Unit 8 Pythagorean Theorem And Irrational Numbers — Unit Plan

Title	Takeaways	Assessment
Lesson 1 The Areas of Squares	—	It's a Square (1 problem) Find the area of square $ACEG$ . Show Solution 100 square units
Lesson 2 Side Lengths and Areas	—	Area Estimate (1 problem) 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? (1 problem) 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 (1 problem) 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 $\sqrt{18}$ (1 problem) 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 (1 problem) 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 $c^2$? (1 problem) 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? (1 problem) 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 (1 problem) 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? (1 problem) 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 11 Applications of the Pythagorean Theorem	—	How High Up? (1 problem) An 11.5 m support pole is attached to a vertical utility pole to help keep it upright. The base of the support pole is 4.5 m from the base of the utility pole. How high up the utility pole does the support pole reach? Assume the vertical utility pole makes a right angle with the ground. Show Solution Approximately 10.58 m
Lesson 12 More Applications of the Pythagorean Theorem	—	Diameter of a Cone (1 problem) 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 (1 problem) 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 (1 problem) 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 (1 problem) 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 $x$ . 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 (1 problem) 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.
Lesson 17 Infinite Decimal Expansions	—	Repeating in Different Ways (1 problem) Let $x=0.147$ and let $y=0.\overline{147}$ . Is $x$ a rational number? Is $y$ a rational number? Which is larger, $x$ or $y$ ? Show Solution Yes, $0.147=\frac{147}{1,000}$ is rational. Yes, $0.\overline{147}=\frac{49}{333}$ is rational. $y$ is larger than $x$ since $y=0.147147147 . . .$ and $x=0.147000000 . . .$ .
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