Activity Sheet
Using the Box Method to
Discover the FOIL Method
Teacher Tips
Have students complete the Area review. The purpose of this warm up is to access students’
prior knowledge, so that they may apply it to a new situation. This will “set the scene” for the
multiplying binomials activity sheet.
• Remind students that a definition of area is NOT a formula for the area of a
rectangle.
• Talk about the area as the amount of surface space or the amount of unit squares
that will fit on an object or the rectangle (as in #2 of the warm-up).
• For #3 , explain that the large rectangle holds 13 squares and the smaller one will
hold 11 squares , so together they must hold 24 squares. You want to set in the
idea of adding the areas to get the total area of the composite figure for later.
• For #4, most students will add the side lengths and then multiply 8*9 to get the
area. Also discuss the method of finding the area of each small rectangle and then
adding them to get the same result. When using the box method, students won’t
be able to add the small side lengths because one of them will be a variable.
After going over the warm-up, complete the multiplying binomials activity sheet. The
purpose of this activity sheet is to guide students thinking so that they may observe, on their
own, how to multiply the binomials without having to draw the boxes (or to construct their
own understanding of the FOIL method, or some version of it).
• explain that students will be finding the product of two binomials.
• They will be using an area model (box method) to find the product.
• They will be finding the area of a rectangle to model/represent the product.
• The side lengths of the rectangle will be represented by the two binomials.
Therefore, the area of the rectangle will be the product. Students will find the
areas of the smaller rectangles and then add them to find the total area which will
be the product.
• **Since the binomials are only “representing” the side lengths of the rectangle,
negatives are included (most students won’t even notice that some of the side
lengths are negative so you probably won’t need to address this.)
After completing side one of the activity sheet, students should be able to multiply the two
binomials without using the box method. Hopefully, they have derived the FOIL method, or
some version of it on their own . They should be able to describe this method in writing.
For students who need extra help describing the method, ask them to look at each row of
the rectangle/box individually and ask them how they got the answers that should be in each
box. Discuss this similarly with the entire class when you go over the activity sheet, to make
sure everyone understands.
**NOTE: I do not teach the FOIL method prior to this lesson, the purpose of
this lesson is for students to derive the FOIL method or some version of it all
on their own.
© 2012 The Enlightened Elephant
Name _____________________________________________________ Date ___________ Per _______
Area Review
1. Define Area. ___________________________________________________________________
__________________________________________________________________________________
2. Determine the area of the rectangle.
Total Area _______________
3. What is the total area of the composite shape below?
13 square
inches
11 square
inches
4. Find the area of each box.
Then find the total area of the figure below.
2 in
7 in
3 in
5 in
© 2012 The Enlightened Elephant
Total Area _______________
Warm-up
KEY
1. Define Area.
The amount of space on the surface of an object.
2. What is the area of the rectangle.
35 square units
3. What is the total area of the composite shape below?
13 square
inches
11 square
inches
24 square inches
4. Find the total area of the figure below.
2 in
7 in
3 in
72 square inches
5 in
© 2012 The Enlightened Elephant
Name _____________________________________________________________ Date _______ Per_____
Multiplying Binomials
Steps for the box method
1. Find the area of each small box.
2. Find the area of the large box by combing the areas of all four small boxes.
A. EXAMPLE: Use the box method to find the product of (x+4)(x+2) by finding the area of each
individual box and then adding them together. Each binomial has two terms. The side lengths of the
small boxes will be represented by the terms from each binomial.
x
+ 4
x
1. determine
the area of
each box
+
2
2. Combine all four small boxes together by
addition to find the total area of the large box
___________________________________ = (x+2)(x+4)
B. As you work through the problems below, think about what is happening mathematically with the two
binomials. How is the area of each small box determined? Try to recognize patterns or repeated
procedures and attempt to derive a procedure to multiply the binomials that does not require the use
of boxes.
Directions: Use the box method to find the product of the binomials.
1. (x+ 3)(x+ 5)
2. (x+ 4)(x− 3)
x
+ 5
x
x
x
+
3
+
4
Answer=
3. (x+ 1)(x− 7)
Answer=
© 2012 The Enlightened Elephant
Answer=
4. (x− 2)(2x− 6)
Answer=
- 3
C. Looking back at your work from part B, the box method, think about any patterns or repeated
procedures that you did. For example, How did you find the area of just one box?
Describe a procedure that does not require the use of boxes to find the product of the two
binomials below.
To multiply (x+4)(x+6) _______________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
D. Apply the method you described in part C to find the products below. DO NOT use the box method.
a. (x+6)(x+7)
b. (x+2)(x-5)
c. (3x+1)(x-3)
d. (x-2)(x-4)
e. (2x+5)(x-6)
f. (4x+8)(2x-3)
© 2012 The Enlightened Elephant
Key
Area Model (Box Method)
I. Use the area model (box method) to find the product of the binomial below.
(𝑥+ 2)(𝑥+ 4)
*Find the product by finding the total area of the boxes.
1.Each binomial will represent the lengths of each side of the large box, so you’ll have a “x +2” by
“x+4” rectangle.
2. Each binomial has two terms. The side lengths of the small boxes will be
represented by the terms from each binomial.
3. Find the area of each small box.
4. Find the area of the large box by combing the areas of all four small boxes.
+ 4
x
x
x
2
4x
+
2
2x
8
x 2  6x  8
(x+2)(x+4)=_
II. As you work through the problems below, think about what is happening mathematically
with the two binomials. Try to recognize patterns or repeated procedures and attempt to derive
a procedure that does not require the use of boxes.
Use the area model (box method) to find the product of the binomials.
2. (𝑥+ 4)(𝑥− 3)
1. (𝑥+ 3)(𝑥+ 5)
x
+ 5
x
x2
5x
+
3
6x
15
2
Answer= x 11x 15
x
- 3
x
x2
 3x
+
4
4x
12
Answer=
4. (𝑥− 2)(2�− 6)
3. (𝑥+ 1)(𝑥− 7)
- 7
x
2x
x
x 2  7x
x
+
1
x
7
--2
Answer=
x 6 x 7
2
© 2012 The Enlightened Elephant
x 2  x 12
- 6
2x 2  6x
 4x
12
2
2x
10x 12
Answer=
III. Looking back at your work/steps from part II using the box method, think about
any patterns or repeated procedures that you did.
Describe a procedure that does not require the use of boxes to find the product of
the binomial below.
(x+4)(x+6)
Answers will vary. Students should write something similar to the FOIL method. For
example, distribute (multiply) the x in the first binomial to the terms in the other
binomial. Then distribute (multiply) the 4 from the first binomial to the terms in the
other binomial.
IV. Apply the method you described in part III to find the products below.
a. (x+6)(x+7)
x 2 13x  42
© 2012 The Enlightened Elephant
b. (x+2)(x-5)
x 2  3x 10
c. (3x+1)(x-3)
3x 2  8x  9
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