Purpose In Professor Pancake, students learned to use concrete models and arrays to solve 2-digit by 1-digit multiplication problems. In this activity they connect the arrays to area models.
Prepare Animal Band, Problem #1 (PG. 32) so that it can be projected using your classroom technology.
Make 1 copy of Animal Band (PG. 32–37) for each student.
Write the journal question on the board: How does an array connect to an area model for multiplication? Other materials:
□
Base-ten blocks: 1 set for every pair of students□
Colored pencils
Teacher-facilitated w/ Small Student Groups
Small Group
Tutoring/Intervention Centers
Journal Anchor chartSetting Up For Instruction
• Can you create a rectangle that shows the problem? • How can you break the rectangle up in groups of 10? • How many are in each group?
• How many are in each row? • How many are in each column?
• Does your concrete model match your drawing? • Can you simplify your drawing?
• What are the partial products?
• What is the total of the partial products? • What is the product?
CONCRETE MODELS AND ARRAYS
Goal: Use a concrete model and an array to solve 2-digit by 1-digit multiplication 1. Put students in pairs and hand out materials.
2. Project Animal Band, Problem #1. Have students use base-ten blocks to build a model for the problem and then fill in
the array.
What operation do you need to use to solve the problem? How do you know? Multiplication; Maria Raccoon practices 3 hours each day for 14 days; 14 groups of 3
What are the dimensions of the rectangle? Why? 14 by 3; 14 is the number of groups and there are 3 in each group. What size rectangle will you draw in the grid? 14 by 3
What are the two equations needed to solve the problem? 10 × 3 and 4 × 3 How many hours does Maria Raccoon practice? 42 hours
AREA MODELS FOR MULTIPLICATION
Goal: Connect arrays and area models for multiplication
3. Have students use a different color pencil to trace around the two rectangles, 10 by 3 and 4 by 3.
4. Say: Instead of using an array with all the squares, we can make a simple drawing of the array, using only the rectangles that you traced.
Have students draw the two rectangles of the area model. What are the dimensions of the big rectangle? 3 by 10
What is the area of the big rectangle? Why? 30 because 3 × 10 = 30. Show students how the big rectangle matches the 3 longs, 10 + 10 + 10.
What are the dimensions of the small rectangle? 3 by 4
What is the area of the small rectangle? Why? 12 because 3 × 4 = 12. Show students how the small rectangle matches the 12 units.
What is the sum of the areas of the two rectangles? 42 How many hours does Maria Raccoon practice? 42 hours
5. Have students work together to solve the rest of the problems, building their solutions with base-ten blocks, and drawing the arrays and area models.
WRAP IT UP
Goal: Solidify the connection between arrays and area models
6. Hand out sticky notes. Ask students to discuss the journal question and then write their own thoughts on the sticky
note.
How-To Guide
In Professor Pancake (PG. 7), we looked at how students can use concrete models of equal groups and arrays to represent two-digit by one-digit multiplication. Now students will extend their understanding to the use of area models.
The table below shows 23 × 4 represented as equal groups and an array.
CONCRETE MODELS
23
23
23
23
ARRAY
23
4
As students move into exploring area models, it may be helpful to start with an array to demonstrate how to break apart (decompose) the two-digit number based on place value. The number 23 is equivalent to 20 + 3. This decomposition is shown in the array below.
20
4
3
4
+
Decomposing 23 into 20 + 3 results in two rectangles: one with the dimensions of 20 by 4, and the other with dimensions of 3 by 4. Adding the areas of these two rectangles together will give us the product of the entire 23 x 4 figure.
We can shift our array to grid paper to use square units with no overlap or gaps to make the connection between what students know about arrays and what they know about finding the area of a rectangle.
20
4
3
4
+
Modeling Two-Digit by One-Digit Multiplication Using Concrete Models, Arrays, and Area Models (1 of 2)
In 3rd grade, students learn that the area of a rectangle is related to the number of rows times the number of unit squares in each row. This can be represented by the equations shown below.
20
4
3
4
+
20 × 4 =80
3 × 4 =12
Finally, the areas of the two rectangles are added together to find the area of the entire 23 x 4 figure.
23
4
92
A firm understanding of the area model will help students make the connection to partial products, which we’ll explore in our next activity.
Modeling Two-Digit by One-Digit Multiplication Using Concrete Models, Arrays, and Area Models (2 of 2)
Directions: Make a model and fill in the array to match the model. Then draw an area model and solve the
problem.
1
Maria Raccoon is learning guitar. She wants to start a band with her friends. She practices guitar 3
hours each day. How many hours does she practice in 14 days?
Fill in the array to match the model.
3
10
+
4
3 × 10 = 30
3 × 4 = 12
30
+ 12
42
Draw an area model.
3
10
+
4
3 × 10 = 30
3 × 4 = 12
14 = 10 + 4
× 3 = 3
12
+ 30
42
Solution:
42 hours
2
Maria asks Cora the crow to sing in the band. When Cora squawks, it wakes up all her neighbors.
There are 12 houses near her, and each house has 4 people in it. How many people does Cora wake
up?
Fill in the array to match the model.
4
10
+ 2
4 × 10 = 40
4 × 2 = 8
40
+ 8
48
Draw an area model.
4
10
+ 2
4 × 10 = 40
4 × 2 = 8
12 = 10 + 2
× 4 = 4
8
+ 40
48
Solution:
48 people
ANSWER KEY
ANIMAL BAND
(PG. 2 OF 6)
3
Next, they ask Elena the Eagle to play piano. Elena loves music. She learns 17 new songs on piano
each week. How many songs does she learn in 2 weeks?
Fill in the array to match the model.
2
10
+
7
2 × 10 = 20
2 × 7 = 14
20
+ 14
34
Draw an area model.
17 = 10 + 7
× 2 = 2
14
+ 20
34
2
10
+
7
2 × 10 = 20
2 × 7 = 14
Solution:
34 songs
4
Last they ask Lila the elephant to play drums. Lila hits the drums so hard that all the pictures fall
off the walls. There are 18 pictures in each room, and there are 6 rooms in the house. How many
pictures fall?
Fill in the array to match the model.
6
10
+
8
6 × 10 = 60
6 × 8 = 48
60
+ 48
108
Draw an area model.
18 = 10 + 8
× 6 = 6
48
+ 60
108
6
10
+
8
6 × 10 = 60
6 × 8 = 48
Solution:
108 pictures
ANSWER KEY
ANIMAL BAND
(PG. 4 OF 6)
5
The girls decide to play a big show on the beach. They practice 5 hours a day for 15 days. How many
hours do they practice?
Fill in the array to match the model.
5
10
+
5
5 × 10 = 50
5 × 5 = 25
50
+ 25
75
Draw an area model.
5
10
+
5
5 × 10 = 50
5 × 5 = 25
15 = 10 + 5
× 5 = 5
25
+ 50
75
Solution:
75 hours
6
The show was great! Everyone loved it. The band played 9 songs. Each time they started a song, 19
more animals started dancing. Nobody stopped dancing until the end. How many animals danced?
Fill in the array to match the model.
9
10
+
9
9 × 10 = 90
9 × 9 = 81
90
+ 81
171
Draw an area model.
19 = 10 + 9
× 9 = 9
81
+ 90
171
9
10
+
9
9 × 10 = 90
9 × 9 = 81
Solution:
171 animals
ANSWER KEY
ANIMAL BAND
(PG. 6 OF 6)
Directions: Make a model and fill in the array to match the model. Then draw an area model and solve the
problem.
1
Maria Raccoon is learning guitar. She wants to start a band with her friends. She practices guitar 3
hours each day. How many hours does she practice in 14 days?
Fill in the array to match the model.
Draw an area model.
2
Maria asks Cora the crow to sing in the band. When Cora squawks, it wakes up all her neighbors.
There are 12 houses near her, and each house has 4 people in it. How many people does Cora wake
up?
Fill in the array to match the model.
Draw an area model.
Solution:
3
Next, they ask Elena the Eagle to play piano. Elena loves music. She learns 17 new songs on piano
each week. How many songs does she learn in 2 weeks?
Fill in the array to match the model.
Draw an area model.
4
Last they ask Lila the elephant to play drums. Lila hits the drums so hard that all the pictures fall
off the walls. There are 18 pictures in each room, and there are 6 rooms in the house. How many
pictures fall?
Fill in the array to match the model.
Draw an area model.
Solution:
5
The girls decide to play a big show on the beach. They practice 5 hours a day for 15 days. How many
hours do they practice?
Fill in the array to match the model.
Draw an area model.
6
