Algebra I Lesson

Advanced Factoring: Unlocking ax² + bx + c (The a·c Method)

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8th or 9th GradeBuilding Thinking ClassroomsCCSS.MATH.CONTENT.HSA.SSE.A.2,CCSS.MATH.CONTENT.HSA.SSE.B.3a60 min

Learning Objective

Students will factor quadratic trinomials of the form $a x^{2} + b x + c$ (where $a > 1$ ) completely using the $a \cdot c$ box/grouping method, including recognizing an overall GCF and identifying prime polynomials.

Lesson Overview

Students discover the box/grouping method for factoring trinomials with leading coefficients greater than 1 by first reversing the expansion process, then building a 4-step algorithm through thin-sliced practice at vertical whiteboards. The lesson moves from positive coefficients to negative signs, hidden GCFs, and prime polynomials.

Materials

Vertical non-permanent surfaces (whiteboards) or large chart paper
Dry-erase markers (one per group)
Erasers or socks
Random grouping cards or tool
Anchor chart paper for consolidation
Individual mini-whiteboards or paper for exit slip

Scaffolded Task Progression

1Task 1

Low-Floor Launch — Reverse the Grid:

Draw a 2×2 grid on your board. Write

2 x^{2}

in the top-left cell and +3 in the bottom-right cell. The outside dimensions are

(2 x + 1)

across the top and

(x + 3)

down the side.

1. Fill in all four cells by multiplying the dimensions.
2. Add all four cells together to get one trinomial.
3. Now look at the two diagonal cells (top-right and bottom-left). Multiply those two terms together. Also multiply the corner cells

2 x^{2} \cdot 3

. What do you notice?

2Task 2

Low-Floor Launch — Reverse the Grid:

Draw a

2 \times 2

grid on your board. Write

2 x^{2}

in the top-left cell and +3 in the bottom-right cell. The outside dimensions are

(2 x + 1)

across the top and

(x + 3)

2 x^{2} \cdot 3

. What do you notice?

Word bank:
- Grid: a box with rows and columns used to organize multiplication.
- Diagonal cells: the boxes that are across from each other, not touching.
- Trinomial: an expression with three terms.
- Dimension: a side length of the rectangle or box.
- Multiply: to find the product of numbers or terms.

3Task 3

Slice 1 — First Reversal (Hint Provided):

Now reverse it. Factor

2 x^{2} + 7 x + 3

.

- Draw a 2×2 box. Place

2 x^{2}

in the top-left and +3 in the bottom-right.
- Multiply them:

2 \cdot 3 = 6

. Find two numbers that multiply to

6

and add to

7

(the middle coefficient).
- Place those two numbers as

x

-terms in the remaining diagonal cells.
- Factor out the GCF from each row and each column to find the outer dimensions.
- Write the final factored form.

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