Represent problems with diagrams

A mathematical diagram translates a verbal situation into a labeled representation—such as a bar model, number line, table, coordinate-plane sketch, or geometric figure—so that quantities, units, knowns, unknowns, and relationships are explicit. The representation distinguishes additive, multiplicative, fractional, ratio, percent, measurement, and geometric relationships and supports forming or interpreting an equation; it need not be drawn to scale, but its labels and structure must preserve the mathematics. Formal multivariable modeling and advanced geometric representations are outside this scope.

Detailed Explanation: Represent problems with diagrams

A diagram turns the words in a problem into a picture with labels. To make one:

Identify the whole, the parts, and the unknown.
Write the units, such as cans, dollars, or miles.
Choose a diagram that matches the relationship.
Label the known quantities and use the diagram to form an equation.

Example

A class collected $(48)$ cans. Of the cans, $\frac{3}{8}$ are aluminum. How many cans are not aluminum?

Step 1: Identify the whole and the parts

Whole: $(48)$ cans
Aluminum: $\frac{3}{8}$ of the cans
Not aluminum: unknown

Because the problem uses a fraction of a whole, draw a bar divided into $8$ equal parts:

All cans: 48 cans
+------+------+------+------+------+------+------+------+
| Al   | Al   | Al   | Not  | Not  | Not  | Not  | Not  |
+------+------+------+------+------+------+------+------+
    3 equal parts          5 equal parts

The $8$ equal parts represent the denominator, $8$ . The $3$ aluminum parts represent the numerator, $3$ . The remaining $5$ parts are not aluminum.

Step 2: Find the value of one part

Since $(48)$ cans are divided into $8$ equal parts:

48 \div 8 = 6

Each part represents $6$ cans.

Step 3: Find the unknown amount

There are $5$ parts that are not aluminum:

5 \times 6 = 30

So, $(30)$ cans are not aluminum.

The diagram helped show that the number of non-aluminum cans is the remaining $\frac{5}{8}$ of the total.

Learn by doing: Represent problems with diagrams

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Algebra Weights - 3 Scales, 2 Shapes (Simple Substitution, Simple Answer), to Answer

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