Relate area to the operations of multiplication and addition.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Tiling a rectangle row by row reveals a shortcut: a rectangle 4 units wide and 6 units long has 4 rows of 6 squares, so its area is 4 × 6 = 24 square units. Third graders first tile and count, then notice that multiplying the side lengths always gives the same answer, and finally use multiplication directly to solve real problems, such as finding how much carpet covers a 7-foot by 9-foot floor.
The connection runs the other way too. Any product can be pictured as the area of a rectangle, which makes area models a powerful thinking tool. Splitting a 6-by-8 rectangle into a 6-by-5 part and a 6-by-3 part shows that 6 × 8 = 6 × 5 + 6 × 3, the distributive property in picture form. Area is additive, so students can find the area of rectilinear figures, such as an L-shaped room, by cutting them into non-overlapping rectangles and adding the areas. Four strands live in this one standard (tiling, multiplying side lengths, area models for the distributive property and composite shapes), and together they tie measurement tightly to multiplication.
Students may add 4 + 6 to get 10 for the area. Tile the rectangle to show 24 squares and link the count to 4 × 6.
When splitting an L-shaped figure, children sometimes count the corner square in both rectangles. Shade each part in a different color.
Some side lengths of rectilinear figures are not labeled. Students need to work them out from the labeled sides before multiplying.
A student may draw a split rectangle but write an expression that does not match. Label each part with its dimensions and its area.
An L-shaped room can be split into a rectangle 8 feet by 5 feet and a rectangle 3 feet by 4 feet that do not overlap. What is the total area?
Answer: The total area is 52 square feet.
Move from tiles to partly drawn grids (only the first row and column shown) to plain rectangles with labeled sides. This progression helps students see the rows and columns even when they are not drawn. Grid paper floor plans make rectilinear area problems concrete and engaging.
Assessments include area of a rectangle from its sides, missing side length problems, choosing an expression that represents a split area model, and area of composite figures. Multi-part items often combine several strands.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 63 (also accepted: 63 square feet)
Multiply the side lengths: 7 × 9 = 63 square feet.
Answer: B) 6 × 5 + 6 × 3
Each part's area is 6 times its width, so the total is 6 × 5 + 6 × 3 = 30 + 18 = 48.
Answer: 8 (also accepted: 8 meters)
4 × ? = 32, so the length is 32 ÷ 4 = 8 meters.
Answer: 36 (also accepted: 36 square units)
5 × 6 = 30 and 2 × 3 = 6. 30 + 6 = 36 square units.
Answer: C) He added the side lengths instead of multiplying
3 + 7 = 10, but area is 3 × 7 = 21 square units.
Answer: 20 (also accepted: 20 square feet)
4 × 5 = 20 square feet.
Answer: 56
8 × 5 = 40 and 8 × 2 = 16. 40 + 16 = 56.
A full lesson with slides, activities and an exit ticket on relating area to multiplication and addition, pitched to grade 3 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 3.MD.C.7 with an answer key, ready in about a minute.
Make a worksheet →Turn relating area to multiplication and addition into a quiz students answer online that marks itself, with a class summary for you.
Build a test →(a) tile a rectangle and show the area equals the product of the sides; (b) multiply side lengths in problems; (c) use area models to represent the distributive property; (d) find areas of rectilinear figures by adding rectangles.
A shape whose sides all meet at right angles, like an L-shape or a T-shape. It can always be split into rectangles.