As a child in the 80’s I received what I now consider to be very surface level math instruction. I learned the rules. I followed the rules. The rules worked. The end.
With this surface level training, I managed to successfully earn a high school diploma and a college degree without ever having to think about why I carried a one when I was adding or why a 2 became a 12 when I borrowed. I did not enjoy math or think it was fun, but I survived.
I became a parent in 2003. Throughout the aughts and 2010’s parents like me were losing their minds over “new math.” It’s not that math itself changed. The answer to 10+10 will always be 20. It was how math was being taught that went through an evolution. Instead of kids being expected to learn and execute the rules, they were expected to understand why the rules were the rules in the first place. They were also learning to break numbers apart and put them back together in ways mom and dad had never seen before. A common parental chorus was, “We had one way and it worked. Why fix something that isn’t broken?”
I’ve come to believe it was broken, but I’ll save that discussion for another day.
Fast Forward to 2025, and parents who missed out on “new math” as a child are still banging their heads on the table. If you are one of those parents, let me share with you a few things you need to know to make homework time a little easier.
Standard Form
The normal way we see and write numbers is called standard form. 8, 27, and 593 are examples of numbers in standard form.
Expanded Form
Expanded form tells us the value of the individual digits in the number.
Let’s put the number 241 in a place value chart.
- The 2 is in the hundreds place.
- The 4 is in the tens place.
- The 1 is in the ones place.
One way to visualize the value of each digit in the number 241 is to fill out ten frames.
- The 2 in 241 represents 2 one hundreds. Therefore the value of the 2 is 200.
- The 4 in 241 represents 4 tens. The value of the 4 is 40.
- The one in 241 represents 1 one. The value of the 1 is 1.
When we write a number in expanded form, we show the value of each digit, and we put a plus sign between each digit.
Standard Form: 241
Expanded Form: 200+40+1
Different ways to Represent 241
- Craft Sticks
- Base 10 Blocks
- Money
- Ten Frames
- Place Value Chart
Regrouping
The largest number that can go in any given place value column is 9. We can have 9 ones, but as soon as we get to ten, we now have one ten and zero ones. This is why we “regroup.”
- Only single digits (0-9) can fill a column on our place value chart. As soon as we get to two digits, we’ve made a group of ten. The one group of ten is placed in tens column.
- Standard form: 10
- Expanded form: 10+0 (one group of 10 and zero ones.)
- Standard form: 100
- Expanded form: 100+0+0
Base 10
When we regroup, we are trading 10 units of a given place value for one bundle of the next higher place value.
- ten singles= one bundle of ten
- ten bundles of ten= one bundle of 100
- ten bundles of 100= one bundle of 1000.
This is why we refer to our numbers as being part of a “base 10 system.”
- Ten units get traded for one rod of 10.
- 10 rods get traded for one flat of 100.
- 10 flats get traded for one cube of 1000.
Regrouping/Carrying the 1
65+9=?
Problem: 596+9
- Step One: Show 563 on the place value frames.
- Step 2: Add 9 ONES in the ones column. (I’ve shown these in red.)
- 563+9
- Step 3: Since one entire top frame in the ones column is full, we made a group of ten.
- We have to regroup. We exchange the 10 ones for 1 group of 10. We cross off the full frame, and and put a new 10 in the tens column.
- Step 4: Count up your answer.
Subtraction
Example: 563-7
- Begin with 563
(500+60+3) -7
- We only have 3 units in the ones column. We do not have enough to subtract 7.
- When we borrow, we are taking one of our bundles of 10 and breaking it into 10 ones. Now we have 5 hundreds, 5 tens, and 13 ones.
(We still have a total of 563.)
Borrowing
- Now that we have 13 ones, there are enough ones to subtract 7.
Our answer is 556.
I hope this information is helpful. I’ll be back with more math help in the future!