Getting started

This is mostly applicable to parents of young children (preschool). The idea is to introduce math fundamentals in normal play.

Tools

The main tool is set of blocks. We eventually want enough to demonstrate 5 cubed, so we need 125 blocks. Get some 2x2's and cut them into cubical blocks. Remember:

A 2x2 isn't actually 2 inches by 2 inches
The saw kerf takes up space

Therefore measure your wood and make the cuts to get near cubes. They should be good enough to have under 1/4" tolerance stack in 20 cubes. Sand the edges and corners. Put them in a grocery bag.

Next, make a number line. Take a 6 foot 1x2, and mark "0" in the middle. Measuring with your actual blocks, label tickmarks for -12...0...12. Your blocks should line up nicely along the numberline, no matter how they are turned (original sides or your cuts).

Make a number plane. Take a 2 foot by 2 foot chunk of 1/4" plywood. Mark number lines for x and y axes of -2..10.

Put all the Tools in the living room or wherever you spend time together. They should always be available, and a normal part of daily life.

Exercises

Use the number plane as a flat surface to build towers. Who can make the tallest tower? Count the number of blocks used. Use the number line to help count blocks. (You will probably find it works best to stack towers using the originally milled surfaces instead of your own cuts.) Knock the towers over.
Build forts and castles with the blocks. Borrow blocks from each other to complete sections of the forts. Who has the most blocks? Is it a fair sharing?
Draw dots on some of the blocks to make dice. (Use real dice as examples. Explain about opposite sides adding up to 7). Use the dice for games. In general, allow any kind of scribbling on the blocks.
On the number line, use blocks to show addition. The game is to guess the answer, and then check by using the blocks. E.g., "How much is 2+3?". Keep all answers under 10. Then do commutativity and associativity. E.g., "Is 2 blocks plus 3 blocks the same as 3 blocks plus 2 blocks?"

Enumeration
Push the blocks into one row and check it on the number line
Identity
3 blocks plus 0 blocks is just 3 blocks
Commutative
Switch the set of 2 with the set of 3 -- same answer
Associative
Slide 1 block from the set of 3 to the set of 2 -- same answer
Use the number line to show subtraction. Again the game is to guess the answer and check it with the blocks. E.g., lay out 5 blocks and remove 2 blocks. First work only with natural numbers (answers all come out 0 or more). Show you can add the same amounts to get back to the original number.
Show negative numbers by subtracting too much (5 blocks minus 6 blocks). Show the pattern of subtracting being the same as moving to the left -- and keep going. Then, since you are in negative territory, adding blocks doesn't get you to zero right away. -- you have to fill in the negative spaces first.
Use the number plane to show multiplication. Show:

Enumeration
Push the 2 rows of 3 blocks into one row and check it on the number line
Identity
One row of 3 blocks is just 3 blocks.
Commutative
Grab the whole set of 2 rows of 3 blocks and turn it so it is 3 by 2.
Distributive
Slide blocks around so 2x3=2(2+1)=2x2+2x1

Moving to paper

This is appropriate for K-6. The problem with existing math texts is that they are full of trivia (who needs "casting out nines"?). The essentials aren't that complex. A reasonably attentive parent, one-on-one with a child, can do a much better job in far less time.

Tools

The blocks, number line, and number plane.

A pad of paper and a pencil with a good eraser.

Exercises

Show how to add in columns, e.g.:
```
      12
      34
     ---
      46
```
Check answers by counting up blocks.
Show how to carry 10's to next column:
```
      27
      36
     ---
      63
```
Demonstrate the idea by adding blocks until there are more than 10, then replacing them with a bag containing the 10 blocks. Do this for enough tens to make a 100's bag full of 10's bags. [Now you see why we needed > 100 blocks]
Demonstrate how to do multiple rows, and what to do if some have fewer digits:
```
    2345
     127
    1198
   -----
```
Now do a really long addition:
```
    12356876433455667
                  237
    23468976543334442
   ------------------
```
The payoff is to tell the kid he/she can now do bigger problems than a calculator.
Now we hit a problem. We absolutely must memorize the single digit additions. Write up practice sheets of 100 randomized practice problems and do timed tests.
```
  1   4   9   5    ....
  6   3   7   2    ....
 --  --  --  --   
```
Require 100% correct in 2 minutes. It is very easy for this to be an emotion killer. To keep it a game:
1. The coach/parent/teacher has to do them too -- but twice as many in the same time. [Of course you can't...which is the whole point as far as the kid is concerned.]
2. Cheer for incremental improvement. Keep track and maybe even draw a graph showing progress.
3. Mix with other activities.
As soon as the goal is reached, stop doing the work sheets. Ceremonially burn the remaining copies.
Similarly demonstrate and practice subtraction. Single column, then multicolumn, then very large multicolumn. Follow up with tests of 100 random single-digit problems in 2 minutes.
Similarly multiplication. Single column, then multicolumn, then very large multicolumn. Follow up with tests of 50 random single-digit problems in 2 minutes.

Exponents

Tools

Blocks, numberline, number plane, paper and pencil.

Exercises

Make square and cube numbers. Introduce exponential notation as just a fancy way to write multiplication.
```
     1
    3  = 3

     2    (1+1)   1  1  
    3  = 3     = 3 *3  = 3*3 = 9 

     3    (1+1+1)    1  1  1  
    3  = 3        = 3 *3 *3 = 3*3*3 = 27 
```
Note that anything to the zeroth power must be 1:
```
     1    (1+0)    1  0
    3  = 3      = 3 *3  = 3*X -->X = 1
```
NOTE: You have enough blocks for 5 cubed, but the idea is obvious by 4 cubed.

Rational Numbers

This should be a brief series of sessions. Cover the basics, gain mastery and move on.

Tools

Same blocks, numberline and numberplane, paper and pencil as before.

Cut some more 2x2 stock to make fractional parts. Can't use an existing block, because the saw kerfs would leave it too small. E.g.:

  +---------+
  |  1/3    |
  |---------|
  |  1/3    |
  |---------|
  |  1/3    |
  +---------+

Make:

2 halves
3 thirds
4 quarters
5 fifths
6 sixths (thirds cut top to bottom)
8 eights (quarters cut top to bottom)
10 tenths (fifths cut top to bottom)

These are rather fragile, so save them is a separate bag. Use them to demonstrate ideas, but work mostly on paper.

Exercises

Add with same denominator:
```
  1/3 + 1/3 = (1+1)/3 = 2/3
```
Convert to same denominator:
```
  1/3 + 1 = 1/3 + 3/3 = (1+3)/3 = 4/3
```

Find common denominator via cross multiplication

  1/3 + 1/2 = (2/2)*(1/3) + (3/3)*(1/2) = 2/6 + 3/6 = 5/6

Simplify via factoring
```
  4/6 = (2/2)*(2/3) = 2/3
```

Real Numbers

Tools

Here we leave the blocks behind. Instead, add a calculator, with the paper and pencil.

Exercises

Solve rationals via long division. Check answers on the calculator. Start with 1/4:
```
    0.25
   ----------
  4|1.0000
      8
    ----
      20
      20
    -----
       0

  1/4 = 0.25 on calculator
```
Then do 3/4. Keep doing examples until the student gets the hang of long division. Carefully avoid examples where repeating fractions are the result.
Then do 1/3. Act surprised when it keeps repeating. Explain the overbar notation for repeating.
```
                       _
  1/3 = 0.33333333 = 0.3
```
[Don't try to explain why moving from one base to another sometimes gives repeating strings]
Go through the rational number exercises again, solving in long division (thus decimal notation) and checking with the calculator.
Test with 25 long division problems. These should be 1 or 2 digits into 1 or 2 digits:
```
    ----------        -----------
  3 |20           12 | 1.5          ....
```
The time limit is a judgement call. The idea is that student is writing fairly quickly, and gets 100% correct. Depending on the specific problems given, such a test may take many minutes.

Creator: Harry George
Updated/Created: 2004-06-26