just used words!) As per usual: 1. What do you think the puzzle is? 2. What is your response?
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Classic:
A circular wall has height that the varies smoothly as you go around the circle. Explain why there must be two points 180 degrees "apart" along the circle where the wall is the same height.
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A horizontal line is drawn between the lines y=0 and y=1, dividing the graph of y = x^2 into two regions as shown. At what height should that line be drawn so that the sum of the areas of these two regions is minimal?
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One end of a stick, 1 unit long, moves along the perimeter of a convex polygon with the stick 90 degrees to each side as it goes along. At corners, it swivels as shown. If the perimeter of the polygon is P, what is the length of the path traced by the other end of the stick?
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Classic:
There are essentially 9 different ways to draw four non-intersecting circles on a page.
How many essentially distinct ways for five?
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What value of x gives an orange square of half the area?
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Partitive versus Quotative division? Or is it Quotative versus Partitive? (Really unenlightening K-12 education jargon!) Have I got it right with my pics?
My question: In what sense are these the same? How explain to a student in a convincing/meaningful way?
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2666/6665 = 2/5. In fact, any equal number of 6s in numerator and denominator give a fraction equivalent to 2/5.
199999/999995 = 1/5. In fact, any equal number of 9s in numerator and denominator give a fraction equivalent to 1/5.
Other similar examples?
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Been asked a couple of times in a row this week to share my defn of what mathematics is & what a mathematician does. I responded as below.
Today's "puzzle:" What is your defn of each? (Do my attempts have any merit?)
(Also: Did you know the about the circle numbers in the Δ?)
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What fraction of the whole square is a single blue square in the second diagram? Third diagram? 100th diagram?
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A circular wall has height that the varies smoothly as you go around the circle. Explain why there must be two points 37 degrees "apart" along the circle where the wall is the same height.
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I am often asked what my favourite math puzzle is. It's the wordless one below. This video explains why it is my favourite . Have you a formative math moment?
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Here's a famous--and mighty curious--way to multiply two numbers: "parabolic multiplication."
My question, just for intellectual curiousity (not pedagogy) is just how much does this model "explain"?
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Where across a circle could one place a pair of orthogonal lines so that the sum of the two shaded areas shown is as small as possible? (What is that smallest value?)
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Would taking classic and tempting errors "head-on" be a worthwhile classroom activity and discussion?
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I just published “Just teach my kid the <expletive> math”
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Love this little gem from Lill, 1867. (Google "Lill's method" to see a slew of materials on a curious geometric approach to solving polynomial eqns. But can you first see on your own how each slope here satisfies am^2 - bm + c = 0?)
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This is making the rounds: My only request: In learning the guitar you know what beautiful guitar music is & can appreciate wanting to internalize its practice & form. Please give the fluency/practice in math its context too of beauty, story, meaning, joy.
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A HS quadratics unit should be conducted so that this comes as a complete & utter shock--because it is mind shocking: Adding heights of matching data points in a symmetric U-shaped graph and a linear graph produces a perfectly symmetrical U-shaped graph. Just wow!
@GlobalMathProj
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How many essentially distinct ways are there to draw five circles on a page?
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Is it not shocking that the standard long multiplication algorithm is commutative, that, no matter the order one computes the product the final sum on the bottom line is sure to be the same? Today's puzzle: How would you explain why this is so to a student, or just for yourself?
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It is impossible to draw an equilateral triangle on a square lattice of dots with each vertex on a dot. (Why?) But can we get close?
Is it possible to have two vertices on a dot and the third within 0.00000001 units from a dot?
(The picture is not equilateral.)
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Do you like any of the approaches below for computing 1005-787? Other?
Math is for each of us to own &do in whatever good way suits us best.
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Classic: Draw parabola y = x^2 and for integers a<-1 and b>1 connect the points (a,a^2) and (b,b^2) on the graph. Which integers on the y-axis are missed and why?
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Cup A contains 100 mL of juice. Cup B is empty.
I move 10% of the juice of cup A into cup B, and the 10% of the contents of cup B back into A. That leaves 91 mL in cup A.
What happens if I keep repeating this 2-step process? Will I empty cup A? Will amount of juice in A stablise?
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Loads of fun finding products in standard graphs. (Can you explain why the y-intercepts are what they are?)
But does the fun stop with just y=x^2 and y=x^3?
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I've had requests to make my COLLEGE ALGEBRA FOR HUMANS notes available as hard-copy print versions. Each chapter turns out to be book length (!), so here are eight chapters each available on (search for the title). Of course, free PDFs are here too:
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A square of "radius" 1 is centered about the origin.
Rays are drawn from the origin and, on each ray, the point on the ray, outside the square, and a distance 1 from the square is marked.
What shape do these points outline?
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Classic: Is it possible to draw a square of area 3 on a grid with vertices on grid points? Squares of areas 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, ...?
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A parabola is the set of all points equidistant from a circle of radius zero (a point) and a circle of infinite radius (a line). What is the curve of points equidistant from two circles of finite radius? (What is "distance from a circle"?) Circles intersect? One inside other?
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Numbers 1 thru 10 written on a board. Two numbers a & b chosen at random and used as the legs of a right triangle. Hypotenuse h is computed. Then h is added to the board and a & b each erased. Process repeated 8 more times until a single number remains. What is that number?(Why?)
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Today's Puzzle: Assuming I could draw neatly, and assuming I could draw forever, what fraction of the big square is shaded? What infinite sum am I seeing?
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Interesting conversation going on in FB right now started by John Abreu. Are you familiar with the citardauq formula? It clearly gives the correct answers in this case (and actually does so in all cases!)
@daveinstpaul
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I just published “It’s Time to Let Go of Antiquated Edicts in the Mathematics Classroom”
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A circular pen has a fence whose height varies continuously. Is it, for sure, possible to find two points on the fence "60-degrees apart" so that a rod placed at those points will be horizontal?
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A grasshopper jumps along a line, first 1 inch, either left or right, then 2 inches, either left or right, then 3 inches, either left or right, and so on up to 50 inches, either left or right.
Could the grasshopper end back its start?
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A circle rolls inside a parabola. Does the center of that circle also trace a parabola?
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Given an a-by-b rectangle, a square of the same perimeter has side the arithmetic mean of a & b; the same area, the geometric mean of a & b; the same diagonal, the quadratic mean of a & b. A square of side the harmonic mean of a & b has what in common with the original rectangle?
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I just published “7 times 8 is 54: The Call for Math Memorization”
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There are 1000 purple dots in total in this multiplication table . (These are the dots in the rightmost column & bottommost row.)
How many dots are there of each other color?
Can you explain what you notice?
Who knew the basic multiplication table was full of beautiful mystery!
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MVThm: For a differentiable function on [a,b], there is a value c such that f'(c) = average slope.
For f(x) =x^2, we have c = arithmetic mean of a & b.
For f(x) =1/x, we have c = geometric mean of a & b.
Is it obvious that geometric mean is always less than the arithmetic mean?
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The arithmetic mean (a+b)/2 and the harmonic mean (reciprocal of arithmetic mean of reciprocals) naturally appear in a trapezoid/trapezium w sides a & b. Might there be a natural appearance of the geometric mean sqrt(ab) or the quadratic mean sqrt((a^2+b^2)/2) or other such too?
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Another Math Without Words (of a similar theme). Two puzzles: 1. What is the puzzle? 2. What is your response to the puzzle?
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Typical school approach to fractions is muddled, to say the least.
"A fraction" is a portion of pie.
"Multiply = of" (2/3 x 4/5 is "2/3 of 4/5")
"Divide=keep, change, flip" (2/3÷4/5 is 2/3x5/4). HUH?
PUZZLE: Draw a convincing picture of pie to show why 2/3÷4/5 is "2/3 of 5/4."
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A rope the same length as the circumference of a circle is wrapped around the circle as shown. (The rope starts pulled taut tangent to the circle and remains pulled taut as it is wrapped.) If the circle has area 1, what is the area of the region swept out by the rope?
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School math says: To factor ax^2+bx+c, look for integers p & q such that pq =ac and p+q=b. If such p & q exist, then writing ax^2+bx+c as ax^2+px+qx+c is sure to lead to factorisation IN INTEGERS. Can you identify the deep number theory-not explained-that ensures this will be so?
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A horizontal line is drawn between the lines y=0 and y=1, dividing the graph of y = x^n into two regions as shown (n>0). At what height should that line be drawn so that the sum of the areas of these two regions is minimal? Does that height depend on the value of n?
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Classic: Choose any two (square) numbers on main diagonal of a multiplication table. Find their sum & their difference. Also, use the two square numbers to make a square in the grid. Sum the two entries in the remaining corners. Prove you are sure to have a Pythagorean triple!
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In this infinite tree, to get to the number 25, say, start at 1 and step right, left, left, right. Let's say the "tree code" for 25 is 1RLLR. What's the tree code for the number one hundred? For one million?
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The sequence of triangular numbers begins 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, ...
What is the first triangular number that ends with a 2?
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What horizontal line drawn between y=0 and y=1 on the graph of y=2^sqrt(x) - 1 minimizes the sum of the two shaded areas shown?
[The answer to all these questions is y=1/2 if we're dealing with a strictly increasing function. The real question is how to best explain this.]
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Here's the story of the VINCULUM. I really love the vinculum and think this historical story has so much pedagogical value for the classroom.
@GlobalMathProj
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Dots labeled 1, 2, 3, 4 on a vertical line. Starting on one side of the line walk a path crossing dots in turn to opposite side. The order of dots crossed gives a permutation of 1234. Are all 24 permutations possible? What proportion of them are? 5 labeled dots?
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[Classic] Place the numbers 1, 2, 3, .., 15 in a ring so that any two consecutive numbers sum to a perfect square.
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Mathematics: the most creative of arts that leads to timeless truths. Behold a proof and touch eternity. Wow! A lovely potent piece by
@fjmubeen
.
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Oh YES! Math classrooms should be nothing but white boards! Getting up and standing and pacing while thinking is so important. The emotional safety of not being committed to anything you write (it can be erased) encourages thinking and experimenting and trying. YES YES YES!!!
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The work of an educator is to pave the way for epiphanies.
(Isn't that what teaching math is really about?)
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How obvious is it that (x-a1)(x-b1)+(y-a2)(y-b2)=0 has to the equation of the circle with AB as diameter, where A=(a1,a2) and B = (b1,b2)? Does one have to grind through the algebra to see this is so?
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Did you know? Draw a triangle inside a circle of diam 1. If its three angles have measures A, B, C, then its side lengths are sin(A), sin(B), sin(C)? [In fact, you can take this to be THE statement of the Law of Sines if you wish.]
@JenniferWathall
#trigmadehuman
@GlobalMathProj
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An equilateral triangle moves about inside another maintaining its matching orientation. What can you say about the sum a+b+c of distances shown?
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The graph of x^3 + y^3 > x + y is curious.
Can you predict what the graph of x^2 + y^2 > x + y will be?
How about that of x^100 + y^100 > x + y?
How about x^100 + y^100 > x^2 + y^2?
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It is known that all triangles with corners on a square lattice containing no interior lattice points have the same area. Is the same true for all triangles drawn on a triangular lattice that contain no interior lattice points?
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I know this is, by default, self-promotion. But I do want to publicly thank
@amermathsoc
/
@maanow
/
@mathmoves
for doing such a tremendous job with my latest book. Color even! If you are thinking of publishing a math book, publish with them!
@GlobalMathProj
@CmonMattTHINK
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Actually: If anyone is interested in my thoughts on memorising math facts (which folk typically take to mean multiplication facts) here's a piece I wrote a while back:
@Mathgarden
@JenniferWathall
@DynamicsSIAM
@GlobalMathProj
@suevanhattum
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One of my favourite puzzles: Place 3 pennies and a dime anywhere on a table top. Record starting position of the dime. Leapfrog the dime over each of the pennies in turn, then again each penny in turn the same order. Lo & behold ... the dime is back at start! (Works in 3D too!)
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In the sequence 0, 1, 1/2, 3/4, 5/8, 11/16, ... each term is the (arithmetic) average of the previous two terms. What is a formula for the Nth term of the sequence?
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A little gem from Hippocrates I believe. (ca 450 BCE). What is the area of the lune formed by two circular arcs with centers shown?
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The pizza my father ordered last night made me wonder: Can you cut a pizza into 9 pieces of equal area with two V and two H cuts? If no, how do you know? How close can you get? (i.e. How small a difference between areas of max and min pieces can you get, say, assuming radius 1?)
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How might you explain why, in a standard multiplication table, the entries in each L-shape (gnomon) sum to a perfect cube?
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Writing a little book about dots (of course!). Including a picture like this in it. How many dots are there of each colour?
@GlobalMathProj
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Sixteen points (dots of no dimension) are drawn on a page. Is there, for sure, to be a straight line you could draw across the page that separates the dots into two sets of eight dots, eight on each side of the line?
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Use lots of copies of the same scalene triangle to tile the plane, making lots of equilateral triangle spaces. Place a dot in the middle of each space and see regularity. What does this say about the centers of three equilateral triangles drawn on the sides of a scalene triangle?
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Three shops A, B, & C situated on an empty plane. They form a triangle. I will drive to the shops, park my car somewhere, get out, walk to one shop, then a next, then to the third, then back to my car. Any advice on where should I park to minimize the total amount of walking?
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a||b is the maximal count of intersections that can arise if one draws a dots on one line, b on a parallel line, and connects each point on one line with each and every point on the other line with line segments. What's 100||200?
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Can a power of two be the hypotenuse of an integer right triangle?
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The volume of a circular cylinder is Base Area x height. Pinch the top to a line it is half this. Pinch the top to a point, one third this. Is there a shape. a cylinder morphed a wee bit in a way easy to describe, with volume some other fraction of base area x height?
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A wordless puzzle today. (Well, two puzzles, really. The first is: what is the puzzle? And the second is: What are your responses to it?]
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Classic: Suzzy has 2 coins. Suzie has 3. They each toss their coins. Whoever gets the most HEADS wins. In case of an equal count of heads, Suzzy wins.
What are Suzzy's chances of winning?
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0.1|3|6|10|15|21|28|... = 100/729
0.1|4|9|16|25|36|49|...= 110/729
What (unusual) decimal equals 120/729? Equals 130/729? etc.
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It's been a while. Just wrote up a blog essay "COVID-19 Exposes Mathematics Education Inadequacies: A modicum of (secret) relief for Educators."
All the usual math teaching norms are out the window right now. That's actually opportunity for profound good.
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Two squares are constructed as shown in two congruent right isosceles triangles.
Which square is the largest and what is the ratio of their two areas?
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Label three faces of a die with the fraction 1/2, two with the fraction 1/3, and one with the fraction 1/6. Roll the die. Then the probability of seeing the number you see is the number you see. Are there alternative ways to label the die so that this is again true?
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A draft of Chapter 3 of COLLEGE ALGEBRA FOR HUMANS is done. (Here are chapters 1, 2, and 3.)
I am rewriting and tightening up all my algebra notes and thoughts into one tome. I think I properly tightened up making sense of the AREA MODEL for arithmetic
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6 circles stacked along the diam of a big circle. The sum of small diams =large diam. Surprise that sum of Ps of small circles = P of big circle. (Whoa!)
The sum of areas, however, can vary. No upper bound: can be really close to 100% A of big circle. But is there a lower bound?
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