**Getting Started ****.................
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****How to Use It****
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****Example****
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****The Controls****................
................. ................. ................. .................****Complements**

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**The
Ten Circle Counter**

*This is not actually a game.*

*It is an online tool to help you become familiar*

*with an excellent subtraction strategy.*

*It puts the process into pictorial form.*

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**Just press "Start"**

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This tool will visually explain one of the shorthand methods

for easy addition and subtraction.

Use the mouse to click the yellow (top) + arrow three times

After 3 clicks we have counted to 3.

If you do 12 clicks

you will fill the inner circle

then jump to the next outer ring

and land on the 12:

the green circle

tells you what number you are on.

*The top red arrow *

*will undo the count*

*taking you back one
at a time*

**Calculate 23 - 4 = 19**

Begin by clicking the top yellow arrow

until you reach 23.

If you accidently overshoot

click the top red arrow to go back to 23.

Click the lower (shaded) minus button

to start subtracting from 23:

(a) We began at 23

(b) After one back-click we arrive at 22

(the shaded area is one step backwards)

(c) The sum was 23 - 1 and the last digits of that are just 3 - 1

(d) No surprise here. The working is just 3 - 1 = 2 for the last digit.

Repeat that process for two more back-clicks.

This takes us from 23 back to 20:

Not much happening here either.

We can see we have made 3 shaded steps back from 23

to arrive at 20.

The calculation on the right just shows 23 - 03 = 20

by virtue of subtracting 3 - 3 to 0.

Now things start to get a bit more interesting!

Click the lower red shaded negative button again.

Now we have gone backwards 4 steps from 23 to 19:

Let's see if we can interpret what this is all about!

The explanation is given below:

(a) we started on 23

(b) we went backwards 4 steps (the shaded area)

(c) we arrived at 19

(a) The units digit is worked out as 3 - 4 coming somehow to 9

(b) Astoundingly we see "3 - 4" is equal to "3 + 6" which admittedly is 9.

But how can we explain why we can do that?

Here's how:

(a) We need to explain that 3 - 4 is the same as 3 + 6 inside the 10-circle

(b) we start at **3** in the 10 circle because we began at 2**3**.

(c) The green inside arrow counts FORWARDS 6 steps.

(d) it arrives at 9

Hence the 3 + 6 = 9

but how do we know which number to pick?

There are two paths around any circle.

A forward path (adding) and a backward path (subtracting)

This shows that starting from 3 and counting back 4

is the same in the 10-circle as

starting from 3 and counting forward 6:

3 - 4 = 3 + 6 = 9

inside 10-circle.

This always works

and 4 and 6 are **complements** in 10-circle.

That is, they both add together

to make 10.

So we can be sure for example that

23 - 8 = 15

because

8 and 2 are complements (add to 10)

and so

3 - 8 = 3 + 2

inside 10-circle

which comes to 5.

We get the

**LAST DIGIT**

of the answer.

The first digit

we can **guess**.

**ADDITION WORKS IN A SIMILAR WAY:**

18 + 3 = 21

Start by counting to 18

then add another 3 using the shaded "+" arrow:

Here's the explanation:

(a) The last digits of 18 + 3 are just 8 + 3

(b) Somehow we need to show this is the same as 8 - 7 (which is 1)

(notice that 7 and 3 add to 10)

(c) We started at 18

(d) We added on 3 (the shaded portion)

(e) We arrived at 21

(f) The green arrow moves BACKWARDS 7 STEPS

from 8 to 1 inside the 10-circle.

**Note:**

**Instead of adding 3**

**we took away 7**

**7 and 3 make 10**

**This is a perfectly general**

**way to get the last digit of
the answer**

**when an addition goes**

**over 9.**

**9
and 1**

**8
and 2**

**7
and 3**

**6
and 4**

**5
and 5**

**and**

**0
and 10**

To subtract below zero:

ADD the complement.

Example:

Guess it's in the 20's

then:

36 - 9 = 2 (6 + 1) = 27

To ADD over 9

SUBTRACT the complement.

Example:

32 + 9

Guess it's in the 40's

then:

32 + 9 = 4 (2 - 1) = 21

**Click to count up or count down**

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**Once you have arrived at a
starting number**

**click either of these to ADD or
SUBTRACT from it**

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**Reset back to zero**

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**SPEED UP = UP ARROW**

**SLOW DOWN = DOWN ARROW**

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**JUMP ROADBlOCKS = SPACEBAR**

Not ready?

Just jump over the road blocks.

The same ones will come around again.

**CHANGE MUSIC = ENTER**

**SUPER CHARGER GAME**

Game too slow for you?

You can vamp it up by left clicking the top line

(the red arrow is pointing to it)