Back to workshopTeacher answers · Choose the same paper size in the print dialog.

Fraction Classroom · 2.3.0 · Teacher answers & responses

Cover an area without gaps or overlaps

Build a rectangle with equal square units, then connect a complete covering to rows and multiplication.

Three rows of five equal unit squares cover the rectangle completely. Each small square represents one square unit.5 units across; 3 units tall
Three rows of five equal unit squares cover the rectangle completely. Each small square represents one square unit.

Build the idea

model. A rectangle has 3 rows with 5 unit squares in each row. How much area does it cover?

  1. Agree on one unit square. Every tile must cover the same amount of space; mixing large and small tiles does not give one consistent unit.
  2. Cover the rectangle completely. Move overlapping tiles and fill gaps before counting. A count describes area only when the covering is complete and non-overlapping.
  3. Trace one row of five squares, then count the three rows. Mark each counted row so no square is counted twice.
  4. Record 3 × 5 = 15 square units. Turn the page: five rows of three give the same covering and the same area.

Worked example: 15 square units. The three complete rows contain 5 + 5 + 5 squares, so 3 × 5 = 15.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. A rectangle has 4 rows of 3 unit squares. Find its area in square units.

Answer: 12

Teacher response: Four complete rows each contain three squares.

p2. A rectangle has 2 rows of 7 unit squares. Find its area in square units.

Answer: 14

Teacher response: Count two equal groups of seven square units.

p3. A tiled rectangle has area 20 square units and 4 squares in every row. How many rows are there?

Answer: 5

Teacher response: Five groups of four cover all twenty squares.

e1. Independently draw 6 rows of 3 unit squares. State the area in square units and explain each factor.

Answer: 18

Look for:

  • Names square units rather than just units.
  • Explains that the covering has no gaps or overlaps.

Teacher response: Six names the rows; three names the unit squares in each row.

Transfer the idea

transfer. Two drawings each contain 12 equal unit squares, but the squares are arranged differently. Must their areas be equal?

Answer: Yes, if the squares have the same unit size and do not overlap. Each arrangement covers 12 square units; its outline may differ.

Run the 15-minute workshop

teach-1. 3 minutes: Ask learners to tile a 3-by-5 outline. Pause if tiles overlap and have them repair the covering.

teach-2. 4 minutes: Compare counting every tile with counting complete rows. Label one row and ask what the other factor counts.

teach-3. 5 minutes: Complete the first three practice items. Require a sketch or a row explanation alongside each number.

teach-4. 3 minutes: Use the independent check, then ask whether turning the paper changes the number of tiles.

Respond to the learner

misconception. There are 5 squares along the top and 3 down the side, so the area is 8.

Teacher response: Those counts name the row length and the number of rows. Trace the squares inside the rectangle: adding the two side counts misses most of the covering.

support. Draw row boundaries and number one complete row before counting the remaining rows.

extend. Build two different rectangles with 12 tiles and compare their areas before comparing their boundaries.

Teaching notes