*SECTION* — EVENS AND ODDS

Prerequisite: Comfort counting to 10, some comfort counting to 20, add and subtract 1 and 2 easily

— MULTIPLYING BY 2 —

Very gradually, introduce multiplication to your child by a small change in language — start referring to doubling a number as taking "two of it" or as "multiplying it by 2."

GAME — NIM — DOUBLING THE LIMIT

Set a starting total, say 20. Let your child choose whether to go first or second. During the first turn, a player chooses to subtract 1 or 2 from the current total. After the first turn, a player may subtract any number from 1 up to twice the number used on the last turn. The first person to reach 0 wins (an alternative rule is they lose). Once children learn to play this without writing anything down, it is a fun travel game.

GAME — COUNTING EVENS AND ODDS

Use a small collection of Number Cards involving some small quantities. Start with combinations of three cards and work your way up to more cards.

Suppose the numbers are 1, 2, and 3. The question is: If you randomly pick two cards and add them, are you more likely to get an even or odd number? Count how many ways there are of getting an odd number versus an even number. For example, in the case of using 1, 2, and 3, there is one way to get an even number (1 + 3) and two ways to get an odd number (1 + 2, 2 + 3). So the odd number sums are more likely.

To make a game of it, let one player be Even and the other player be Odd. See who has the most successes after a dozen trial runs.

ACTIVITY — SKIP COUNTING BY 2's

In addition to being good practice, skip counting is a faster way to count a collection of things, such as toes, than counting them one at a time.

Count by 2's while pushing your child on the swings. Start by alternating with your child — you say 1, your child says 2, you say 3, your child says 4, and so on. After establishing the pattern, one of you can say their part without the other person saying anything.

While traveling, find something fun to skip count like yellow cars. Skip count up or down by 2's ending by 20. Initially, do this starting at 0 or 20, but eventually, start at any number.

GAME — HALVES AND HALF NOTS

Two players agree on a target number, say 20, set their running total to 0, and choose who will go first. A turn starts by generating a number using one die or the sum of two dice. If the number is odd, the player must double it. If the number is even, the player may take half of it as many times as is desired as long as the numbers being halved are even. The player then adds that final result to the running total as long as that does not put the total over the target — if the final result cannot be used, the turn is skipped. The player who brings the total exactly to the target number wins.

There are a few variations. You can allow the player not to change the initial number. You can allow halving at most once. Finally, you can practice subtracting by starting at the target number and subtracting your way down to 0.

*SECTION* — NUMBER BONDS

Prerequisite: Some comfort adding and subtracting small single-digit numbers

PUZZLE — SUM GROUPS

Use a grid of numbers with a target sum. Find groups of two or three numbers that add up to the target. The members of a group must share sides. Use tokens, such as different types of food items, to identify each group within the puzzle. When complete, the entire puzzle will be made up of identified groups.

Create these puzzles by starting with an empty grid and working your way around the grid using pairs and triples that add up to the target sum.

GAME — GO FISH WITH SUMS

Using a target sum that your child is comfortable with, remove cards at or above that target from the Number Card deck. If there are more than two players and you have restricted the numbers significantly, you may need to use several decks.

Play starts by dealing 5 cards to each player. Put the remaining cards in a common draw pile. Players take turns "fishing" for cards whose number will sum to the target sum with cards they already have.

For example, a player might ask a player, "Do you have any 4's?" If that player has some 4's, they are handed over and the original player gets another turn. However, if that player does not have any 4's, then the player says "Go Fish!" and a card is drawn from the draw pile. If the drawn card matches with a card they have, the player may have another turn; otherwise, the turn is over and play continues to the left.

When a player has a pair of cards that sum to the total, the player puts that pair on the table in front of them. The game is over when all cards are in pairs. The player with the most pairs wins.

To create some variety, allow players to use more than two cards to create a group of cards that sum to the target sum. Another possibility is to say that two cards match when their difference is a specified target difference.

ACTIVITY — WHICH NUMBER AM I

Depending on how many children are playing, there are two ways to play this.

Adult with two children: Each child draws a card and places it on their forehead facing out without seeing it. You announce the sum of the cards and the children are challenged to figure out their own card by looking at the other child's card.

Adult with one child: Create several pairs of cards in advance where each pair of cards has the same total known to all. The unused cards are moved to the side. Pick up one random pair of cards, place the cards on your foreheads, and figure out each card by looking at the other person's card.

*SECTION* — ORDER IN THE HOUSE

Prerequisite: Can count up and down from 0 to 20

GAME — GOING UP

Use one deck (two if there are several players) of cards that go from 0 to 20. Each player is dealt four cards face up in the order they are dealt, and the remaining cards form a draw pile.

You can either play that a player must select one of their existing cards in advance or, the friendlier version, that the card can be selected after seeing the new card. Either way, one of their cards is replaced by the top card of the pile and the replaced card is put on the draw pile bottom.

The first player to get their cards in order wins. Make this easier by allowing duplicate cards. Make this harder by using more cards in a hand.

GAME — GOING UP SOME MORE

Each player draws a staircase with anywhere from 4 to 10 steps (more steps for older players).

Create a number using Number Cards numbered from 0 to 9. The first selection will be from the cards 0 to 1, and this is the tens digit. Then select a ones digit from all the cards. The two digits together form the number. If possible, the player must place this number on their steps. The first player to fill their steps in increasing order from bottom to top wins. Make it easier by allowing neighboring steps to have the same value.

GAME — CREATING ORDER

Start with a deck of Number Cards that go from 0 to 40 (higher with more players). Deal ten cards to each player (fewer cards for younger players). The dealt cards are placed in front of each player in the order they are dealt. The remaining cards are placed face down in a draw pile. The top card is placed facing up as the first card in a discard pile. The goal of the game is to get the cards in ascending order from left to right.

During a turn, a player may select either the top card of the discard pile or the draw pile - this card must be used to replace one of the cards in front of the player, and the replaced card is put face up on the top of the discard pile.

You can play that the first player to get their cards in order wins, and that's all there is to it. Or, you can use a point system for each round of play. In this system, the winner receives 15 points. The other players receive one point for every card they have in order starting with their lowest card.

*SECTION* — SMALL ADDING AND SUBTRACTING

Prerequisite: Some comfort adding and subtracting small single-digit numbers

GAME — CLOSEST TO 10

Use a deck of cards from 1 to 10. Choose a target sum, say 10. Each round starts by dealing five cards face down on the table. One player chooses three of those cards and turns them over. Two cards are chosen from those three to add up as close to the target as possible. The unused three cards are given to the other player to choose two cards to get close to the target. The player closest to the target from either side wins a point.

Have a number line handy to use for discussions about which player is closer to the target sum.

One variation is to use subtraction instead, in which case you would use a lower target.

GAME — ZOO RESCUE

Use two dice or two sets of number cards going from 1 to 6. Each player has 6 tokens — animal tokens are perfect for this game if you have them. Each player also has a piece of paper with boxes numbered from 0 to 5. Each player decides where to put their 6 tokens — it is okay to put more than one token in a box.

During a player's turn, two numbers are created by rolling the dice or picking two cards, and the difference of those numbers is used. A player can free one of their tokens if they have one in that box. The first player to rescue all their tokens wins.

Alternatively, use cards numbered from 1 to 10 and have 10 boxes numbered from 0 to 9.

An important part of this game is choosing good boxes to place tokens in to start with. With practice, your child will realize that some boxes are more likely to occur than others.

INVESTIGATION — COMMON SUMS

Make a sheet of paper with 12 rows. In each row, put 8 squares. The leftmost column of squares has the numbers from 1 to 12 written in the squares. Put 1 token on each of the 12 numbers. Start rolling a pair of dice. After each roll, move the token for the sum of the dice one square to the right. The goal for each token is to be the first to get all the way to the right across the page.

Let your child come up with some questions to investigate. Some natural questions are:

• Which token will win and why?
• Which tokens do well and which ones do poorly?
• Which token is the worst?
• How will the winners change if the rows are changed to have fewer squares or more squares?

Have your child explain their ideas about the answers to these questions, and then investigate their ideas by running experiments.

Add a competitive element to this by guessing which token will win before the round starts.

*SECTION* — SUDOKU VARIATIONS

Prerequisite: Comfort counting to 10, some comfort counting to 20, add and subtract 1 and 2 easily

— INTRODUCTION —

All puzzles on this page start with the basic Sudoku rule that a square grid is filled with numbers so that each number appears exactly once in each row and column.

Make these puzzles by starting with a completely filled puzzle, removing many or all the numbers, and creating subregions and the appropriate extra information for that type of puzzle.

To save space, the examples here are all 4 by 4. You can make larger puzzles yourself, or you can find larger versions of these on the internet.

PUZZLE — EVEN-ODD AND KROPKI SUDOKU

These two puzzles have the usual additional Sudoku restriction that the numbers must occur exactly once in each marked subregion.

For Even-Odd Sudoku puzzles, the squares for even numbers are greyed in. This makes them generally easier than regular Sudoku and it reinforces even and odd numbers.

Kropki Sudoku is the same as regular Sudoku except two types of dots placed between cells are added. If the dot is hollow, then the two numbers are one apart. If the dot is filled in, then one number is half the other number.

PUZZLE — SUMDOKU, DIFFDOKU, SUMDIFFDOKU

In addition to the usual Sudoku rules, these puzzles are broken into subregions that have a target number assigned to them. Unlike standard Sudoku, it is allowed for a number to be repeated in a subregion as long as that does not break the rule about no repeats in a row or column. If a subregion has just one square in it, then the target number will be the value of that square.

In Sumdoku, the sum of all the numbers in a subregion is the target number. In Diffdoku, all subregions have one or two squares. If a subregion has two squares, then the difference of the two numbers is the target number.

In a Sumdiffdoku puzzle, both addition and subtraction are used. The subregions are marked with a "+" or a "-" to indicate whether to take a sum or difference.

To vary the math calculations, use different groups of numbers instead of the usual 1 to 4 for a 4 by 4. For example, use the numbers 1, 3, 5, and 7. If you do this, list the numbers above the puzzle so your child will know what to use.