Using Models to Add Up to 20

When numbers get bigger than 10, it can be hard to just picture them in your head. A model gives you a drawing that shows exactly how two numbers combine to make a total, so you can see the addition instead of only guessing at it. Two of the most useful models for addition up to 20 are the bar model and the part-whole model. Both models show the same idea: two smaller numbers, called parts, join together to make one bigger number, called the whole.

A bar model draws the whole amount as one long bar. The bar is then split into sections, one for each part being added. The length of each section shows roughly how big that number is compared to the others, and the whole bar shows the total.

Suppose you want to add \( 9 + 6 \). Draw one bar to stand for the total. Split it into two sections: one section labeled 9 and the other labeled 6. The whole bar, from end to end, stands for the answer.

Because the whole bar is made of both sections joined together, you can read the model as \( 9 + 6 = 15 \). The bar model also shows why addition order does not change the total: if you drew the 6 section first and the 9 section second, the whole bar would still be the same length, so \( 6 + 9 = 15 \) too.

A part-whole model uses circles or boxes instead of a bar. One shape at the top stands for the whole amount, and it connects with lines down to two shapes that stand for the parts. This model is especially good for showing that a total is made of exactly two pieces, and it works well when a word problem gives you two amounts and asks for the total, or gives you a total and one part and asks for the missing part.

Reading this model gives the same fact as the bar model, \( 9 + 6 = 15 \), just drawn a different way. Once you can draw either model, you can pick whichever one matches the problem best: bars are often easier for comparing lengths or amounts, while part-whole circles are often easier for showing "these two pieces make this total."

Models are most helpful when a word problem does not give you a plain addition sentence right away. Read the problem first and decide what the whole is and what the parts are.

Example: "There are 8 red marbles and 7 blue marbles in a jar. How many marbles are in the jar altogether?" Here, 8 and 7 are the two parts, and the total number of marbles is the whole. Draw a bar split into a section of 8 and a section of 7, or draw a part-whole model with 8 and 7 as parts. Either model leads to the same addition sentence, \( 8 + 7 = 15 \).

Once you are comfortable placing numbers into a model, it is worth practicing the same kind of problem using a number line to add up to 20, since jumping forward on a number line shows addition in yet another way. As the numbers get more familiar, you can also work on addition facts up to 20 so you recognize totals like \( 9 + 6 \) or \( 8 + 7 \) quickly, without needing to draw a model every single time.

Bar and part-whole models are not limited to two parts. A bar can be split into three sections instead of two, and a part-whole diagram can connect a whole to three parts instead of two. If you want to see how this works with sums like \( 5 + 4 + 6 \), take a look at adding with 3 numbers up to 20, which builds directly on the same modeling idea used here.

• Decide first whether the problem is asking for a total (the whole) or a missing part.
• Label every section of your bar or every circle in your part-whole model with a number.
• Keep the sections in a bar roughly in proportion, so a bigger part looks bigger than a smaller part.
• Check your model by writing the matching addition sentence underneath it, such as \( 9 + 6 = 15 \).

Related lessons

• Math 1