This is mostly applicable to parents of young children (preschool).
The idea is to introduce math fundamentals in normal play.
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
- 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?"
- Push the blocks into one row and check it on the number line
- 3 blocks plus 0 blocks is just 3 blocks
- Switch the set of 2 with the set of 3 -- same answer
- 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.
- Push the 2 rows of 3 blocks into one row and check it on the number line
- One row of 3 blocks is just 3 blocks.
- Grab the whole set of 2 rows of 3 blocks and turn it so it is 3 by 2.
- Slide blocks around so 2x3=2(2+1)=2x2+2x1
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.
The blocks, number line, and number plane.
A pad of paper and a pencil with a good eraser.
- Show how to add in columns, e.g.:
Check answers by counting up blocks.
Show how to carry 10's to next column:
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:
Now do a really long addition:
The payoff is to tell the kid he/she can now do bigger problems than a
- 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:
- 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.]
- Cheer for incremental improvement. Keep
track and maybe even draw a graph showing progress.
- 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.
Blocks, numberline, number plane, paper and pencil.
- Make square and cube numbers. Introduce exponential notation
as just a fancy way to write multiplication.
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
This should be a brief series of sessions. Cover the basics, gain
mastery and move on.
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 |
- 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.
- 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
Here we leave the blocks behind. Instead, add a calculator, with the
paper and pencil.
- Solve rationals via long division. Check answers on the
calculator. Start with 1/4:
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
- 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
- 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.