T128 Mathematics Success Grade 6 [OBJECTIVE] The student will solve problems involving finding percent as a rate per 100, finding the whole given a part, and the percent. [PREREQUISITE SKILLS] concepts of ratio and rate, equivalent fractions [MATERIALS] Student pages S61 S69 70 centimeter cubes per student pair Colored pencils [ESSENTIAL QUESTIONS] 1. How can I define percent as a ratio? 2. How can I find a percent when given a part and the whole? 3. How can I find the whole when given the part and percent of a number? [WORD WALL WORDS] part ratio, percent, whole = x 100 [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the role of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Graphic Organizer, Pictorial Representation, Concrete Representation [WARM-UP] S61 (ANSWERS ARE ON T137.) Have students turn to S61 in their books to begin the Warm-Up. Students will solve rate problems involving unit pricing and constant speed. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review answers as a whole class. {Verbal Description, Algebraic Formula} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [1 day (80 minutes) M, GP, IP, WG, CP]
Mathematics Success Grade 6 T129 SOLVE Problem (WG, GP) S63 (Answers on T138.) Have students turn to S62 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to solve problems involving percent as a rate per 100, finding the whole, given a part, and the percent. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description} Percent as a Rate Using Centimeter Cubes (CP, WG, M, GP) S62 (Answers on T138.) M, GP, CP, WG: Have students turn to S62 in the books. Distribute 70 centimeter cubes to student pairs. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Concrete Representation} Percent as a Rate Centimeter Cubes Step 1: Direct students attention to the grid on S62. Partner A, identify the number of squares on the whole. (100) Partner B, place 25 centimeter cubes on the grid. Partner A, how many cubes out of 100 have been placed on the grid? (25 out of 100) Partner B, determine if the number of cubes could be represented as a ratio in fractional form. (Yes, 25 Step 2: Step 3: Partner A, add 10 more cubes to the 25 cubes already on the grid. Partner B, identify the ratio of cubes to 100 in word and fraction form. (35 out of 100, or 35 Partner A, remove 4 of the cubes showing on the grid. Partner B, identify the ratio of cubes to 100 in word and fraction form. (31 out of 100, or 31
T130 Mathematics Success Grade 6 Percent as a Rate - Pictorial Percent as a Rate - Pictorial (10 minutes CP, WG, M, GP, IP) S62 (Answers on T138.) M, GP, WG, CP: Students will continue to work on S62. They will be using the information from the concrete modeling of percents to create pictorial models. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Concrete Representation, Pictorial Representation} Step 1: Refer students back to the grid on the top of S62. Partner A, place 10 centimeter cubes horizontally at the top of the page. Partner B, remove each of the cubes and shade the squares using a blue colored pencil. Partner A, identify the ratio of the blue squares to 100. ( 10 Partner B, how many squares per 100 are shaded on the grid? (10) Partner A, if 10 shaded squares equal the percent of squares shaded out of 100, what is the percent of 10 shaded squares? (10%) Record. Step 2: Partner B, add 15 centimeter cubes to the 10 squares already shaded. Partner A, remove each of the cubes and shade the squares using a green colored pencil. Partner B, identify the ratio of the green squares to 100. ( 15 Partner A, determine the percent of green squares to 100. (15%) Record. Step 3: Have student pairs complete the activity with the centimeter cubes by following Step 2 and adding 20 orange cubes and then 30 red cubes and filling in the ratios and percents for each color. Step 4: Have student pairs look at the shaded centimeter grid and determine the number of squares in the grid that are not shaded. Partner B, how many squares are not shaded? (25) Partner A, how can we write that value as a ratio? ( 25 Partner B, how can we write that value as a percent? (25%) Record. Step 5: What does the word percent mean? (out of 100) Record.
Mathematics Success Grade 6 T131 Percents and Ratios - Pictorial Percents and Ratios - Pictorial (10 minutes CP, WG, M, GP, IP) S63 (Answers on T139.) M, GP, CP, WG: Have students turn to S63 in their books. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer} Step 1: Direct students attention to Problem 1 on S63 Partner A, what is the problem asking us to find? (40% of 100) Partner B, what is the value we know? (40%) Partner A, identify the total number of squares on the grid. (100) Partner B, determine what you are looking for. (the number of squares that equal 40%) Partner A, identify the ratio for the shaded part of the grid. ( 40 Record. Partner B, how many shaded squares equal 40% of the grid? (40) Record. Step 2: Have students look at Problem 3 and discuss how it is different from Problem 1. Partner A, how is Problem 3 different form Problem 1? (Problem 1 gives the percent and asks for the part of the grid that is shaded. Problem 3 gives the value of shaded squares and asks for the percent.) Partner B, identify the total number of squares on the grid. (100) Partner A, identify the part that is shaded. (25) Partner B, explain how we can write the value of the shaded squares as a ratio. ( 25 Partner A, determine the percent of shaded squares. (25%) Partner B, justify your answer. (The shaded squares equal the percent of squares shaded out of 100, or 25%.) Step 3: Have students look at Problem 5 and discuss how it is different from Problem 3. Partner A, how is Problem 5 different form Problem 3? (Problem 3 gives the value of shaded squares and asks for the percent. Problem 5 gives the value and the percent and asks for the total value.) Partner B, identify the total number of squares on the grid. (100) Partner A, identify the part that is shaded. (15)
T132 Mathematics Success Grade 6 Partner B, explain how we can write the value of the shaded squares as a ratio. ( 15 Partner B, determine the whole number of squares to which 15 is compared. (100) Record. This means that 15 is 15% of 100. IP, CP, WG: Have students work with their partners to complete Problems 2, 4, and 6 on S63. Monitor closely to make sure students are using the appropriate vocabulary. Have students come back together as a class and share their results. {Verbal Description, Pictorial Representation} Percent as a Rate - Abstract with Graphic Organizer (CP, WG, M, GP, IP) S64, S65 (Answers on T140, T141.) M, GP, CP, WG: Have students turn to S64 in their books. They will be working with percentages and using the pictorial model to create a graphic organizer. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer} Percent as a Rate Abstract with Graphic Organizer Step 1: Direct students attention to the grid and questions for Problem 1. Partner A, what is the problem asking us to find? (50 is what percent of 100?) Partner B, what is the value we know? (50) Partner A, identify the total number of squares on the grid. (100) Partner B, determine what you are looking for. (the percent of squares that equal 50 out of 100) Partner A, identify the ratio for the shaded part of the grid. ( 50 Record. Partner B, 50 is what percent of 100? (50%) Record. Step 2: Look at the graphic organizer for Problem 1. Partner A, identify what you are looking for in Problem 1. (percent) Circle the word percent in the first box and in the unknown box. Partner B, using the grid, identify what is known in the problem. (the part of 100 that is shaded, and the total number of squares on the grid, which is 100) Partner B, identify the total number of squares on the grid. (100)
Mathematics Success Grade 6 T133 Step 3: Have students look at the second box in the graphic organizer that is labeled Unknown. Partner A, what is the value of the denominator of the second fraction? (100) Partner B, why is 100 the denominator when finding a percent? (When finding a percent, a number is always compared to 100.) Partner A, the numerator for this fraction is percent. When finding a percent, to what number is the percent compared? (100) Partner B, the numerator for the first fraction is part. With your partner, determine what part of the whole is using the grid. (the number of squares shaded to represent the part of the whole that is shaded) Partner A, the denominator for the fraction is whole. With your partner, determine what a whole means using the grid. (the entire number of squares, including the shaded and non-shaded squares) Explain to students that there is a proportional relationship they can use to find missing values when working with percents. part whole = % 100 Step 4: Look at the Equivalent Fractions box. Partner A, identify the part of the grid that is shaded. (50) Record for the numerator in the equivalent fractions box. Partner B, identify the number of squares in the grid. (100) Record for the denominator in the equivalent fractiosn box. Do we know the percent? (No) Record as an x to represent the unknown value. Step 5: Partner A, what can we multiply 100 by to get 100 in the denominator? (1) Justify your answer. Record. (Whatever we do in the denominator we must do in the numerator.) Partner B, what is the product of 50 by 1? (50) Record for the numerator in the solution box.
T134 Mathematics Success Grade 6 Step 6: Direct students attention to Problem 2 and have student pairs compare Problem 2 to Problem 1. (There is no grid.) Partner A, what is the problem asking you to find? (the percent) Circle the word percent on the graphic organizer in the first and second boxes. With your partner, set up an equivalent fraction problem. Partner B, what is the part of the whole? (2) Record. Partner A, what is the whole number? (4) Record. Partner B, what is the unknown for the equivalent fractions? (Percent) Partner A, how can we represent the unknown value of percent? (with a variable) Record. With your partner, find the equivalent fraction. What is x equal to? (50) Record. What is the final answer? (2 is 50% of 4.) Record. Step 7: Direct students attention to Problem 3. Partner A, what is the problem asking you to find? (the part) Circle part in the first and second boxes of the graphic organizer. With your partner, set up equivalent fractions. Partner B, what is the whole for this problem? (4) Record. Partner A, what is the percent for this problem? (75%) Record. Partner B, how will the unknown part be represented? (with a variable) Record and solve to find the x. What is the final answer? (3 is 75% of 4.) Record. Step 8: Direct students attention to Problem 4. Partner A, what is the problem asking you to find? (the whole) Circle whole in the first and second boxes of the graphic organizer. With your partner, set up equivalent fractions. Partner B, what is the part for this problem? (1) Record. Partner A, what is the percent for this problem? (25%) Record. Partner B, how will the unknown part be represented? (with a variable) Record and solve to find the x. What is the final answer? (1 is 25% of 4.) Record. IP, CP, WG: Have students work with their partners to complete Problems 1-4 on S65. Monitor closely to make sure students are using the appropriate vocabulary. Have students come back together as a class and share their results. {Verbal Description, Pictorial Representation, Graphic Organizer}
Mathematics Success Grade 6 T135 Percent as a Rate Abstract (CP, WG, M, GP, IP) S66 (Answers on T142.) M, GP, CP, WG: Have students turn to S66 in their books. Students will be applying what they have learned about percents and rates without a graphic organizer. Make sure students know their designation as Partner A or Partner B. {Verbal Description, Graphic Organizer, Algebraic Expression} Percent as a Rate - Abstract Step 1: Direct students attention to Problem 1 Partner B, are we finding the part, the whole, or the percent? (the whole) Partner A, explain how you know this. (The part and the percent are given information.) With your partner, complete the graphic organizer using the steps from S64 and S65. Step 2: Have student pairs look at Problem 2 and discuss how it is different from Problem 1. (It has no graphic organizer.) Partner A, what is this problem asking you to find? (the whole) Partner B, explain how you know this. (The part and the percent are given information.) Partner A, explain how to set up equivalent fractions. (Use the part whole = % formula and substitute in the given values.) 100 Partner B, what value do we know in the first fraction? (the part) Record. Partner A, what information do we know in the second fraction? (the percent and the denominator which is always 100) Record. Partner B, how can we represent the unknown in the denominator of the first fraction? (using a variable) Record. Find the equivalency and record. 30 x = 30 30 100 100 = 30 100