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Lesson 1The Areas of Squares and Their Side Lengths

Let’s investigate the squares and their side lengths.

Learning Targets:

  • I can find the area of a tilted square on a grid by using methods like “decompose and rearrange” and “surround and subtract.”
  • I can find the area of a triangle.

1.1 Two Regions

Which shaded region is larger? Explain your reasoning.

Two quadrilaterals, labeled “A” and “B,” on square grids. Both quadrilaterals are not aligned to the horizontal or vertical gridlines. Quadrilateral “A” is on a grid that has 2 rows of 4 squares. The quadrilateral is drawn starting at the left most vertex. The second vertex is 2 squares to the right and 1 square up from the first vertex. The third vertex is 2 squares to the right and 1 square down from the second vertex. The fourth vertex is 2 squares to the left and 1 square down from the third vertex. The first vertex is 2 squares to the left and 1 square up from the fourth vertex. Quadrilateral “B” is on a grid that has 3 rows of 3 squares. The quadrilateral is drawn starting at the left most vertex. The second vertex is 1 square to the right and 2 squares up from the first vertex. The third vertex is 2 squares to the right and 1 square down from the second vertex. The fourth vertex is 1 square to the left and 2 squares down from the third vertex. The first vertex is 2 squares to the left and 1 square up from the fourth vertex.
 

1.2 Decomposing to Find Area

Find the area of each shaded square (in square units).


  1. 4 right triangles are shown that create a large square. The side lengths are 9 and 1. The hypotenuse of each triangle is the side length of a smaller inner square.

  2. 4 right triangles are shown that create a large square. The side lengths are both 5. The hypotenuse of each triangle is the side length of a smaller inner square.

  3. 4 right triangles are shown that create a large square. The side lengths are 7 and 3. The hypotenuse of each triangle is the side length of a smaller inner square.

Are you ready for more?

Any triangle with a base of 13 and a height of 5 has an area of \frac{65}{2} .

Two right triangles are shown. Both are made up of the same 4 different shapes.

Both shapes in the figure have been partitioned into the same four pieces. Find the area of each of the piece and verify the corresponding parts are the same in each picture. There appears to be one extra unit of area in the right figure than in the left one. If all of the pieces have the same area, how is this possible?

1.3 Estimating Side Lengths from Areas

Squares A, B, and C are graphed on a grid.
  1. What is the side length of square A? What is its area?
  2. What is the side length of square C? What is its area?
  3. What is the area of square B? What is its side length? (Use tracing paper to check your answer to this.)

  4. Find the areas of squares D, E, and F. Which of these squares must have a side length that is greater than 5 but less than 6? Explain how you know.

    Squares D, E, and F are graphed on a grid.

1.4 Making Squares

Use the applet to determine the total area of the five shapes, D , E , F , G , and H . Assume each small square is equal to 1 square unit.

open applet in presentation mode

Lesson 1 Summary

The area of a square with side length 12 units is 12^2  or 144 units2.

The side length of a square with area 900 units2 is 30 units because 30^2 = 900 .

Sometimes we want to find the area of a square but we don’t know the side length. For example, how can we find the area of square ABCD ?

One way is to enclose it in a square whose side lengths we do know.

Square ABCD is graphed on a grid.

The outside square EFGH has side lengths of 11 units, so its area is 121 units2. The area of each of the four triangles is \frac12 \boldcdot 8 \boldcdot 3 = 12 , so the area of all four together is 4 \boldcdot 12 = 48 units2. To get the area of the shaded square, we can take the area of the outside square and subtract the areas of the 4 triangles. So the area of the shaded square ABCD is 121 - 48 = 73 or 73 units2.

Square EFGH is graphed on a grid. Square ABCD is graphed within Square EFGH and has points that fall on the side lengths.

Lesson 1 Practice Problems

  1. Find the area of each square. Each grid square represents 1 square unit.

    Squares A, B, C, and D are graphed on a grid.

  2. Find the length of a side of a square if its area is:

    1. 81 square inches
    2. \frac{4}{25} cm2
    3. 0.49 square units
    4. m^2 square units

  3. Find the area of a square if its side length is:

    1. 3 inches
    2. 7 units
    3. 100 cm
    4. 40 inches
    5. x units

  4. Evaluate (3.1 \times 10^4) \boldcdot (2 \times 10^6) . Choose the correct answer:

    1. 5.1 \times 10^{10}
    2. 5.1 \times 10^{24}
    3. 6.2 \times 10^{10}
    4. 6.2 \times 10^{24}

  5. Noah reads the problem, “Evaluate each expression, giving the answer in scientific notation.” The first problem part is: 5.4 \times 10^5 + 2.3 \times 10^4 . Noah says, “I can rewrite 5.4 \times 10^5 as 54 \times 10^4 . Now I can add the numbers: 54 \times 10^4 + 2.3 \times 10^4 = 56.3 \times 10^4 .” Do you agree with Noah’s solution to the problem? Explain your reasoning.

  6. Select all the expressions that are equivalent to 3^8 .

    1. (3^2)^4
    2. 8^3
    3. 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3
    4. (3^4)^2
    5. \frac{3^6}{3^{\text-2}}
    6. 3^6 \boldcdot 10^2