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Mathematical Area Problems





 

Video solutions to help Grade 7 students learn how to use the area properties to justify the repeated use of the distributive property to expand the product of linear expressions.

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Common Core For Grade 7

New York State Common Core Math Grade 7, Module 6, Lesson 21


Lesson 21 Student Outcomes


• Students use the area properties to justify the repeated use of the distributive property to expand the product of linear expressions.

Lesson 21 Summary


• The properties of area, because they are limited to positive numbers for lengths and areas, are not as robust as properties of operations, but the area properties do support why the properties of operations are true.

Lesson 21 Classwork

The objective of the lesson is to generalize a formula for the area of rectangles that result from adding to the length and width. Using visuals and concrete (numerical) examples throughout the lesson will help students make this generalization.

Opening Exercise
Patty is interested in expanding her backyard garden. Currently, the garden plot has a length of 4 ft. and a width of 3 ft.
a. What is the current area of the garden?
Patty plans on extending the length of the plot by 3 ft. and the width by 2 ft.
b. What will the new dimensions of the garden be? What will the new area of the garden be?
c. Draw a diagram that shows the change in dimension and area of Patty’s garden as she expands it. The diagram should show the original garden as well as the expanded garden.
d. Based on your diagram, can the area of the garden be found in a way other than by multiplying the length by the width?
e. Based on your diagram, how would the area of the original garden change if only the length increased by 3 ft.? By how much would the area increase?
f. How would the area of the original garden change if only the width increased by 2 ft.? By how much would the area increase?
g. Complete the following table with the numeric expression, area, and increase in area for each change in the dimensions of the garden.
h. Will the increase in both the length and width by 3 ft. and 2 ft., respectively, mean that the original area will increase strictly by the areas found in parts (e) and (f)? If the area is increasing by more than the areas found in parts (e) and (f), explain what accounts for the additional increase.



Example 1
Examine the change in dimension and area of the following square as it increases by 2 units from a side length of 4 units to a new side length of 6 units. Observe the way the area is calculated for the new square. The lengths are given in units, and the areas of the rectangles and squares are given in units2.
a. Based on the example above, draw a diagram for a square with side length of 3 units that is increasing by 2 units. Show the area calculation for the larger square in the same way as in the example.
b. Draw a diagram for a square with side length of 5 units that is increased by 3 units. Show the area calculation for the larger square in the same way as in the example.
c. Generalize the pattern for the area calculation of a square that has an increase in dimension. Let the side length of the original square be a units and the increase in length be by b units to the length and width. Use the diagram below to guide your work.

Example 2
Bobby draws a square that is 10 units by 10 units. He increases the length by x units and the width by 2 units.
a. Draw a diagram that models this scenario.
b. Assume the area of the large rectangle is 156 units2. Find the value of x.

Example 3
The dimensions of a square with side length x units are increased. In this figure the indicated lengths are given in units, and the indicated areas are given in units2.
a. What are the dimensions of the large rectangle in the figure?
b. Use the expressions in your response from part (a) to write an equation for the area of the large rectangle, where A represents area.
c. Use the areas of the sections within the diagram to express the area of the large rectangle.
d. What can be concluded from parts (b) and (c)?
e. Explain how the expressions (x + 2)(x + 3) and x2 + 3x + 2x + 6 differ within the context of the area of the figure.


 

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