Understanding Bar Models in Mathematics
Bar models help students solve problems by making it easier to see the relationships between numbers, often helping them decide whether they need to add, subtract, multiply, or divide to solve the problem.
Bar models are especially useful for comparison problems, whether they involve addition or multiplication. These problems can be tricky because sometimes the larger quantity is already known, and students need to subtract or divide to find the smaller quantity. Other times, they might need to figure out the larger quantity or even the comparison itself, such as “How many times larger is 12 than 4?” Bar models help students visualize whether they are finding a smaller or larger quantity, or the comparison, making these problems easier to solve.
Why Are Bar Models Important?
Bar models are versatile visual tools that represent quantities and their relationships as rectangular bars. They are commonly used to solve addition, subtraction, multiplication, and division problems, as well as more complex concepts like fractions, ratios, and word problems.
Bar models help students develop a deeper understanding of mathematical concepts by focusing on the relationships between parts of a problem rather than just the calculations.
Types of Bar Models
Part-Part-Whole
These models show how smaller parts combine to form a whole, making them ideal for addition and subtraction problems.
For example, for the problem, “Sarah has 8 red apples and 5 green apples. How many apples does she have in all?” draw two bars of the same size, one on top of the other.
The top bar represents the total, which is unknown. The bottom bar is divided into two parts to show the two types of apples. Label one part with a value of 8 and the other with a value of 5. Students can now see that the top bar represents the total of 8 and 5 combined, or 8 + 5 = ?
Similarly, given the problem, “Sam has 15 stickers. Eight of the stickers are glitter stickers, and the rest are scented stickers. How many scented stickers does Sam have?” draw two bars of the same size, one on top of the other.
The top bar represents the total, 15. The bottom bar is divided into two parts to show the two types of stickers. Label one part with a value of 8, and leave the other part as unknown. Students can now see the problem as 8 + ? = 15 or 15 – 8 = ?
Comparison
Comparison bar models show the relationship between two quantities. They are especially helpful for problems where students need to figure out the smaller or larger quantity or the comparison itself. For example, “John has 12 marbles, and Maria has 8. How many more marbles does John have?”
In this example, the bar model is used to represent how much larger 12 is than 8, in an additive sense. This can be interpreted as 8 + ? = 12 or 12 – 8 = ? The model helps students see how both subtraction and addition are connected, and how both operations can be used to solve for the unknown.
For the problem, “How many times larger is 12 than 4?” the bar model is used to represent how much larger 12 is than 4, in a multiplicative sense. This can be interpreted as 12 ÷ 4 = ? or ? × 4 = 12. The model helps students see how many smaller bars, each valued at 4, are needed to equal the length of the bar valued at 12.
Multiplicative
These bar models represent multiplication and division problems, often showing equal groups or repeated addition. For example, “Sarah has 3 cups of juice, and her friend Emily has 4 times as much juice as Sarah. How many cups of juice does Emily have?”
In this example, the smaller quantity (3 cups of juice) and the multiplicative relationship to the larger quantity (4 times as much) are known, but the larger amount is unknown. The model helps students understand that the larger quantity can be found by multiplying the smaller quantity by the comparison, 3×4.
“Max has 16 pencils, which is 4 times as many pencils that Luke has. How many pencils does Luke have?”
In this example, the larger quantity (16 pencils) and the multiplicative relationship to the smaller quantity (4 times as many) are known, but the smaller amount is unknown. The model helps students understand that the smaller quantity can be found by dividing the larger quantity by the comparison, 16 ÷ 4, and to see the clear connection between multiplication and division.
Developing Mathematical Communication About Bar Models
Using Bar Models to Solve Word Problems
Bar models are particularly useful for breaking down word problems, but students need clear, explicit instruction when learning how to draw accurate bar models that represent the problem and how to use them to find solutions. Guiding students with step-by-step questions can help them navigate the process. Over time, they will start to ask themselves these questions independently. Here’s a list of possible guiding questions to help students work through using this representation:
- “What are the objects or quantities in the word problem? What is being compared?” Draw a bar model and label the parts.
- “Does the word problem give us the total?” or “Does it give us the larger quantity?” If yes, label it. If not, place a question mark in the larger bar.
- “Does the word problem provide the value of any smaller parts?” or “Does it tell us the smaller quantity?” If yes, label it. If not, place a question mark in the smaller bar(s).
- “Based on the information in the bar model, how can we solve the problem?” or “What equation(s) does the bar model represent?” Write the equation and solve.
Common Misconceptions About Bar Models
Misconception: Bar Models Are Only for Addition or Subtraction
While often introduced for simple operations, bar models are equally useful for multiplication, division, fractions, and ratios. Encourage students to use them across a variety of problem types.
Misconception: Bar Models Always Look the Same
Students might assume all bar models follow the same format. Emphasize that the structure depends on the problem type, such as comparison versus part-part-whole.
