Elementary Multiplication

## Printables, DIY, and Supply List

#### Supply List

- Large bead frame
- Checkerboard
- Checkerboard tiles
- Decanomial bead box
- Golden bead frame
- Graph paper

#### Bank Game Printable

Print out all the pages. Use the scraps to make three tickets that say: Cashier, Customer, and Bank.

If you printed the ink saving version, you can be color in the cards with colored borders with the corresponding color. Use some color pencils to color them in.

#### Large Bead Frame DIY

#### Checkerboard DIY

Checkerboard DIY

#### Checkerboard Tiles

These will be used with the checkerboard. Store in a plastic compartment box like this one: https://amzn.to/3g5hd6z

Store each number in a separate compartment.

If you don't want a printed version, these are pretty easy and simple to DIY. Just purchase some wood tiles: https://amzn.to/31pOnJW. Paint them gray and white and write down the numbers with a sharpie. You need 4 of each number from 1-9 on each color for a total of 72 number tiles.

#### Decanomial Beads Printable

#### Flat Bead Frame DIY

#### Graph Paper

## Large Bead Frame

#### Large Bead Frame Introduction

The hierarchical materials should be introduced before the large bead frame because of the addition of the new hierarchies. While a child may understand the idea without the hierarchical materials to an extent, the materials leave a huge impression on the child. Go back to the Passage into Elementary Module and refer back to the lesson on the hierarchical materials.

The large bead frame is no different than the small bead frame in the way that it works.

The only difference is that that there are more hierarchies to work with you can and solve some longer and more difficult problems.

This material can be used for addition, subtraction, and multiplication.

So let's get started!

Bring out the bead frame. Show your child the hierarchies and relate them back to the hierarchical material and small bead frame.

**"Remember how we used the small bead frame and it went up to one thousand? Well the large bead frame goes up to a million, like the hierarchical material. We're going to imagine that each one of these green beads *point to the green units of million beads at the bottom of the frame* represents the large green million cube. And there are TEN of them. That is quite a lot!"**

Go back up to the top and start counting the unit beads. When you get to the 10th bead, say, **"Hm, we got to 10 units. What can we do?"**

After solid work with the small bead frame and hierarchical material, they will know that 10 units means one ten. Pause, and let them think.

**"Yes! 10 units is equal to one ten."** Slide the units back and exchange for one ten bead. Do the same with the tens. When you get to the 10th ten, ask the same question again.

Once they've answered, slide the ten beads back to the left and exchange for a hundred bead.

Continue like this with the rest of the hierarchies, exchanging for one of the next category once you've reached 10 of the previous category. You can go all the way up to 10,000,000 (ten million).

#### Using the Large Bead Frame Paper

Going through all the hierarchies and show how to write on the bead frame paper: 1 - 1 million showing the zeros.

Start with the units, slide one unit

#### Forming Numbers

Building numbers - nothing needed here. just dictate a number, and build. Build and say the number... Use a combination of big and small quantities. Skip categories.

First you compose the number.

Then you write a number and the child builds it (you may use the task cards for this)

Then the child writes a number, reads it, and builds it. You can use some dice to compose numbers

Then the child builds a random number and writes it on the large bead frame paper.

#### Addition with the Large Bead Frame

Addition with the large bead frame is done in the same way as addition with the small bead frame, but with larger numbers. Refer back to the Remedial Math module to learn how to do addition with the small bead frame.

This lesson is the last lesson to the passage of abstraction for addition. After this, the child can do addition on paper, mentally, without the need for a manipulative.

#### Subtraction with the Large Bead Frame

Subtraction with the large bead frame is done in the same way as addition with the small bead frame, but with larger numbers. Refer back to the Remedial Math module to learn how to do subtraction with the small bead frame.

This lesson is the last lesson to the passage of abstraction for subtraction. After this, the child can do subtraction on paper, mentally, without the need for a manipulative.

#### Basic Multiplication with the Large Bead Frame

Before we get to big work with the large bead frame, we're going to start with some small problems to gain familiarity.

Start with a simple problem. 7 x 2.

Show how to take seven two times. Slide seven units down to the right. **"Seven taken one time."**

Slide again. You will have only three left.

**"1, 2, 3..."**

Note that you have to exchange for a 10 and continue counting.

Slide all the green back to the left and then slide one ten to the right.

Then continue counting the units.** "4, 5, 6, 7..."**

Read how much you have on the frame.

**"One ten and four units. 14."**

Keep practicing this way. Go up to the higher categories as well, such as **"four tens taken four times" **and "five hundreds taken three times"

#### Multiplication with a One Digit Multiplier

write problem on left side of paper

now break up on the right side. write multiplier in brackets to the right of that.

show how to multiply and exchange and record answer.

#### Magic Zeros

After working a bit with the large bead frame, doing problems with a one digit multiplier, and before beginning some work with problems with two-digit multipliers, you can introduce the idea of magic zeros - which make multiplication with big numbers so much easier!

#### Multiplication with a Two Digit Multiplier

write problem on left side

break up by multiplier

multiply by unit

then show how to multiply by tens place - magic zero - would be too much to multiply a small multiplicad with a big multiplier

multiply

record on left side

#### Multiplication with a Three Digit Multiplier

same. but now its broken up into three separate parts

## Checkerboard

#### Checkerboard Introduction

Like with every new material, you're going to start by introducing it. You'll need the checkerboard mat for this lesson.

**Note:** After completing this lesson, you can choose to move on to the next lesson (Building Quantities) if you notice an interest and readiness. Otherwise, come back to it another day. Prep yourself for both before presenting.

Take out the checkerboard mat, lay it out.

Ask "Do these colors look familiar to you? Let's see what this board tells us."

You'll notice that the colors follow the pattern of the hierarchy.

The green squares represent units of, the blue squares represent tens of, and the red square represents thousands of.

Point to the first green square at the bottom right corner.

"This square represents units." Point to the 1 written on the right side and the one written at the bottom. "One taken one time is one."

Move on to the next square. Again, point to the 1 on the right side and the 10 beneath the square. "One taken 10 times is 10. So this square represents 10."

Do the same with the rest of the squares in the row.

When you get to the green square of the thousands, notice that it is green. What does that mean? It's units of thousands, Then the blue is tens of thousands and the red is hundreds of thousands and then it moves on to the millions family. You may start to notice the pattern on the board.

Moving over to the next row, notice that it starts with a blue square instead of a green square.

Point to the 10 to right of the square and the 1 at the bottom row. "10 taken one time is 10. That's why this square is blue. The 10 on the side here tells us that everything on this row is going to be 10 times greater than the first row."

You'll repeat the same process, going through the entire row, noticing the patterns, discussing, and observing what they notice in the process. Allow space for self discovery!

Moving on to the next row, you'll do the same. Point out that the 100 on the side means that this row is going to be 100 times greater than the first row. Go through the entire row. You'll notice that this row goes into the billions!

Show how excited you are to see such big numbers. "Wow! Those are some REALLY big numbers!"

**Extensions:**

While not a necessity at all, you can do some of these extension exercises if you notice that your child needs some extra practice internalizing the pattern of the checkerboard. You can also do these extensions in parallel to the next lesson.

- Drawing out the pattern on graph paper
- Providing colored squares and a poster board to glue on squares and create the pattern. This one is a bit more involved obviously as you'll need to cut out squares - my suggestion would be to use construction paper or any paper that is already colored, instead of having to print it out.
- You can also use these labels to reinforce the value of each square.

#### Using Bead Bars to Create Quantities

In this section, you'll show your child how to build and read quantities on the checkerboard. The checkerboard is much more abstract than any of the other materials the child may have worked with and reading quantities on the checkerboard can be a little tricky for some children.

You will need the decanomial bead box for this lesson.

"Did you know... that we can use these beads to make some really big numbers on the checkerboard?"

Take a bead bar from the decanomial bead box. Any bar works, except for the golden 10 bar. Let's use five for our example.

Put it on the the unit square.

"When I put it the five bar on the unit square, it means five units." Move it over to the tens square. "When I move it to the tens square, it means fifty. If I put it on the hundred square, it means five hundred." Move along the row. If they start to understand, they may pitch in and tell you what the quantity is as you move along the row.

"Now I'm going to show you something interesting..."

Take it back to the unit square. "Over here it means 5 units. If I move it up to the tens square, now it means fifty..." Move it up the the blue square and then to the hundred and the thousand each time saying, "If I move it up here, it stands for 500... and then 5000."

Now it take it back to the unit square. "Now I'll show you another trick.

Take it back to the unit square. "Here it's five units. Now I'm going to move it up here to the tens, so now it's fifty. Now it's going to slide down! And it's still fifty! How interesting is that?!"

You can do a bunch of variations like this.

Put the bead bar on the hundred square on the first row, and say "This is five hundred."

Take it up diagonally to the right. It will land on the red square on the second row.

"Here it's two hundred as well." Take it up once diagnally more."And it's two hundred here too! So let's play a game. Let's put this somewhere randomly on the checkerboard and let's find out what the value will be"

Place a bead bar randomly on the mat. Let them try to figure out what is stands for. If they're having great difficulty in figuring it out, you can say, "Sometimes it's easier if I slide it down to the first row."

**On the same day or on a different day:**

The next step is to start making numbers with multiple digits. Take two to three random bars and place them on the units, tens, and hundreds squares.

Say, "This number has x in the units place, x in the tens place, and x in the hundreds place.

I can read this as.... *whatever the quantities are*."

Continue practicing putting beads and reading the values.

Once you feel confident that they can make, recognize, and read values and quantities on the checkerboard, it's time to start multiplying!

#### Multiplication with a One-Digit Multiplier

For this lesson, you'll need: the white and gray number tiles, checkerboard mat, decanomial bead box, paper and pencil. Graph paper is the best option for math as it helps keep all the place values aligned.

Write down a problem, either one that you make up on the spot or from the task cards.

**7,534 x 3**

"The white cards are for the multiplicand." Make the multiplicand with the white tiles. The multiplicand will go over the numbers at the bottom of the checkerboard.

image of checkerboard with multiplicand arranged

"The gray tiles are for the multiplier. Our multiplier is 3." Use the gray tiles for the multiplier - the number that represents the amount of times you will take the multiplicand. You'll take a gray tile with "3" written on it and put it over the "1" on the right side of the checkerboard.

"Now I have to take four units, three times. Four taken three times."

Place three four bars on the unit square. Continue on to the next category.

"Three taken three times..." Put three three bars in the tens square.

Do the same with the hundreds and thousands.

"I can't read the product like this! I need to simplify it first!"

**ADD ENTIRE PROCESS HERE**

4 x 3 = 12 put a two bar in the units place and a one bar in the tens place

Go back to the unit square. "4 taken 3 times is 12." Remove the three 4 bars and exchange them for thier product, 12: a 1 bar and a 2 bar. " Put the two bar in the unit square and carry over the one bar to the tens square.

Now move to the tens square. "Three taken three times is nine plus one. That makes 10 tens. I'll have nothing in the tens and one in the hundreds" Remove all the beads from the ten square and put a one bar in the hundreds square.

Move on to the hundred square "Five taken three times is 15, plus one so I have 16 hundreds." Remove all the beads from the tens square and put a six bar in the hundred square and a one bar in the thousands square."

Go on until you reach the end of the problem.

Now read out the product and record your answer.

Record starting from the units so you can make sure everything is lined up properly.

#### Multiplication with a Two-Digit Multiplier

Write down your problem on graph paper.

**4,754 x 14**

Make the multiplicand with the white cards and the multiplier with the gray cards. You're going to start multiplying with the four, so flip over the gray cards that says 1. You do this so you can focus on the multiplier that you are working on and not get confused.

First you'll multiply the multiplicand by the four in the units place, just like you did in the previous lesson.

Flip over the gray four since you are now done with it and unflip the one in the tens place as you will now be multiplying by it.

**Complete multiplication with the second digit in the same way you did with the first digit.**

**first you simply each row, then you slide them down to the first row.**

---------rewrite-------Now, do you remember how this blue square means ten? I will put all my tens together." Slide the bar diagnally down to the tens place on the first row. Do the same with the rest of the beads in the second row.

"Now I can simplify and see how much I have!"

Now read out the answer and record the answer on the paper.

**You can do this same process with a problem with a three and four digit multiplier as well, simplify and slide everything down diagnally to the first row.**

Introduce the next lesson and then work on multiplication with three and four digit mulipliers.

#### Using Multiplication Facts

For this lesson, you'll need: the white and gray number tiles, checkerboard mat, decanomial bead box, paper and pencil. Graph paper is the best option for math.

Write down a problem, either one that you make up on the spot or from the task cards.

**7,534 x 36**

"The white cards are for the multiplicand." Make the multiplicand with the white tiles. The multiplicand will go over the numbers at the bottom of the checkerboard.

image of checkerboard with multiplicand arranged

"The gray tiles are for the multiplier. Our multiplier is 3." Use the gray tiles for the multiplier - the number that represents the amount of times you will take the multiplicand. You'll take a gray tile with "3" written on it and put it over the "1" on the right side of the checkerboard.

"Now I have to take four units, three times. What is four taken three times?"

*If the child isn't fluent with their times tables, they can use the decanomial beads to find the answer. Or any other creative way to visualize the multiplication facts (like Waldorf multiplication flowers, flashcards, etc).*

image of waldorf multiplication flower

--**-------do carrying over when you put down the beads------**

"Four taken three times **(4x3)** is 12." Put a one bar and two bar to represent twelve . Continue on to the next category.

"Three taken three times... is nine." Put a nine bar in the tens square. Emphasize that because the three is in the 10's place, it's actually thirty and that you are taking thirty - three times, which means 90.

Do the same with the hundreds and thousands.

"I can't read the product like this! I need to simplify it first!"

Four taken three times was 12, that means two stays in the units place and one carries over to the tens place because 12 is one ten and two units.

Do same with the rest, find the value, then perform the necessary simplification.

"Three taken three times... is 9, plus the one is 10 tens... that means I have none in the tens place and one in the hundreds place. Put one red bead in the hundreds place."

"Five taken three times is 15, plus one so I have 16 hundreds. I'll put the six in the hundreds place and I'll carry over the one to the thousands place."

Go on until you reach the end. Now flip over the unit multiplier cards and unflip the tens multiplier card and complete the multiplication.

Once you're done multiplying the multiplicand with the multiplier in the tens place.

Slide down all the beads from the second row down to the first.

Simplify where ever needed. Each square should only have one bead bar at the end.

Now read out the product and record your answer.

Record starting from the units so you can make sure everything is lined up properly.

**Use the same process with problems with a three or four digit multiplier. The same rules apply. You will simply slide down all the partial products to first row. **

#### Multiplication with Partial Products

The next step is to break the pieces apart and pay attention to the process of multiplication with a multi-digit multiplier. This is one of the steps taken towards abstraction and being able to complete a multiplication problem on paper without a manipulative.

You can show this step once your child practice with doing multiplication on the checkerboard with multiplication facts.

Start the multiplication, multiplying all the numbers in the multiplicand with the unit multiplier

Once you've completed multiplication with the unit multiplier, do the necessary simplification and record your first partial product.

Now multiply with the next multiplier and repeat the process of simplification and then record the the next partial product.

Make note of the fact that the first answer in the second row is NOT a unit and therefore, you will not write that value in the units place. Make sure you write it under the tens.

Slide the beads down to the first row and do the necessary exchanging to make sure there's only one bead in each row.

Add the two partial products on paper to get your answer and then check the beads on the checkerboard to see if it was correct.

**You will do the same with problems with three and four digit multipliers.**

#### Geometrical Form of Multiplication on Graph Paper

## Flat Bead Frame

#### Flat Bead Frame Introduction

#### Building the Quantities

Did you know... that we can use these beads to make some really big numbers on the checkerboard.

We read the numbers value depending on where we put it on the board.

Get a two bar... this is a two. When I put it on the units square, it stands for two units. If i move it to the tens square, it stands for twenty. Move it along the hundreds, thousands and say "here it stands for *quantity*.

Now I'll show you something else that's very interesting.

Take it back to the unit square. "Here it's two units."

Take it up to the blue tens square on top. "Here it's two tens... or twenty."

Bring it down diagonally (to the left) to the blue square on the first row"... "This is also twenty!"

Put it on the hundred square on the first row... "This is two hundred..." take it up diagonally to the right (it will land on the red square on the second row). "here it's two hundred as well..." Take it up once more... "And it's two hundred here too."

So let's play a game. Let's put this somewhere randomly on the checkerboard and let's find out what it stands for.

Place it randomly... somewhere.

Sometimes its easier if I bring it down to the first row. (go down diagonally).

I can also make numbers with multiple digits.

take three random bars, and place them in the units, tens, and hundreds place.

This number has x in the units place, x in the tens place, and x in the hundreds place

I can read this as.... *whatever the quantities are*.

Continue practicing putting beads and reading the values.

#### Multiplication with a One Digit Multiplier

#### Multiplication with a Two Digit Multiplier

#### Partial Products

#### Multiplication with a Three Digit Multiplier

## Abstraction

#### Abstract Multiplication on Paper

Materials List

- Large Bead Frame
- Checkerboard
- Checkerboard tiles
- Decanomial bead box/checkerboard beads
- Flat bead frame