Things to Remember! Area – the number of square units that covers a two-dimensional figure Rectangle – a four-sided figure with four 90° angles Square – a rectangle with four equal side lengths Distributive Property – breakdown one or two factors of a multiplication problem into its addends, multiply each by the other factor, and then add the products together to get the whole answer Examples: 54 x 2 = (50 + 4) x 2 38 x 12 = (30 + 8) x (10 + 2) = (50 x 2) + (4 x 2) = (30 x 10) + (30 x 2) + (8 x 10) + (8 x 2) = 100 + 8 = 300 + 60 + 80 + 16 = 108 = 456 3 units x 5 units is read 3 units by 5 units. u2 is read units squared. in2 is read inches squared.
OBJECTIVES OF TOPIC (
C
Find the area of rectangles with whole-by-mixed and wholeby-fractional number side lengths by tiling, record by drawing, and relate to fraction multiplication. Find the area of rectangles with mixed-by-mixed and fraction-by-fraction side lengths by tiling, record by drawing, and relate to fraction multiplication. Measure to find the area of rectangles with fractional side lengths. Multiply mixed number factors, and relate to the distributive property and the area model. Solve real world problems involving area of figures with fractional side lengths using visual models and/or equations.
Module 5: Addition and Multiplication with Volume and Area
This topic begins with students using tiling to find the area of rectangles. Tiling is a strategy used to find area of rectangle by covering the entire figure with square units and fractional parts of square unit.
Example of tiling
Example Problem: Randy made a mosaic using different color rectangular tiles. Each tile measured 3 inches x 2 inches. If he used six tiles, what is the area of the whole mosaic in square inches? The drawing below resembles an area model used in earlier modules when students multiplied whole numbers and decimal fractions. Now the area model has fractional parts. The 3 is thought of as 3 + . Using tiling, each whole square represents 1 square inch. To represent inch, the whole square is cut in half and only half is showing in the model. There are 6 whole squares and two s. 3 inches
in
two s equal 1 whole
2 inches
The area of one tile is 7 square inches. Since there are 6 tiles, the area of the whole mosaic is 42 square inches or 42 in2 (6 x 7). Algorithm using the distributive property:
3 x 2 = (3 + ) x 2 = (3 x 2) + ( x 2) = 6 + 1 = 7 Algorithm without using the distributive property; the mixed number is changed to an improper fraction:
3 x 2 =
x 2=
=
=7
Eventually students will just record partial products rather than draw individual tiles. Example Problem: Francine cut a rectangle out of construction paper to complete her art project. The rectangle measured 4 inches x 2 inches. What is the area of the rectangle Francine cut out?
7
12 8 ft Wall
in
4 inches 2 in
Application Problem: John decided to paint a wall with two windows. Both windows are 3 ft by 4 ft rectangles. Find the area the paint needs to cover.
Window
𝟐
2x4 = 8 in2
2 x = 𝟐 = 1 in2
Window
8 ft
in 𝟒
Wall:
x = in2
x 4 = 𝟒 = 1 in
2
8
Add the partial products together to find the area. 8 in2 + 1 in2 + 1 in2 + in2 = 10 in2
12
96
7
= 103 ft2
The area of the wall is 103 ft2.
The area of the rectangle cut out is 10 square inches. Algorithm using the distributive property: 4 x 2 = (4 + ) x (2 + ) =(4 x 2) + (4 x ) + ( x 2) + ( x ) = 8 + 1 + 1 + = 10
Window:
Algorithm without using the distributive property; mixed numbers are changed to improper fractions: 4 x 2 = x
=
= 10
**The algorithm is provided so students are exposed to a more formal representation of the distributive property. However, students are not required to be as formal in their calculations. Using an area model to keep track of their thinking is sufficient. Problem: Find the area of a rectangle that measures km x 2 km. Draw an area model if it helps. km 𝟔
2 km
km2 or 1 km2
2x=
km2
x=
km 1 + = 1 + + =1
𝟒
𝟐𝒙𝟑 𝟒
𝟔
𝟐
𝟒
𝟒
= = 1 = 1
The area of the rectangle is 1 km2.
4
3
12
𝟑 𝟐
2
= 2 + = 2
𝟏
𝟏
𝟐
𝟑
= 12 + 𝟐 = 12 + 1 = 13
𝟏
𝟑
13𝟐 + 2𝟒 = 13𝟒 + 2𝟒 = 15𝟒 15 x 2 windows = (15 + ) x 2 = (15 x 2) + ( x 2) 𝟔
= 30 + 𝟒
𝟐
𝟏
= 30 + 1𝟒 = 31𝟐
31 ft2 is the area of two windows. 103 - 31 = (103 – 31) - = 72 - = 71 The paint needs to cover 71 square feet.
Application Problem: John decided to paint a wall with. two windows. Both windows are 3 ft by 4 ft rectangles. Find the area the paint needs to cover. 12. 8.
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