Curated Ideas for Engaging Students from a Distance

MTBoS BLAUGUST

 Today I saw a slide show how to engage students in distance learning.  There are quite a few ideas curated from various blogs and such.  The presentation is not original with me.  I am sharing it with permission of the author.  The Facebook group where I saw it is a public group called Amazing Educational Resources. 

I hope that some of the ideas in the presentation are helpful to all of you working so hard to capture the hearts and minds of your students while delivering required content ... from a computer screen.  No easy feat for sure!

If you want to download the presentation, click here and make a copy.

Just for fun ... brain teasers

MTBoS BLAUGUST

Awesome Shelli asked early today if anyone has a slide show of brain teasers that could be used as an "away" screen on Google Meet.

Her ask resonated with me ... felt like a challenge that I wanted to tackle today.

So ... I pulled together a few brain teasers today that might be interesting to a wide range of students.

You are welcome to copy this slide show, use it, modify it, add to it!

The last slide has links to the resources from which I borrowed these teasers.  They are not original with me.  I don't have solutions for most of htem.

Exploring Patterns ... extending thinking

 

Patterns are EVERYWHERE! Some suggest that the study of mathematics is the study of patterns ... identifying them, categorizing them, generalizing them.

Five universal generalizations are true about patterns:

• patterns have segments that are repeated
• patterns allow for prediction
• patterns have an internal order
• patterns may have symmetry
• patterns are everywhere

In fact, patterns make a great curriculum organizer for interdisciplinary work since patterns are evident in language, science, history, music, art, and more!

Here are a few ideas for exploring patterns in math in introductory ways ... 

1). Read children's books!  I love to connect with students using a read aloud ... talking through a picture book before getting started with math.  AND yes!  It works well even in middle and high school!

Some of these are more sophisticated than others ... and the last one (bottom right corner) is a coloring book.

2)  Explore polygonal or figurate numbers.  I have a task I've used with students ... you may find it helpful.  "Explore Numbers in Shapes."

3). Check out the activities, explorations in YouCubed ... there are more than a dozen pattern activities ready to implement in your classroom.

4). Visual Patterns is an excellent site ... set up to use regularly as a key activity in your math program.

5). Mathigon has a great series of lessons, interactive, free ... ready to use to enrich students' understanding of patterns.

6) NRich is another excellent source for problem solving, stretching students' thinking and exploring topics.  They have numerous pattern activities at all levels ... perfect for differentiating!

I am sure there are other great resources!  What sources to you recommend for exploring patterns with students?

Always Sometimes Never ... which is it for you?

Always, Sometimes, Never are statements that allow students to justify thinking using prediction, inference, and testing hypotheses.  Several summers ago I began curating ASN statements.  Today I updated the Almost Sometimes Never blog site to make it more user friendly.

Originally, I set up the site to be used one slide at a time.  There are 179 statements in 13 categories: Absolute Value, Conic Sections, Exponential Functions, Financial Literacy, Functions, Geometry, Linear Equations, Number Properties, Polynomials, Real Number System, Systems of Equations, Trigonometry, and Trig Identities. 

But I realized that you may want several slides and it's a pain to copy them one by one.  So today, I created a page with LINKS to SLIDE SHOWS ... one for each topic.  You can make a copy of my slide shows ... embed them in your assignments.  I hope this will make the site more useful to everyone.

I've been thinking about expanding the site.  @PamJWilson tweeted about using ASN statements and wrote about using NON-Math ASN statements today in her blog post ... and is creating a shared document.  What a great idea to continue collecting, sharing, posting ASN statements ... mathy ones, non-mathy ones!

When using ASN statements ... 

• encourage students to talk math with one another, agreeing, questioning each other, helping others to think through their reasoning
• suggest that students consider working through examples, use modeling, tables, graphs, equations to justify their work
• ask probing questions to help students clarify their thinking or to develop their vocabulary
• possibly review expected vocabulary before beginning the activity
• obviously the key is for all students to explain their thinking!

What is a favorite Always, Sometimes, Never statement or instructional routine that you might share with us?

Fun with Exponential Functions

 

I enjoy teaching exponential functions!  Here are some ideas for exploring exponentials ...

-

Mini Poster Projects

CHOOSE JUST ONE of the FOLLOWING Ideas (A, B, C, OR D) and create a mini poster in a class
slides presentation. Include a title, table of values, graph, and art to illustrate the problem.

Access the activity here.

-

Always, Sometimes, Never Exponential Statements

Access the slide show here.

-

A few critical thinking questions

Access the document here.

#MTBoSBlaugust To Infinity and Beyond! An enrichment project idea

A favorite picture book is,Infinity and Me, by Kate Hosford.  If you haven't read it, do so!  If you teach math, buy it for your classroom.  Many of us teach sets of numbers early in the year.  We talk about infinity in domain and range.  Infinity is one of those topics that makes for a great enrichment project!

Here is an example of a "choice board" that you could use as a project in your class ...

You can access the Google Doc here ... just copy it to your Google Drive.

To INFINITY, and BEYOND!

Complete three activities making a tic-tac-toe from the Choice Board.  Use at least three of the following resources. Create a reference page for the resources you use.

• Angier, N. (2012, December 31). The Life of Pi, and Other Infinities. Retrieved August 24, 2016, from http://www.nytimes.com/2013/01/01/science/the-life-of-pi-and-other-infinities.html
• Chartier, T. (2014, April 14). Mime-matics - the infinite rope. Retrieved August 24, 2016, from https://www.youtube.com/watch?v=--nbhhuabHo
• Hosford, K., & Swiatkowska, G. (2012). Infinity and me. Minneapolis: Carolrhoda Books.
• How big is infinity? - Dennis Wildfogel. (n.d.). Retrieved August 24, 2016, from http://ed.ted.com/lessons/how-big-is-infinity
• The Infinite Hotel Paradox - Jeff Dekofsky. (n.d.). Retrieved August 24, 2016, from http://ed.ted.com/lessons/the-infinite-hotel-paradox-jeff-dekofsky
• Vi Hart. (2014, June 6). Retrieved August 24, 2016, from http://vihart.com/how-many-kinds-of-infinity-are-there/
• Weltman, A. (2014, February 20). Talk Like a Computer, Infinite Hotel, and Video Contest. Retrieved August 24, 2016, from https://mathmunch.org/2014/02/20/talk-like-a-computer-infinite-hotel-and-video-contest/

To INFINITY, and BEYOND! Choice Board

Define infinity for a 1st grader.  Define it for a 6th grader.  Define it for a high schooler.  How do your definitions change?	Explain the infinity symbol.  What is it’s name?  How did it originate?	What does infinity look like to you? Draw a picture that captures infinity.
N. Angier says in her New York Times post, “... there are infinities, multiplicities of the limit-free that come in a vast variety of shapes, sizes, purposes and charms.”  What is meant by “different” infinities?	Select either the infinite hotel paradox or Zeno’s Paradox.  Write a story, poem, or essay about the one your choice.	What is a doppelgänger, and why does the existence of an infinite universe suggest yours might exist?
Make a list of quantities that you believe are infinite and present that list to a group of classmates. Invite your classmates to debate whether or not the quantities are finite or infinite, and if infinite, which kind of infinity.	Mime Tim Chartier uses movement to explore and express the idea of infinity. Watch his youtube video.  Then, create your own skit or mime to convey how you see infinity.	Is it possible to find the sum of an infinite series?  Explain.

Reference:  Teaching the Mathematics of Infinity. (0604). Retrieved August 24, 2016, from http://learning.blogs.nytimes.com/2013/01/30/teaching-the-mathematics-of-infinity/

Creating a Working Document for Quadratics Unit

 

I decided to try out a hyperlinked document to experiment with how to put together a unit for students.  I started with a familiar one.  I am working on a series of enrichment lessons for gifted middle schoolers who are already accelerated in basic math.  I found this "notebook" style to be challenging and yet, it may be useful.  

I'm curious how you are organizing lessons?

The part I struggled with is how students would keep notes, share thoughts and so on.  In the document above I refer to Padlet a couple of times.  And I do like that site for sharing whole class.

BUT just now I saw how one school created a Google Doc for a whole unit ... as a "scrapbook" for students to keep all of their unit work together.  I can't publish theirs here but in the next few days I want to create that kind of document for this unit.  We'll see how that goes :). (If you want to track down an example of the Google Doc scrapbook, check out this link ... high school integrated math 1.

Structuring Lessons for Virtual Instruction

Structuring Lessons for Virtual Instruction

I'm working with a group of teachers about planning a differentiated lesson.  While discussing lesson planning, I realized that we go about that process differently, organize our work differently, create different styles of lesson plans.

On top of that, many teachers are planning for online delivery.  How might that look different from f2f classes?

While I have been teaching online for a university for 16 years, I do not have experience delivering classroom instruction K12 online.  So take my thoughts with a grain of salt :)

The first thing that is significant to me is organization.  Teachers and students are both flustered by technology.  How might lessons be organized so no one is guessing what to do, when to do it, where to find it?

A Hyperdoc might be a good idea at this point.  Hyperdocs can look different. I'm going to suggest a couple of ways of organizing lessons.

Borrowed from https://www.aoptech.org/michele-kiss.html

I've used three different sets of labels to set up a basic Hyperdoc model.

• 5E model 
• EELDRC or Enroll, Explore, Label, Demonstrate, Review, and Celebrate
• Upside Down Version to I Do, We Do, You Do

In each of these models, the teacher plans out lessons inviting students to participate by accessing links, videos, forms, online tutorials, online practice, and more.  In addition students can snap and insert photos, respond and/or reflect right in the document itself.  It becomes not only an organizer for the lesson plan but also could be an organizer for brief notes for the student.

If you want to improve upon those Hyperdoc ideas ... check out Slides Mania's Hyperdoc Notebook ... You could create a notebook for each unit.  Her work is amazing and it is shared freely!

The last suggestion for organizing lessons is using Deck Toys!  This site is new to me but I enjoyed playing with it.  I created a sample lesson on Babylonian Numbers ... it's not great - just my first time.  By the way, you can embed video, require answering before continuing, embed Desmos, and Delta Math!  I also found several already created for secondary math!  You will want to check those out! Anyway ...if you are looking for a way to organize lessons this may be a great one!

How are you building lessons online?  What structure are you using?  What ideas might you share with a first year teacher?

By the way, I wrote about Lesson Planning Structures in a previous blog post.  You might find it interesting if you are new to lesson planning!

Resources to Extend Thinking with Quadratics

 

Here are a few activities that I've used to extend thinking in our work with the Quadratic Function ...

----------------------

True False Statements Sorting Activity - Characteristics of Quadratics

----------------------

Odd One Out Activities give students opportunity to analyze, build vocabulary, and develop a discerning eye for mathematics.

---------------------

Partner Discussions or "He Said She Said" Activity asks students to analyze 2 problems, discuss them and then debrief.

----------------------

Always, Sometimes, Never statements are great for helping students think deeply about quadratics.

-----------------------

Last ... Here is a collection of questions to promote critical thinking:

1) Write three different quadratic functions whose graphs have the line x = 4 as an axis of symmetry but have different y - intercepts.

2) The points (2, 3) and (-4, 3) lie on the graph of a quadratic function.  Explain how these points can be sued to find an equation of the axis of symmetry.
3) Write a quadratic function that has as roots, x = 2a and x = -5b.

4) For what nonzero whole number m does the quadratic equation x^2 + 3mx + (9/2)m = 0 have exactly one real solution for x?

5) What characteristics are most easily determined from a quadratic function written in standard form?  In factored form?  In vertex form?

6) The point (1, 5) lies on the graph of a quadratic function whose axis of symmetry is x = -1.  Your classmate says the vertex could be (0, 5).  Is your classmate correct?  Explain.

Extending Thinking with Systems of Equations

 

Systems of Equations can be fun to explore.  

Here are some ideas for extending thinking...

-------------------

Odd One Out is always a great discussion starter!

-------------------

Another critical thinking activity ... "Partner Talk or He Said, She Said" ... students analyze problems on their own, pair up to discuss, and then debrief as a class:

-------------------

This Tic Tac Toe board offers students opportunities to explain their understanding of systems of equations.  How might you use something like this in distance learning?

-------------------

My Analysis Task asks students to complete several activities to demonstrate mastery of systems of equations:

-------------------

Last ... if you like using "Always, sometimes, never" statements, here are a set of 10!

More Data Collection Ideas ... Old and NEW

Data Collection Activities are great for introducing functions.  I wonder if you were teaching remotely, how it might work to collect and analyze data?  I wonder if students were given time to assemble the supplies and collect data at home, if then you could analyze it together online?

A couple of ideas new to me

M & M's, Balloons, Cups and More  is a pdf of several activities published several years ago by NCTM. This is a series of six different activities where students collect data, graph the data points, determine which family of functions it belongs to and find an equation for the data. The functions include linear, quadratic, and exponential.

By the way, I found that activity on this site, Mathematics - Algebra for All.  There are a number of great activities linked there.  Definitely a place to bookmark!

Walk the Plank - When one end of a wooden board is placed on a bathroom scale and the other end is suspended on a textbook, students can "walk the plank" and record the weight measurement as their distance from the scale changes. The results are unexpected— the relationship between the weight and distance is linear, and all lines have the same x‑intercept. This investigation leads to a real world occurrence of negative slope, examples of which are often hard to find.

And from a previous post ... 

Linear (and Quadratic) Function: Pass the Ball

All the information needed for this activity is in this blog post. Students pass a ball to one person, timing the event. Then to two people, three people, etc. This is a very easy lab to set up.

Another activity is the message/whisper chain.  Students will enjoy this lab!

The Wave Lab is also a great introductory lab suitable for linear functions!

Pass the Book is similar to the previous two!

If you teach Algebra 2 and want to review both linear and quadratic functions in a single activity ... consider this looking through a tube idea!

Quadratic Function: Stacking Starbursts and Kangaroo Conundrum

Both of these activities are simple, table top activities that result in quadratic patterns. I provide instructions in this blog post.

The Water Flow Lab looks fascinating ... 

And EVERYONE loves a catapult!  Check out this data collection opportunity!

Square Root Functions: Inclined Plane Data Collection

A copy of the instructions can be found here. Students roll a marble on an inclined plane - varying the distance of the roll, and measuring the time it takes to reach 0.

You can model the movement of a metronome here using a bottle and spring!

Here's another version of a pendulum lab!

PhET has a pendulum virtual lab worth investigating if you don't have materials!

Exponential Functions: M 'n M Data Collection

This is popular for obvious reasons - students love to eat the m 'n ms after the experiment is complete. We do both parts ... exponential growth and decay. The handouts with instructions can be found here and here.

Paper folding also works for exponential functions ... and requires little preparation or materials!

Mathy Cathy explains how she uses the Sierpinski Triangle to model exponential functions!

Rational Functions: Spaghetti Cantilevers

Instructions are online.  Students bundle 2, 3, 4, 5, 6 pieces of spaghetti and hang a weight from the end without breaking the spaghetti.  They collect the data and analyze it.  The rational function is the model for the cantilever. 

MORE IDEAS!

Several labs are shared in this conference handout!

Also check out this Twitter thread where folks shared ideas for labs!

Parent Function Ideas!

I'm feeling inspired by my teacher friends on Twitter ... they are the BEST of the BEST!

Anyway ... someone on Twitter was asking about Parent Functions.  My activities are "old-school" ... but maybe some of these ideas will inspire creativity in lesson planning.

One of my goals in studying parent functions is to engage students in identifying key characteristics, and then comparing/contrasting the various functions to build fluency with them.

So first we build a parent function "notebook."
Then we identify key characteristics.

Next we compare/contrast.

And last we explore some questions that hopefully get students to thinking!

These activities are online here.

 Possible Synthesis Questions

1. Order the parent functions we are studying from least to greatest by the rate at which f(x) increases as x increases for x > 1.  Explain your thoughts.
2. Use the set of points {(-1,-1), (0,0), (1, 1)} to answer each question.
1. What parent function best describes the set of points?
2. If the points (-2,8) and (2, 8) were added, what parent function would best describe the set?
3. If the point (1, 1) were replaced with (1, -1) what parent function would best describe the set?
4. If the point (-1, -1) were replaced with (4, 2) what parent function would best describe the set?
5. Select 1 or 2 points to change or to add to create a different function than those already described.  Explain your selection and the parent function that would best describe the set.
3. What functions have traits in common? Can you identify a trait that is unique to just one function?
4. Create “Who Am I” Riddles for 5 of the parent functions.  For each riddle use 4 to 6 clues.  Here is an example:  My graph is continuous.  My graph has an intercept at (0,0).  My domain is the set of all nonnegative real numbers.  My range is the set of all nonnegative real numbers.  The shape of my graph is sometimes referred to as an eyebrow.  What parent function am I? (The synthesis questions have been collected over time from various textbooks, activities and such. They are not original with me but I did not keep track of the sources.)

Concept Cards ... sorting can be helpful as students are working out the relationships in parent functions.