# Category Archives: Decimal system

## Misconceptions in math

If you have visited this blog for a while, you may have noticed my continuous quest in helping children deepen their understanding in math. I share here what I do as a parent with my own children, hoping that it could help other parents as well. At home, I use games, math discussions,  explorations of word problems, etc. But I also try to identifying any misconceptions they may have in their learning.

Example.

My son Tom, 5, has learned how to count by Fives and Tens, while playing hide-and-seek with older kids.

– “Ten, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety, One hundred ! Ready or not, here I come !”.

I was quite surprised, initially, to hear him count by Ones, Fives, and Tens, as it is not a concept often mastered at a young age. So one day, as he was playing with his cars, I asked him if he could count them.

– “Sure !”, he said, pointing one car at a time, “One, two, three, four, five, six, seven, eight, nine, ten !”.

– “Could you count them by five?”, I asked

– “Sure !”, and pointing one car at a time as he did while counting by Ones, he counted : “Five, Ten, Fifteen, Twenty, twenty-five, thirty, thirty-five, forty, forty-five, fifty !”.

– “Could you count them by Tens ?”, I asked

– “Sure ! Ten, Twenty, Thirty, Forty, Fifty, Sixty, Seventy, Eighty, Ninety, One hundred !”, pointing again one car at a time.

He could root count by Ones, by Fives, by Tens up to a hundred and more. But he did not understand that counting by Fives means counting by groups of five, counting by Tens means counting by groups of tens. For him, it was just three independent ways of counting. You have ten cars when you count by Ones, fifty when you count by Fives, a hundred when you count by Tens. But since you “usually count by Ones”, you have ten cars. Seems logical in Tom’s world.

With Tom, 5, a discussion is usually the best way to assess his understanding, with a “tell me about what you are doing”, or “what does it mean to …”. With Rosie, 8, I like to use pretend-playing : she is the teacher, I am the child. I do not do it often during the school year, but Summer break is a fun time to do so. And since Summer break starts tomorrow… you should hear more about it pretty soon.

Stay tuned !

## Playing with the Base-10 blocks to practice addition

“Mom ! Could we play that game again?”.

Over Summer,  Rosie, my 6 year old daughter has enjoyed exploring the decimal system and operations playing with the Base-Ten blocks (see my previous post here if you want to know more about these blocks).

Since she asked me to play our latest game first thing in the morning,  it is probably worth sharing it.  So here I am.

Material:

• Base-Ten blocks (I usually use eNasco (website here) when I order math-related material, but you can find these Base-Ten blocks on Amazon as well). Each player gets:
• 1 Hundred (a plate of 100 Units, also called Flat),
• 10 Tens ((a bar of 10 Units called Rod or Long),
• 10 Ones (little cubes called Units)
• Die 0 to 9 (I love fancy dice, you can find them online (eNasco !), at a children’s store, etc, for 20-50 cents each). You can otherwise make a deck of 10 cards, numbered from 0 to 9.

How to play : Here come Woody and Buzz again for the demonstration !

• Both Woody and Buzz have a Hundred in front of them. They get 10 Tens, and 10 Ones. The goal? Covering the Hundred by adding Ones and Tens.
• Woody starts. He rolls the die/draws a card. He gets a 4. He adds 4 Ones to start covering his Hundred
• It is Buzz’s turn. He gets a  3. He adds 3 Ones on his Hundred.

• It is Woody’s turn. He gets a 7. Let’s the fun begin ! He uses his 6 Ones left to go to 10, trades the 10 Ones  for a Ten. And add 1 more One to make 11.

Woody gets a 7

Woody uses his 6 Ones left to go to 10

Woody trades his 10 Ones for 1 Ten

•  And so on until Buzz and Woody cover their Hundred.

What I like about the game :

• It gives Rosie plenty of opportunities to explore addition with a result reaching the next Ten. “e.g. I have 7, I get 6, I need 3 to reach 10. And add 3 more.”
• The game can be played at several levels
• practicing adding Ones and trading 10 Ones for a Ten, without formally keeping track of how many blocks are covering the Hundred
• modeling the addition of  two 1-digit numbers and  the addition of a 1-digit number to a 2 digit number
• connecting each turn to an equation – Ex: 8 + 9 = 8 + 2 + 7 = 10 + 7 = 17
• It gives me plenty of opportunities to model and express what I am doing,  even if I do not expect Rosie to do so at this point with upper numbers. By the way, I like to call these blocks Ones, Tens, or Hundreds, instead of Units, Rods, and Flats. I think it helps Rosie learn the nomenclature of  the decimal system.
• “Let me count how many Ones I have at this point. I have 3 Tens, it means 10, 20, 30 Ones, and I have 5 more Ones, so I have 35 Ones total !”
• “I have 35, I get 8, if I decompose 8 into 5 + 3, I reach 40, and add 3 more. I have 43 !”
• It can be adapted to a cooperative game, if you want to avoid to have a winner and a loser. Both players work together, taking turns, and see if they can cover the Hundred in less than 20 turns, for instance.
• On a side note, Rosie said at one point : “That’s funny, it is actually easier to add numbers when you are in the 20s or 30s, than when you are in the 10s. You have 24, you can really hear the 4, so you know you need 6 to reach 30 ! But with 12, or 13, you don’t hear the 2 or the 3. It is hard !”.  Now, how could I have guessed she would ever say that?

Gotta go ! Gotta play, again !

## The base-10 blocks: a must-have at home?

I do not remember learning how to count. I believe I learned the pattern first, started counting up to 10, then 100, and  so on. For my 6 year old daughter Rosie, however, it may have been different, as she early on set up a goal to reach: 100. Indeed, in K,  it is a big deal Mom to know how to count to 100.  Once she reached her goal, though, I started wondering if she hadn’t given too much importance to 100 as such. Like when a few weeks ago, she started counting by 2s in the car. 1…3…5….//…43….45… 47…….. until …. 97… 99…  and 100 !!!!! As if there was a wall: I have reached 100, I am good now, I can stop.

So I make sure she has plenty of opportunities to explore the decimal system. To 100. And up. And a set of manipulatives that I find quite educational is the base-10 Blocks.

The set comprises Ones (little cubes called Units), Tens (a bar of 10 Units called Rod or Long), Hundreds (a plate of 100 Units, called Flat), and Thousands (a big cube comprising 1000 Units, called 1000 Blocks).

And Rosie enjoys “playing” with them,  trading blocks i.e. 10 Ones for a 1 Ten, or 10 Tens for a 1 Hundred, using them to solve word problems, addition (see picture hereafter), even “creating” large numbers and see what they would look like (see 1358 for instance, on the picture below).

Just google Base-Ten blocks, and you will find tons of activities using these blocks, depending on your child’s grade level, and the concept you want to explore (place value, addition, subtraction, multiplication, etc). If you do not have a set, you can even find online 2D templates to print out (would be helpful for the kids to see the 3D version at least once, though, before switching to a 2 D version).

I found these blocks extremely helpful. I am a fan. And you can count on me to write posts regularly about them.

## Starting our exploration of the decimal system with a card game

My daughter’s last day of School was on Friday. On Saturday, we started our Math journey. Just to show Rosie, sooner rather than later, that I will do my very best to make it enjoyable.

A deck of cards on the kitchen table did the trick.

Even if the card game, based on adding up pairs of numerals to make 10, was a subtle step towards our exploration of the decimal system, and its magic number 10 (i.e 10 digits, 0 to 9), the base number of our system of numeration . First, we explored the concept, then, we played.

Exploring…

Now, your child may already know by heart all pairs of numerals that make 10. But I was not sure about Rosie. Throughout the year, she brought home worksheets with equations balanced correctly (e.g. 4+6=10), but I wanted to check she understood what was underneath each symbol, each digit, each equation.

1) She first explored ways to make 10, on her own, using her beloved Legos®.

What I found interesting is that Rosie quickly mentioned that whether she started with 8 Legos® and completed with 2 Legos®, or started with 2 Legos®, and completed with 8 Legos®, she ended up with 10 Legos® . Away from any symbol, without knowing it, she was building up fondations of the commutative property of addition (e.g. 8 + 2 = 2 + 8).

2) Then, we added some symbols. And finished with the equation. This step is fundamental to me: connecting the numbers of Legos® (i.e. how many Legos® she had in front of her) to the numerals (i.e. the symbol representing the number of Legos®) of the equation.

3) Finally, we decided to take pictures of all her combinations. Another option would have been to draw the combinations.

No need to say that she was ready to play.

… and playing

I have never written card game rules before, so please, write a comment or contact me if you need some clarification. Fortunately, Buzz Lighter and Woody came to my rescue, allowing me to take pictures as they were playing.

Players and Cards

• The game is played by 2-4 people, with a deck of 40 cards (Ace,  2 to 10).
• The goal for the players is to get rid of all their cards in hand, by pairing cards so that the sum of their numerals equals 10 (e.g. an Ace with a Nine, a Eight with a Two, etc).

Rules

• All players receive 5 cards, displayed face up in front of them.
• The players look at their hands, and discard the pair(s) of cards making 10 (e.g. Buzz Lighter can make a pair with a Four and a Six. Woody cannot.)
• The first player draws a card from the  stock pile.
• If he/she can combine the card with one of his/her cards to make a 10, he/she withdraws both cards and plays again (e.g. Buzz Lighter draws an Seven and combine it with a Three to make a 10. He withdraws both cards).
• If he/she cannot , he/she just withdraws the card.
• It is the second player’s turn to play. And so on until one player gets rid of all his/her cards.

Playing with a rising 1st grader, I kept the rules quite basics on purpose, but here are some examples of modifications you could do with upper graders:

• Players can make 10 by combining several cards (e.g. a Two, a Five and a Three).
• When a player draws a card, he/she puts it on the table so that the card is available to anybody, i.e. the fastest player to see a pair, gets the card
• A player who cannot make a pair has to keep the card, i.e. the game goes longer, as each player can get more and more cards
• and… anything you may see that can add some enjoyment !

Have fun doing Math !