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Fraction Classroom · 2.0.0 · Student pages

Find the overlapping part of a whole

Build a two-direction area model to connect fraction multiplication with overlap and square units.

Anonymous group code: __________   Date: __________

Quick diagnosis

Try all three. Explain with words, a sketch, or objects.

d1. A whole rectangle has 3 equal columns and 4 equal rows. What fraction of the whole is one cell?

d2. In a 3-column by 4-row grid, shade 2 columns vertically and 3 rows horizontally. How many cells have both shadings?

d3. Should 1/2 × 2/3 be greater or smaller than 2/3? Explain the meaning of the first factor.

Worked examples

Study the example that matches the difficulty. Keep the other for a second session.

w1. Use an area model for 2/3 × 3/4.

  1. Draw one rectangle and label it one whole.
  2. Split it into three columns and select two.
  3. Split the same rectangle into four rows and select three.
  4. The overlap has 2 × 3 = 6 cells out of 3 × 4 = 12 equal cells; six twelfths is one half.

Worked example: 6/12 = 1/2 of the original whole.

w2. A rectangular mat is 3/4 meter long and 2/5 meter wide. Find its area with a unit-square model.

  1. Draw a 1 meter by 1 meter reference square.
  2. Divide its length into fourths and width into fifths, making twenty equal cells.
  3. Select three fourths in one direction and two fifths in the other.
  4. The mat occupies six cells, each 1/20 square meter, for an area of 3/10 square meter.

Worked example: 6/20 = 3/10 square meter.

Practice

For a short session, choose three tasks. Use the others for a second try or follow-up.

p1. Draw a grid showing 1/2 × 3/5. Label the whole and count the overlap.

p2. A rectangle is divided into 4 columns and 3 rows. Two columns and one row are selected. Write the product and its value.

p3. A student says 2/3 × 1/4 = 3/7 by adding both pairs of numbers. Explain using a grid.

p4. Find the area of a rectangle measuring 2/3 meter by 1/2 meter. State the unit.

p5. Does switching 2/3 × 3/4 to 3/4 × 2/3 change the area? Explain by turning a grid.

p6. A rectangle has 5 columns and 4 rows. An overlap covers 3 columns and 2 rows. Give its fraction of the whole in simplest form.

Transfer check

Try one independently. Save the other for a later check.

t1. A garden uses 3/4 of a rectangular plot. Herbs occupy 2/3 of the garden. What fraction of the entire plot contains herbs? Name both wholes in the story.

t2. A sign has side lengths 5/6 meter and 3/4 meter. Find its area, then explain why the result is less than 5/6 square meter.