Dear Parents,

Here are two more addition and subtraction strategies we are teaching the second graders: tape diagrams and number bonds. You are already familiar with number bonds, but we thought we’d explain it again to clear up any doubts your child may have.

Tape Diagrams

A tape diagrams is really just a model or a drawing of a word problem! As we know, a picture is worth a thousand words. It is incredibly helpful for children to actually see a diagram of a word problem before they try to do any calculations. It helps them see the “big picture” or the whole idea, and gives them clarity as to just what it is they need to figure out. As we will see in the examples below, having an illustration of the relationships between the numbers involved in the word problem can help children take a look at how reasonable their answers are.

Tape diagrams are also known as strip diagrams, bar models, fraction strips or length models. We are calling them tape diagrams at our school, because they look like segments of tape:

Up to this point, the children have, of course, been making visual representations of problems, drawing apples, trees, or base ten blocks to represent the numbers and give them a better sense of what they need to do to solve problems. Now the drawings become more abstract, showing quantities as opposed to actually showing groups of objects.

If a word problem calls for adding 13 apples to 37 apples, when using tape diagrams, children are drawing something more abstract, not one group of 13 apples and another of 37, but rather two rectangles (tape segments) that bear a relationship to one another, the “13 apple tape” being shorter than the “37 apple” one.

**Part-Whole Model for Addition and Subtraction**

In the problems below, we need to think about what the WHOLE is, and what the PARTS are.

Sara brought 4 apples to school. After Mark brings some more apples to school, there are 9 apples altogether. How many apples did Mark bring?

9 apples = the WHOLE, shown here with with parenthesis at the top of the problem.

4 apples brought by Sara = just a PART. We still need to figure out what PART Mark is bringing.

This is a very simple example! But when children see that the top part of the drawing is going to be 9, they can see that Mark bringing 12 apples is a very unreasonable answer, because it’s way too big! (12 is bigger than the WHOLE of 9 apples.)

**Comparison Model for Addition and Subtraction**

1. Mateo has 5 toy cars. Josiah has 2 more than Mateo. How many cars do Matteo and Josiah have altogether?

In this problem, we are comparing the PARTS. We are comparing the amount of Mateo’s cars to Josiah’s cars. We know the smaller quantity. To find the larger quantity, that is, the amount of cars Josiah has, we add 5 + 2. Later, to get the WHOLE, we then add up the PARTS.

Here it is very useful to have a diagram that shows us the relationship between the amount of cars that Mateo and Josiah have. Since we are making a **comparison between how many they each have,** **it is helpful to have the PARTS of the problem one above the other.** In Mark and Sara’s apple problem above, in which there was no comparisons of the PARTS, the tape diagram was set up differently. The PARTS were side by side, with the WHOLE stretching above them.

2. Sam has 45 stamps. He has 20 more stamps than Joe. How many stamps does each boy have? How many do they have in all?

Again, it is very useful to have a diagram to show us the relationships between the quantities of stamps that Joe and Sam have. Since we are making a **comparison** between how many they each have, we have have the PARTS of the problem one above the other, and thus we see how much bigger one part is than the other.

We finish by adding up the parts together to get the whole.

This is just a start! Our students will need time and practice to understand how they need to set up the diagrams, depending upon whether the word problems call for a comparison of the parts, or for taking a part from the whole, or adding up the parts.

The second graders will be using tape diagrams in third, fourth and fifth grade, and beyond. Tape diagrams can be set up differently for multiplication and division. They help to give “a thousand words” more to word problems, and thereby aid children’s understanding.

**Number Bonds**

We worked on number bonds at the beginning of the school year. To review, a number bond is a visual way to see a number and the parts that can make up a number. In this way, number bonds help children visualize addition as putting things together and subtraction as taking things apart.

As we do addition and subtraction problems in second grade, we create number bonds by decomposing or “breaking up” larger numbers into smaller parts, so that we can add and subtract more easily.

**Addition with number bonds**

When doing addition, we often break up numbers into tens and ones, then add tens to tens, ones to ones, and then everything together.

**48 + 23 = _______**

a. We can break up the numbers into tens and ones:

b. Then we add the tens together, the ones together, and then both parts together:

**Subtraction with number bonds**

**61 – 27 = _______**

Subtraction will be a bit trickier. When we look at the tens and ones in this subtraction problem, we see that we need to take away more ones (7) than there are in the first number ( only 1 in 61). When the second graders use base ten, they know that in order to be able to take away 7 ones, they’ll need more ones to do so – – and so they exchange a ten.

a. Here, they will need to break 61 up in such a way that there will be enough ones to subtract from. They will need to be careful about the number they place in the red circle: **Here is where most of the second graders make their mistakes!**

b. With number bonds, we can be creative when breaking apart or decomposing numbers, and we don’t NEED to stick to tens and ones. Here is an example:

c. You can break up 61 this way (50 and 11) and you could just as easily do it as 51 and 10. It’s up to you. You just need to make sure there is a number larger than 7 (in this case) in the red circle.

Continuing on, we will subtract 50 – 20, and 11 – 7.

The tricky part to remember is that we then **ADD UP the two partial answers**, because the 30 and the 4 represent the PARTS of the initial equation of 61 – 27.

d. The way we have just done 61 -27 with number bonds connects with the way in which the children have done subtraction with the place value (HTO) chart. Here, we have decomposed 61 into 50 and 11.

e. We continue to solve the problem: 11 – 7 = 4 (in the ones column) and 50 – 20 = 30 (in the tens column). We end up with 30 + 4, or 34.

Our goal is for our students to have a large repertoire of strategies at their fingertips; we want them to have lots of tools in their tool box to choose from! Later on, with lots of practice and equipped with strong mental representations of these strategies they’ll have loads of flexibility and be much better able to do the math mentally.

Here are two videos you can watch, to see how the second graders are using number bonds.

https://learnzillion.com/lessons/3142-find-unknowns-by-creating-number-bonds

https://www.engageny.org/resource/grades-pk-2-math-supporting-coherence-across-grades-number-towers-number-paths-and-number

Please let us know if you have any questions! Thanks for all the help you are giving our students!

The Second Grade Team