Column Algorithms (Place Value)
updated September 10, 2020
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This page has helpful links and LOADS of free resources to help you plan your second grade year. Enjoy!
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Column Algorithms (Place Value)
for Class 2
Pinterest - Renee Schwartz
My curated collection of visuals! Browse sample main lesson book pages, watercolor paintings, chalkboard drawings, etc. for
Note: Of all the math blocks that we teach over the years, I think this one is the most important to do with the
Montessori materials, to help children understand place value, carrying, and borrowing
in a deeply hands-on way. I do think in this case that they are absolutely essential. Get the hands-on materials,
not just the place value stamps.
Montessori materials that I see as compatible with the Waldorf Column Algorithms block:
Golden Bead Material
Number Cards, 1 to 9000
There are lots of YouTube videos made by Montessori schools which show how to present and use these materials.
For Montessorians: If you do this block first, and then the Shapes & Number Patterns block (where I recommend using the Colored Bead Material),
students will be in a good position to then move into the Golden Mat and Checker Board work.
Other Helpful Links
Activity One - Gems / Pouches / Basket
Many years ago I was at a homeschool training at Barbara Dewey's and a woman named Misty
shared with me what she had done to introduce place value. A group of families worked together to set
this up for the children in their homeschool group. They laid out 100
individual gems in a long line on the ground. The children counted as they picked them up: 1, 2, 3, etc.
When they got to gem 9 their hands were beginning to get full. Right beside the tenth gem was a small pouch. They made
a group of ten gems in a pouch, then picked up the eleventh gem, etc. After nine pouches (each of ten) and nine
individual gems, they finally reached gem 100! For this they were provided the tenth pouch and a basket to put all ten pouches in.
It was a lovely idea and she said they always understood it thereafter.
To do this
in a way that is compatible with the Montessori color coding for the digits, use green glass gems, small blue pouches
for the 10, and a large red basket for the 100.
1 lb. green glass gems (100+)
for small blue pouches of 10:
Etsy white linen favor gift bag
beautifully made 100% white linen and so easy to dye blue
LinenForest sells these as a set of 10 (and it would be fun to dye these with your child first and keep a mystery what you were going to use them for...)
red willow basket with folding handles
To introduce this
place value activity with a story you could use The White Snake from Grimm's
(organizing the grains of millet into sacks)
Activity Two - Place Value Color Coding
Practice writing numbers to the hundreds place with the digits in the correct color
units - green
tens - blue
hundreds - red
Activity Three - Introduce Golden Bead Material
Read Redwoods by Jason Chin
Some useful numbers you can make from this story are
2,000 ; 200 ; 29 ; 2 ; 1,600 ; 370 ; 379
It is important to see that the digit 2 in different places represents different things. As one litte girl remarked to me,
"It's still two but they count in a different way."
For an SWI connection, you can show the base < Titan > and the word sum for < Titanic >
Titan + ic
Introduce Small Wooden Number Cards with Box
(if they ask why the thousands place is green, you can simply say
"hmmm, I wonder...")
Introduce unit beads, ten bars, hundred squares, and thousand cubes
Small Wooden Number Cards with Box
100 Golden Unit Beads
45 Golden Bead Bars of 10 with Box
10 Wooden Hundred Squares
9 Wooden Thousand Cubes
100% Cotton Small Working Mat
this is placed under your work as you build and work with the Number Cards and Golden Bead material; using a mat to delineate your work space is an important
part of the Montessori philosophy and teaches the student to show care and respect for the materials
Practice making a quantity with the number cards and then representing it with the beads
Stack the number cards to show standard form; slide them apart to show expanded form and to make it easier for your child to understand and make the given quantity
Note: Writing a number in color and building the quantity vs. building a quantity and then writing the number in color vs.
saying a number aloud and then writing it in color and making it...
these are all very different skills so
do many variations.
You can also have them say a number and you write it in color and they tell you whether the colors are correct!
A simple story that can give you lots of inspiration for making and writing numbers (how many animals hopped onto that raft? 47? 1,210?)
is Bear Came Along by Richard Morris. I also chose this for more practice with SWI, and we did the
SWI Investigation of < water >.
Activity Four - 45 Layout
Activity Five - Crocodile Game
Build both numbers, then compare them
Go to the thousands place, using this as the opportunity to talk about why thousand is green (it is the Units place of the Thousands Family, thus
For some children the comparison is easier to see with the Stamp Game
You can also make the numbers with BOTH materials: this is 2712 compared to 2722
Activity Six - Dice Game
Go to the millions place, and teach how to read large numbers
Talk about the names of the families: the Millions Family, the Thousands Family, and the Simple Family (which
has no last name)
When you get to the comma, you say the last name of the family
Definitely read A Million Dots by Andrew Clements
Activity Seven - Infinity Street
A nice book to introduce this lesson is Is a Blue Whale the Biggest Thing There Is?
by Robert E. Wells
Here is a helpful blog post with photos for the Infinity Street lesson
We have houses and mailboxes up to septillion, and slips of paper with the names of all of the families up to novemdecillion
This is a fantastic extension for children who are learning about prefixes (which happens in Second Grade).
I like to give them the slips of paper
after septillion and have them figure out how to put them in order.
Of course you can refer to the months of the year.
I have also found Zin! Zin! Zin! A Violin by Lloyd Moss to be extremely useful in this lesson,
especially for sliding under the radar the prefix
< sex > as representing six (trio, quartet, quintet, sextet, septet,
Note: They see why September / October / November / December are months 9 / 10 / 11 / 12 and not
months 7 / 8 / 9 / 10 (as would be more logical) next year in the Calendar study.
Activity Eight - Adding with Regrouping
Journey to Cahokia: A Boy's Visit to the Great Mound City by Albert Lorenz, or another book
that involves barter or trade (the trade must be a fair one, so don't choose "Hans in Luck")
Here is a sample problem. Make and add with the GBM, 213 + 128
Write the digits in color and arrange them vertically on the paper
Have the child take ten individual unit beads back to the store and trade them for a ten bar. Tell them
that you're trying to make the number in the simplest way possible. ALWAYS add beginning
with the units place and then move to the left.
You should then have 1 unit bead, 4 ten bars, and 3 hundred squares. Write the answer in color: 341. DO NOT
show your child how to do this problem on paper. The paper is the physical representation of the active work.
Other addition problems we've used (always write digits in color for problem & answer):
679 + 120
1214 + 65
Play Crocodile Game using two addition problems; the crocodile eats the larger sum:
236 + 23 ____ 242 + 7
And don't forget to work with multiple addends. This is especially helpful if a child is trying to doing things
with mental math and you want to bring him/her back to the material.
Students need to have completely internalized
the concepts through repeated experiences using the material, or you will find that they suddenly stumble when the problems get more difficult
or you move to a new operation. If a child can carry with addition but can't figure out borrowing in subtraction, they never
really understood regrouping in the first place!
Crocodile Game with two addition problems with multiple addends:
326 + 12 + 4 + 1 ____ 314 + 3 + 2 + 18
It's also fun to build something with the golden bead material, like a castle or a mandala design, and then calculate its value. This often
leads to practice in regrouping (10 beads is a ten bar, 10 ten bars is a hundred square, 10 hundred squares is a thousand cube)
Activity Nine - Subtracting with Regrouping
You can create practice problems with or without a story. Math problems can be found everywhere!
The Titanic: Lost... and Found
by Judy Donnelly
page 18 static subtraction: 2,227 - 1,100
page 37 dynamic subtraction: 2,227 - 705
page 43 static subtraction: 1985 - 1912
Activity Ten - Multiplying with Regrouping
Activity Eleven - Dividing with Regrouping
The whole idea behind division is "sharing out fairly." It's fun to read a story like The Little Mouse, The Red Ripe Strawberry, and The Big Hungry Bear
by Don and Audrey Wood.
Division is the only operation where we begin by sharing out the LARGEST pieces first (thousands), since to regroup
here you have to go down to the next value piece. With all of the other operations you begin solving the problem at the units place.
If you don't have the skittles from the Stamp Game to use for this you can
also use peg dolls or even wooden animals. Do not go past dividing by 9 at this point. Two- and three-digit divisors are covered
in the Stamp Game (Decurian Division and Centurian Devision).
Activity Twelve - Introduce Stamp Game
Continue to work on daily math practice and adding / subtracting / multiplying / dividing with regrouping throughout the school year.
This final activity, Introducing the Stamp Game, will come at some point in the school year and will vary per child depending on when
he/she has fully internalized the Golden Bead material. Dr. Maria Montessori was famous
for all of the in-between math materials she created to take the child sequentially on the journey from concrete
to abstract. Do not just jump from the Golden Bead Material to paper & pencil. The Stamp Game is the next step.
You can also make a Stamp Game yourself using small squares of colored construction paper. That would be completely fine (and when Maria
Montessori invented it she did use actual paper stamps). Since it is the next step in abstraction, keep the color coding intact. And
your child can also help make the squares.
I do NOT suggest making your own Golden Bead Material. It needs to be perfectly proportionate,
and I think that having it be store bought & special is important as well.
If you are a Waldorf homeschooler or classroom teacher interested in using Montessori materials
in the Waldorf environment, and you have questions, I'm happy to schedule a phone or Zoom call! Click on
Consulting to email me.
My blog posts from teaching this topic: