Fly on Math Teacher’s Wall – Put the Value in Place Value

A group of math bloggers have gotten together to join forces in ‘squashing’ some math misconceptions.  For the inaugural blog hop we are tackling the big idea of Place Value.  So, read through my thoughts and then at the bottom is a link to another blogger and their thoughts on Place Value.

 

Before we start, answer this question (I’ll come back to it later, but get your answer now):

How many tens are in 243?

 

For far too long textbooks have only focused on the ‘place’ a digit is in when writing lessons for place value.  Typical questions sound like this, “In the number 64,235, what place is the 4 in?”  Questions like that may be important, but they do not emphasis Place Value.  Even questions that ask students to tell the value of a digit do not emphasize Place Value.  I suggest that they might actually cause misconceptions surrounding the big idea of developing Place Value as a whole, i.e. questions like “What is the value of the 2 in 64,235?” are leading children to see digits in isolation without developing an understanding of how the digits around it compare.  When we break Place Value down into its parts and focus on them in isolation, focusing on the place a digit is in or focusing on the value of the digit, it does not give children a full picture of what Place Value is about.  The Number and Operations in Base Ten progressions document, makes the point that the underlying understanding we need to help children comprehend is the relationship between the values of the places; “the value represented by each place is always 10 times the value represented by the place to its immediate right” (p.2).  The digits within each place represent 0 through 9 of those units and when we get 10 of those units it makes one of the next highest unit.  Van de Walle (2013) says it like this; we need children to be able to understand how 100 can be 10 tens or 100 ones.

Let me share a story about my son.  One night he was helping my husband and I count out some cash that we were going to be depositing and he counted out “1 hundred, 2 hundred, 3 hundred, 4 hundred, 5 hundred, 6 hundred, 7 hundred, 8 hundred, 9 hundred, 10 hundred, 11 hundred…” (See TMWYK’s Ten Hundred Doras as another example of children describing numbers in this way.) As he was counting out the physical hundred dollar bills it made complete sense to call it 10 hundred, NOT 1 thousand.  But when it is written as 1000, how often do we call it 10 hundred?  We don’t.  Not on numbers like 1000, but we do on numbers like 1500 (did you say that as 15 hundred or as 1 thousand 5 hundred???).  It’s an interesting thing to think about…we ask kids to do all kinds of bundling activities (count the items and when you reach 10 you bundle them up and now it is called a 1 of the next highest unit).  But how often do we let them use their informal, yet mathematically solid, counting of those bundles and continue on with their counting?  Usually we don’t, we stop them, and have them turn those 10 hundreds into 1 thousand, then they can continue counting.  When we stop their connections to cool patterns they are noticing, it creates children, and adults, who, when asked how many tens are in 243 respond with ‘4.’ 

 

The answer to “How many tens are in 243?” is not 4.  If I had 243 t-shirts and decided I wanted to bundle them into rolls of 10 t-shirts, would I only have 4 rolls?

 

Cathy Fosnot’s Contexts for Learning Mathematics has a fabulous T-shirt factory unit.  In which, students have to help Grandma Eudora organize all the t-shirts she is making (and along the way start to understand place value).  If Grandma Eudora had 52 t-shrits laying around unbundled, you have the kids bundle the t-shirts into rolls of 10 to make them more organized and easier for Grandma to count.  As they do each bundle of 10, record the status of the t-shirts in a T-chart:

This activity does more than just develop place value; it also helps you facilitate some awesome discussions about composing and decomposing numbers.  As you move into larger numbers of t-shirts, you can use the chart to help you lead discussions about the patterns they see and connect them to the ideas of place value.

 

 

Okay, so, back to our original question; How many tens are in 243?  If Grandma Eudora had 243 t-shirts, when we bundle them up there would be 24 bundles of ten in those 243 t-shirts. i.e. 24 tens in 243.

Let’s try another.  How many hundreds in 2000? Hint: it is not 0.  I also like to encourage kids to think in terms of money when working on place value.  For example, if you have $2000, how many hundreds would you need to make that amount of money? How many tens are in $2000?  How many ones? How many tenths (or dimes)?

 

Doing activities in which students look for patterns and relationships helps them develop a more cohesive understanding of place value that is not developed in the types of tasks most textbooks use for Place Value.

 

Want to try another one?  How many tenths are in 1036.5?  Hint: it is not 5.

I will leave you with these questions to help you with your state of disequilibrium (or maybe these will put you into a state of disequilibrium)…

How many thousands are in 1036.5?

How many hundreds are in 1036.5?

How many tens are in 1036.5?

How many ones are in 1036.5?

How many tenths are in 1036.5?

How many hundredths are in 1036.5?

 

Post your thoughts in the comments about how you figured out the number of tenths in 1036.5, and then continue on in your journey to increase your understanding of Place Value by visiting Teaching Math by Hart.  She is next in the blog hop.