Are You Ready? Ratios

Transcription

1 Ratios Teahing Skill Objetive Write ratios. Review with students the definition of a ratio. Explain that a ratio an be used to ompare anything that an be assigned a number value. Provide the following examples: number of boys in the lass to number of girls; number of students taking Algebra 2 to number of students taking Geometry; and height of one partiular student to that of another. Ask eah student to desribe one ratio that they an think of to write. Review eah of the different ways in whih a ratio may be written: word form, ratio form, and fration form. Ask: Is it possible to simplify a ratio and if so, when? (Yes, if the terms of the ratio share any fators other than 1, then the ratio an be simplified, just like simplifying a fration.) Review the example. Point out that units are not inluded in a ratio. PRACTICE ON YOUR OWN In exerises 1, students use a triangle and a table to write a variety of ratios. CHECK Determine that students know how to write ratios. Students who suessfully omplete the Pratie on Your Own and Chek are ready to move on to the next skill. COMMON ERRORS Students write the terms of the ratio in the wrong order. To avoid this, enourage students to write the words first, then the numbers. Students who made more than 2 errors in the Pratie on Your Own, or who were not suessful in the Chek setion, may benefit from the Alternative Teahing Strategy. Alternative Teahing Strategy Objetive Write ratios using a dek of ards. Have students work in pairs. Give eah pair of students a standard dek of ards. Have eah pair of students do the following: Mix the deks of ards. Cut the dek so that eah student has roughly half the dek. The dek does not need to be divided exatly in half. Have eah student reate the following table on a piee of paper. Distribution of Cards Hearts Diamonds Spades Clubs Total Have students ount the number of ards they have for eah suit and reord the results in the table. Review eah of the different ways in whih a ratio may be written: word form, ratio form, and fration form. Have eah student use the information they reorded in the table to write the following ratios in three different ways: 1. Number of hearts to diamonds 2. Number of spades to lubs 3. Number of red ards to blak ards 4. Number of hearts to total number of ards (Students answers will vary depending on the ards that are in their half of the dek.) When all the students have written their ratios, instrut them to exhange their half of the dek with their partner. Partners should hek eah other s answers by ounting their ards and writing the orret ratios. 35 Holt Algebra 2

2 Name Date Class Ratios Definition: A ratio is a omparison of two or more numbers, alled the terms of the ratio. Ways to Write Ratios Words Ratio Fration 2 to 3 2:3 2 3 Example: Write the ratio of the measures of angle A to angle B in triangle ABC below. A B C Step 1: What are the terms of the ratio? angle A and angle B Step 2: What is the measure of angle A? 65 What is the measure of angle B? 0 Step 3: Write the ratio in the order requested: 65 to 0 or 65:0 or 65 0 Step 4: Simplify if possible: 13 to 16 or 13:16 or Pratie on Your Own Use ABC to write eah ratio three different ways. Write your answers in simplest form. A AB to AC 2. AC to AB 3. length of shortest leg to hypotenuse C 5 B 4. perimeter of ABC to area Use the table to write eah ratio in simplest fration form. 5. red ars to blak ars 6. red ars to white ars 7. red ars to not red ars. red ars to total ars Cars in the Parking Lot Red 20 Blak 43 White 10 Chek Use DEF to write eah ratio three different ways. Write your answers in simplest form. E 9. DE to EF 10. DE to DF D F 11. DE to perimeter of DEF 36 Holt Algebra 2

3 29 Classify Triangles Teahing Skill 29 Objetive Classify triangles (right, aute, obtuse). Begin the lesson by reminding students that angles an be lassified as right, aute, or obtuse. Ask: What is a right angle? (an angle that has a measure of 90 ) Draw a right angle on the board. Remind students that an aute angle is one that has a measure less than 90, and an obtuse angle is one that has a measure greater than 90. Point out that lassifying triangles is similar to lassifying angles. Review eah of the types of triangles. Ask: Why does an obtuse triangle only have one obtuse angle? (The sum of the angles annot be greater than 10.) Point out that an angle may look like a right angle, but it is not a right angle unless one of the angles has a measure of 90 or the symbol in the orner of the two legs indiates that it is a right angle. Have students omplete the pratie exerises. PRACTICE ON YOUR OWN In exerises 1, students lassify triangles as aute, right, or obtuse triangles. CHECK Determine that students know how to lassify triangles. Students who suessfully omplete the Pratie on Your Own and Chek are ready to move on to the next skill. COMMON ERRORS Students may onfuse aute and obtuse triangles beause they do not pay attention to all three angles. Students who made more than 1 error in the Pratie on Your Own, or who were not suessful in the Chek setion, may benefit from the Alternative Teahing Strategy. Alternative Teahing Strategy Objetive Classify triangles (right, aute, obtuse). Materials needed: game board shown below Review with students the three lassifiations of triangles. Tell students they have a very diffiult problem to solve. Hand out the following problem: The angles of a partiular obtuse triangle have the following properties: the measure of the largest angle is 5 times the measure of the smallest angle; the measure of the angle that is not the smallest or the largest is 40 bigger than the smallest angle and 40 smaller than the largest angle. What is the measure of the smallest angle? Tell students that they will find the answer to the problem by orretly lassifying the triangles on their game boards, and that the answer to the problem is: the number of aute triangles times the number of obtuse triangles, plus the number of right triangles. The first student to arrive at the orret answer wins. (20 ) If time permits, have students try to find the measures of the other two angles using algebra. (60 and 100 ) 69 Holt Algebra 2

4 Name Date Class 29 Classify Triangles Right Triangle Aute Triangle Obtuse Triangle one right angle three aute angles one obtuse angle Example: If a triangle has angles with measures of 9, 52, and 30, what kind of triangle is it? Answer: Sine 9 90, the triangle has one obtuse angle; the triangle is an obtuse triangle. Pratie on Your Own Tell whether eah triangle is aute, right, or obtuse Chek Tell whether eah triangle is aute, right, or obtuse Holt Algebra 2

5 30 Triangle Sum Theorem Teahing Skill 30 Objetive Use the Triangle Sum Theorem to find the measures of missing angles. Have students read the Triangle Sum Theorem. Point out that the theorem is easily stated in words or using symbols. Ask: What is the sum of the measures of the angles of a right triangle? (10 ) An aute triangle? (10 ) An obtuse angle? (10 ) Does the kind of triangle determine the sum? (No) Draw a right triangle on the board. Be sure to inlude the symbol whih indiates that the triangle is a right triangle. Ask: Sine one of the angles is 90 and the sum of all the angles is 10, what is the sum of the other two angles? ( ) Work the example. Then have students look at, but not solve, problem 4. Point out that students will need to use algebra skills when there is more than one unknown angle. Write the following example on the board: x 3x 140. Remind students how to ombine like terms and then solve the equation. PRACTICE ON YOUR OWN In exerises 1, students find the value of x using the Triangle Sum Theorem. CHECK Determine that students know how to use the Triangle Sum Theorem. Students who suessfully omplete the Pratie on Your Own and Chek are ready to move on to the next skill. Alternative Teahing Strategy Objetive Use the Triangle Sum Theorem to find the measures of missing angles. Materials needed: multiple opies of game piees and game boards (enlarged) Review the Triangle Sum Theorem with students. Then hand out the game piees. Game piees (first round) Game board (first round) 75 42? 0??? 75 4? ? Tell students they are going to play Find that Triangle using the Triangle Sum Theorem. When you say Go students should try to math the orret game piee with the missing angle for eah triangle. The first student to orretly math all the game piees wins. (Answers from left to right: 15, 45, 57, 5, 105, 60 ) For the seond round, tell students they need to find the value of x to math the game piees to the triangles. Remind them to set up equations and solve for the variable. Game piees (seond round) COMMON ERRORS Students may add or subtrat inorretly and arrive at the wrong angle measure. Students who made more than 2 errors in the Pratie on Your Own, or who were not suessful in the Chek setion, may benefit from the Alternative Teahing Strategy. Game board (seond round) (Answers from left to right: 61, 2, 45, 36, 70, 30) 71 Holt Algebra (x 30) 2

6 Name Date Class 30 Triangle Sum Theorem Triangle Sum Theorem: The sum of the measures of the angles of a triangle is 10. A m A m B m C 10 C B Example: Find the value of x Answer: 0 70 x x 10 x x 30 Pratie on Your Own Find the value of x (x + 20) Chek Find the value of x Holt Algebra 2

7 31 Pythagorean Theorem Teahing Skill 31 Objetive Find the length of the hypotenuse of a right triangle. Have students read the Pythagorean Theorem. Restate the theorem in words, as follows: the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Emphasize that the hypotenuse of a right triangle is ALWAYS the side that is opposite the right angle. Ask: If the lengths of all three sides are found orretly, whih side will always be the longest side? (the hypotenuse) Point out that it does not matter whih leg is represented by a and whih is represented by b, but the hypotenuse must always be represented by. Work the example, stressing that you must square the legs first before you add them. Sine most numbers are not perfet squares, tell students that they may need to simply radials. Work a few examples to remind them of the proess. PRACTICE ON YOUR OWN In exerises 1 6, students find the length of the hypotenuse of several right triangles. CHECK Determine that students know how to use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle. Students who suessfully omplete the Pratie on Your Own and Chek are ready to move on to the next skill. COMMON ERRORS Students may add the lengths of the legs before squaring them. Students who made more than 1 error in the Pratie on Your Own, or who were not suessful in the Chek setion, may benefit from the Alternative Teahing Strategy. Alternative Teahing Strategy Objetive Verify the Pythagorean Theorem using a ruler. Materials needed: several piees of lined paper and a ruler Remind students that the Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Tell students they are going to verify the theorem. Have students take one piee of lined paper and fold it arefully in half (vertially), making a distint rease in the paper. Instrut them to unfold the paper. Instrut students to use a ruler to draw a vertial line up the rease inhes long, and a horizontal line at the bottom of the vertial line, 6 inhes long. Next, have students onnet the two lines with a diagonal, forming a right triangle. Using a ruler, students should arefully measure the length of the hypotenuse. Instrut them to label the lengths of the legs, a and b (6 and ), and the length of the hypotenuse, (10). Ask: Aording to the Pythagorean Theorem, how are a, b, and related? (a 2 b 2 2 ). Have students onfirm this by substituting their values into the equation. Repeat the exerise above on separate sheets of paper using the following measurements: 1) vertial line 4 inhes; horizontal, 3 inhes (hypotenuse should equal 5 inhes) 2) vertial line m; horizontal, 5 m (hypotenuse should equal 13 m) 3) vertial line 15 m; horizontal, m (hypotenuse should equal 17 m) When you feel omfortable that students know how to use the Pythagorean Theorem, move on to examples that do not require measurements. 73 Holt Algebra 2

8 Name Date Class 31 Pythagorean Theorem Pythagorean Theorem If a right triangle has legs of lengths a and b, and a hypotenuse of length, then a 2 b 2 2. a hypotenuse legs b Example: Find the length of the hypotenuse of the right triangle. Answer: a 2 b The length of the hypotenuse is Pratie on Your Own Find the length of the hypotenuse in eah right triangle. If the length is not a whole number, give the answer in simplest radial form Chek Find the length of the hypotenuse in eah right triangle. If the length is not a whole number, give the answer in simplest radial form Holt Algebra 2