Decimal Multiplication with Grids: Models and Missing Factors

A hundredths grid makes decimal multiplication visible: one whole grid has 100 equal squares, so each small square is 0.01. To complete a sentence such as 0.2 × □ = 0.4, count how many groups of 0.2 fit into 0.4. Two groups fit, so the missing factor is 2.

Set the value of the whole

For these models, the entire 10 × 10 grid represents 1. One full row or column contains ten small squares and represents 0.1. Twenty squares represent 0.20, which is the same number as 0.2.

Keep that unit fixed throughout the calculation. A count of 12 small squares means twelve hundredths, or 0.12—not 12 wholes.

Find a missing factor: 0.2 × □ = 0.4

The blue columns show one group of 0.2. The orange columns show another group of 0.2. Together, they cover 40 of the 100 squares.

1. Identify the size of one group: 0.2, or 20 squares.
2. Identify the total: 0.4, or 40 squares.
3. Count the groups: 40 ÷ 20 = 2.
4. Check the completed sentence: 0.2 × 2 = 0.4.

For 0.5 × □ = 1, each group covers half a grid. Two halves cover one whole, so the missing factor is also 2.

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Multiply two decimals: 0.3 × 0.4

Repeated groups work well when one factor is a whole number. When both factors are less than 1, use an overlapping area model. Here, multiplication means taking three tenths of four tenths.

1. Shade 3 of the 10 columns to represent 0.3 of the width.
2. Shade 4 of the 10 rows in a second color to represent 0.4 of the height.
3. Count only the overlapping squares: 3 × 4 = 12.
4. Each square is one hundredth, so 0.3 × 0.4 = 0.12.

The written method agrees: 3 tenths × 4 tenths = 12 hundredths. The product is smaller than either positive factor because each factor is less than 1. Multiplication does not always make a number larger.

If a product exceeds 1, use more than one whole grid. For example, 0.6 × 2 = 1.2 covers 120 small squares: one full grid and 20 squares of another.

Practice and check

Complete each original multiplication sentence. Draw equal groups if it helps, then check by multiplying.

1. 0.3 × □ = 0.6
2. 0.4 × □ = 0.8
3. 0.1 × □ = 0.5
4. 0.6 × □ = 1.2
5. 0.7 × □ = 1.4

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For more place-value and decimal review, explore the Grade 5 Math Online Center. The Grade 5 number and base-ten standards connect decimal operations with concrete models, drawings and written reasoning.

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