In Module 4, students will learn place value strategies to add and subtract within 100; they’ll do one- and two-step word problems of varying types within 100; and they will develop conceptual understanding of addition and subtraction of multi-digit numbers within 200. Starting from concrete, then moving to pictorial and lastly to a more abstract approach, students will use manipulatives and math drawings to help deepen their understanding of the composition and decomposition of units.
They will not be taught the traditional algorithm (carrying and borrowing to add and subtract), but rather other strategies that help them develop their conceptual understanding of math.
Many of us adults were taught the PROCEDURE of adding and subtracting with “carrying and borrowing.” We were not taught the CONCEPT, that is, the why of adding and subtracting, only how to do it using a shortcut.
When we saw that there were too many units in the ones column, we were taught how to “carry a ten” – – pass it over to the tens column. We were also taught how to “borrow a ten” – – get numbers from the tens column to put in the ones column, in case there were not enough units in the ones column for use to carry out a subtraction problem. It was if the number 35 could start having a conversation with itself, with one part asking the other to give it something!
Giving children this algorithm is like telling them how to get somewhere by taking a shortcut: go left, take the second right, go three streets, turn left again, etc. etc. They will need to memorize all the ins and outs in order to get to their destination.
Giving children addition and subtraction strategies based on a strong conceptual knowledge of place value is giving them the understanding of how to use a map to arrive at a destination: they are given the big picture and guided to discover and use multiple pathways to get to their destination. Memorization is not required.
In second grade we are striving to teach the students the CONCEPT of addition and subtraction. What is really happening is that we are REGROUPING numbers by composing and decomposing them, that is, showing numbers in different ways in order to make addition and subtraction easier to do. The children are encouraged to be flexible thinkers and to learn that 100 can also be shown as 10 tens, 234 could be shown as 23 tens and 4 ones, or 22 tens and 14 ones, for that matter!
In class, we also us the term exchange to emphasize the fact that 10 ones is the same as one ten, 10 tens is equal to 100. We can exchange ten 10 € bills for one 100€ bill. We study how an exchange can be made for units of equal value. Here are two excellent videos to watch to help understand how to explain this process to second graders:
Here is an excellent article titled, “Nix the Tricks: A Guide to Avoiding Shortcuts that Cut out Math Development” by Tina Cardone et al.
Below are some simple drawings which we hope you will find useful to see the WHY and HOW of what we are doing in class. When your child does the homework, he or she will need a pencil and a pen or marker in a contrasting color. Below, we have used red to show contrast, but any color is fine. We are DRAWING the numbers in base ten, in a way that simplifies the shape of base ten blocks:
For example, 48 will be 4 tens (4 lines) and 8 ones (8 circles or dots).
48 + 24 = ________
DRAW the numbers in base 10 (meaning, lines for the tens, and circles or dots for the ones) on the place value chart. Ask your child to organize the numbers; tens like tally marks, ones like dots on a ten frame, as shown below). The two numbers are placed in middle part of mat, the first number above the second one. (The answer will be placed below the line.) We start out by adding up the ones. Ooops! Too many ones for the ones column (there are 12).
- Take a red pen, and circle a group of 10 ones (you are composing them into a group of ten).
- With the red pen, draw an arrow over to the tens column, where you will be putting your group of ten. Remember to cross off the circle of ones, so you don’t count it twice!
WHAT’S THE DIFFERENCE?
When we subtract, we are actually looking at two quantities to see the difference between them, that is, we are trying to figure out how much more one group has than the other.
- We have two groups here.
2. Before human beings had numerals with which to subtract symbolically, they could figure out which of two groups was bigger by taking away one from one group, one from the other, and so on, till one group disappeared and the larger group remained. Here, we have taken away one from the group of flowers , and one from the group of butterflies.
3. We continue until the smaller group (butterflies) is entirely crossed off. We see that the group of flowers still has 14 flowers. It is larger by 14. The difference between the two groups is 14.
- When we subtract, we use the above rationale. DRAW 35 and 19 in base ten on the place value chart. We know that 35 is the larger of the two numbers, but we don’t know by how much. We can see that there are only 5 ones in the top part of the table – – not enough to be able to successfully cross off all those 9 ones in the bottom number ! (See HTO chart below).
- We need to break 35 apart (decompose 35) in such a way that there will be enough ones in the top group to be able to successfully cross off all those 9 ones in the bottom group. So, with a red pen we’ll take a group of ten, cross it off (so we don’t accidentally count it later), draw an arrow to the ones column, and draw 10 ones in red in the ones column.
- Now in red we will cross off the 9 ones (from the 19), and also cross off 9 ones (from the 35). (Remember the procedure from “what’s the difference?“)
- Now we will cross off the 1 ten (from 19), and also the 1 ten (from 35). (Remember “what’s the difference?“)
- Finally we draw what is left in base ten: (1 line and 6 circles), and we write that number in the number sentence: 35 – 19 = 16
If we teach our students the shortcut (the traditional algorithm of “carrying” and “borrowing”) for doing addition and subtraction without previously giving them the rationale for their actions, learning this shortcut will be meaningless to them.
But by understanding the rationale behind addition and subtraction, these operations become meaningful, and easy to visualize and understand. The traditional algorithm will come later.
If you have any questions please don’t hesitate to ask us!
The Second Grade Team