By the time students get to 4th and 5th 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.
Happy (Place Value) Teaching!!
Click here for a complete, ready-made math unit that teaches more difficult place value concepts for upper elementary students.