Multiplying polynomials with an area model
MathematicsAlgebraAges 13–14
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Algebra
A rectangle with length a + b and width c + d is split into 4 cells. Students drag the handles on the edges to change a, b, c, d and see that each cell is a product (a monomial), with the sum of the four cells equal to (a + b)(c + d). Switch to (x + b)(x + d) mode to see x² + (b + d)x + bd.
Lesson: Polynomial multiplication
What it shows
The area model turns polynomial multiplication into geometry. A rectangle with sides a + b and c + d splits into four cells with areas ac, ad, bc and bd, so (a + b)(c + d) = ac + ad + bc + bd: every term of one factor multiplies every term of the other, the distributive property applied twice. With sides x + b and x + d the cells are x², bx, dx and bd, and the two middle cells combine into the like term (b + d)x. The picture also exposes the common error ac + bd, which leaves out two cells.
How to use
Drag the round handles on the edges, or use the Segment a to Segment d sliders, to resize the four cells. Begin with Numbers so students check that the four cell areas add up to the whole rectangle, then switch to Letters to see the general identity. The (x + b)(x + d) notation shows the two middle cells collecting into one term. Ask students to predict each product before changing a length.
Parameters you can change
- Segment a 1–8 unit
- Segment b 1–8 unit
- Segment c 1–8 unit
- Segment d 1–8 unit
- Notation Numbers, (x + b)(x + d), Letters a, b, c, d
Questions to explore
- How many partial products appear when a two-term polynomial multiplies a three-term polynomial, and what would the area model look like?
- In (x + b)(x + d), why can the cells bx and dx be combined while x² and bd cannot?
- Why is (a + b)(c + d) not equal to ac + bd, and which cells does that answer leave out?
