# Multiplying in parts (partial products)

This is a complete lesson with explanations and exercises about multiplying in parts, also called partial products algorithm, with two-digit numbers. It is meant for fourth grade, and works as a stepping stone before students learn the regular multiplication algorithm. In a nutshell, students learn to break two-digit numbers into two parts, and to multiply the parts separately. They actually use the distributive property, but we do not need to explain that to 4th grade students.

Multiply 3 × 46

Break 46 into two parts: 40 and 6.

Then multiply those two parts separately by 3:
3 × 40 is 120, and 3 × 6 is 18.

Then add these two partial results: 120 + 18 = 138.

Here is another way of showing the same thing, using bundles of ten.

 46 46 46

3 × 40 = 120

3 × 6 = 18

Lastly, add 120 + 18 = 138.

Study these examples. Multiply the tens and ones separately, then add:

 8  ×  13        (10 + 3) 8 × 10  and  8 × 3   80  and  24 = 104 5  ×  24        (20 + 4) 5 × 20  and  5 × 4   100  and  20 = 120 7  ×  68        (60 + 8) 7 × 60  and  7 × 8   420  and  56 = 476

1. Multiply the tens and ones separately. Then add to get the final answer.

 a. 6  ×  27             (20 + 7) 6 × ____ and  6 × ____   ______ and ______ = ___________ b. 5  ×  83             (         +      ) 5 × ____ and  5 × ____   ______ and ______ = ___________ c. 9  ×  34             (         +       ) 9 × ____ and  9 × ____   ______ and ______ = ___________ d.  3  ×  99 3 × ____ and  3 × ____ = ___________ e.  7  ×  65 7 × ____ and  7 × ____ = ___________ f.  4  ×  58 4 × ____ and  4 × ____ = ___________

The picture illustrates the area of a rectangle with sides 8 and 24. It is also divided into two rectangles.

The area of the WHOLE rectangle is 8 × 24 square units. We can find that by calculating the areas of the two rectangles, and adding.

The area of the first rectangle is 8 × 20 = 160 square units.
The area of the second rectangle is 8 × 4 = 32 square units.

So, the area of the WHOLE rectangle is the sum 160 + 32 = 192 square units.

 8 × 24 =  8 × 20  +  8 × 4 = 160 + 32 = 192

2. Fill in the missing numbers. Write the area of the whole rectangle as a SUM of the areas of the smaller     rectangles. Also find the total area.

 a.   ___ × ______  =  ___ × ______  +  ___ × ___       = _________ b.   ___ × ______  =  ___ × ______  +  ___ × ___       = _________ c.   ___ × ______  =  ___ × ______  +  ___ × ___       = _________

3. It's your turn to draw. Draw a two-part rectangle to illustrate the multiplications, like in the previous problem. You do not have to measure the sides to make them exactly so long, a sketch is good enough.

 a.  7 × 16 =  ___ × ______  +  ___ × ___    = _________ b.  5 × 21 =  ___ × ______  +  ___ × ___    = _________ c.  8 × 34  =  ___ × ______  +  ___ × ___    = _________

4. Break the second factor into tens and ones. Multiply separately, and add.

a.  6 × 19

 6 × 10  →6 × 9    → 6  0+   5  4   1  1  4

b.  3 × 73

 3 × ______  →3 × ______  → +

c.  4 × 67

d.  5 × 92

e.  9 × 33

f.  7 × 47

5. Multiply in parts. You can write the partial products under the problems, if you wish.

 a.  5 × 13 = _______ b.  9 × 15 = _______ c.  5 × 33 = _______ d.  8 × 21 = _______ e.  4 × 22 = _______ f.  7 × 51 = _______

6. Compare. Write < , > , or  =  in the boxes between the number sentences.

 a.    10 × 10 9 × 11 b.    6 × 12 5 × 14 c.   8 × 22 5 × 27

7. Solve. Write a number sentence for each problem, not just the answer.

 a. Jack bought eight shirts for \$14 each. What was his total bill?     _______________________________________________ b. Mary and Harry set up nine rows of seats in the school auditorium,     with 14 seats in each row. After that, they still had 56 seats left in     the storage that they didn't use. How many seats are there in total?     _______________________________________________ c. A small hammer costs \$17. Another, much better one, is three     times as expensive. Find the cost of the more expensive hammer.     _______________________________________________

The video below also explains this same idea: first students are taught to multiply two- or three-digit numbers in parts (such as multiplying 3 × 89 as 3 × 80 and 3 × 9, and adding those) as a preparation for learning the usual multiplication algorithm.

This lesson is taken from my book Math Mammoth Multiplication 2. The lesson addresses the following Common Core standards for 4th grade: 4.OA.1, 4.OA.2, and 4.NBT.5.