Distributive Properties of Geometric Models

Distributive Properties of Geometric Models

Area of a Rectangle

Opening

Area of a Rectangle

  • Watch the video about finding ways to express the area of a rectangle when the length and width are unknown.
  • Write two expressions for the area of this rectangle.

VIDEO: Area of Rectangle

Math Mission

Opening

Write two equivalent expressions and show they are equivalent using the distributive property.

Garden Plots

Work Time

Garden Plots

Six students have garden plots of various sizes in a garden. All the plots are rectangles and the garden itself is a rectangle.

Denzel’s plot has an area of bc.

Mina’s plot has an area of be.

Jason’s plot has an area of ad.

Rosa’s plot has an area of ae.

Chen’s plot has an area of ac.

Emma’s plot has an area of bd.

  • Make a sketch showing how the garden might be divided. Label each plot with the name of the student who is in charge of the plot, followed by the area of the plot.

Hint:

  • Start by drawing Denzel’s garden plot. Label the length b and the width c.
  • Whose plot can you draw next to his?

Area of Garden Plots

Work Time

Area of Garden Plots

For each of these problems, write two expressions. Use your garden diagram and the distributive property to show that your expressions are equivalent.

  • If Chen gives Denzel his plot, what will the area of Denzel’s new plot be?
  • If Chen keeps his plot but Emma and Mina give Denzel their plots, what will the area of Denzel’s new plot be?
  • What is the area of the entire garden?

Hint:

  • Look at your sketch of the garden. Outline the area of Denzel’s new plot after he gets Chen’s plot. What is the length and width of Denzel’s new plot?
  • Look at your sketch of the garden. Find Denzel’s original plot. Add the areas of Emma’s and Mina’s plots. What is the length and width of Denzel’s plot now?
  • What expression represents the length of the entire garden? What expression represents the width of the entire garden?

Prepare a Presentation

Work Time

Prepare a Presentation

  • Explain how you wrote the two expressions for each area.
  • Explain how you used the distributive property, a(b + c) = ab + ac, to show that the two expressions are equivalent.

Challenge Problem

  • Write a problem that involves area and the distributive property, and find the solution.
  • Then exchange problems with a partner and solve each other’s problems.

Make Connections

Performance Task

Ways of Thinking: Make Connections

Takes notes about your classmates’ approaches to solving the problems.

Hint:

As your classmates present, ask questions such as:

  • What strategy did you use to sketch the garden?
  • How does your sketch of the garden compare to those of your classmates?
  • Is there only one correct garden sketch? Explain.
  • How do your two expressions represent the area of
    this plot?
  • Where do you see the length of the plot in the sketch? Where do you see the width?
  • How is the distributive property represented in your garden sketch?

Expressions and Properties of Operations

Formative Assessment

Summary of the Math: Expressions and Properties of Operations

Read and Discuss

  • The distributive property enables you to rewrite an expression as an equivalent expression.
  • It works for algebraic expressions in exactly the same way that it works for whole numbers. You have already used the distributive property to help you make mental math calculations, such as determining that: 12 · 32 = 10 · 32 + 2 · 32.
  • With algebraic expressions, the distributive property works the same way: a(b + c) ab + ac
  • This rectangle is a geometric model of the distributive property. The area of the rectangle can be found using two different expressions: a(b + c) or ab + ac. The equality of these expressions, a(b + c) = ab + ac, is the distributive property.

Hint:

Can you:

  • Explain the distributive property?
  • Show how to represent the distributive property using a visual model?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

I like using the distributive property because …