Representation in: Place Value and the Base Ten System
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1. Let's do some math! Subtract 73-35 and 123-89.
Did you use the algorithm? How would your students feel like they should approach this problem?
Do you have any mental math strategies for these problems? Do
Are you able to estimate for reasonable answers?
What do you need to understand in order to arrive at a correct answer?
What could go wrong?
2. Now do the math with a representation.
Try doing the two subtraction problems using pictures or virtual base 10 blocks. If you haven't used base 10 blocks before, Subtracting with Base-10 Models gives instructions for one approach.
Other possible representations to introduce, depending on your student's place in the CRA model:
Unifix cubes, or these more adult-leaning centimeter cubes
For more virtual manipulatives, check out Polypad.
3. Dig a little deeper.
Read this article from The Kentucky Center for Mathematics about bundles and sticks
Here is a student-facing representation-to-abstract level chapter on place value from Adult Literacy Fundamental Mathematics: Book 2.
The Complete Guide to Using Base 10 Blocks from brainingcamp
All about Place Value Charts from the National Centre for Excellence in the Teaching of Mathematics
If you have some funds for manipulatives, these centimeter snap cubes are a bit more adult than regular-sized snap cubes found in elementary classrooms.
4. Enduring Understandings: What conceptual understanding are we seeking to help our students develop?
4. Enduring Understandings: What conceptual understanding are we seeking to help our students develop?
a. The value of a digit depends on its place. A 5 in the ones place represents 5, while a 5 in the tens place represents 50. Students need to see numbers as quantities made of units of ones, tens, hundreds, etc.
a. The value of a digit depends on its place. A 5 in the ones place represents 5, while a 5 in the tens place represents 50. Students need to see numbers as quantities made of units of ones, tens, hundreds, etc.
b. Our base-ten system is built on groups of ten. Ten ones can be composed into one ten; ten tens can be composed into one hundred. Likewise, one ten can be decomposed into ten ones when needed.
c. A number can be represented in more than one equivalent way. For example, 42 can be thought of as 4 tens and 2 ones or 3 tens and 12 ones. The quantity has not changed—only its composition has changed.
d. Regrouping does not change the total value. When we trade 1 ten for 10 ones, we are not “borrowing” something from another place. We are renaming the same quantity using different units.
e. Subtraction means finding the difference between quantities or removing a quantity. Students should connect the symbols to an action or relationship, rather than seeing subtraction primarily as a sequence of procedural steps.
f. Regrouping is based on the structure of our number system, not a trick or rule. The important conceptual shift is from “I need to borrow” to “I can decompose one ten into ten ones because those quantities are equivalent.”
The algorithm should eventually feel like a concise recording of reasoning students already understand, rather than a new procedure they have to memorize.
5. Let's build some collective understanding through discussion. We know that reflection and connection helps to deepen knowing. Please take a few minutes to open up the Google Doc below and share your thoughts.
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