Tuesday, September 28, 2021

Using Virtual Manipulatives to Strengthen and Extend Your Student's Math Learning

Using Virtual Manipulatives to 

Strengthen and  Extend 

Your Student's Math Learning

Enrichment Workshop for Parents of K-5 Students at Providence St. Mel School, Chicago

    Wednesday, September 29, 2021

Marsha Landau, Ph. D.      mathmarsha@gmail.com

Introduction to Virtual Manipulatives

We'll start by looking at a few of the virtual manipulatives that are included in the REVEAL Math e-Tools kit.

Note: Students should be able to see and access the etools, as soon as they log into their account. They can be found on the landing page. There is an eToolkit directly below the "To Do" list. After clicking on the eToolkit, they will see a menu that includes "Add eTools." When they select that, they are given an option of tools to use.

The complete list:

Today's focus, time permitting:

Base 10 blocks



        Make fourths on the geoboard. Record here:

    (from a unit by Tom Scavo)

Pattern blocks


Websites with many virtual manipulatives:

polyominoes; tangram; measuring tools; Tantrix tiles, under Patterns and Art; number tiles; multiplication grid; fraction bars and circles; algebra tiles; spinners; polyhedral dice; coordinate system; tables. There is a tutorial

geoboard; fractions; area/perimeter explorer; interactive clock; stopwatch; spinner (up to 9 regions); dice; coin flip; tangram; pattern blocks; scale; place value chart; hundreds chart; two color counters; color counters; graph color bars (Cuisenaire rods); graphs; cube; number line; calculator (4 operations only); thermometer (0-100 Celsius)

    Unifix cubes; ten-frames; number lines; two-color counters; color tiles; base ten blocks; math balance; dice; spinners; pattern blocks; geoboard; algebra tiles. 

fractions (fraction bars and fraction circles); geoboard; clock; money pieces; number frames; number line; number pieces (base ten blocks); pattern shapes (pattern blocks); partial product finder (multiplication arrays)

coins, counters, centimeter rods, Dienes (base ten) blocks, dice, fraction bars, geoboard, hundred square, pattern blocks, pentominoes, tangrams

Virtual Manipulatives for Elementary Math - YouTube 

start watching at 10:50, featuring Math Learning Center

at 24:30 doing addition

NCTM Illuminations

coin box and equivalent fractions manipulatives

among the apps: factorize; geometric solids; grouping and grazingadjustable spinner; Bobbie Bear; cube nets; concentration (match different representations of numbers, shapes, etc.); ten frame; area tool; cubes (for finding volume of a box); fraction gamehandshakes problem; bunny times

most give feedback to student--student can do this without the parent present

Coin box--some built-in tasks

Five Frame--for the little ones--show small numbers--lots of sound effects and audio support

Geogebra   at about 38:00 in the video

lots of tools posted by teachers

Video goes to a tool for adding fractions; use expand tool to find equivalent fractions that work to combine the fractions

how-to videos from The Math Learning Center  Each video is about 3 minutes
    The Number Line App (old video--website is now more user-friendly) shows how to use it; very clear!

For the geoboard there is a second video with updates explaining new tools. Very cool!

Pattern Shapes App (pattern blocks in the correct colors) drag, rotate, change color, duplicate, delete; blank workspace or square or isometric grid. Can choose silhouette outlines to fill in like a puzzle, or make your own (?). I can choose an outline, share the link, share the code, and everyone can try to fill it in with shapes.

Arithmetic Practice -- 4 basic operations

Saturday, March 20, 2021

Enrichment for Grades 2 and 3

 Tasks for exploring math that are likely outside the school curriculum

Task 1.    What can you do with 4 identical isosceles right triangles?

Go here: Mathigon Polypad.

Add four identical isosceles right triangles to the work space:

Explore what you can create when you place the triangles in such a way that they have a side in common.

For example, this is a hexagon:

Sketch your results. How many different designs can you make? 

Did you make a single triangle? a square? a non-square rectangle? a non-special parallelogram?  a trapezoid? 

Task 2.    Sharing a cake

Go here: 25-pin geoboard

Jessica pretended to make a cake in this shape on her geoboard:

Show all the ways she can "cut" the cake into two equal shares for herself and her brother James.

Task 3.    Towers

Go here: Unifix

Here are some examples of a unifix tower 5 cubes tall:

Find all the ways you can make a tower 3 unifix cubes tall when you have two colors, brown and yellow, available.

Then write a letter to a student who was unable to attend this session. Describe all of the different towers that you built that are three cubes tall, when you had brown and yellow cubes available to work with. Why were you sure that you had made every possible tower and had not left any out?

Task 4.     Tetris and beyond

Go here:  Color tiles

Explore the shapes you get when you put squares together in such a way that they have a whole side in common. The figure below shows two blue TRIOMINOES (composed of 3 squares) and two examples of green QUADRIMINOES (composed of 4 squares). Two shapes are the SAME if you can slide, flip, or turn one of them so it fits exactly on top of the other.

Can you make any DIFFERENT triominoes from the two shown? What about quadriminoes? Sketch your findings.

Challenge: Make all the different PENTOMINOES that are possible. (Hint: There are more than 10.) Sketch your findings.

Wednesday, October 21, 2020


Parent Resources for Supporting Math Learning at Home

Interactive Manipulatives Online

Virtual Manipulatives from Glencoe -- algebra tiles, attribute blocks, base ten blocks (through thousands), pan balance, centimeter cubes (base ten blocks for ones, tens and hundreds), clock, color tiles, connecting cubes (similar to Unifix), currency, fraction circles, fraction tiles, geoboard, geometric solids, number cubes (similar to dice), number line, pattern blocks, spinners, tangrams, thermometers, two-color counters [requires Adobe Flash] [Note: Adobe will be taken down at the end of December]

mathsbot.com offers a vast array of virtual manipulatives, including two-color counters which are easy to flip between red and yellow. a recent find! (link added 11-15-2020)

Counters – GeoGebra

Areas with Geoboard – GeoGebra

Mathigon Polypad offers pentominoes and tangrams in addition to repeating manipulatives already listed: polygons, number tiles, number bars, fraction bars, algebra tiles.Scroll down on the left hand side to access all available manipulatives.

Free Virtual and Online Manipulatives from hand2mind is really an offer for a 30-day free trial of the virtual manipulatives plus downloadable printable manipulatives and activities. Example: Pattern Blocks.

Using Khan Academy as an In-Home Tutor

Suppose you need help  in order to help your third grade student with the distributive property.
You can find relevant pages at Khan Academy by doing a Google SITE SEARCH for that topic:

Open Google and search for       site:khanacademy.org "distributive property"

Here are a few of the top results:

Distributive property explained (article) | Khan Academy

Distributive property when multiplying (video) | Khan Academy

Intro to distributive property (article) | Khan Academy

    (I really liked this one--the page title has the word "Intro" which suggests it will start teaching the concept from the beginning. We can look at this page together."

So... I would start with the Intro, then try the video.
The video is part of a sequence of short lesson pieces that do a good job with the topic.
If your student needs more, return to the search and do some exploring.

Khan Academy Kids
Khan Academy Kids is a free online education program for children ages 2 - 7. The mobile app was designed by child development experts at Stanford University and engages kids in core subjects like early literacy, reading, writing, language, and math, while encouraging creativity and building social-emotional skills. Always free, no ads, no subscriptions, with an offline version available for when internet access is difficult.

Supporting Your Student's Math Learning


A "frame of reference"-type article for those who are interested: Addressing Unfinished Learning After COVID-19 School Closures  50 pages
From the Office of Education, two resources:
and Where Can Teachers, Caregivers, And Parents Find Free Digital Resources To Support Early Math Learning? This resource from REL West addresses frequently asked questions about where teachers, caregivers, and parents can find digital resources to support early math learning. (Links to other resources)
Wide Open School:  
I looked at grade 2; includes links to online books--cute!  Example: Addtron
 An excellent collection--check it out!

 with links to free worksheets

Dad's Worksheets This links to a sample--mixed addition and subtraction word problems, suitable for grade 2. There are thousands of worksheets, with answers, on all computation topics.

for a free ebook of math puzzles, grades 1-5 (and beyond) (requires an email address). Note--the Critical Thinking Co. also offers free samples from every product, downloadable PDF files.

For your students (with your guidance):

Be creative! What happens if we dip a paper towel roll into water and unroll it? What shape do you predict you get? Try it! (Same for toilet paper roll) See this: demonstration
I explored the app for 4-year-old--comfortable, slow pace, nice graphics, appropriate content.

practice skills; excellent sets of mixed word problems

MathPickle puzzles and games organized by grade

Scope & Sequence – Curriculum – Zearn  A new find--some of it tailored to e-learning.
There is a Learning Recovery Packet for each grade, K-5. The site has connections with New York State Common Core math curriculum and assessments (including solutions). Great for practice.

 Family card game booklet  for older preschool through early elementary grades. All you need is a standard deck of cards. I got my start with numbers playing old-fashioned 7-card and then 10-card Rummy and Fantan.Later I played Pisha Pasha, a variant of the classic War, but much more fun. 
(for middle school) Math Mapper offers free diagnostic practice problems and assessments designed to evaluate middle school students’ mathematical progress and immediate, item-by-item feedback for students to help them continue working at their ability level until they fully understand a mathematical concept. The "Resourcer" takes you to other free web sites for additional instruction--for example, this page at Illustrative Mathematics  about function rules.
Some of the resources reach down into the intermediate grades as well--for example, this page addressing factors, multiples, and common factors: Counting Cogs.
(beyond K-6) 
a free, topic-aligned tutorial video library, each with a corresponding video notetaking guide

Strengthening Your Own Internet 'Sleuthing" Skills

This time we are looking for   "Math Expressions" "grade 4" "equivalent fractions"
Be sure to use quotation marks.
I generally look for sites that appear to school districts--teachers often post links to large chunks of the textbook and/or ancillary materials.

For this search I found a few excellent prospects:

These two pages look the same at first glance, but they are NOT.
The second link says it is for grade 3; I'm looking for grade 4. So I'll start with the first link.
Scrolling down, this looks like material intended for teachers--if I'm trying to help my student at home, at the moment I AM the teacher, so I'm interested.
I see lots of excellent explanation, and it is geared to the Math Expressions series, so what I learn here and guide my student to learn, should be consistent with what is being taught at PSM. 
Now comes the sleuthing part.
I want to find out what other resources this site might include, or lead me to.
First, I need to study the URL:  
I know this is a Unit 7 Overview, so in the URL the %20 characters are to be ignored.
Now I see where the 7 shows up. What if I want to look back at Unit 6?
All I have to do is change the 7 to a 6 and see where that new URL takes me. And it works!
How can I find comparable materials for grade 2?

I can do a Google SITE SEARCH for Grade 2 trying shorter bits of the URL that worked for Grade 4.
I can try:   site:https://www.svsd.net/cms/lib5/PA01001234/Centricity/Domain/361/  "Grade 2" "Math Expressions"

Poking around the internet, I finally came to what I think should have popped up much earlier: the publisher's website.

Now I can tinker with this URL and explore all the pages that are available online.
One of my first discoveries:        http://eduplace.com/math/mthexp13/g4/probtypes/pt_g4_01.pdf

I still want to look at the first URL that looked interesting:

This is a useful collection of resources that go beyond the textbook to other websites that include virtual manipulatives, games, etc. Worth a look.

Following my instincts from link to link, within a half hour I found a website with complete high school math textbooks, available free. These are all by Ron Larson and Laurie Boswell, two big names in math textbooks.
Free Easy Access Student Edition

Many of the following resources were featured in the 2019 Parent Enrichment session:

from Ways a Parent Can Help with Math

Websites and Readings for Parents

mathmarsha blog: CTD Parent Seminar April 18, 2015 focused on apps for mobile devices

mathmarsha blog: CTD Parent Workshop 11/3/12 addressed math and literature, projects, games, manipulatives, etc.

Helping Your Child Learn Mathematics With activities for children in preschool through grade 5      (One of the few resources for preschool math activities)

Strength in Numbers | National Numeracy Challenge helps adults learn the math needed for everyday life. A good place to start if you yourself are not comfortable with math.

High School Math Review Tutorials Problems - a free resource with examples, problems, explantions and exercises.  Parents can use these free resources and problems to help your child learn math.

Maths activities for children | Family Maths Toolkit has collections of activities for age ranges 3-13. Topics include: around the house, math and money, games, out and about, books and TV. From UK.

Prekindergarten Mathematics Family Newsletters | EngageNY

Maths Genie - 1-9 GCSE Specification Revision  British page with links to videos explaining all the major mathematics topics from grades 1-9. There are some language differences, but helpful nevertheless. More British videos, lessons, and exercises here: The Maths Teacher



Here's an example of an app that's available online from NCTM. It's pitched at grades 3-8, but I wouldn't hesitate to share this with much younger children--it provides an exploration of algebraic thinking about equations using a balance scale:

Pan Balance – Shapes

Here's a screenshot:


If you remember LOGO or Turtle Geometry from the early days of home computing, you'll like Turtle Pond. This introduces children as young as pre-K to rudimentary coding.

I also like Pick-a-Path (one of my favorites--plays fanfare for correct solutions! Suitable, at the higher levels, for grades 3 and up), Isometric Drawing ToolTessellation Creator.

Several good online games are available at PBS Kids Odd Squad Games. In particular, I liked "Down the Tubes." The underlying math includes number stuff (different sums to make a particular target length of tube) and geometric transformations (turns and flips). The levels build slowly, but eventually get mathematically interesting. The graphics are sufficiently eye-catching so your child will have the patience to continue to a challenging level.


NOT free:

DragonBox - Discover the game of Math 

The game claims it "secretly teaches algebra," which is true, but I think there are advantages to having a parent or teacher ask the occasional question or suggest recording what is happening in the game on paper so the algebra can be made more explicit. A good description of the game variations can be found here: DRAGONBOX ALGEBRA - The game that secretly teaches algebra.

Mystery Math Museum  $3.99 on iTunes.  (It has occasionally been offered at no charge. I was lucky that way!)  My granddaughter asks to return to this one. I'll show you why I like it enough to spend $ on it.

Greater Gator on the App Store $.99 Currently (as of August 29, 2020)
(Available as a box of cards doe $10.95): https://www.24game.com/p-58-single-digits-ages-9.aspx)

Make 24 using each of the four numbers exactly one time. 
For example, with the numbers {1, 2, 3, 4} I could get
(1 x 2 ) x ( 3 x 4) = 24 and say 2 x 12 = 24
or I could write (2 x 3)  x (4 ÷ 1) = 24 and say 6 x 4 = 24.

Math Puzzles - Room 703
Note: the number sit on the box. This card shows 1, 2, 6, 6.

Math Puzzles - Room 703

Here the numbers are 2, 2, 6, 9.
Math Puzzles - Room 703  

*Links to free mobile device versions of pick-a-path, equivalent fractions, and others from the National Council of Teachers of Mathematics here.

Big Seed is one of my favorites. It develops visual-spatial skills as you flip shapes to cover the presented puzzle. I find it much more challenging than my granddaughter does!

There is a tangram app for iPhone and iPad available for free:

See post featuring manipulatives for physical tangrams.

Khan Academy Free.

GeoGebra on the App Store  Free. Dynamic geometry and algebra.


https://www.nnchallenge.org.uk A British website with support for students at many levels. Login required, but free site.


Maths bingo (mobile app)

Everyday Activities

Cooking uses numbers

Act like an architect: Draw your bedroom | GreatKids

Math and Literature

mathmarsha blog: Math and Literature: Books and Suggestions/Links for How to Use Them

Math and Children's Literature -- a GREAT resource!

Math Read-aloud Books

List of recommended titles

Fruit fractions | GreatKids

Good Books for Problem-Posing Extensions


Please, read stories written by me and illustrated by my granddaughter. They have been "published" online by the website freekidsbooks.org.

A Fractionated Fairytale – A maths lesson story

Steps of Wonder – Advanced Maths Story

Yes, That’s My Mother! – Maths stories for elementary school

And one more available on this blog:

Not Your Average Halloween

Stuart J Murphy has written a plethora of easy readers for young children that address math topics. I'm not wildly enthusiastic about them, but they do contain pages of activities and games for parents to use to go beyond the stories. His website has free pdfs.

A popular title is:

Math Curse hardcover about $10 from amazon; also a video on amazon prime, "free" for subscribers.

How to access Math Expressions online:  start here and follow the directions for your child's grade level book and resources.

Holidays and Seasonal

Pumpkin Math Estimates (grades 8 or 9)

Halloween Math Worksheets, Exercises for Preschool, Kindergarten, First to Sixth Grades

Christmas Math Worksheets, Exercises for Preschool, Kindergarten, First to Sixth Grades

Note: Alway open and look over each page before sharing with your child. When I checked the links on Sept. 22, 2019 the ads on some of the pages were for underwear!

Skill Practice

Problem Solving and Puzzles

Problem-Solving Strategies (click "see inside")