Math Links for Week Ending Apr 16th, 2021

Featured Links

Given the shear amount of information we are bombarded with regarding C19 stats, it's important to make sure we talk to our students about how information is presented, even if it's just in text form. Context can be everything
Curriculum Tags: Gr7, Gr8, MDM4U
https://twitter.com/dtkung/status/1382848416636661765

Great question for a #MathForSocialJustice class:

What's misleading about the original @cnn tweet?

What's misleading about @PatFinelli's retweet?

(Don't know about CNN, but the replies make it clear that Pat knows what's wrong with his.) pic.twitter.com/uEmBf3Xzhv

— Dave Kung (龚仲孝) (@dtkung) April 16, 2021

Resource Links

I'm always looking for new ideas for doing unit reviews with students 
Curriculum Tags:All
https://twitter.com/howie_hua/status/1382491625813811200

For review sessions, I make study guides on Google Slides where each slide has a few questions on a topic. Students comment or post pictures of their solutions so they can help each other out. Rather than the teacher choosing what to review, students can choose what to work on.

— Howie Hua (@howie_hua) April 15, 2021

I think we can never have enough visuals of the proof of Pythagorean Theorem. I think that this one would be a good one to discuss with students
Curriculum Tags: Gr8, MPM1D, MFM1P
https://twitter.com/panlepan/status/1382636350596333571

Pythagoras proof animated with @geoegebra #shearing #translationhttps://t.co/GcpY0Ekl4y pic.twitter.com/QEiYAY168R

— Vincent Pantal🍩ni (@panlepan) April 15, 2021

I have created activities that use this visual proof of the sum of the angles in a triangle but I like this demo file that @TimBrzezinski created
Curriculum Tags: Gr8, MPM1D, MFM1P
https://twitter.com/TimBrzezinski/status/1381614982392123393

Messing around in @Desmos. Shout out, T/Y, & credit to @von_Oy for her insight re: defining the sector parametrically (in 3 lines): https://t.co/YijEXM8WMJ @AlgebraDesmos #MTBoS #ITeachMath #geometry #elemmathchat #msmathchat pic.twitter.com/1FqEG6xpY2

— Tim Brzezinski (@TimBrzezinski) April 12, 2021

Video Links

So many math famous people in this tribute video, can you name them all?
Curriculum Tags: All
https://twitter.com/edsouthall/status/1259937974676045825

So I asked my favourite mathematicians and maths educators if they'd help me make a tribute to Tom Lehrer...

Hope this brings you a smile! pic.twitter.com/b7esn5iHGG

— Ed Southall (@edsouthall) May 11, 2020

Image Links

A couple of math jokes Do you get them?
Curriculum Tags: Gr7
https://twitter.com/pickover/status/1382120320845369348

This is a mathematics joke involving a PIN (personal identification number). pic.twitter.com/5BXo4GB24L

— Cliff Pickover (@pickover) April 13, 2021

Curriculum Tags: MHF4U
https://twitter.com/pickover/status/1381788789220249603

This is a mathematics joke. https://t.co/efDbvEuQcE pic.twitter.com/Jn9SscMD5H

— Cliff Pickover (@pickover) April 13, 2021    

Math Links for Week Ending Apr. 9th, 2022

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This reminds me of my Percent Guesser but it's way better. 
Curriculum Tags: Gr7, Gr8
https://twitter.com/kurt_salisbury/status/1379069206977777672

How good are you at estimating fractions? @desmos #iteachmath #mtboshttps://t.co/Uwke3OsJSi pic.twitter.com/kx4VA9wrAO

— Kurt Salisbury (@kurt_salisbury) April 5, 2021

Resource Links 

I have always loved the ability to show math relationships with physical constructions. In this case, from @DPScher, we investigate the sum of the angles in a triangle. I have a @Desmos version here but this is way more hands on. 
Curriculum Tags: Gr8, MPM1D, MFM1P
http://www.sineofthetimes.org/triangle-angle-sum-embodied/

I don't think Marilyn Burns has invented this game but it's a good starter that can be used at any age. It seems like it's easy and it is easy to play. But to consistently win, you have to create a strategy. Try it out
Curriculum Tags: All
https://marilynburnsmath.com/race-for-20%e2%80%95a-counting-game-for-all-ages/

Something that most kids can probably relate to is the tooth fairy. So why not a step function representation of tooth fairy earnings? With this interactive, you can actually play with the parameters and see how things might change.
Curriculum Tags: MCR3U, MHF4U
https://flowingdata.com/2021/04/08/tooth-fairy-exchange-rate/

Video/Audio Links 

I have recently read @TimHarford's new book The Data Detective and there is one story in it about Florence Nightingale and her famous graph that told the story of the infections happening and why increasing sanitary conditions would lower the death rate in the military hospitals. Before you hear that story you might want to watch this to get a visual of the actual graphs she was making
Curriculum Tags: MDM4U
https://www.youtube.com/watch?v=VTdVPNvwULM

And now here is @TimHarford's story from his book and on his podcast Cautionary Tales. In it he digs way deeper into the story and actually talks about some more hidden sides. 
Curriculum Tags: MDM4U
https://timharford.com/2021/03/cautionary-tales-florence-nightingale-and-her-geeks-declare-war-on-death/

A few weeks ago I posted a video from @standupmaths about proving there was cheating happening on a record breaking speed run on MineCraft. Here's a shorter audio version from the @BBCMoreOrLess podcast.
Curriculum Tags: MDM4U
https://www.bbc.co.uk/programmes/p09ch7qv

Image Links 

A good way to bring the real world into proving things in analytic geometry
Curriculum Tags: MPM2D
https://twitter.com/MrValencia24/status/1374434972585496577

Today geometry ss at #WeAreChappaqua used @geogebra to prove/disprove a real object in their house was a parallelogram using a method of their choice 🙂#MTBoS #ITeachMath #MathisFun #MathRocks #MathPlay #MathEd pic.twitter.com/XCZ4ohUVES

— Libo Valencia (@MrValencia24) March 23, 2021

Math Links for Week Ending Apr. 2nd, 2021

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I have a pile of @Desmos activities that are useable but not ready for primetime (if you are interested here's a link) but I am slowly cleaning them up and releasing them. In this case, it's a way to introduce the Pythagorean theorem by focusing on the area model, well before just looking at the abstract relationship. 
Curriculum Tags: Gr8, MPM1D, MFM1P
https://teacher.desmos.com/activitybuilder/custom/5ef94d7221f7d778901d9338

I've also learned some Pattern Matching in Desmos CL from @MrChowMath and released a couple of others. One on investigating Pi and the other on Scientific Notation (with two versions - one with only positive exponents the other with both positive and negative). 
Curriculum Tags: Gr7, Gr8
https://teacher.desmos.com/activitybuilder/custom/603c2e023614df0d4f67c7aa
https://teacher.desmos.com/activitybuilder/custom/5f713da629b5b60b2f6a5aac
https://teacher.desmos.com/activitybuilder/custom/5f6c7c97a9176f3ebf55f986

[New @Desmos Activity] If you have done an informal activity where students measure the circumference & diameter of various circles to estimate Pi, then this short activity helps to solidify that it's actually 3.14https://t.co/JUNFclWxKi#MathChat #MTBoS #iTeachMath pic.twitter.com/i9lFDB14E6

— David Petro (@davidpetro314) March 28, 2021

[New @Desmos Activity] Thanks to @mrchowmath and his pattern matching tutorial videos, I finally have this set of scientific notation practice problems working wellhttps://t.co/JUNFclWxKi#MathChat #MTBoS #iTeachMath pic.twitter.com/7TFk6SGazM

— David Petro (@davidpetro314) March 28, 2021

Resource Links
I was just talking about this problem in class in the context of Pi Day. Now we have a way to do multiple experiments. 
Curriculum Tags: MDM4U
https://twitter.com/Christophe_Sew/status/1376843025784008710

Toothpicks ✅
Paper ✅@desmos activity ✅

All ready for some Buffon's Needle tomorrow. 🧮https://t.co/S33WXtOwLm@davidpetro314 #ImproveMyAB #ITeachMath pic.twitter.com/QfUSAkxIhs

— Mr. S (@Christophe_Sew) March 30, 2021

Have you heard? There was a ship stuck in the Suez Canal. This article has so much good information, I dare you not to come up with at least one related math problem. 
Curriculum Tags: All
https://www.washingtonpost.com/world/2021/03/25/faq-suez-canal-ever-given/

Lots of information here that you could easily create some #math problems out of https://t.co/dMIF0qnLaK

Pythagorean Theorem, Ratios, Trig, angles and more#MathChat #MTBoS #iTeachMath

— David Petro (@davidpetro314) March 28, 2021

I can't get enough of these #SlowRevealGraphs from @jennalaib. Be sure to check out the entire site for all the posts. 
Curriculum Tags: Gr7, Gr8, MBF3C, MDM4U
https://slowrevealgraphs.com/2021/03/29/speedy-animals/

I like skyscraper puzzles, an now a @Desmos version
Curriculum Tags: All
https://twitter.com/TMonark/status/1378074463141519363

I edited @TheErickLee 's awesome #SkyScrappersPuzzle with a "Reset" button with the help of @mrchowmath and offered it to my Geometry class before we start doing #TriangleCongruence and it was fun and full of #LogicalThinking #MTBoS https://t.co/lImwtjpqWj pic.twitter.com/EYF3sYzfm3

— Monark Trivedi (@TMonark) April 2, 2021

Video Links

This is older but I hadn't seen it before. Good for probability discussions including the need for multiply trials in order to have experimental probability match theoretical. Thanks to @robinmath3 for pointing this one out. 
Curriculum Tags: Gr7, Gr8, MDM4U, MBF3C
https://www.youtube.com/watch?v=FRlbNOno5VA

I had to look up what the Euclidian Algorithm was but thanks to this video, I have a better idea. 
Curriculum Tags: Gr7, Gr8

https://twitter.com/howie_hua/status/1369476859092557826

Tactile version of the Euclidean Algorithm pic.twitter.com/Hjj5TEO5Q3

— Howie Hua (@howie_hua) March 10, 2021

A couple more pattern matching videos from @MrChowMath for those Desmos CL-Heads out their
Curriculum Tags: All
https://www.youtube.com/watch?v=oPAevo7CaHU
https://www.youtube.com/watch?v=JTAGsw1InOg

Image Links

Here's a good reminder of why this classic thinking problem is a good one
Curriculum Tags:All
https://twitter.com/MrValencia24/status/1375499464509681667

Inspired by the work of @pgliljedahl, today ss at #WeAreChappaqua experienced a different approach.

Engagement ✔️
Collaboration ✔️
Knowledge Mobility ✔️
Gallery Walk ✔️#ThinkingClassroom ✔️#MathPlay ✔️#MTBoS #ITeachMath #MathisFun #MathRocks #MathArt #MathEd pic.twitter.com/fKPb5JjO02

— Libo Valencia (@MrValencia24) March 26, 2021

And Ever Given keeps giving us math problems
Curriculum Tags: Gr8, MPM1D, MFM1P
https://twitter.com/howie_hua/status/1375660413065437187

Math teachers: "Okay class, given line m is parallel to line n and angle a is 135 degrees, find the measure of the missing angles." pic.twitter.com/7dB25JYxTt

— Howie Hua (@howie_hua) March 27, 2021

This seems like a pretty complex way to calculate root 2 but I like it.

Curriculum Tags: All
https://twitter.com/pickover/status/1375493243266289671

This is mathematics. https://t.co/0pMybzHYfP pic.twitter.com/Qe0s71LpHk

— Cliff Pickover (@pickover) March 26, 2021

I've always like this story about Ramanajan
Curriculum Tags: All

https://twitter.com/fermatslibrary/status/1376525060488454153

1729 was the number of the cab that G.H. Hardy took when he visited S. Ramanujan in the hospital.

Hardy thought it was a "dull number" and Ramanujan replied "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways" pic.twitter.com/OFc0UATSXR

— Fermat's Library (@fermatslibrary) March 29, 2021

Math Links for Week Ending Mar 26, 2021

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Though these are not specific to math, I think that any help with assessment is good for teaching. Here @Wheeler_Laura compiles some ideas from the Twitterverse
Curriculum Tags: All
https://mslwheeler.wordpress.com/2021/03/10/%f0%9f%9a%bd-learningintheloo-efficient-grading-practices/

 Resource Links

Here's a nice interactive to demonstrate common denominators with adding fractions. 
Curriculum Tags: Gr7, Gr8
https://www.geogebra.org/m/zg6dvrh9

I made this fraction thingy in @geogebra https://t.co/yuAXaI91RA pic.twitter.com/86fOwXoQUD

— Steve Phelps (@MathTechCoach) March 20, 2021

Video Links

In this 3-Act Task, from Dane Ehlert, we look at rate/ratios with real numbers (and dominos). Below is the first act. 
Curriculum Tags: Gr8, MPM1D, MFM1P
https://whenmathhappens.com/2021/03/22/dominoes/

For those of you into Desmos CL, then you know that the pattern matching code is both interesting and powerful but perhaps a bit of mystery. In these three videos @MrChowMath walks us through some samples. 

Curriculum Tags: All
https://www.youtube.com/watch?v=NtDJdVW5uMk

https://www.youtube.com/watch?v=EEfBweAWVEM
https://www.youtube.com/watch?v=9TNf4sZ2iD4

Image Links

It turns out you can think about math too much
Curriculum Tags: All
https://twitter.com/pickover/status/1374142930689089537

Mathematics, Feynman, physics.

The divorce complaint of Richard Feynman's second wife, hater of calculus. pic.twitter.com/bzwDeGnH8k

— Cliff Pickover (@pickover) March 22, 2021

Math Links for Week Ending Mar. 19th, 2021

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Last summer I shared the site Mathigon.org as a new "textbook of the future". More recently, I have started to use their virtual manipulative repository Polypad. There is so much here - algebra tiles, fraction tiles, prime factor circles, customizable spinners and so much more. You can create a class and share things within your class and via Google Classroom (among other places). Just check out their YouTube channel for all kinds of ideas.
Curriculum Tags: All
https://mathigon.org/polypad 

Resource Links

More work from @TimBrzezinski, this time on equivalent ratios
Curriculuum Tags: Gr7, Gr8, MPM1D, MFM1P
https://twitter.com/TimBrzezinski/status/1371947548441784320

Creating = ratios with an @openmiddle theme. How many proportions can we create? Which digit(s) are not as user-friendly as others? Why? 🤔 https://t.co/NJfqzTypUg @geogebra @robertkaplinsky #MTBoS #ITeachMath #math #ElemMathChat #MSMathChat pic.twitter.com/4od6OQ04JO

— Tim Brzezinski (@TimBrzezinski) March 16, 2021

Having good math starters is a great way to keep students engaged. @SteveWyborney has a lot on his site and one he calls Esti-mysteries. I like them because not only are they low floor because all kids can estimate but then he goes on to give clues based on the answer so kids can start using their mathematical skills to refine their guesses. Totally check them out but @RNHung has turned a bunch of them into a @Desmos activity so take a look at that too.
Curriculum Tags: Gr7, Gr8
https://teacher.desmos.com/activitybuilder/custom/5fc84f717124020d13b2d9a4

The BEST conversations in breakout rooms this year. Esti-mystery tasks are all created by middle school students, bundled into a @desmos activity. Thank you @stevewyborney @mr_stadel @edcamposjr @Cbustamante2222 #FAME20 @maeputt #teach180 https://t.co/Yz8bSqt2yA pic.twitter.com/IF4LXP8WE7

— Richard Hung (@rnhung) December 18, 2020

Article Links
Another link to add to you Pi Day collection
Curriculum Tags: All
https://twitter.com/stevenstrogatz/status/1370750077661290500

Pi Day is tomorrow. Here was my attempt, a few years ago, to explain what all the fuss *should* be about. https://t.co/0p8Z7fpWOV

— Steven Strogatz (@stevenstrogatz) March 13, 2021

Video Links

This high school student has mad function skills
Curriculum Tags: MCR3U, MHF4U
https://twitter.com/Desmos/status/1372307145220370434

Let's go #BehindTheGraph with "Hummingbird" by Veronika Price from Berkeley High School! Throw a 💚 down for Veronika! https://t.co/XJEyDHWqsU pic.twitter.com/cRKEpjv7lT

— Desmos.com (@Desmos) March 17, 2021

Image Links

Here is an interesting data visualization
Curriculum Tags: MDM4U 
https://twitter.com/simongerman600/status/1369211417321148416

People calculate the weirdest things. Which celebrity based on their date and location of birth, was most likely to have been conceived at the 1969 Woodstock festival? Apparently the answer is Tina Fey. Source: https://t.co/UfkeiZa3JA pic.twitter.com/SAyEGGnt0V

— Simon Kuestenmacher (@simongerman600) March 9, 2021

A simple premise but definitely some good problem solving needed to determine the answer here. 
Curriculum Tags: Gr7, Gr8, MPM1D, MFM1P
https://twitter.com/jamestanton/status/1370858260010463232 

It's already pi-day in some parts of the world. So here's tomorrow's puzzle now: a tweak on a classic. 10 cm of perimeter length added to a square, an equilateral triangle, a circle. Which has smallest gap width? [Or am I being foolish: it all depends on the sizes of the shapes?] pic.twitter.com/rKU7tXcwJc

— James Tanton (@jamestanton) March 13, 2021

Math Links for Week Ending Mar. 12th, 2021

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It seems that @TimBrzezinski has really been on the @Openmiddle train lately. Here are two more to add. One on the distributive law and the other on order of operations.
Curriculum Tags: Gr7, Gr8, MPM1D, MFM1P
https://www.geogebra.org/m/suwcgzyd
https://www.geogebra.org/m/amukryav

Building a distributive property expression with an @openmiddle theme. My hope is that this will actually help Ss remember to distribute in the future. Where would we NOT want to place certain digits? Why? 🤔 https://t.co/3m33zqFiIC @geogebra @robertkaplinsky #MTBoS #ITeachMath pic.twitter.com/KxGIBD3t0V

— Tim Brzezinski (@TimBrzezinski) March 12, 2021

Re: order of operations, excited to be on an @openmiddle theme for my MS Ss. Feedback provided, yet they’ll still need to reason. Create TONS of versions by having Ss set their own goal value (lower left): https://t.co/p9EK6FjREB @geogebra #MTBoS #ITeachMath #math #RemoteLearning pic.twitter.com/Aai123c9EM

— Tim Brzezinski (@TimBrzezinski) March 7, 2021

Resource Links

Many students look at our normal everyday math problems as impossible.  Most are not, but you might want to give them some insight into some real impossible problems of geometry, topology and other topics. 
Curriculum Tags: All
https://www.quantamagazine.org/when-math-gets-impossibly-hard-20200914/

Many of the most celebrated results in mathematics are proofs of impossibility. https://t.co/7DKyhG0rJm pic.twitter.com/Lt0EKHYgU4

— Quanta Magazine (@QuantaMagazine) March 7, 2021

It's Pi Day on Sunday. Here are all of my PiDay resources. I am trying to suggest that The Rolling Stones are secret Pi lovers. Do you see it?
Curriculum Tags: All

https://www.pinterest.ca/davidpetro314/pi-day/

Opinion Links

Getting kids talking is the focus of the book Five Practices for Orchestrating Productive Mathematics Discussions. It's a great book with a lot of great ideas. Extending some of those ideas is @GeoffKrall in this post. Check it out for more thoughts on expanding the conversation in your math class. 
Curriculum Tags: All
https://emergentmath.com/2021/03/11/the-next-steps-in-classroom-conversation-orchestration/

What's AFTER the "five practices for orchestrating productive math discussions." Great, from @geoffkrall. https://t.co/Y8Mc1OjouN

— Dan Meyer (@ddmeyer) March 12, 2021

Video Links

I think there is some great stuff on this YouTube channel (especially if you love Legos) but in this video they make a machine that creates a gearing ratio of 1 Googol:1. Before you get to it in the video, have your students figure out how long it will take for the last gear to rotate once. 
Curriculum Tags: Gr7, Gr8
https://www.youtube.com/watch?v=QwXK4e4uqXY

A musical interlude for your pleasure
Curriculum Tags: All

https://twitter.com/MarkChubb3/status/1370392504722452486

Mathematical disagreement... turned into song for our entertainment: https://t.co/86Xgx5B8eC

— Mark Chubb (@MarkChubb3) March 12, 2021

Image Links

One way to remember the digits of Pi is to make a poem with words with the same number of letters as the digits. 
Curriculum Tags: All
https://poetrywithmathematics.blogspot.com/2021/03/math-girl-gives-us-pi.html

Soon it will be Pi-Day (3/14); celebrate by speaking in Pilish! For examples, visit https://t.co/2aAAB2I6v8 #PiDay2021 pic.twitter.com/zHPvyc5dBx

— JoAnne Growney (@MathyPoems) March 11, 2021

Math Links for Week Ending Mar. 5th, 2021

Featured Links

I like the idea of puzzles that promote mathematical thinking. These grid puzzles do just that. These were developed by @CaroleGreenes and Tanner Wolfram. The Alge-grids has a 3x3 grid with algebraic expressions in each box. They all have the same variable, a, and the object is to fine the one value of a that gets the numbers 1-9 appear in the boxes. The Pattern-grids requires you to put 3 Xs, circles and triangles in the grid in a certain way based on the hints shown. These are the PDF digital copies of the books with over 50 puzzles in each book. 
Curriculum Tags: All
Alge-Grid
Pattern-Grid

Resource Links

I like these @openmiddle type problems and this factoring one from @MathEqualsLove is a good example. 
Curriculum Tags: MPM2D, MCR3U, MCF3M, MBF3C
https://mathequalslove.net/factoring-puzzle/

I really like this open-middle style factoring puzzle from Public Schools of North Carolina's Resources for Algebra Blackline Masters. I recreated it in both Google Slides and Jamboard. https://t.co/jfyQQOyFxv #mtbos #iteachmath #iteachalgebra pic.twitter.com/N0vrfXIDZg

— Sarah Carter (@mathequalslove) February 26, 2021

Systems of equations with balances via @Desmos
Curriculum Tags: MPM2D, MFM2P
https://teacher.desmos.com/activitybuilder/custom/5db31fdacbf6600c2fc16005?collections=5f8a43dd06b0d9a8bd84c3d0%2C5f8a442006b0d9a8bd84c3d4

Dynamic hangars! Help your students understand systems of equations using “Make Them Balance,” an activity from our new middle school math curriculum. https://t.co/2n5Q7SSMYh pic.twitter.com/ae96EOiKN2

— Desmos.com (@Desmos) February 26, 2021

Here's a great foldable to connect geography and geometry
Curriculum Tags: MPM1D, MFM1P
https://twitter.com/idm314/status/1365395922516082692

We made a foldable version of the Fuller map in our poster. Print it and build your own icosahedral globe.https://t.co/K7WtHCBPFU pic.twitter.com/Y9Dc5gBis1

— International Day of Mathematics (@idm314) February 26, 2021

Opinion Links

I think this is a great idea for assessment protocols. It should certainly take away some anxiety. 
Curriculum Tags: All
https://twitter.com/wheeler_laura/status/1366566694110519296

My first step in this direction was to use @howie_hua's strategy of reading the test over as a group. Then w/o pencils or calculators they get 5 mins to go talk to anyone they'd like. After 5 mins everyone comes back to their seat & writes test individually. #ThinkingClassroom https://t.co/wVrkQA1OEl

— Laura Wheeler (@wheeler_laura) March 2, 2021

Video Links

PiDay is coming up
Curriculum Tags: All
https://www.youtube.com/watch?v=BdHFLfv-ThQ

Image Links

Do you have enough geometric skills to figure this out?
Curriculum Tags: MPM1D, MFM1P
https://twitter.com/panlepan/status/1364891666457522176

The octagon that thinks it's David Bowie.
What fraction of this regular octagon is coloured in red ?#GeometrySnacks #ShowYourWork #WFIS pic.twitter.com/UwrRkL0qoR

— Vincent Pantal🍩ni (@panlepan) February 25, 2021

How pointy is your angle? A math pickup line, perhaps?
Curriculum Tags: MPM1D, MFM1P
https://twitter.com/jamestanton/status/1366387133934489600

The "pointiness" of an angle of measure x degrees is
(180-x)/180.
Eg a straight angle is flat & has 0 pointiness. A small angle is sharp & has pointiness close to 1. The total pointiness of a polygon is the sum of the pointinesses of all its corners. What do you notice? pic.twitter.com/dw11LdFjUD

— James Tanton (@jamestanton) March 1, 2021

There is math all around us. You just have to notice it
Curriculum Tags: All
https://twitter.com/davidpetro314/status/1366952230620852225

Clearly we really like these chips at my house. Three different people bought three different sized bags today. There is a math problem here somewhere #MathChat #MTBoS #iTeachMath pic.twitter.com/S2OxZ90nuP

— David Petro (@davidpetro314) March 3, 2021