Using the Montessori Peg Board

Here are two examples of pre-algebra Montessori “work”

The Peg Board

The Peg Board has many uses, here is the example we worked on this week.  The purpose is to take a square and make it into a smaller square that is equivalent to the larger square. 

For this example, we took the number 625.

First, my son "D" takes the units pegs (units are always identified as the green color, tens are blue and hundreds are red.)  At this age, he is already well prepared to have the amounts represented by just a color – so there is some abstraction in this work that has come with years of being in the Montessori classroom using these representations.

Here is the peg board with the units pegs in a square that is equal to 625. 

Second, he now counts over 10 pegs from the side, and replaces each row of 10 with a blue peg (which represents 10.)

 

Third, he continues with this process of replacing rows of 10 pegs, with a blue peg.

Fourth, he then begins to replace rows of “10s” pegs with the 100’s peg (red) where he is able.

This process continues until the smallest square possible is built – equivalent to the original square of 625!  From here we can see that 625 is equal to 25²

He learns to document this process as well as he goes along.  Which you'll see in the next example.

Working with Squared Numbers

For this one, I’ll use the same numbers for sake of ease.

Materials needed include:

• Montessori Decimal Bead set (up to the ten bars)
• Montessori number tiles (I take them from the checkerboard multiplication set we have)
• Pipe cleaners – used for the “()” in the problem
• Cuts of paper with “+” and “X” on them.
• Montessori “hundreds” squares (each represent 100 beads)

Working to find the answer to 25²

First, we agree on a few things:

We agree that 25² = (20 + 5)² = (20 + 5) X (20 + 5)

Now he can work on the problem visually to come to his answer.

Step 1: He lays out the problem using the actual amounts shown by the beads. (See part of the problem built out visually below)

The "2" is in blue which represents the 10s place, making the 2 stand for "20".

Step 2: Once he has the two rows of the problem both written and shown visually with the beads and tiles, he begins to use the hundreds squares and beads to find and show the answer to the problem.

As he finds the answers, he finds he will need 4 hundreds boards (represented by the squares) and then he finds how many 10 bars he will need - and places them in the binomial square shape (which he is already familiar with from previous works.

The binomial square!

So he finds that he needs 400 + 100 + 100 + 25

400 - represented by the 4 100s squares

100 + 100 represented by the 10s bars

25 represented by the 5 bars (light blue bars have five beads on them)

His written notation looks like this:

From here he can work through the answers to multiple problems, practicing the steps over and over as well as practicing writing out the problems in his math notation workbook.

I didn’t know how to do this until Jr. High school, (and even then it was pretty shaky) but perhaps if someone had shown me how all squared numbers actually make a square – I might have found it interesting and concrete enough to try a bit harder.  I’m looking forward to practicing this myself so I can help my daughter prepare for algebra the same way!

If you'd like more examples of this with more detailed instructions or pictures, feel free to contact me and I'll be happy to share examples to help you along!

Montessori Grammar Symbols

Maria Montessori, knowing that children respond well to shapes and colors, used a system of colored shapes to represent the different parts of speech.

These are taught during the early years in the children’s house – introduced as the 3-D shapes initially.  Then, in 1^st grade and beyond, they are introduced again and used in many ways in the classroom.

By the time the child begins sentence analysis, these are already very familiar and the names of the words are quickly and easily memorized – making grammar study not so intimidating (not to mention not so boring!)

(Picture borrowed from www.montessoriforeveryone.com)
Each shape and color has a meaning:

1. The Noun is represented by a black triangle. The triangle stands for the pyramid, one of the first human structures, and black is for carbon, believed to be the first mineral discovered by humans.

2. The Adjective (small dark blue triangle), Article (smaller light blue triangle), and Pronoun (large purple triangle) are part of The Noun Family and so use the triangular shape with different colors.

3. The Verb is a circle, to represent the shape of the sun which gives life. The red color also symbolizes life (blood). The verb gives life to the sentence. The Adverb is a smaller circle and is often pictured orbiting the verb like a planet; it depends on the verb for existence.

4. The Conjunction is a small pink bar which represents a link in a chain; the Preposition is a green crescent to symbolize a bridge. It bridges the noun to other words in the sentence.

5. The Interjection is a gold triangle with a circle on top; it combines the symbols of the noun and the verb together

Montessori Language Sample Lesson – Symbolizing:  The Predicate

In this lesson, my daughter is already familiar with Nouns, Pronouns, Verbs and Articles.  I’ve just given a lesson on Predicates and now she will practice what she has just learned.  This will take a lot of repetition with various activities – but this is the first step I use with her because it keeps her interest high and her failure rate to a minimum!  (Another motivating factor for her is being able to self-correct.)

To add interest, I use cut out words which are color-coded to match the Montessori grammar symbols.  I laminate them so they will last.  This helps her come up with creative sentences to use for her first attempts at symbolizing sentences.

Later, I'll give her a page with sentences and eventually, a whole paragraph from a story or book to symbolize.  I may just ask her to start with symbolizing 1 or 2 types of words.  My 4th grader can symbolize a paragraph with nouns, verbs, articles, predicates, interjections, adjectives, adverbs and conjunctions.

You don't have to purchase expensive Montessori materials for this, you can simply create your own or get them online and laminate them.  I did purchase  small template with the "official" shapes so the kids can make their symbolizing more neat - and it's a bit more fun as well!

Montessori Algebraic Pegboard - Square Roots

Here’s my first attempt. I'm starting with something very easy! This example uses some affordable materials that can be repurposed for many different math concepts. In this example, the goal is to help the child understand how to find a square root of a number with two periods.

 

FINDING THE SQUARE ROOT OF 625 ON THE ALGEBRAIC PEGBOARD

Preparing the board to find the square root of 625... Note that Montessori kids already understand that each color represents a value. Green = units, blue = 10s and red = hundreds

We put 6 red beads (6 hundreds), 2 blue beads (tens) and 5 green beads (units) in their respective dishes.  Starting with the red beads we make a square.

Below: Building the largest possible square with 100s pegs, 4 X H square. The student will not be able to add another row to the bottom and right sides with the 100s pegs. So they will need to exchange the remaining 100s pegs for 10s, giving 20 more 10s, for a total of 22.

Below: Building the largest possible rectangles with 10s pegs

Below: Completing the square of 625 with units pegs. They have now built a complete binomial square. They find the square root by counting the values of the pegs along the bottom or right side. Each of these sides has two 10s and five units. They review that these values can be written as a binomial (20 + 5) or as a monomial (25).

Below: The square root of 625 is 25

We repeat this activity with several numbers between 100 and 9,999 that have square roots without remainders, and then advance to numbers with square roots with remainders.

If you're thinking, "Well, that was a lot of work when there are many shortcuts that could have been taken." I'd say - you're right. But the purpose behind all Montessori materials is to first be sure the child has a deep understanding of what is happening in an operation, before he is shown any shortcuts. This develops a deeper understanding that will last a lifetime, rather than a memorization of steps that last only until the test on Friday.

The saying is really true... "To teach is to learn twice." Although, I'm not sure I ever learned the first time, so I'm thrilled to be able to learn it now! Particularly since these materials make it fascinating and fun. I can't count how many times I've said, "OH, so THAT'S why they taught me to do the steps that way in school!"

Montessori Sentence Analysis - Lesson 1

Here is one of the most basic Montessori Language Lessons on  Sentence Analysis.

This material is presented after the child is already familiar with the Montessori shapes that represent the different words.  For example:  A red circle represents the action and the person or thing is represented by a smaller, black circle, etc.  Later, once these concepts are mastered, the child is given the names for each of these words such as ‘verb’ or ‘noun’.  They are quickly memorized at this point.

I’m still working with “K” (my daughter) on these but also starting sentence analysis at the same time.

I prepare a simple sentence on a strip of paper using a noun and a verb.  “Fred eats cheeseburgers.”

I ask her to read the sentence aloud. 

Next steps:

1)      I ask her to find the first action word.

2)      She cuts out the action:  “eats”

3)      I have her place the action “eats” at the top of the table and have her symbolize it using the Montessori symbols she is already familiar with.  (The red circle represents the verb.)

4)      She reads the remaining words. (“Fred and cheesburgers”)

5)      I ask her, “Who eats?”

6)      I tell her that for this, we have a new symbol.  I place the black arrow (with the point away from the action) next to the first action.  I tell her that this is how we symbolize “the thing (or person) that does the action.”

7)      I have her place “Fred” to the left of the black arrow.

8)      Then I ask her “what” Fred eats.  She places the word “cheeseburgers” to the right of the verb.

NOTE: Both nouns are placed on black circles because in Montessori, the black circle represents the "person, place or thing" - the noun.

We can then play around with the words to see that they only sound correct when they are in this order.

This is a very simple operation, but this will build into large and complex sentence dissection in the future.  (I’ll share some of those soon!)

From here, she has some follow-up work that includes writing some of her own sentences which use two nouns and a verb.

This operation will build and build into complex sentence analysis.

When complex sentence analysis is started at this age, then when the child is required in 6^th or 7^th grade to do this type of work it will come easy to her! 

Montessori Single-Digit Multiplication Board

The development of the mathematical mind starts with practical life and sensorial exploration in toddler level.  Once a child has journeyed through primary Montessori, their skills are refined and developed, bringing it all together in the lower elementary level.

A wonderful multiplication tool for early learners is the Multiplication Bead Board.  Using this, we can draw a clearer picture of how symbols on paper can only make sense to a child after sensorial experience.  When school removes real objects from learning math concepts, the study can become dry and meaningless.  Montessori felt, by feeding the child’s natural interest in all aspects of mathematics, we serve by giving sensorial experiences first and only then the representatives on paper.

Montessori materials are certainly ingenious, so I’m going to continue to share all I can!

This is for children just starting to learn how to multiply single digit numbers (for example, 8 X 5), start with the Montessori Multiplication board:

If you look at the board's layout, the problem was 2 x 5 -- children mark 2 places on the top of the board, put down 5 beads for each place, and count the result. This hands-on practice helps children internalize exactly what happens for equation, allowing them to make the transition to solving written equations quickly and easily.

Another ingenious thing that Montessori materials do – is teach other skills almost subconsciously.  For example, this bead board also teaches the child about squared numbers.  Because each time the child multiplies a number by itself – she will see a square shape on the board.  This information will come to use in the upper elementary classroom when she begins learning about square roots.  She will go back and say “ah ha!  I remember that from the multiplication board!”

The way I use this at home is to give the child several rectangle cuts of paper, each one with all of the possible multiplication problems on it.  Starting with 1s, then 2s, then 3s and so on.  As she finds the answers to each one, she writes it down.  We use those to make a multiplication “booklet” that she can go back to again and again.   Or sometimes make a second booklet as part of her process of memorizing the work.  She also uses this to see the patterns of skip-counting, which we also practice with Montessori Bead Chains.

I let the child do the work as many times as she would like.  Once I am sure she understands the work well, then we strive to start memorization of the answers. 

From there, we work on 5-minute math sheets, with the goal of her eventually answering 100 questions in 5 minutes.  Perhaps not until much later in her school career – but memory and speed do come!

Montessori Checkerboard Multiplication

One of the many materials made for learning multiplication in the Montessori classroom is the checkerboard. This one is a favorite in our house - even with my youngest who normally doesn't "like" math. Even she asks to do this work just for fun some days.

The colorful checkerboard materials build upon child's prior knowledge. When children begin with the familiar, there is no reason to panic when learning more challenging concepts.

The colors green, blue, and red representing the ones, tens, and hundreds place values are repeated from previous materials to give the child a sense of mastery.

The beads are the same colors and represent the same amounts as the bead cabinet found in Montessori preschools and early elementary classrooms.

The multiplicand is placed on the bottom and the multiplier is placed along the side.

The child sets out three bead-bars, four times.

The child sees that 3x4=12 and exchanges by putting a two bead-bar in the ones place and carrying a one bead to the tens place.

My youngest rolls a die to come up with her own problems to solve at this first level of multiplication. Eventually moving up to multipling large numbers - up to the millions!

Here is an example of her performing 214 X 3 =

She sets up the checkerboard with beads representing the numbers she's selected.

Now her board is ready for her to do some exchanging. We always start with the "units" place so her first exchange comes with those beads in the green square.

If she knows that 3 X 4 = 12, she can simply do the exchange.

If she has not yet memorized this answer, then she can count the beads to come to the answer of "12".  (That's the lovely thing about Montessori materials - the teacher doesn't have to correct the child, because the material is self-correcting!)

With the checkerboard, the number 12 would be shown with a 1 in the tens place (blue square) and a 2 in the units place (green square).

  

She documents her work as she goes.

My daughter is still using other materials to learn her multiplication facts, but I've found that there is no need to wait to perform these larger operations, while learning her facts at the same time.  It makes it more fun for her as she feels like she's performing "big kid" work - and she is!


The checkerboard can be used for bigger, more advanced operations too. These multiple digit divisors use what we call the "magic slide".

Stay tuned for an example of that once we get there!  Here is a good video to help you out in the meantime: http://www.youtube.com/watch?v=pmougAurzOk

History.... and Art!

We've been working on Story of the World history for 2 years now, with both kids working in different time periods.  I finally decided to back up "D" for a review of Ancient times along with "K".  We just finished Ancient times and even though we've been building our timeline books, we decided to do a visual timeline.  The leather bound history time lines each child has been making will be great keepsakes.  But there is just something about being able to be visually reminded of what took place and when by seeing it up on a wall all in one place.  It makes history more fun when you can see visually what was happening in say, India and China at the same time. Or to see who else was alive at the time Shakespeare lived.

Our wall art: History Timeline brings history to life!

The yellow tape represents Ancient times and the blue tape indicates where the Middle Ages begins.  We'll keep adding colors as we move along - and keep adding more historical information as we find it.

Daddy wasn't thrilled about the new wall decor, but it was fun making it and we're looking forward to adding to it as we learn more.

Art Show Prep
We've also been preparing for the homeschoolers art and talent show.  This will be our first time and K was hesitant to enter.  But she's really done some wonderful work in her art class - that talent gene must have skipped a generation (me) and gone right to her!  We're looking forward to the fellowship with other home schooling families, as well as seeing all of the art work the other home schoolers have done this year!

K's fawn in woods entry

K's water color and cutout drawing.

The weather was so warm last week, we had our lunches outside on the patio!