Prior to the lesson, students should understand and have practiced:
- Shading in multiplication arrays on 10 × 10 grids. If students struggle with this, you may want to use the Rectangle Multiplication tool in Grouping mode as an opening activity.
- Decomposing factors into addends, such as 47 = 40 + 7.
- Multiplying by powers of 10. For example:
|40 × 300||= (4 × 10) × (3 × 100)|
= (4 × 3) × (10 × 100)
= 12 × 1,000
2 × 1 Multiplication Arrays
Begin with 2 × 1 problems. Gauge your students' understanding in this activity to determine the pacing for the 2 × 2 activity.
Work through the example problem 14 × 6 to introduce the
solution method. Before you begin showing students arrays, have them
estimate the answer. This helps develop mental math skills and gives a
benchmark to judge accuracy when they complete the problem. Since 14 is
closer to 10 than 20, approximate the answer to be about 10 × 6 = 60.
Since we rounded down, the actual answer will be a bit bigger than 60.
Model the partial product arrays for students using the 14 × 6
example. Model the following steps, using the illustration as a guide:
- Decompose the first factor into tens and ones to give (10 + 4) × 6.
The problem can now be solved using the distributive property.
- Draw an array that is 14 × 6.
- Break the 14-unit horizontal line into a tens piece and ones piece.
In this case, the pieces will be 10 and 4. Draw a vertical line through
the rectangle at this point, creating two rectangles. Label the length
and width of both rectangles. Label the 10 × 6 rectangle as 60, and the
4 × 6 rectangle as 24, as shown.
- The rectangles represent 2 partial products. Ask the students what
to do with the 60 and 24. It is important that students know that partial
means each array represents just part of the problem. They should also
understand that to get the total product you need to add. The solution
is 60 + 24 = 84.
As you model, have students participate by coming up to help build
the model. Next, demonstrate solving another problem while students
create their own base 10 block or paper grid models.
Have less capable students work with a partner. To transition
to more individualized work, have individuals and pairs compare their
work with that of others and describe their partial arrays aloud. Do as
many problems as needed to support your variety of learners. Keep in
mind that large 2-digit numbers get unwieldy to work with; numbers
between 10 and 30 work the best for modeling.
2 × 2 Multiplication Arrays
Model the problem 13 × 16. Once students understand the process,
they can work problems at your direction for continual practice.
Important points for 2-digit by 2-digit multiplication include:
- The problem must be decomposed. It becomes (10 + 3) × (10 + 6).
- Draw an array that is 13 by 16.
- Decompose the factors by breaking the array into smaller pieces.
Break the 13-unit horizontal line into 10- and 3-unit long pieces.
Then, break the 16-unit vertical line into 10- and 6-unit long pieces.
- Draw lines to separate the tens from the ones (10 from 3 and 10 from 6). This creates 4 rectangles.
- Label the outside dimensions.
- Students should be able to see the 4 partial products of this problem: (10 × 10), (3 × 10), (6 × 10), and (3 × 6).
- Label the inside areas of the 4 partial products.
- Finally, add the partial products together: 100 + 30 + 60 + 18 = 208.
After students show understanding and fluency with the models, show
them how the models are reflected in a partial product method for
recording multiplication. Showing the array model and the distributed
recording method side by side helps students see the connection between
Students use 2 × 1 Multiplication activity sheet and 2 × 2 Multiplication
activity sheet. Each activity sheet has one example, workspace provided
for additional problems, and an area to draw arrays. Teach students how
to complete the tables using the examples already solved.
Tips about using this multiplication method:
- Label the entire array model before you begin calculating.
- Perform your calculations beside the array model.
- While the decomposed factors can be multiplied in any order, direct
students to use the order in the examples to begin. List the products
in the order of the traditional algorithm.
- As you compute partial products, locate them on the array model to make the connection between the array and the calculations.