By the time students get to 4

^{th}and 5

^{th}grades, they should already have strong conceptual understanding of the following place value concepts:

-counting by ones,
fives, tens, hundreds, thousands, etc.

-each place value
is 10 of the place value to the right.

-writing numbers
in words and numerals.

-working with
expanded form for smaller numbers.

-comparing and
ordering smaller numbers.

-rounding whole
numbers.

-using place value
understanding when adding and subtracting.

-multiplying by
ten.

Hopefully, the
students have been taught to use these concepts, and not just to apply ‘tricks’
to get the answers. If they’ve been
using shortcuts (eg, ‘just add a 0’ for multiplying by 10, or ‘circle the
place, look at the number to the right, and change everything to 0’ for
rounding, or ‘just carry the 1’ when regrouping in addition), then helping
students understand harder place value concepts in upper elementary is going to
be a real struggle. If they’ve truly
developed the conceptual understanding for these place value concepts, then the
more complex ideas in the upper grades should build naturally on what they have
already learned. (And the same going
forward- what they will learn in middle and high school should build naturally
on the place value concepts from upper elementary, making them even more
important.)

In upper
elementary, the focus shifts to larger/ smaller numbers, including decimals,
and to generalizing and applying the base ten concepts. Many of the concepts are the same ideas, just
now applied to larger numbers and decimals.
In lower grades, students learn to compare numbers to the thousands
place. In upper grades, they extend this
understanding to comparing larger, multidigit numbers and decimals. In fourth and fifth grade, students work on
place value concepts such as:

-understanding
that the place value to the right is 1/10 of the one to the left, and the one
to the left is 10 times the one to the right.

-use rounding and
estimation fluently and apply them as a check on the reasonableness of a number
or answer.

-comparing and
ordering decimals and larger numbers.

-working with
expanded form for decimals and larger numbers.

-apply place value
understanding when performing arithmetic.

-applying patterns
of zeros and ‘moving the decimal’ when multiplying or dividing by ten.

-beginning to work
with exponents (an extension of multiplying by ten)

To help develop
these concepts in upper elementary, try:

-relating numbers
to money. This works for both very large
numbers (a billionaire is worth ONE HUNDRED millionaires), and for decimals
(dollars and cents are a very concrete way for kids to understand ones, tenths,
and hundredths.)

-working with
calculation problems where it makes more sense to apply place value concepts
than to use the standard algorithm.
1,000- 999 is a perfect example of this.
Many students will struggle trying to borrow from the 1 in the thousands
place, across all those zeros, but students with strong place value
understanding will see that 999 is only one away from one thousand.

-do a lot of estimating…
especially with money. A great way to
practice this is by estimating the total grocery bill for a list of items.

-continue working
with number lines. (They aren’t just for
learning to count from 0-100!) Making a
decimal number line from 0.00 to 1.00, counting by tenths and hundredths is a
great way for students to make sense of how tenths, hundredths, and ones
connect and why the zeros at the end of decimals are not always necessary. Try having each pair or group make a decimal
number line between two different whole numbers (one group does 0.00-1.00, one
does 1.00-2.00, etc.), so students can see how decimals connect to whole
numbers, and will help rounding to the nearest whole make more sense.

-using real life
examples of large numbers and decimals. It
can be hard to make these numbers concrete for kids, and finding real life
examples can help.

-do math talks and
strategy talks where students share ideas and strategies based on place value
understanding. If your students aren’t
at that level yet, start with modeling it for them.

-encourage
students to use strategies other than the standard algorithms. The standard algorithms are very practical,
and students definitely need to be able to use them. But we also have calculators that can
replicate the standard algorithms for us.
What calculators cannot replicate is the more flexible thinking and
understanding that comes from solving problems using a variety of strategies
and the strong place value concepts that develop from working with numbers in
this way.

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