Celcius vs Fahrenheit. Formulas vs Graphs

I usually teach converting temperatures from Celsius to Fahrenheit and the other way using the standard formulas.

If you want to go from Celsius to Fahrenheit you use the C = 9/5 F + 32   while F = 5/9(C-32).  These work ok until you ask students at what temperature are both Celsius and Fahrenheit the same.

This is when I promptly lose them because they don't realize they can determine the answer in one of two ways.  They can either set the two formulas to be equal or they can graph.

As you can see if you replace the C or F with x and y, you can graph the two linear equations and they cross at the point (-40, -40).   This means when the Celsius temperature is -40, the Fahrenheit is -40 and vice versa.    I got the idea the other day as I was looking at the formulas and realized the formulas are basically linear and I could easily create a visual representation using Desmos.  You see the results in the photo above.

The other way is to set them equal to each other and solve:

9/5 x + 32 = 5/9(x-32)

9/5 x + 32 = 5/9 x - 160/9

9/5 x + 288/9 = 5/9 - 160/9

9/5 x = 5/9 x - 448/9

9/5 x - 5/9 x = -448/9

81/45 x - 25/45 x = -448/9

56/45 x = -448/9

56 x = -2240

x = -40

This means the value has to be -40 for the two temperatures to be equal.

You could have done each equation separately by subbing x in so it looks this way

x = 9/5 x + 32
-4/5 x = 32
x = -40

or

x = 5/9( x - 32)
x = 5/9 x - 160/9
-4/9 x = -160/9
(-9/4) (-4/9x) = -160/9 * -9/4
x = -40

So there you have it.  -40 Celsius is -40 Fahrenheit and a way to prove it.  Have a great day and enjoy yourselves.  Let me know what you think, I'd love to hear.

Warm-up

If it takes 16 oranges to make 59 ounces of juice, how many oranges do you need to make 1 gallon of juice?

Warm-up

The average tree produces 325 oranges per year.  If your orchard has 90 trees in it, what is the total crop for the year?

Slowmo, stop motion, and Animation in Math.

 I have been trying to think of ways to improve instruction and learning in my math class.  In math, we say if a student can explain their thoughts and how they came up with the answer, they are showing understanding but it seems to me that if they can create a video showing their thinking, that works.

Animation is a great way to create equations and move things around without you standing there and explaining every step. Instead, you can have the terms move on screen by themselves, or move to the proper spot.

I have used keynote to create animation showing how two shapes were congruent using transformations.  I used keynote because that is the presentation software I know well.  Right now, I am working on learning to use Google slides for the same thing.  I have several ideas I want to bring to life using animation.

1.  Instead of talking about slope, why not create animation so the word positive climbs diagonally up towards the top, then the word negative is going down to the starting level, the zero slope is a flat area and undefined is a cliff but I'm seeing the words appearing one letter at a time.

I made a quick animation showing the idea for slopes.  My students do not always stop to think about which slope is associated with upwards, downwards, flat, or straight down.  My art isn't the best so forgive me.  I used animation HD for kids.  It was free and so easy to use.


2.  Showing how to solve various equations using animation.  You can have the terms move or you could arrange for a train or other vehicle to move the terms around.  In addition, it is possible to put music in the background or voice over to explain why.

3. Stop motion can be used to show how the area of a circle comes from the circumference of a circle.   If you cut the circle into certain shaped pieces and rearranged, you can rearrange the sections into a parallelogram.

4. Students can create their own characters who talk about a problem on the board, each step done via stop motion or slow motions.

This video was made with Toontastic on my iPad.  I used the characters from it and the three part storyline.  It was an easy app to use.  If you notice, my characters spoke about slope in general as seen or experienced in real life.  It was not that technical in terms of finding the slope itself and it explored the idea that a slope could be positive or negative based on which direction you are going.

The above are just a few suggestions plus two short videos I made using two different apps.  Let me know what you think, I''d love to hear.  I hope these videos work properly.  Have a great day.

Merry Christmas

Christmas Infographics

This is the perfect time of year to sneak in reading or creating infographics.  The holiday season has so many different possibilities for learning to read and interpret infographics.

This is a skill we need in today's society because society has moved to sharing information via infographics rather than the standard graphs.

This site has a something like 12 different Christmas related infographics.  My favorite one is the one that looks at the best selling toy each decade beginning with 1910.  The graphic includes prices for each toy adjusted for inflation and the prices are in pounds so students have to convert the unit of currency.  In addition, the toys are divided into genders.  The other thing is this infographic records the change of consumer tastes over the time.

Another infographic addresses Christmas and pets which looks at the number of families who see pets as members of the family, how many presents they buy their pets, and which gender is more generous.  One of the infographics I find most interesting is the one outlining why being Santa is the most dangerous job out there.  The infographic looked at everything from number of children and stops to how fast Santa has to travel to do it all, and snacks left for Santa in various parts of the world.

This site has 9 infographics which are different from the first one.  There is one infographic on the ugliest sweaters at Christmas.  Although it looks at ugly sweaters, this infographic actually breaks down the number of sweaters purchased by region and around the world from this place.  Another infographic explores the safety of Christmas trees from buying them, to caring for them, to disposing of them.  The infographic even took time to look at what causes Christmas tree fires.  Unfortunately, not all the links in the above two sites are live.

This site is one big infographic on the economics of the holiday season.  It looks at economic growth, mortar growth, global e-commerce growth, target audience and spending,  Towards the end is a section where they give hints to vendors on ways to improve their chances of increasing their sales.  It is a very interesting graphic.

This site has tons of Christmas infographics including the most popular recipes, the history of Christmas trees, Christmas dinners, greatest holiday movies, and so many others.  Most of these infographics are fairly short with 5 dinner infographics, 5 destination infographics, but they all have some nice information.  I checked out the one on Christmas cards.  It looked at the anatomy of a Christmas card, one on showing the insight into holiday marketing for businesses, facts about Christmas cards, the 12 days of Christmas, and breaking down the gift card trend.

I love the way this last site is done because you can break the class up into smaller groups, assign groups to interpret each category, or individual infographics in each category.  Students can present their findings via Flip-grid, Google Slides, or any other digital method.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

You just paid a Christmas Tree decorator $213 to place your lights, ornaments, and tinsel on the tree.  It took her 3 hours to complete the job.  How much did you pay her per hour?

Warm-up

If the average cost of a real Christmas tree in 2019 is $81, and they sold 32.5 million trees nationwide, how much did people spend nationally on this many trees?

The Cost of The 12 Days of Christmas.

 The 12 days of Christmas is one of the more popular songs heard at this time of the year and every year at least one media outlet announces the current cost of obtaining all the items.

The overall cost of the 12 items have increased over the past few years.  In 2016, the cost ran $34,363.49 which went up to $34,558.65 in 2017, and in 2018, they were $38,926.03.  This year, the cost will run just a bit more.


The break down this year is as follows:

1 partridge in a pear tree is 210.17 down 4.5 percent.

2 turtle doves is $300.00 down 20 percent.

3 french hens is still running $181.50 which has not changed in price since last year.

4 calling birds didn't change price since last year and is still $599.96.

5 golden rings will set you back $825.00 which is a 10 percent increase.

6 geese-a-laying is $420.00 up 7.7 percent from the previous year.

7 swans-a-swimming is one of the most expensive items at $13,125 which didn't increase at all.

8 maids a milking didn't change and still costs only $58.00

9 ladies dancing didn't change from $7,552.84.

10 lords-a-leaping still runs $10,000, the same as the previous year.

11 pipers pipping $2748.87, up must 8/10th of a percent from last year.

12. drummers drumming will set you back only $2972.25, up just 0.8 percent.

The total is $38,993.59 for all of these items but if you gave the number sang in the song with all the repeats, it would cost you $170,298.03 or $781.94 more than the previous year.  The numbers quoted are for obtaining all the items locally.  If you had to order them online, it would cost you $42.258.91.

It wouldn't take much to turn this information into a graphing excursive.  The data from 2018 and 2019 can be found here for both local and online purchases. Students can translate the information into a graph using a spreadsheet and then produce various graphs from the program. This page has the total cost for all the gifts beginning in 1984.

The same graph for the total cost also allows people to look at the change in the price of each individual items.  The price of a partridge in the pear tree ran $32.52 in 1984 and is now $210 in 2019.  This graph makes it possible for students to calculate the percent increase or decrease each year in order to create graphs showing the information.  I think it would be interesting to see how the cost of each item changes over time.

I hope you explore the second site, it is cool.  Let me know what you think, I'd love to hear.  Have a great day.

PBS Learning Media + Math

 The other day, I discovered the PBS Learning Media site filled with so many cool things you can use in your math class.  This site has videos, interactive lessons, interactive, lesson plans, documents, and all sorts of other things.  This site has a search engine to find lessons on a specific topic or grade.  Most of the material has information on the grades each activity is meant for. I searched mostly for high school material since that is what I teach but there is material for all grades from prek and up.

I looked at the video section and found a nice video on using probability in search and rescue operations.  Search and rescue is something quite real out in the bush of Alaska because sudden storms blow in causing people to get lost, or it warms up and the ice thins so people fall into the cold water and die.  This is a relevant topic for us.

The unit includes the video complete with teaching tips so the instructor knows the best way to teach this.  In addition, there is a page listing all standards associated with the material.  The teaching tips include questions for students to answer as they watch the video so they have to focus on the material rather than enjoyment.  Furthermore, it is possible to download the unit as a zip file.

Next I checked out their interactive lessons and found two that immediately attracted my attention.  One was on Math in Fashion which included Project Runway, and Math in Special Effects. The Math in Special Effects is geared for grades 7 to 10 and includes a teachers guide, transcript, and says it is a student directed lesson.  Students sign into the site to do their work and have it saved.  The lesson also has a page listing common core standards this unit meets.

The site also has several activities that are interactive such as the one where students get to engineer a jetliner.  The activity has the student how to adjust the wings and location of the engine for the best performance for four different situations.  In addition to the usual teaching tips, there is material to provide both background and further reading, and the answer key.  Of course, the activity does include the necessary common core standards to make life easier.

The lesson plans are focused on specific mathematical topics.  Some of the lessons are targeted to teach a topic such as ratios,  or a general lesson on dance.  I checked out the everyday algebra targeted lesson which begins by giving a summary, estimated time to complete, and the learning objectives.  The next page divides the lesson up into three sections, the introductory activity, the learning activity, and the culminating activity, each section has a video with information on what the teacher should focus on.  It includes further information and lists the standards.

The documents section has several interesting topics including Math's hidden women, and one on the risk of flying.  I looked at the document on the risk of flying and found it quite interesting because it had both graphics and written words.  The author even took time to explain how the risk is calculated, in two different ways, per mile, per flight, or average risk for a person.  The explanations are quite clear and easy to follow. 

Last there are galleries which focus on one topic.  I looked at the one on living wage vs minimum wage.  The gallery has five videos, a lesson plan with information on each clip and questions to go with each clip.  One activity has students find the cost for several items in their community so they can calculate the living wage for their location.  I like this because it is everything costs more in the villages of Alaska so having figures based on local prices makes the exercise more realistic.

If you need more material for your class, check this site out.  It is well organized, has great support materials, and topics that are interesting.  Let me know what you think, I'd love to hear.  Have a great day.

Digital Compass for Geometry

I do not like using compasses and straightedges in my Geometry classes because most compasses I end up with break, or the pencil holder dies, or something else so I can't use them.  I've also had some students who would see them as a good weapon so I'd rather not provide them with that temptation.

Unfortunately, I had not found anything I could easily use to  recreate certain geometric drawings.  I looked for digital possibilities but didn't find any until the other day when someone on twitter recommended Robocompass.  It is a web based program and comes highly recommended by a variety of people.

As soon as I opened the program, it took me to a blank slate.  On the left side is the space where you type in the commands.  At the top left side there are several examples such as tessellation's using rotations or translation, bisecting an angle and several other things.  On the right side, there is a help button which lists all the commands, form, and what they do.

As you type in one of the standard commands, a help box pops up, showing you the general command and a real example with numbers so you can figure out what to do.  I had fun exploring it and I created a small piece of pie.

I made a video of the program I made so you could see it in action.  Once you have finished your project, you hit the play button and it shows each step of the process by showing each line and highlighting it so you can connect the command with the action.

It is easy to use.  The help page was clear with the generic commands and specific examples to show how it works.  I could readjust the numbers until I actually had the final product I wanted.  I plan to play with it more.

This free program is fun and best of all, it works on my iPads too so it can be used on computers or iPads.  I hope you check it out.  Let me now what you think, I'd love to hear.  Have a great day.

Warm-up

An Icelandic horse weighs about 350 kg while a regular horse weighs about 475 kg.  The Icelandic horse weighs what percent of a regular horse?

Warm-up

The Icelandic horse has a height of 140 cm versus the 165 cm of a regular horse.  What percent smaller is the Icelandic horse vs the regular horse.

Ways to Show "Applied Mathematics."

The other day, I started class by showing a clip from a new station showing a trebuchet they built to chuck pumpkins.  I talked about how much of the math taught in class is more theoretical but that we see math being applied in real life.

I decided it is time to show students actual situations of applied math in situations without discussing the actual mathematical equations, just the applications in general terms.

I've been trying to promote the idea that the equations they study in class are ways to describe everything we see or experience in the world.  The kids really liked being the Trebuchet send the pumpkin over 2400 feet.  The person in charge spoke about how they built model, how they failed, tried again, eventually getting it to work.  Everything we talk about in class.

In the future, I plan to show clips of various sports, especially basketball, as a way of having students name where they see math, or look at a clip from various car races, dog sled races, or snow machine races to identify the math.  At this point it is not important to identify the actual equations but it is important to look at identifying situations.

Another thing to do would be look at various buildings to show how architects use geometric shapes, parallel and perpendicular lines, arcs, etc as they design buildings.  This would be a perfect situation to bring up historical buildings such as the pyramid of Giza, the Roman Colosseum, or other ancient buildings to look at the math behind them.  Aside from showing the application of mathematics, it also ties other subjects to math, reading a cross curricular connection.

Since the only way in and out of the village is by plane, I could easily bring up the idea of the math involved in scheduling the flight from the cost of the plane, calculating the price of the individual ticket, employee salaries, benefits, the cost of fuel, and scheduling planes because one plane flies several times everyday.

This would be a perfect place to also discuss how professionals prepare estimates when you take in your car or other vehicle in for repair, or you are having the house painted, the roof redone, the floors done or anything like that.  I don't know if the local businesses use them here but I know in most places they do.  This activity also introduces the importance of having accurate estimations so you land the job.

I found a site on the internet with some free videos and some that require a subscription to view but all take time to discuss math and various situations such as designing sunglasses, playing the drums,  designing buildings in earthquake areas, making faster bikes, dancing, and so many other topics.  In addition, each video has worksheets to accompany each video and there are quite a few educator resources.

I'm hoping this helps students connect mathematical equations with real life.  Let me know what you think, I'd love to hear.  Have a great day.

Visualization and Word Problems in Math

Visualization is an important skill in mathematics because it helps people see problems more clearly.  Although we teach students to draw pictures or diagrams but we don't always stress the diagrams can clarify the situation so much better than using only words.

In addition, visualization helps students turn mathematical language into something they can understand and apply.  It turns situations into understanding and students can process their learning.

Perhaps when we introduce word problems we might ask students how we'd represent the situation visually.  The problem with concentrating on key words is that some words such as per can mean multiplication or division depending on the context.  If students look at word problems as if they do when reading a book, they might do better.  In reading, they are told to picture the scene in their minds, why don't we ask them to produce a drawing of it on paper, on their iPads, or even create something in VR/AR or in animation.

Rather than working through umpteen word problems, why don't assign each student or group of students one or two word problems to do.  Before they start, ask them how they could represent the word problem visually, where have they run across this type of situation in real life, and how could they present the problem and its solution through video, Desmos, Geogebra, or a sequence of images or pictures.

Recent research indicates that it is important for the brain to use both symbols such as numbers, and visualizations because mathematical learning is optimized.  When students use symbols, they access one part of the brain and when they create visualizations, they are using a different part of the brain, so when they use both, the two parts of the brain communicate and their learning improves.

Furthermore, creating visualizations also makes math more exciting and improved student performance.  In addition, creating visualizations involves the brain using higher level thinking skills, and shows people there is creativity in math and math is not just numbers.

So back to word problems.  It is important for the student to create a visualization of the problem first so they understand the problem and the context of the mathematics.  The visualization can be as simple as drawing a picture or it could be more complex using stop motion animation, creating a cartoon, a comic strip, or a video.  Just some way for students to "see" the problem.

Once students are comfortable with creating visualizations for word problems, they can take the next step of creating their own word problems with associated visualization to show others the problem.  Visualization is the key to learning to do word problems.  Let me know what you think, I'd love to hear.  Have a great day.

How Lava Lamps Create Random Numbers

I watched a NCIS episode the other day where one of the people had to destroy a bunch of lava lamps used for computing to save the day.  I honestly thought that was a plot device but I checked it out and indeed there is at least one company who uses lava lamps for this purpose.

This company acts as a gate keeper for data on the internet.  In order to maintain security of the data, they have to produce a bunch of random numbers, completely random and totally unpredictable.  To do this, they use the "lava" in lava lamps to help randomly generate numbers.

They do this by recording the bubbling of over 100 lava lamps in the lobby of their main office in San Francisco.  This video is then fed into an advanced algorithmic program which takes the bubbling and changes it into random numbers that are more random than can be produced by humans.

Most random numbers are created by programs written by humans so they are really not that random.  The numbers produced can often be guessed or figured out by hackers but the numbers based on lava lamps is much more random and secure.  There is less chance for these numbers to be cracked.

This lava lamp randomness is actually quite secure.  As stated, the lava lamps are in the lobby of the company's San Francisco office for everyone to see.  By having the lamps out in the lobby for everyone to see, it adds another layer of security to the production of random numbers.  When people walk around, the vibrations change the rate of bubbles or if they block the light it changes the temperature inside the liquid, changing the bubble rate.  Every little change, makes the randomness even more random.

You may wonder where random numbers come into play for computers and the internet.  Well, it's like this.  The minute you log into a website, the computers assign you a random number for identification.  This is to prevent hackers from impersonating people and getting their money, etc. Unfortunately, most random numbers are not random because they are produced by algorithms but the strings can be figured out if the first number is known.  

This is where lava lamps come in.  Their blurps, and bubbles are much more random because the mixture of oil, wax, and water is effected by changes in temperature, vibrations, and other things.  This means the rate that bubbles are produced is never constant.  

Thus the arrangement of bubbles and blurps rising through the liquid is constantly changing making the production of random numbers much more random.  These rates are changed into random numbers which are used in cryptographic keys for extra security.  This unpredictability is the key to randomness in this whole scheme.

I don't think the inventor of the lava lamp ever envisioned his creation being used for computer security.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

A turtle can travel at 0.09 m/s.  How many miles per hour is that?

Warm-up

How many total square are there in a chessboard?

Chess + Coordinate Plane = Transformation

At one point in my life, I learned chess.  I didn't learn it because I had a burning desire to learn it but my brother decided to learn the game and needed someone to play against.  I didn't care for the game because no one took time to explain one needs to think about the consequences of various moves.  I had no problem with the movement of each piece. because every move could be explained in terms of movement.

Just the other night, I realized chess has so much in common with the coordinate plane.  When my brother taught me chess, he neglected to mention the notation system used in it.  A board is 8 squares by 8 squares or a total of 64 squares.  Assume the chessboard is in quadrant one of the coordinate plane.  The 8 squares along the x-axis are given letters a to h, while the 8 squares lining up along the y-axis are given numbers from one to eight. When you hear someone say "Knight to c3" it means the person is moving his night to the third column along the x axis, then up three units to the third row.  If we were thinking in terms of normal coordinates, we'd say (3,3).

I suspect the use of letters for one coordinate makes it less confusing but this notation system is referred to Algebraic notation.  It is customary to use this system to write down all the moves made during a game so the serious players can go back and review every move, looking for errors and mistakes.  This is the way they improve their playing ability.

Another thing about chess is that all pieces pretty much move in a linear direction.  Some pieces move along a line with a slope of 2 or -2 while others move along a line with a slope of 1 or -1.  Then there are pieces that can move only one square or one jump while others move across the board in one smooth line with a slope of zero or undefined.  Every move can be defined as a transformation.

Chess players use another geometric concept called the "rule of square".  They visualize a square to determine if a pawn will get through the other players defenses.  This has been used since the Middle Ages as a way to make a judgement without using a lot of mathematics.

Furthermore, mathematicians have created two problems  based on chess pieces and a chess board.  One is the Eight Queens problem where people try to place eight queens on a chessboard so that none of the queens threaten each other.  The problem was thought of in 1848 by Max Bezzel in Germany and it wasn't solved until 1972 with the help of computers and lots of work.

Another problem is the Knight's Tour problem where you try to have the knight visit every single square on the board.  This problem is directly related to the Hamiltonian path problem in graph theory.  This problem dates back to Arabic manuscripts from the 9th century.  Mathematicians found several different solutions including Euler which he presented at the 1759 meeting of the Berlin Academy of Sciences.

I love that chess is mathematically related  and I've learned there are several high ranking chess players who are mathematicians.  I wish I'd known more about the chessboard when I first learned chess.  It might have made it easier for me to learn more about it.  Let me now what you think, I'd love to hear.  Have a great day.

Glogs and Math

I am just finishing up a flipped learning class for recertification.  One of the chapters in the book mentioned glogs so I had to look them up because I've never heard of them.  Glogs are a shortened form of graphical blogs where the readers can interact with the "poster" on the blog.

The graphical poster is digital and can have videos, pictures, sounds, and so much more.  In other words, instead of reading the blog, you are interacting with the graphical interface.

Think about having students create this interactive digital posters to share information on certain topics.  In addition, creating the interactive poster encourages collaboration when students work together and they use higher order thinking skills to organize the information into a presentation.  It is also a valid method for assessment because it allows students to show their learning.

The cool thing is that there are at least two sites on the web where one can both find and make interactive posters for use in glogs.  Both of these sites have a free level and if you want more you can pay.

1.  Glogster is one possibility.  It allows people to insert text,  graphics, images, walls, audio, video, web , 3-D and VR, along with clip art.  They also have a library of over 40,000 blogs in their content library which includes math and templates teachers or students can use to create something new by simply replacing the blanks with the appropriate videos and text as needed.

Students often ask "How do I start?" and these templates give them a place to start.  It is their work, everything from pictures to text to videos but the arrangement is there.  For students who are good at starting with an empty palette, they can begin with a totally blank canvas and build what they see in their mind.

There is a free version for one teacher and up to 30 students that comes with limited editing and 3-D or media content, no access to premium features and no help desk but for free, it comes with enough to work with.

2. Thinglink is another possibility.  Thinglink allows the teacher to create interactive images or create virtual tours.  I've used this to create interactive pictures for my hyperdocs.  I created an interactive picture on slopes.  There are places on the picture that identify zero, undefined, positive and negative slopes within the ride.

The free version is primarily for the teacher to create interactive pictures for the public mode but for a small amount per year, it is possible to upgrade to the premium account.

The premium allows you to have up to 35 students use the account to create their own interactive posters via assignments using collaborative editing, and allows the teacher to grade the work and provide feedback.  I've never used the premium but I know other teachers who love this and have used it.

If you haven't implemented Glogs into your repertoire, give these sites a shot to determine which one is best for you.  Let me know what you think, I'd love to hear.  Have a great day.

See, Think, Wonder in Math.

I'm finishing off a class on flipped learning and as part of one assignment, I had to create a lesson.  As part of the lesson, I included having students look at a set of graphs, mappings, tables, and coordinates as an introduction to the topic.  They will explain what they see, what they think, and what they wonder as they look at them and then they will record their thoughts in Google Slides.

As I wrote the lesson, I realized it is easy to include the What do I see, What do I think, and What do I wonder in the math lesson.  It is perfect as a way of sparking previous knowledge, mathematical thought, and curiosity.  The picture does not have to be of a math equation, it might be something like the picture above that is filled with arcs.  Lots and lots of arcs used as part of the sculpture.  When I get ready to do slopes, etc I'll post pictures of mountains, or perhaps a picture of the Stockmarket.  Then when it is time to do three dimensional shapes, I can use pictures of buildings like the tower of Pisa, to introduce the topic.

This particular routine encourages careful observations while taking time to think about their interpretations of the picture.  As stated earlier, this routine can be used to introduce a topic or to provide a connection during the unit between real life and the math being studied.  It can also be used toward the end of a unit to make students think about extending the topic or starting the transference of knowledge.

This works best when putting the I see.... I think.... I wonder.... together as one unit and it is important to have them include reasons for their  thinking because one of the math standards is to communicate their thinking.  If you use this routine with visible patterns, start with asking "What do you see?" which has them describing what they see such as shapes, patterns, etc.  Next ask "What do you think?" which could have them thinking about the next step in the pattern, or what math is associated with what they see.  It takes them to the next step in the process of thinking.  The "What do you wonder?" asks them to take things a step further so they might figure out a way to predict the pattern after 10, 20 or 100, or maybe they'll wonder about the mathematics behind something.

Another use of see, think, and wonder is with graphs published in newspapers.  It might be a graph of  the latest pickup truck, the price of the average thanksgiving dinner, or other public graph found in an advertisement.  This activity has students explain what they see, what they think is going on, and what does it make them wonder so they get a better grasp on the idea that sometimes graphs can be misleading.

In addition, this can be used when beginning a new unit.  To do this, have students look through the new material while asking them what they see, what they think they will learn in the unit, and what questions they have about the unit.  If you place a templet online using Google Slides, students can record their thoughts for each question so you can go in later and read their responses. Furthermore, as the teacher, you can go into Desmos and create your own See, Think, and Wonder activities complete with places for students to record their thinking.

It is important to have students discuss their thinking because thinking leads to understanding.  In addition, this activity helps students improve their mathematical communication and increases their critical thinking skills.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If 33.5 percent of the $6.22 billion dollars were made from mobile devices in 2018, how much in sales was that?