# Math Virtual Manipulatives

Today’s post is a recommdendation of several math websites with FREE virtual manipulatives.  Each site has their advantages and disadvantages, and you really just need to play around with them to decide which ones have the features and manipulatives you like best.

• Use of these help children of all grade levels with this math progression (from concrete to pictorial, to abstract).
• They are FREE, so no need to order and pay for them via a math catalog.
• If you load them onto your students’ ipads or chromebooks, there is no need to get out the tubs of manipulatives in the class (which we all know get dirty, and it takes half of the class period to distribute them IF there are even enough).
• Students can have their own set right in front of them.
• You can give the links to parents for students to use at home.
• While these are virtual on the screen, students can still manipulate them which is almost as good as the concrete objects.
• The teacher can save time without having to draw geometric shapes, rulers, base ten, etc. on the board.
• They are great for problem solving projects:
• How many different ways can you use the square tiles to show an area of 24 in the shape of a rectangle?
• How many different ways can you show 1/2 with the fraction bars or circles?
• Create shapes with angles that measure _____, _____, and _____ degrees.
• With the balance scale, show 3 x 6 on one side and balance it with another multiplication expression.
• How many different geometric shapes can you make using same size triangles? (example: 2 triangles can make a square)
• What are some you’d like to share with others in this blog???

Didax virtual manipulatives

Math Learning Center apps

This is a new one I just found out about via an online math conference. It has some really cool features. The activities and lessons are more for upper elementary, but the manipulatives are for any age group.

# More Number Talk Ideas – Part 2

As I mentioned in my last post (More Number Talk Ideas – Part 1), there are many ways to conduct Number Talks with your students. The last post focused on Picture Talks and Which One Doesn’t Belong (WODB). This week I am focusing on Estimation Mysteries and Data Talks.

Esti-Mysteries

Steve Wyborney has been super gracious to share his math estimation mysteries with educators via the nctm.org blog and through his website: https://stevewyborney.com/

What are they?  Each esti-mystery features a clear container with identical small objects (cubes, dice, marbles, manipulatives, etc.). Students estimate how many are in the container, then proceed through 4-5 clues revealed one at a time in a ppt format.  Clues give information dealing with number concepts such as even/odd, less than/greater than, place value, multiples, prime, composite, etc. With each clue, students can then revise their estimate to try to eventually match the actual amount revealed on the last slide. Different clear containers are used with each mystery to include ones with irregular shapes.

It’s really interesting for students to share how they arrived at their estimate, to list possible answers, then defend their choice.  And of course, the rejoicing when/if their estimate matches the revealed amount!

Data Talks

You may have heard of the youcubed website (https://www.youcubed.org) which is partially commanded by well-known Stanford mathemetician Jo Boaler and her co-hort Cathy Williams. You will be amazed at all of the math resources and task ideas for all grades at this site. I witnessed another webinar hosted by these scholars regarding the increased need for data science. While professionals surveyed rarely actually use the algebra, geometry, and calculus learned from high school or college courses, there is definitely an increase in the need for data science / statistics as noted in almost everything we do. So youcubed has made it their mission to ramp up data science resources. One already in the works is “Data Talks” which provides some real-world, interesting, thought-provoking data presentations ready for class discussion.  The link is right here:  https://www.youcubed.org/resource/data-talks/

You will find graphs and tables of all types (some very creative ones), with topics such as these:

• Steph Curry’s shooting and scoring % shown on a basketball court diagram
• Social media use
• Paper towel hoard in 2020
• Dice combinations

Before diving into the data presented, get students to notice first . . . “I noticed . . .”  and follow analysis with “I wonder . . .”  The “I wonder” questions promote ideas about trends and change in data.  Here’s a sample graph regarding possible outcomes when adding 2 dice (graphic from google, not youcubed.org):

Possible noticing and wondering:

• I noticed the graph goes up and then down symmetrically.
• I noticed there are 11 possible sums using 2 dice.
• I noticed the bar for 7 is the highest.
• I noticed numbers on the left side go up by .02 each increment.
• I wonder why 7 is the highest? What are ways to roll a sum of 7?
• I wonder what a graph would look like when actually rolling 2 dice numerous times? Will it be similar to this one?

I highly encourage you to check these out! I will add the sites to my resource list (top bar of my blog) for easy access.

Till next time . . .  Cindy

# Interactive math lessons and activities on NCTM

Resource – http://illuminations.nctm.org

This is a math resource I absolutely love! It is a product of the National Council for the Teachers of Mathematics (NCTM).

This site includes lesson plans and interactive activities. Search in several ways: by topic, by standard, or by grade level. Need some strategy games? Check out “Calculation Nation” (some of which can be played against other players), and “Brain Teasers.” I have just added this link to my Resources page (on my blog home page).  Pass this along to parents for them to use with their children at home!

Many of the lessons connect to exploration projects and literature. The interactive features are outstanding!! These are perfect for the smartboard, on laptops, or tablets. Doing Zoom lessons? Then these are also wonderful for sharing the screen to introduce or review concepts. Once you are on the home screen, click the Interactives box (right side) and then the desired grade level. There are dozens of great applets, but here are a few you might really like. I have linked them for easy reference, so just click on the  title and you’ll be there:

Dynamic Paper: Customize graph paper, number lines, spinners, nets, number grids, shapes (to include pattern blocks, color tiles, and attribute blocks), and tessellations. You can also choose inches or cm. These can be customized, saved and printed as jpeg or pdf. I created the spinner shown here from this application.

Five Frame and Ten Frame tools: Geat activities to build number sense using five or ten frames. These may take 1-2 minutes to load.

Cubes: Build a rectangular prism one cube, or row, or layer at a time and then compute the volume or surface area.

Coin Box: Drag and exchange coins. There is also a feature I like (the grid at the bottom right corner), which puts coins in blocks (by 1s for pennies, 5s for nickels, 10s for dimes, and 25s for quarters). This really helps see the value of the coins. Want more info about coin blocks? Once on the Coin Box page, click on the “Related Resources” tab.

Try these for fractions: Fraction Models (which includes decimal and percent equivalencies) and the Fraction Game.

Geometric Solids: Create a shape (either transparent or solid) and swivel it around to see all of the faces, vertices, and edges. It has a cube and pyramid as far as basic 3D shapes are concerned. I wish it had more that 3rd-5th students would encounter.

Here’s a nice multiplication game:   Two players (or a player vs. the computer) choose factors from the bottom bar to create products shown on the game board grid to get 4 in a row (and try to block your opponent from getting 4 in a row).  Be sure to see the directions included.

Some of the interactives require an NCTM subscription.  The ones I have listed above should be okay to access without membership. I have subscribed for years and just paid \$94 for this coming year. Well worth it if you plan on using their site extensively.  This subscription also entitles you to a print and and online journal, blog capabilities, and more.

Enjoy these and so many more!!! Let us know if there are others you recommend.  I’ll highlight more on my next post.

# Excellent FREE Online Reading and Math Sites – My favorites!

Are you looking for a place to find great quality FREE online stories / books that fit your child’s or students’ reading levels? What about instructional math videos just right for explaining math concepts, and virtual math manipulatives (especially if students don’t have any at home)?  I have 4 I highly recommend and will highlight below.

1. https://www.readworks.org/  Readworks.org is a very high quality non-profit site which can be used with individual children or the whole class.  You will find articles, ebooks, and a library. Search by grade level, lexile level, genre, skill, etc. Most articles have an audio function and can be presented digitally or printed. The text selections really help build background knowledge that a lot of students are missing. Be sure to check out their “Article a Day” program.  As a teacher or parent, you can create a class and make assignments.  Comprehension questions and free response questions are included. ALL of it is FREE.  They have webinars available to learn about all of the features. Well worth your time!
2. https://www.wilbooks.com/wilbooks-free-resources  If you are looking for leveled books for PreK-3rd grade students, then Wilbooks may definitely meet your needs.  There is a good selection of fiction and non-fiction leveled by grade level or guided reading levels A-M.  Levels A and B have around 30 titles each. Not all of the levels have that many, as it varies. The back cover of each book tells the grade level, guided reading level and word count in case you want to do a running record. If you want access to their entire collection, the price is VERY reasonable. I haven’t purchased it myself YET, but it states \$1.99 for a monthly individual account. I have been using these books and the students like them!
3. https://learnzillion.com/p/  This is a really good site for math instructional videos and lessons.  Learn Zillion used to be totally free, but like others you now have to purchase a subscription to get everything they have to offer.  BUT, by creating a free account you still get access to about 1000 videos. You can search by grade level, standard, key word, etc. The instructional videos are done very well and are easy to follow (at least the one’s I have viewed).  And they are short and concise.  These would be great to use with a zoom lesson in your class or as a parent who is searching for the right way to explain a strategy.  The objectives are clearly stated, videos are often also available in a slide format so you can explain it yourself, and you have the option to make assignments as well.

So there you have it, 4 great websites well worth your investigation!!  Do you have some to recommend? Just respond by clicking the little speech bubble.

P.S.  If you are interested in any of the following to meet your professional or personal needs, please go to the bottom of my “About Me” page for more details (black bar at top of this blog).

• Professional devlopment – private, job-embedded, workshop, or webinar
• Working as an online tutor
• Referring a student for online tutoring

# Virtual math tools

Every once in a while you come across something wonderful, and you want to share with your friends.  Well, I am doing that with this FREE website.  It is https://www.mathlearningcenter.org/resources/apps

Here is what you will find.  Click the i on each app and you get great visual instructions about the tool bar at the bottom of each app.  These can be used on your Smartboard as well as installed as an app on a laptop or ipad, etc. A few of the apps have a share / copy feature (a box with an arrow coming out). All of them have a writing tool to accompany the app.

• Fractions: Fraction bars or circles
• Geoboard:  3 different boards, put stretchy bands on (no more worries about breaking them with this app), use for area, perimeter, shapes, arrays, area of irregular shapes
• Clock: Program the hands and the clock (Roman numerals, minute guide), shade parts of the clock, show elapsed time
• Math Vocabulary Cards:  Great for review or quiz. Adjustable for different math topics and grade level. 3 parts on each review question:  Term, definition, picture
• Money Pieces:  Display and hide coins.  The coins can be shown as part of a block to relate to base ten blocks. The coins do seem a little small in size, however.
• Number Frames:  5, 10, and 20 frames, 100 grid, counters, and objects.  The 100 grid can be adjusted to make any size array (up to 10 x 10).
• Number Line:  Use for skip counting, addition, subtraction, fractions
• Number Pieces:  This includes base ten pieces. These can also be used to show the area model for multiplication.
• Number Rack (aka Rekenrek):  A great tool for primary grades. Based on use of 5 and 10 as benchmark amounts. Use 1-10 Rekenreks. Count by 5’s, Count by 10’s. Practice sliding the beads – it’s fun!  Here is a link from my blog on ways to use a Rekenrek:
• Pattern Shapes (Blocks): Compose and decompose shapes. Create using the blocks: Duplicate, rotate, change colors! The sillouette shapes enable you / students to use blocks to fill in.  Plus for intermediate grades:  There is an angle measure tool. Measure angles of the polygons presented.
• Partial Products Finder:  Make arrays. Slide the bar on the bottom or side to partition the rectangle into smaller parts. Tap on a section to see a different color.

I will add this link to my instructional resources for future reference.  Enjoy!

I’ll get back to phonics next time.  Have a great week!

# Decimals: Part 1 – The Basics (revised)

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes. Continue reading

# Geometry Websites

There are several great math websites which might help you and your students with geometry and measurement standards such as area, perimeter, volume, surface area, angles, etc.  The ones I am recommending are interactive and often customizable.  Check them out!! (Each title can be clicked to take you directly to the linked website.)

1. Geoboard by The Math Learning Center:  I love the concept of geoboards to help children create polygons and measure area and perimeter.  However, most teachers have ditched their physical geoboards. They are often in boxes relegated to the basement storage areas.  I get it, though.  They take up a lot of shelf space in the class, there aren’t enough rubber bands to go around (aka geobands), the kids misuse them or break them, they don’t stretch far enough, the pegs get broken, etc.

I think you will LOVE this app. Check out the little “i” on how to get the most use out of it, but it has 2 variations for the board size and you can show it with/without gridlines or numbers. There are different colored bands which you drag to the board and stretch to whichever pegs you need. You can shade in areas, copy, and rotate (which is helpful to see if 2 similar shapes are equivalent). There is also a drawing palette in case you want to freehand something or draw lines (and with different colors as well).

What are the possibilities with this?

• Use with primary students to create squares, rectangles, and other polygons. The teacher can elicit different responses with directions such as:  Make a square. Make a different size square. Make a trapezoid. Are any of our trapezoids the same?
• Creations can sometimes be recorded on dot paper – although I would reserve this for less-complicated shapes.
• Count the pegs around the shape to determine perimeter. The teacher might ask students to create a rectangle with a perimeter of 10 (or 12, or another amount). How many different ways are there? Be cautious with diagonal connections because they are not equivalent to vertical or horizontal connections. Think of how you can get students to discover this without just telling them.
• Show the gridlines to help students determine area.  Initially,  students may just count the squares inside the shape. Guide students to more efficient ways to figure this (multiplying, decomposing into smaller sections, etc.).
• This app is also great for creating irregular shapes in which students may decompose into smaller rectangles or triangles. Then check them with the standard formulas.

2. “Cubes” at NCTM’s site (Illuminations):  This one is perfect for volume and surface area.

• Volume:  You can use the gear symbol to select the size (l, w, and h) of the rectangular prism, or use the default ones shown. Then there are 3 tools used to fill the rectangular prism:  individual cubes, rows of cubes, or layers of cubes. I prefer using the layer tool to support the formula for volume as:  area of the base x height.  The base is the bottom layer (which can be determined by looking at the length x the width). The height is the number of layers needed to fill the prism. Once you compute the volume, enter it and check to see if it is correct.
• Surface Area of Rectangular Prism:  To calculate the surface area, you must find the the area of each face of the prism. Again, you can customize the size using the gear tool.  I prefer this as the shapes shown randomly often are too small to see. Yes, there is a formula for surface area — but conceptually we want students to note the surface area can be thought of in three parts. With a click on each face, this app opens (or closes) a rectangular prism into the 6-faced net making it easier to see the equal sized sections:
• Area of the front and area of the back are the same
• Area of the top and area of the bottom are the same
• Area of each side is the same
• Be sure to explore what happens when the prism is a cube.

3.Surface area with Desmos:  This link provides an interactive experience with surface area, using a net. This time, the three visible faces of the prism are color coded, which helps with identifying top / bottom; front / back; and side / side. The prisms on this site are also able to be changed so students can see how altering one dimension affects the surface area.

These three may be more relevant to middle school math standards.  Check them out!!  Also take a look at the “Resources” link (left side of web page).  There are plenty of other good links for arithmetic standards as well – too many to list here.  You may have to create a log-in, but it’s FREE!

Enjoy!  Do you have other websites to recommend? Let us know.

# Eureka Math Blog

I just found this blog for Eureka Math. It has ten very good topics to explore, especially for Lawton, OK users who will likely be directed to the Eureka Math curriculum (also known as EngageNY).  Plenty of good advice for new users. Put it on your list for the summer!!! Click below to get there fast!

https://greatminds.org/math/blog/eureka

I will also add this to my resources list.

Enjoy!  Cindy Elkins

# New OK Math Framework

At last, some help with regard to organization and implementation of the new math OAS (Oklahoma Academic Standards) has arrived!!!!

The OK State Dept. of Education (via their directors of elementary and secondary math) has assembled a great team of math minded teachers and experts to put together a framework of the newly adopted math standards for Oklahoma. Here is the link: OK Math Framework. Look for the following features:

• Introduction video (short) – on the lower right side of home page
• Action and Process Standards
• Suggested Learning Progression
• Objective Analysis

Suggested Learning Progression: This is partitioned off into units, suggested timeline, and objectives. Each unit is presented as a bundle of linked objectives. Many objectives are repeated throughout the year, while some objectives are split so that part of the objective is taught in one unit and completed in a later unit (shown by strikethroughs). Makes so much sense!!! Clicking on the title of the unit (ex: Place Value) will take you to another view with sample tasks.

Objective Analysis: Click on any objective number (ex: 1.N.1.4) and you will see a more detailed explanation of the objective, along with student actions, teacher actions, key understandings, and common misconceptions. Continue reading

# Illuminations NCTM Interactives

Resource – http://illuminations.nctm.org

This is a math resource I absolutely love! It is a product of the National Council for the Teachers of Mathematics (NCTM). This site includes lesson plans and interactive activities. Search in several ways: by topic, by standard, or by grade level. Need some strategy games? Check out “Calculation Nation” (some of which can be played against other players), and “Brain Teasers.” I have just added this link to my Resources page (on my blog home page).

Many of the lessons connect to exploration projects and literature. The interactive features are outstanding!! These are perfect for the smartboard, on laptops, or tablets. Once you are on the home screen, click the Interactives box (right side) and then the desired grade level. There are dozens of great applets, but here are a few you might really like. I have linked them for easy reference, so just click on the  title and you’ll be there:

Dynamic Paper: Customize graph paper, number lines, spinners, nets, number grids, shapes (to include pattern blocks, color tiles, and attribute blocks), and tessellations. You can also choose inches or cm. These can be customized, saved and printed as jpeg or pdf. I created the spinner shown here.

Five Frame and Ten Frame tools: Geat activities to build number sense using five or ten frames. These may take 1-2 minutes to load.

Cubes: Build a rectangular prism one cube, or row, or layer at a time and then compute the volume or surface area.

Coin Box: Drag and exchange coins. There is also a feature I like (the grid at the bottom right corner), which puts coins in blocks (by 1s for pennies, 5s for nickels, 10s for dimes, and 25s for quarters). This really helps see the value of the coins. Want more info about coin blocks? Once on the Coin Box page, click on the “Related Resources” tab.

Equivalent Fractions: Build different fractions in circular or rectangular format. Compare them and see them on a number line. You can manipulate the numerators and denominators to see fractions change right before your eyes! Others for fractions: Fraction Models (which includes decimal and percent equivalencies) and the Fraction Game.

Geometric Solids: Create a shape (either transparent or solid) and swivel it around to see all of the faces, vertices, and edges.

For your graphing needs, check out the Bar Grapher, Circle Grapher, and Data Grapher. With these tools you can create graphs using any of your own data. Some of these need Java installed.

Enjoy these and so many more!!! Let us know if there are others you recommend.