Base Ten Arrays for Multi-Digit Multiplication
Students will be able to use an array to solve multiplication problems with two two-digit factors.
- Review arrays by drawing a simple array (e.g., 3 x 4). Remind students that an array is a drawing, or model, in which the factors determine the number of rows and columns. The product is the total number of items inside the array.
- Tell students that when we draw an array with small factors, like the example above, each factor is represented in single units. But, in order to draw an array with larger factors, we’ll need to represent the hundreds and tens with appropriate units.
- Review base ten notation for drawing models (□ = 100, | = 10, • = 1).
- Draw a number with base ten notation as an example (i.e., 246 = □ □ | | | | • • • • • •).
Explicit Instruction/Teacher modeling(10 minutes)
- Write a two-digit times two-digit problem on board (e.g., 15 x 34).
- Draw one factor with base ten notation in a horizontal line. Then draw the second factor in a vertical line (see related media for an example).
- Create an array by filling in the shape with base ten notation. Point out as you draw, that a 10 rod times a 10 rod would be drawn as a 100 because 10 x 10 = 100. Likewise, a 10 rod times a single unit would be drawn as a 10 rod because 10 x 1 = 10 (see related media for an example).
- When the array is complete, count the value of the 100s squares and write the total. Repeat with the tens and ones.
- Write an addition sentence, adding each unit value (e.g., hundreds + tens + ones). Add to find the product.
Guided Practice(15 minutes)
- Hand out a sheet of graph paper to each student (see related media if needed).
- Guide students through another example (e.g., 42 x 17).
- Give students a problem to try with a partner (e.g., 13 x 28).
- Give students a "try it" problem to solve independently (e.g., 21 x 35).
- Circulate and offer support as needed. Then go over the problem as a class.
Independent working time(15 minutes)
- Write three multiplication problems on the board and instruct students to use arrays to solve (e.g. 52 x 63, 71 x 45, 18 x 34).
- Note: hand out an additional sheet of graph paper if needed or instruct students to use math notebooks to draw their arrays.
- Circulate and offer support as needed.
- Provide additional examples before assigning independent work.
- For independent work, assign problems with smaller two-digit factors, like 12 x 16.
- Assign challenge problems with a three digit-factor, like 132 x 46, and instruct students to solve using an array.
- Hand out a piece of scratch paper or half-sheet of graph paper to each student.
- Write a multiplication problem on the board (e.g., 23 x 56).
- Have students create an array to solve.
- Collect student work as an exit ticket and check for understanding.
Review and closing(5 minutes)
- Ask, "How do arrays help us understand multiplication in a different way than simply calculating the answer with the standard algorithm?"
- Discuss as a class (e.g., drawing the factors and product helps us see each place value separately, adding parts of the product helps us think about breaking numbers into smaller parts).