Unit 8 Pythagorean Theorem And Irrational Numbers — Unit Plan

Title	Assessment
Lesson 1 The Areas of Squares	It's a Square Find the area of square $ACEG$ . Show Solution 100 square units
Lesson 2 Side Lengths and Areas	Area Estimate Mai estimates the area of the square to be somewhere between 70 and 80 square units. Do you agree with Mai? Explain your reasoning. Show Solution I agree with Mai. Sample reasoning: The side length of the square is the same length as the radius of the circle, which is between 8 and 9 units long. That means the area of the square must be larger than 64 square units but smaller than 81 square units, so Mai’s estimate of somewhere between 70 and 80 square units seems reasonable.
Lesson 3 Square Roots	What Is the Side Length? Write the exact value of the side length of a square with each of the following areas. 100 square units 95 square units 36 square units 30 square units For each exact value that is not a whole number, estimate the length. Show Solution 10 units $\sqrt{95}$ units 6 units $\sqrt{30}$ units $\sqrt{95}\text{ units}\approx9.7$ ; $\sqrt{30}\text{ units}\approx5.5$
Lesson 4 Rational and Irrational Numbers	Types of Solutions In your own words, say what a rational number is. Give at least three different examples of rational numbers. In your own words, say what an irrational number is. Give at least two examples. Show Solution Answers vary. Sample responses: A rational number is a fraction, like $\frac12$ , or its opposite, like $\text- \frac12$ . Something like 3.98 is rational too because it is equal to $\frac{398}{100}$ . An irrational number is one that is not rational. It is a number that cannot be expressed as a fraction. $\sqrt{2}$ and $\pi$ are two examples.
Lesson 5 Square Roots on the Number Line	Approximating $\sqrt{18}$ Plot $\sqrt{18}$ on the $x$ -axis. Consider using the grid to help. Show Solution About 4.2.
Lesson 6 Reasoning about Square Roots	Betweens Which of the following numbers are greater than 6 and less than 8? Explain how you know. $\sqrt{7}$ $\sqrt{60}$ $\sqrt{80}$ Show Solution only $\sqrt{60}$ Sample reasoning: Since $6^2 = 36$ and $8^2 = 64$ , the number inside the square root must be between 36 and 64.
Section A Check Section A Checkpoint	Problem 1 Find the exact side length of a square, in units, if its area in square units is: 49 $\frac23$ $\frac{4}{81}$ 0.25 1.29 57 Show Solution 7 $\sqrt{\frac23}$ $\frac29$ 0.5 $\sqrt{1.29}$ $\sqrt{57}$ Problem 2 The numbers $x$ , $y$ , and $z$ are positive where $x^2=5$ , $y^2=23$ , and $z^2=64$ . Plot $x$ , $y$ , and $z$ on the number line. Show Solution Problem 3 Decide whether each number in this list is rational or irrational. $0.\overline{13}$ $\text-\,\sqrt{81}$ $\frac{22}{7}$ $\sqrt{10}$ $\text- 5.867$ $\sqrt{42}$ Show Solution Rational Rational Rational Irrational Rational Irrational

Lesson 7 Finding Side Lengths of Triangles	Does $a^2$ Plus $b^2$ Equal $c^2$? For each of the following triangles, determine if $a^2+b^2=c^2$ , where $a$ , $b$ , and $c$ are side lengths of the triangle and $c$ is the longest side. Explain how you know. Show Solution Sample responses: It is true for Triangle A because it is a right triangle. You can also find the third side length by constructing a square on it and checking. It is not true for Triangle B. You can see this by squaring the side lengths.
Lesson 8 A Proof of the Pythagorean Theorem	What Is the Hypotenuse? Find the length of the hypotenuse in a right triangle if $a$ is 5 cm and $b$ is 8 cm. Show Solution $c=\sqrt{89}$ cm or $c\approx9.4$ cm
Lesson 9 Finding Unknown Side Lengths	Could Be the Hypotenuse, Could Be a Leg A right triangle has sides of length 3, 4, and $x$ . Find $x$ if it is the hypotenuse. Find $x$ if it is one of the legs. Show Solution $x = \sqrt{25}$ or $x = 5$ $x = \sqrt{7}$
Lesson 10 The Converse	Is It a Right Triangle? The triangle has side lengths 7, 10, and 12. Is it a right triangle? Explain your reasoning. Show Solution No. If this were a right triangle, then $7^2+10^2$ would equal $12^2$ . However, this is not the case.
Lesson 12 More Applications of the Pythagorean Theorem	Diameter of a Cone The height of a cone is 12 cm and its slant height is 13 cm. What is the diameter of the base of the cone? Show Solution 10 cm
Lesson 13 Finding Distances in the Coordinate Plane	Lengths of Line Segments Calculate the exact lengths of segments $e$ and $f$ . Which segment is longer? Two line segments labeled e and f are graphed in the coordinate plane with the origin labeled O. The line segment e begins at the point with coordinates negative 2 comma 3 and ends at the point with coordinates negative 1 comma negative 1. Line segment f begins at the point with coordinates negative 1 comma negative 1 and ends at the point with coordinates 2 comma 2. Show Solution The length of $e$ is $\sqrt{17}$ units, and the length of $f$ is $\sqrt{18}$ units. $e=\sqrt{1^2+4^2}=\sqrt{1+16}=\sqrt{17}$ . $f=\sqrt{3^2+3^2}=\sqrt{9+9}=\sqrt{18}$ . Line segment $f$ is longer.
Section B Check Section B Checkpoint	Problem 1 Find the exact value of $x$ . An envelope measures $3\frac12$ inches tall by 5 inches wide. Kiran wants to use it to mail a really cool pencil to a friend that measures 6 inches long. Will the pencil fit in the envelope? Explain your reasoning. Show Solution $x=9$ Yes, the pencil should fit in the envelope if it is put in diagonally. Sample reasoning: Since $3.5^2 + 5^2=37.25$ , the length of the diagonal of the envelope is $\sqrt{37.25}\approx6.1$ , which is a little bit longer than the pencil. Problem 2 Find the distance between the two points. Show Solution 10 units Problem 3 F First of two squares of the same area. This square is divided into the following: A square with side lengths “a”. Two rectangles with side lengths “a” and “b”. A square with side lengths “b”. G Second of two squares of the same area. This square is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths. Complete the explanation for each step of this proof that $a^2+b^2=c^2$ , where $a$ and $b$ are pieces of the sides of the two identical squares in Figures F and G, and $c$ is the length of a side of the smaller square in Figure G. Step 1: $a^2+b^2+2ab$ represents . . . Step 2: $4\boldcdot \frac12 a b + c^2$ represents . . . Step 3: $a^2+b^2+2ab=4 \boldcdot \frac12 a b + c^2$ because . . . Step 4: $a^2+b^2+2ab=2ab+c^2$ because . . . Step 5: $a^2+b^2=c^2$ because . . . Show Solution Sample response: Step 1: $a^2+b^2+2ab$ represents the total area of the 4 quadrilaterals that make up the larger square in Figure F. Step 2: $4\boldcdot \frac12 a b + c^2$ represents the total area of the 4 triangles and the smaller square that make up the larger square in Figure G. Step 3: $a^2+b^2+2ab=4 \boldcdot \frac12 a b + c^2$ because the larger squares in each Figure both have the same total area since they are both squares with side length $a+b$ . Step 4: $a^2+b^2+2ab=2ab+c^2$ because the area of the 4 triangles in Figure G ( $4\boldcdot\frac12ab$ ) is equal to the area of the 2 rectangles in Figure F ( $2ab$ ). Step 5: $a^2+b^2=c^2$ because subtracting an equivalent area from each figure results in an equivalent area remaining.

Lesson 14 Edge Lengths and Volumes	Roots, Sides, and Edges Plot each value on the number line. $\sqrt{36}$ the edge length of a cube with volume 12 cubic units the side length of a square with area 70 square units $\sqrt[3]{36}$ Show Solution 6 between 2 and 3 between 8 and 9 between 3 and 4
Lesson 15 Cube Roots	Different Types of Roots Lin is asked to place a point on a number line to represent the value of $\sqrt[3]{49}$ and she draws: Where should $\sqrt[3]{49}$ actually be on the number line? How do you think Lin got the answer she did? Show Solution Sample response: $\sqrt[3]{49}$ should be between 3 and 4 on the number line. I think Lin placed the point at $\sqrt{49}$ because she thought it was a square root instead of a cube root.
Section C Check Section C Checkpoint	Problem 1 The volume of the cube is 64 cubic cm. Select all values that represent $x$ . A.4 cm B.8 cm C. $\sqrt{64}$ cm D. $\sqrt[3]{64}$ cm E. $21\frac13$ cm Show Solution A, D Problem 2 Write the letter of the plotted point next to the value it matches. $\sqrt[3]{90}$ $\hspace{.1in}\underline{\hspace{.5in}}$ $\sqrt[3]{25}$ $\hspace{.1in}\underline{\hspace{.5in}}$ Point $F$ represents the cube root of what number? $\hspace{.1in}\underline{\hspace{.5in}}$ Show Solution $\sqrt[3]{90}\approx 4.48: L$ $\sqrt[3]{25}\approx 2.92: Q$ 343

Lesson 16 Decimal Representations of Rational Numbers	An Unknown Rational Number Explain how you know that -3.4 is a rational number. Show Solution Sample response: $\text- 3.4 = \text-\frac{34}{10}$ , so it can be written as a negative fraction and is therefore a rational number.
Section D Check Section D Checkpoint	Problem 1 Write $\frac{4}{11}$ as a decimal. Show Solution $0.\overline{36}$ Problem 2 Express $0.\overline{15}$ as a fraction. Show Solution $\frac{5}{33}$

Lesson 18 When Is the Same Size Not the Same Size?	No cool-down
Unit 8 Assessment End-of-Unit Assessment	Problem 1 Select all the numbers that are solutions to the equation $x^3 = 27$ . A. $\sqrt{27}$ B. 3 C. $\sqrt[3]{27}$ D. $27^3$ E. 9 Show Solution B, C Problem 2 Each of the following gives the lengths, in inches, of the sides of a triangle. Which one is a right triangle? A. $\sqrt{3}$ , $\sqrt{4}$ , $\sqrt{5}$ B.3, $\sqrt{7}$ , 4 C. $\sqrt{5}$ , $\sqrt{12}$ , 13 D.4, 5, 9 Show Solution 3, $\sqrt{7}$ , 4 Problem 3 Which of these is equal to $0.2\overline{5}$ ? A. $\dfrac{25}{99}$ B. $\dfrac {23}{90}$ C. $\dfrac14$ D. $2\dfrac59$ Show Solution $\dfrac {23}{90}$ Problem 4 For each number, write the letter of the point that shows its location on the number line. $\sqrt{2}$ $\hspace{.1in}\underline{\hspace{.5in}}$ $\sqrt[3]{25}$ $\hspace{.1in}\underline{\hspace{.5in}}$ $\sqrt{15}$ $\hspace{.1in}\underline{\hspace{.5in}}$ $\sqrt{25}$ $\hspace{.1in}\underline{\hspace{.5in}}$ $\sqrt[3]{8}$ $\hspace{.1in}\underline{\hspace{.5in}}$ $\sqrt{5}$ $\hspace{.1in}\underline{\hspace{.5in}}$ Show Solution $\sqrt{2}$ : A $\sqrt[3]{25}$ : D $\sqrt{15}$ : E $\sqrt{25}$ : F $\sqrt[3]{8}$ : B $\sqrt{5}$ : C Problem 5 Find the exact length of the segment that joins the points $(\text-5, 4)$ and $(6, \text-3)$ . Show Solution $\sqrt{170}$ units Problem 6 Mai’s younger brother tells her that $\frac{10} 7$ is equal to $\sqrt 2$ . Mai knows this can’t be right, because $\frac{10} 7$ is rational and $\sqrt 2$ is irrational. Write an explanation that Mai could use to convince her brother that $\frac{10} 7$ cannot be the square root of 2. Show Solution Sample responses: $\frac{10} 7$ squared is $\frac{100}{49}$ , which does not equal 2. Therefore, $\frac{10} 7$ does not equal $\sqrt 2$ . (With accompanying long division) $\frac{10}7$ is about 1.43, while $\sqrt 2$ is about 1.41, so they cannot be the same number. Minimal Tier 1 response: Work is complete and correct. Sample: $(\frac{10}{7})^2 = \frac{100}{49}$ , which is not 2. Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Response does not explicitly compare $\frac{100}{49}$ to 2; Response compares $\frac{10}7$ to $\sqrt 2 \approx 1.4$ , which is not an accurate enough approximation of $\sqrt 2$ to show that the two numbers are different; arithmetic error in squaring $\frac{10}7$ ; response simply states that $\frac{10}7$ is rational, but $\sqrt 2$ is irrational. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Explanation does not appeal to the work of the unit; no explanation. Problem 7 Elena wonders how much water it would take to fill her cup. She drops her pencil in her cup and notices that it just fits diagonally. (See the diagram.) The pencil is 17 cm long and the cup is 15 cm tall. How much water can the cup hold? Explain or show your reasoning. (The surface area of a cylinder is $2 \pi r^2 + 2 \pi rh$ . The volume of a cylinder is $\pi r^2 h$ .) Show Solution $240\pi$ cm³, or approximately 754 cm³. The diameter of the cylindrical cup is 8 cm, because $\sqrt{17^2 - 15^2} = 8$ . That means the radius is 4 cm. The amount of water the cup can hold is the volume of the cylinder: $V = \pi \boldcdot 4^2 \boldcdot 15 \approx 754$ cm³. Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: $\sqrt{17^2 - 15^2} = 8$ . $V = \pi \boldcdot 4^2 \boldcdot 15 \approx 754$ . The cup can hold about 754 cubic centimeters of water. Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: Omission of units in the final answer; correct mathematical calculations but failure to state how much water the cup can hold; use of 8 cm as the radius of the cup. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: Correct use of the Pythagorean Theorem to calculate the diameter of the cup, but no significant further progress; calculation of the surface area of the cup rather than the volume; work involves visual estimation of the radius of the cylinder rather than calculation; calculation for the diameter of the cup treats this length as the hypotenuse of the triangle. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Work uses neither the Pythagorean Theorem nor the volume formula for a cylinder; two or more error types under Tier 3 response.