Vocabulary: colon, equivalent ratios, fraction, part-to-part, part-to-whole, ratio

Transcription

1 EE8-39 Ratios and Fractions Pages Standards: preparation for 8.EE.B.5 Goals: Students will review part-to-part and part-to-whole ratios, different notations for a ratio, and equivalent ratios. Students will understand ratios as a number of one thing for every of another. Prior Knowledge Required: Can interpret fractions Can produce fractions equivalent to a given fraction Vocabulary: colon, equivalent ratios, fraction, part-to-part, part-to-whole, ratio Review ratios. Draw on the board: ASK: What fraction of the shapes are squares? (3/5) What fraction are circles? (2/5) SAY: If I want to know how many in the whole set of shapes are squares, I can use fractions, just as we did with 3/5. But if I want to compare the squares to the circles, I can t use fractions of a whole, because the squares are not a part of the set of circles. If I want to compare one part to another part, I need to use a ratio. Write on the board: The ratio of squares to circles is 3 to 2 or 3 : 2. Point to the colon (:) and remind students that this is a common way to write a ratio numerically. SAY: The symbol is called a colon and we read it as to. Eercises: Write the ratio for the set. a) circles to squares b) circles to triangles c) triangles to squares Bonus: circles to polygons Answers: a) 1 to 4 or 1 : 4, b) 1 to 3 or 1 : 3, c) 3 to 4 or 3 : 4, Bonus: 1 to 7 or 1 : 7 Ratios can compare parts to wholes. SAY: You can also use ratios to compare parts to the whole. Refer students to the previous eample on the board (3 squares and 2 circles). SAY: In this eample, the ratio of squares to shapes is 3 to 5 or 3 : 5. Write on the board: The ratio of squares to all shapes is 3 to 5 or 3 : 5. The ratio of circles to all shapes is to or :. Point at the blanks and have students signal what number should be written in each. (2 to 5, 2 : 5) F-2 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

2 Eercises: Write the ratio for the set of eight shapes in the previous eercises. Use a colon (:) in your answer. a) squares to all shapes b) circles to all shapes c) triangles to all shapes Bonus: polygons to all shapes Answers: a) 4 : 8, b) 1 : 8, c) 3 : 8, Bonus: 7 : 8 SAY: A ratio is called a part-to-part ratio when it compares one part to another part. A ratio is called a part-to-whole ratio when it compares a part to the whole. Before students do the following eercises, remind them that the letters a, e, i, o, u are vowels and other letters are consonants. Eercises: Find the ratio. Then say if the ratio is part-to-part or part-to-whole. a) vowels in cat to all letters in cat b) vowels in cat to consonants in cat c) consonants in California to all letters in California d) consonants in California to vowels in California e) weekend days to all days in a week f) days in January to days in December g) days in January to days in a year Answers: a) 1 : 3, part-to-whole; b) 1 : 2, part-to-part; c) 5 : 10, part-to-whole; d) 5 : 5, part-to-part; e) 2 : 7, part-to-whole; f) 31 : 31, part-to-part; g) 31: 365, part-to-whole Part-to-whole ratios as fractions. SAY: You can think of part-to-whole ratios as fractions. A part-to-whole ratio is just like a fraction, but with a colon between the part and the whole instead of a dividing line. A fraction is a special kind of ratio, so ratios are more general than fractions. You can compare and order ratios just as you can fractions. Go through the previous eercises as a class and have students write each part-to-whole ratio as a fraction. (a) 1/3, c) 5/10, e) 2/7, g) 31/365) Eercises: Write a ratio and a fraction. a) large circles to circles = : of the circles are large b) white circles to circles = : of the circles are white c) small white circles to circles = : of the circles are small and white Answers: a) 3 : 10 and 3/10, b) 5 : 10 and 5/10, c) 4 : 10 and 4/10 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-3

3 Changing part-to-part ratios to part-to-whole ratios and fractions. Tell students that, if a set has only two parts, such as circles and triangles or girls and boys, then you can find a part-towhole ratio from knowing only the part-to-part ratio. Write on the board: There are 3 girls for every 5 boys. g g g b b b b b So there are 3 girls for every students. ASK: If there are 3 girls and 5 boys, how many students are there altogether? (8) SAY: So, if there are 3 girls for every 5 boys, then there are 3 girls for every 8 students. Write 8 in the blank. Tell students that you just made a part-to-whole ratio from a part-to-part ratio the ratio of girls to students is 3 : 8. ASK: What is the fraction of students who are girls? (3/8) What is the fraction of students who are boys? (5/8) Eercises: Find two part-to-whole fractions from the part-to-part ratio. a) The ratio of girls to boys is 3 : 7. b) The ratio of boys to girls is 4 : 5. c) The ratio of girls to boys is 8 : 7. d) There are 5 boys and 5 girls on the team. Bonus: The ratio of a team s wins to losses is 2 : 5. (Ties are not allowed.) Answers: a) 3/10 of the students are girls, 7/10 are boys; b) 4/9 of the students are boys, 5/9 are girls; c) 8/15 of the students are girls, 7/15 are boys; d) 5/10 of the players are boys, 5/10 are girls; Bonus: 2/7 of the games were won, 5/7 were lost Fill in the first column of the chart below as a class, then have students fill in the rest of the chart. Eercises: Each ratio in the first row describes a different school club. Complete the chart. Ratio of boys to girls 5 : 7 6 : 8 8 : 6 Fraction of students who are girls Fraction of students who are boys Are there more boys or girls? Answers: Ratio of boys to girls 5 : 7 6 : 8 8 : 6 1 : 3 3 : 6 Fraction of students who are girls 7/12 8/14 6/14 3/4 6/9 Fraction of students who are boys 5/12 6/14 8/14 1/4 3/9 Are there more boys or girls? girls girls boys girls girls ASK: How can you tell from each ratio if there are more boys than girls? (if the first number is bigger than the second number) How can you tell from each fraction if there are more girls than boys? (if the fraction that is girls is more than half) F-4 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

4 Eercises: a) If the ratio of girls to boys is 74 : 71, are there more girls or boys? b) The ratio of parks to schools is 19 : 21 in City A and 12 : 11 in City B. Which city has more schools than parks? Answers: a) more girls, b) City A Review equivalent ratios. Tell students that a recipe for an orange-banana smoothie calls for 3 oranges and a banana. Write on the board: O O O B Eplain that you wrote an O for each orange and a B for each banana. Tell students that you want to make lots of smoothies. ASK: If I use 2 bananas, how many oranges would I need? (6) Write another row of letters on the board, as shown below: O O O B O O O B SAY: The recipe looks like it calls for just 3 oranges and 1 banana, but what it actually says is that for every banana you use, you need 3 oranges. The ratio of oranges to bananas is 3 to 1, but it is also 6 to 2. ASK: If I use 1 more banana, how many more oranges would I need? (3) Have a volunteer write another row of letters on the board. ASK: What ratio is showing now? (9 oranges to 3 bananas) Continue writing on the board: O O O B oranges to bananas = 3 : 1 O O O B = 6 : 2 O O O B = 9 : 3 Point out that the ratio 3 to 1 doesn t describe just a single situation, but a whole sequence of possible situations. NOTE: We use the word sequence here to mean a group of ratios that can continue by following a rule. Ask a volunteer to continue the picture to find the net possible situation and ratio. (12 : 4) SAY: All these ratios are equivalent ratios. Point out that, for any description, the order of numbers in a ratio matches the order of what they represent. So, for eample, if we describe a ratio of 3 oranges to 1 banana, we write it as 3 : 1, not 1 : 3. Similarly, if we give a ratio of 19 : 21 to compare parks to schools, we mean 19 parks for every 21 schools, not 19 schools for every 21 parks. Eercises: Write two equivalent ratios for the picture. a) b) circles to squares stars to dots = : = : = : = : Sample answers: a) 2 : 1 = 6 : 3, b) 2 : 3 = 8 : 12 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-5

5 Return to the eample about the orange-banana smoothie. Challenge a volunteer to find the net possible ratio of oranges to bananas without writing the letters. (15 : 5) ASK: How did [volunteer] find this ratio? (from 12 : 4, add 3 more O s and 1 more B) Point out that students can skip count by 3s to continue the first column and by 1s to continue the second column in the column of ratios. Eercises: Skip count to write three more equivalent ratios. a) 5 : 1 b) 1 : 4 c) 10 : 2 Bonus: 3 : 5,000 = : = : = : = : = : = : = : = : = : = : = : = : Answers: a) 10 : 2 = 15 : 3 = 20 : 4, b) 2 : 8 = 3 : 12 = 4 : 16, c) 20 : 4 = 30 : 6 = 40 : 8, Bonus: 6 : 10,000 = 9 : 15,000 = 12 : 20,000 Solving word problems by making sequences of part-to-part ratios. Eplain that you can use sequences of ratios, as in the previous set of eercises, to solve word problems. For eample, a recipe for dumpling dough calls for 8 cups of flour and 3 cups of water. In a restaurant, a cook will make lots of dough, so he will use 40 cups of flour. The cook needs to figure out how much water to use. Write on the board: Dumpling dough recipe: 8 cups of flour for every 3 cups of water Cook uses: 40 cups of flour How much water? Eplain that one way to solve this problem is to write a sequence of ratios, using skip counting. Write on the board: Flour : Water 8 : 3 16 : 6 24 : : : Have students etend both columns. (24 : 9, 32 : 12, 40 : 15) SAY: The cook needs to use 40 cups of flour. ASK: When do you need to stop writing the numbers in the sequence? (when you reach 40 in the flour column) How many cups of water will the cook use? (15) Provide the headings for the ratios in the following eercises to help students who struggle. SAY: Remember that you could keep writing the sequence forever, but you only want to find out the answer to your question. One way to stay focused is to write headings over the columns and circle the column that will give you the information you need. F-6 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

6 Eercises: Write a sequence of equivalent ratios to solve the problem. a) A recipe for granola calls for 3 cups of oatmeal for every 4 cups of raisins. Tina uses 15 cups of oatmeal. How many cups of raisins does she need? b) There are 5 boys for every 6 girls in a class. There are 15 boys in the class. How many girls are there? c) A car uses 7 L of gas for every 100 km. The tank holds 35 L of gas. How far can the car drive on a full tank of gas? Bonus: Maria is traveling to Canada. She knows that there are about 8 kilometers in every 5 miles. A road sign shows that the distance to Montreal is 64 km. How many miles away is Montreal? Selected solution: a) Oatmeal : Raisins 3 : 4 6 : 8 9 : : : 20 Tina needs 20 cups of raisins. Answers: b) boys : girls = 5 : 6 = = 15 : 18, there are 18 girls; c) gas (L) : distance (km) = 7 : 100 = = 35 : 500, the car can drive 500 km; Bonus: km : miles = 8 : 5 = = 64 : 40, 40 miles to Montreal NOTE: If you present the answers above on the board, either write them vertically to match the oatmeal/raisins solution above, or eplain to students that you can write equivalent ratios in a row to save space, but must place equal signs between each pair of ratios to show that they are equivalent, as in the other answers. Write on the board: There are 3 boys for every 2 girls in a class. There are 12 girls in the class. How many boys are in the class? SAY: Ted found a solution. Write on the board: Boys : Girls 3 : 2 6 : 4 9 : 6 12 : 8 There are 8 boys in the class. (MP.3) ASK: Is Ted s solution correct? (no) Why not? (he needs to continue the sequence of ratios until he gets 12 in the girls column, not in the boys column) Have a volunteer correct the mistake (add two rows to the table) and find the right answer. (18 boys) Emphasize how the headings for the columns make it easier to check that we have reached the correct quantity in the correct column. Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-7

7 (MP.6) Eercises: a) There are 3 boys for every 4 girls in a class. There are 12 girls. How many boys are there? b) i) A recipe calls for 2 lb of rice for every 3 lb of carrots. A cook wants to use 15 lb of carrots. How much rice does the cook need? ii) The recipe for rice calls for 1 lb of onions for every 2 lb of rice. How many pounds of onions does the cook need? Bonus: 2 inches are about 5 cm long. 1 foot is 12 inches long. How many centimeters long is 1 foot? Answers: a) 9 boys; b) i) 10 lb of rice, ii) 5 lb of onions; Bonus: about 30 cm Using part-to-part and part-to-whole ratios to solve word problems. Write on the board: There are 4 boys for every girl in an after-school club. There are 25 students in the club. How many boys are in the club? ASK: What is the ratio of boys to girls in the club? (4 : 1) Point out that the 1 is not eplicitly stated, as in 4 boys for every 1 girl, but the 1 is understood. Write on the board: Boys : Girls 4 : 1 ASK: I want to continue the sequence until I see a 25, but where should I look for the 25? PROMPT: Is 25 the number of girls? (no) Is it the number of boys? (no) What is 25 referring to? (the total number of students) SAY: I need to keep track of both the ratio of boys to girls and the total number of students in the club. Add a column for Total. ASK: If there are 4 boys and 1 girl, how many students are in the club? (5) Ask a volunteer to add 4 more boys and 1 more girl to the chart. ASK: How many boys are there now? How many girls? (8 boys, 2 girls) How many is that in total? (10) Tell students that you have to keep adding rows until you get 25 in the Total column. Have more volunteers continue adding rows to the chart until the total is 25, as shown below: Boys : Girls Total 4 : : : : : 5 25 ASK: How many boys are in the club? (20) Eercises: a) There are 4 boys for every 5 girls in a class of 36 students. How many girls are in the class? b) There are 7 boys for every 5 girls in a class of 36 students. How many boys are in the class? c) A bracelet is made of red and blue beads. There are 3 red beads for every 5 blue beads. If there are 96 beads, how many of them are red? Bonus: The ratio of girls to boys in a school is 12 : 13. If the school has 300 students, how many girls are there? Answers: a) 20, b) 21, c) 36, Bonus: 144 F-8 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

8 Etensions 1. Describe the set in three different ways using the ratio 3 : 5. Answer: 3 : 5 = circles to squares = large shapes to small shapes = gray shapes to white shapes (MP.2) 2. Write the ratio of the lengths. Eample: AB to CD = 4 : 5 4 cm 3 cm 5 cm A B C D a) BC to CD b) AB to AC c) CD to AD Bonus: Which ratios are part-to-whole ratios? Answers: a) 3 : 5, b) 4 : 7, c) 5 : 12, Bonus: AB to AC and CD to AD (MP.3) 3. Can there be a set of shapes with the ratio triangles to polygons = 3 : 2? Eplain. Answer: No, because triangles are polygons, so they are a part of the whole, so there cannot be more triangles than polygons. (MP.1) 4. Sun made 20 cups of green paint by miing 1.25 cups of blue paint with every 3.75 cups of yellow paint. a) How much blue paint and how much yellow paint did she use? b) Sun meant to use 1.25 cups of yellow paint for every 3.75 cups of blue paint. She thinks she can correct her mistake by adding some blue paint. How much blue paint does she need to add? Does this idea make sense? Answers: a) Sun used 5 cups of blue paint and 15 cups of yellow paint. b) Sun needs the ratio yellow : blue = 1.25 : 3.75, but she used 15 cups of yellow. If we use? for the number of cups of blue paint she needs, we get equivalent ratios y : b = 1.25 : 3.75 = 15 :?. The missing number? is 45. This means she needs to have 45 cups of blue paint in total; she used 5 cups of blue paint and 15 cups of yellow paint already, so she needs to add 40 more cups to make 45 cups and will have 60 cups of paint in total. If she has a large project and would need that large quantity of paint, her solution makes sense. (MP.1) 5. Find two numbers that add to 12 and are in the given ratio. a) 1 : 2 b) 1 : 3 c) 1 : 5 Answers: a) 4 and 8, b) 3 and 9, c) 2 and 10 (MP.3) 6. Two whole numbers are in the ratio 1 : 3. Rob says they cannot add to an odd number. Is he right? Eplain. Answer: Yes, the second number is three times the first, so their sum is four times the first and the sum is always even. 7. Find two fractions that add to 1 and are in the ratio 3 : 2. Answer: 3/5 and 2/5 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-9

9 EE8-40 Ratio Tables Pages Standards: preparation for 8.EE.B.5 Goals: Students will create equivalent ratios using multiplication, and create and identify ratio tables. Students will determine whether two quantities are proportional from a table comparing their values. Prior Knowledge Required: Can interpret fractions Can produce fractions equivalent to a given fraction using multiplication and division Understands a ratio in terms of for every Can create a sequence of equivalent ratios Vocabulary: colon (:), denominator, equivalent fractions, equivalent ratios, numerator, ratio, ratio table, terms Using multiplication to find equivalent ratios. SAY: Just as you can multiply the numerator and the denominator of a fraction by the same number to get an equivalent fraction, you can multiply both parts of a ratio by the same number to get an equivalent ratio. For eample, I can multiply both parts of a ratio 2 : 3 by 4 and get the equivalent ratio 8 : 12. Write both ratios with an equal sign between them on the board. Eplain that the parts of the ratio are called terms. SAY: The first ratio has terms 2 and 3, and when we multiply both terms by 4, we get a ratio with terms 8 and 12. Eercises: 1. Multiply both terms by 4 to make an equivalent ratio. a) 2 : 5 b) 1 : 3 c) 4 : 7 d) 11 : 9 Bonus: 1,500 : 8 Answers: a) 8 : 20; b) 4 : 12; c) 16 : 28; d) 44 : 36; Bonus: 6,000 : 32 SAY: I would like to check that the ratios are, indeed, equivalent. ASK: How did we find equivalent ratios in the previous lesson? (wrote a sequence of equivalent ratios using skip counting) SAY: Let s check that 2 : 5 and 8 : 20 are equivalent ratios. Have students find the sequence of equivalent ratios using skip counting individually, then have a volunteer write the sequence on the board, as shown below: 2 : 5 = 4 : 10 = 6 : 15 = 8 : 20 Repeat with other ratios from the previous eercises. ASK: Which method is more efficient, multiplying or skip counting? (multiplying) F-10 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

10 Eercises: 1. What number is each term being multiplied by to produce the second ratio? a) 3 : 7 b) 2 : 5 30 : : 25 c) 9 : 4 Bonus: 132 : : : 630 Answers: a) 10, b) 5, c) 8, Bonus: 3 2. Multiply both terms by the same number to produce an equivalent ratio. a) 5 : 7 b) 8 : : : c) 9 : 2 Bonus: 567 : : : Answers: a) 10 : 14; b) 40 : 15; c) 63 : 14; Bonus: 5,670 : 9, What number is the first term multiplied by? Multiply the second term by the same number to produce an equivalent ratio. a) 5 : 11 b) 9 : 7 30 : 45 : c) 8 : 7 Bonus: 240 : : 960 : Answers: a) 6, 66; b) 5, 35; c) 8, 56; Bonus: 4, 1, Multiply the first term by the same number the second term was multiplied by. a) 6 : 11 b) 9 : 8 : 77 : 72 Answers: a) 7, 42; b) 9, 81 Introduce ratio tables. Write on the board: Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-11

11 Eplain that you made this table by multiplying both numbers in the first row by the same number. To fill in the second row, you multiplied both 3 and 2 by 2, so you produced an equivalent ratio. To fill in the third row, you multiplied 3 and 2 by 3, and produced another equivalent ratio. Have students tell you what numbers you need to write in the fourth row. (12, 8) SAY: This table is called a ratio table. Emphasize that, in a ratio table, we get all the numbers in all the rows from the first row, not from the row immediately above, by multiplying both terms in the first row by the same number. SAY: All rows in a ratio table make equivalent ratios. Eercises: Complete a ratio table for the ratio. Multiply the first row by 2, and then by 3. a) 4 : 1 b) 3 : 4 c) 5 : 7 d) 9 : Answers: a) 8 : 2, 12 : 3; b) 3 : 4, 6 : 8, 9 : 12; c) 5 : 7, 10 : 14, 15 : 21; d) 9 : 2, 18 : 4, 27 : 6 Point out that the ratio tables that students produced in the previous eercises are very similar to the lists of equivalent ratios they produced in the previous lesson. The difference is in the way they produced them (by multiplying, instead of by skip counting) and wrote them, not in the ratios themselves. Draw on the board: SAY: In this ratio table, some numbers are missing. But we know that both ratios in the second and third row are equivalent to the ratio in the first row, 4 : 11. ASK: What number do we need to multiply by to get from 4 to 8? (2) SAY: This means the second row was made by multiplying both numbers in the first row by 2. ASK: What number should be in the empty cell of the second row? (22) How do you know? (because 2 11 = 22) Repeat with the third row. (the first row was multiplied by 3 to give 33, so the missing number in the third row is 12, because 4 3 = 12) Eercises: Find the missing numbers in the ratio table. 9 a) 7 1 b) 6 13 c) d) Answers: a) 14, 3; b) 26, 18; c) 9, 21/2; d) 1.2, F-12 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

12 Ratio tables with two rows. Eplain that multiplication is a more efficient way to produce a ratio table than skip counting. Actually, in ratio tables, the rows do not always have to be multiplied first by 2, then by 3, and so on. There might be only two rows, and the number you need to multiply the first row by to get the second row can be any number. Draw on the board: ASK: What number do you need to multiply 4 by to get 20? (5) How do you know? (20 4 = 5) SAY: If this is a ratio table, then the ratios in each row must be equivalent, so we can multiply the other number in the first row by 5 to get the missing number and create a second, equivalent ratio. ASK: What is 7, the first number in the first row, multiplied by 5? What number should be in the empty cell? (35) Eercises: What number is the first row being multiplied by to make the second row? Find the missing number. a) 7 3 b) 8 11 c) Bonus: ,000 Answers: a) 7, 49; b) 9, 72; c) 9, 90.9; Bonus: 40, 200 NOTE: Students who struggle with the eercises above will benefit from adding arrows to the sides of the table and writing the number they use to multiply, as shown below: Using ratio tables to solve word problems. Remind students that, in the previous lesson, they solved problems by finding an equivalent ratio, and that they used a sequence of ratios to do that. Today, they learned a more efficient method to find equivalent ratios, so they can use ratio tables to solve the problems similar to those in the previous lesson. Write on the board: 3 T-shirts cost $10. How much do 15 T-shirts cost? SAY: In this problem, we have two changing quantities, the number of T-shirts and the price. Draw on the board: Number of T-shirts Price ($) ASK: What is the first ratio that I know from the problem? (3 T-shirts for every 10 dollars) SAY: We write this ratio in the first row. Write 3 and 10 in the first row of the table. SAY: Then we need to write the other number we are given, 15, somewhere in the table. ASK: Which column should the number 15 go in? (Number of T-shirts) Why not in the Price column? Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-13

13 (15 is the number of T-shirts; we need to find the cost of those 15 T-shirts) Write 15 in the second row, as shown below: Number of Price ($) T-shirts ASK: What number do we need to multiply the first row by to make this a ratio table? (5) PROMPT: What number do we need to multiply 3 by to get 15? ASK: So, what is the missing number? (50) Write 50 in the empty cell. SAY: This means that 15 T-shirts will cost $50. Repeat with the following problem: 3 T-shirts cost $10. How many T-shirts can I buy for $70? (21) Eercises: (MP.6) 1. There are 5 plums for every 2 apples. Write the quantity you know in the correct column of the ratio table. Then find the quantity you don t know. a) There are 10 plums. How many apples are there? b) There are 10 apples. How many plums are there? Plums Apples Plums Apples c) There are 20 plums. How many apples are there? Plums Apples 5 2 d) There are 50 apples. How many plums are there? Plums Apples 5 2 Answers: a) 10 : 4, b) 25 : 10, c) 20 : 8, d) 125 : Solve the problem by making a ratio table. a) Roy earns $350 every week. How much money will he earn in 4 weeks? b) Kathy saves $25 every 2 weeks. How long will it take her to save $150? c) A motorcycle can travel 100 km on 3 L of gas. Its gas tank can hold 18 L. How far can the motorcycle travel on a full tank of gas? d) A recipe calls for 3 lb meat for 4 portions. A restaurant cook has 75 lb of meat. How many portions can the cook make? e) A recipe calls for 0.5 lb of carrots for 4 portions of salad. How many pounds of carrots do you need for 12 portions of salad? f) Marco saves $12.50 every 2 weeks. How much will he save in 6 weeks? Bonus: Emma reads 7 pages in 10 minutes. Her book is 210 pages long. How many hours will it take her to finish the book? Answers: a) $1,400; b) 12 weeks; c) 600 km; d) 100 portions; e) 1.5 lb; f) $37.50; Bonus: 300 minutes = 5 hours F-14 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

14 Etensions (MP.3) 1. How can you tell immediately, without doing any calculations, that the table below is not a ratio table? ,225 24,525 Answer: 3 < 5, so to make an equivalent ratio, the first number in the second row should be smaller than the second number in the second row. 36,225 > 24,525, so the table cannot be a ratio table. (MP.3) 2. Tessa and Sam are siblings. Tessa is 2 years old and Sam is 6 years old. Sam says that he is three times as old as Tessa, so when Tessa is 5, he will be 15. Is that correct? Eplain. Answer: No, Sam will always be 4 years older than Tessa. So when Tessa is 5, Sam will be 9. (MP.1) 3. The ratio between two numbers is 1 : 6. The sum is more than 50 but less than 60. What are the numbers? Answer: 8 and 48 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-15

15 EE8-41 Graphing Ratios Pages Standards: 8.EE.B.5 Goals: Students will review coordinate grids. Students will graph ratios by converting a ratio table into ordered pairs and plotting the ordered pairs. Prior Knowledge Required: Can interpret fractions Can name fractions equivalent to a given fraction Can create a list of equivalent ratios Vocabulary: aes, ais, colon (:), coordinates, equivalent ratios, first coordinate, ordered pair, origin, plot, ratio, ratio table, second coordinate, skip counting, -ais, -coordinate, y-ais, y-coordinate Materials: BLM Small Coordinate Grids (p. I-2), 2 to 3 copies per student Review coordinates in the first quadrant. Draw the picture below on the board, but without the arrows and labels for the aes and the origin. Point to the aes and tell students that aes is the plural of ais. SAY: The horizontal line is called the -ais and the vertical line is called the y-ais. Tell students that the point at which the two aes intersect is called the origin. Label the aes and the origin, as shown below: 4 3 y-ais 2 1 y origin (0, 0) ais Mark point (4, 3) on the grid and use it to eplain how to describe the location of a point on the grid: From the point (4, 3), trace with your finger down to the -ais and look at the number that is directly below the point. SAY: This point has = 4. Go back to the point and trace with your finger left to the y-ais and look at the number on the y-ais directly to the left of the point. SAY: This point has y = 3. Repeat with the point (3, 4), where the order of the coordinates is reversed. Emphasize that the numbers for the two points are the same, but the points are different. In other words, point out F-16 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

16 that when you write the numbers in a different order, you get different points. SAY: The and y numbers for a point identify its unique location. They are like an address or locator for the point, and they are called the coordinates of the point. The number is called the -coordinate, and the y number is called the y-coordinate. NOTE: Before students look at the conventions for writing an ordered pair, look at more points and review how to locate them, as follows. To practice identifying the - and y-coordinates of various points on the grid, have students copy the grid from the eample on the board. (To save time, students can also use a grid from BLM Small Coordinate Grids.) Have students mark and label any four points they choose on the grid intersections, so that no two points are on the same horizontal or vertical grid line, and all points are to the right and above the aes. Have them label the points and then identify the - and y-coordinates for each point. SAY: Imagine you have to write the coordinates of 100 points. ASK: Would you want to write = and y = 100 times? What could you do to shorten the notation in other words, how could you write it faster? Students might suggest making a chart or writing pairs of numbers with the numbers first and y numbers second, which follows alphabetical order. Eplain that mathematicians have a convention, which is to epress the place of a point with two numbers in brackets and a comma in between. The is always the first number, on the left, and the y is always the second, on the right: (, y). Point out to students that they already know that the order is important and that (4, 3) and (3, 4) are different points. To emphasize that the order is important, such a pair of numbers in brackets is called an ordered pair. NOTE: Technically, the term brackets refers to square brackets, and coordinates are written in parentheses. We will refer to brackets instead of parentheses, as the the latter word might present a stumbling block for students, especially English language learners. Have students rewrite the coordinates of the four points on their grids using conventional notation. Eplain that the -coordinate is often called the first coordinate, because it is written first, and the y-coordinate is therefore called the second coordinate. Students can a grid from BLM Small Coordinate Grids to do part b) of the eercises below. Eercises: a) Write the coordinates of each point. y 6 A 5 B 4 3 C 2 D b) Mark the following points on a coordinate grid: E (1, 3), F (5, 6), G (2, 7), H (7, 4) Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-17

17 Answers: a) A (2, 5), B (4, 4), C (6, 2), D (3, 1) b) y 6 G F 5 H 4 E Coordinate grids can include fractions, decimals, and negative numbers. Eplain that the lines on the coordinate grid are drawn parallel to the aes, but the distances between them are usually equal to a whole number. On the grids students have seen so far in this lesson, the lines were one unit apart. SAY: However, we can use skip counting to label the aes for eample, when we need to show points with very large coordinates. Use the grid from the previous eercises to create the grid for the following eercises. Change the -ais to skip count by 2s and the y-ais to skip count by 5s. Tell students that certain points will need to be placed between the grid lines. Plot the point (1, 1) on the new grid to demonstrate how to do it. Point out that the -ais uses skip counting by 2s, so the point with = 1 is eactly halfway between 0 and 2. The y-ais skip counts by 5s, so the point with y = 1 should be one fifth of the way from 0 to 5. Eplain that when you mark a point given by its coordinates on a coordinate grid, you plot the point. When you have a list of points, it helps to cross out each ordered pair as you plot them to make sure you do not miss any. Eercises: a) Write the coordinates of each point. y A D B C b) Plot the following points on a coordinate grid. Cross out the points as you plot them. E (2, 15), F (3, 25), G (6, 7), H (11, 24) Answers: a) A (4, 25), B (8, 20), C (12, 10), D (6, 5) b) y F H E 10 G F-18 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

18 Remind students that fractions, decimals, and negative numbers can also appear on number lines. SAY: This is true for coordinate grids as well; a coordinate grid can show numbers that are fractions (for eample, 1/2), decimals (for eample, 2.5), and even negative values. For negative values, we have to make sure that the area of the grid and the aes allow for the point to be plotted. For eample, point J (1.5, 1) will be halfway between lines for = 1 and = 2, because it has = 1.5, and on the line for y = 1. Draw on the board: y J Before students continue to plot points, look at the grids on BLM Small Coordinate Grids to see how the grids vary and discuss which would be good to use in specific situations, such as with negative numbers or fractions. Have students use the second grid on BLM Small Coordinate Grids for the following eercises. 1 1 Eercises: Plot the points on a coordinate grid: K (2, 3), L (3, 1 ), M (4.5, 2), N ( 5, 2.5) 2 2 Answers: y N L M K Review creating ratio tables. Remind students that to create a ratio table, they must start with a given ratio in the first row, then produce any other row by multiplying both numbers in the first row by the same number. SAY: Usually, we multiply by numbers in order: 2, 3, 4. As an eample, draw the ratio table for 1 : 3 on the board, as shown below: Leave the ratio table on the board for later use. Eercises: Make a ratio table for the ratio. a) 1 : 2 b) 4 : 5 c) 5 : 4 d) 7 : 10 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-19

19 Answers: a) 1 2 b) 4 5 c) 5 4 d) Order matters in ratios. ASK: Is 1 : 3 the same ratio as 3 : 1? (no) Have students eplain how they know. Share the following illustration if something similar does not arise during the discussion: If you mi 3 cups of red paint with 1 cup of yellow paint, the color you get will be a deep orange that is close to red. If you mi 1 cup of red paint with 3 cups of yellow paint, the resulting color will be a light orange that is close to yellow. So, red : yellow = 3 : 1 is very different from red : yellow = 1 : 3. Converting ratios to ordered pairs. ASK: What other mathematical object did we just learn about that has two numbers and where the order matters? (ordered pairs, coordinates of a point) Point out that we can create an ordered pair from each ratio, then draw a point on a coordinate grid using the numbers in the ordered pair as the coordinates. Add a column to the 1 : 3 ratio table on the board, and write the ordered pairs in the new column: (1, 3), (2, 6), (3, 9), and (4, 12). Eercises: Add a column to each ratio table from the previous eercises. In the new column, write the ordered pair for each row. Answers: a) (1, 2), (2, 4), (3, 6), (4, 8); b) (4, 5), (8, 10), (12, 15), (16, 20); c) (5, 4), (10, 8), (15, 12), (20, 16); d) (7, 10), (14, 20), (21, 30), (28, 40) Plotting ratio tables on coordinate planes. Eplain that, to plot a ratio on a coordinate grid, you need to make a ratio table first, make a list of ordered pairs from that ratio table, and, finally, plot the ordered pairs on the grid. Students can use the grids from BLM Small Coordinate Grids for the eercises below. Point out that, for the last two tables, they will need to use the largest grids on the BLM. Eercises: Plot the ordered pairs from parts a), b), and c), of the previous eercises on a grid. Answers: a) y b) y c) y F-20 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

20 (MP.7) Equivalent ratios produce points on the same line through the origin. Ask students to join, or connect, the points that come from the same ratio table on each graph. ASK: What do you notice? (the points always fall onto the same line) Have them etend the lines so that they intersect the aes. ASK: What happens to the lines? (they all pass through the point (0, 0)) What special name does this point have? (the origin) Summarize the facts just discovered: points produced by ratio tables fall on the same line and that line passes through the origin. Eplain that these facts can be used to check whether a ratio table has been plotted correctly. Students can check if the line passes through the origin when continued by placing a ruler alongside it. Students can use another copy of BLM Small Coordinate Grids to do the eercises below. To save time, they can produce and plot just three points for each ratio. Eercises: Create a graph that shows the ratio. Are the points on the same line? If so, does the line pass though the origin? a) 2 : 3 b) 3 : 5 c) 2 : 1 Answers: a) points (2, 3), (4, 6), and (6, 9) should be connected with a line b) points (3, 5), (6, 10), and (9, 15) should be connected with a line c) points (2, 1), (4, 2), and (6, 3) should be connected with a line All lines should pass through the origin. Graphing ratios to solve problems. Eplain that graphing a ratio can help solve certain problems. SAY: For eample, if a train travels at the speed of 70 miles per hour, this means that it travels 70 miles every hour. We can make a ratio table for this train. Write on the board: Time (h) Distance (mi) 1 70 SAY: I know that the train travels 70 miles in 1 hour. ASK: How many miles will it travel in 2 hours? (140) How do you know? (2 70 = 140) Fill in the second row. ASK: How many miles will the train travel in 3 hours? (3 70 = 210) Fill in the third row. Have students copy the table into their notebooks. Add a third column to the table for ordered pairs and have students fill in the third column. The finished table should look like this: Time (h) Distance (mi) Ordered Pair 1 70 (1, 70) (2, 140) (3, 210) Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-21

21 Draw aes on the grid on the board and eplain that you have to use skip counting to mark the vertical ais, because the third ordered pair in your table has the second coordinate y = 210. Eplain that you will skip count by 35s to label the ais, because 35 divides 70 and skip counting like this allows you to make a graph that does not require too much vertical space. Fill in the aes, as shown in the completed graph below. Eplain that the ratio we were given, 70 miles each hour, gives us more information than just numbers. SAY: We know that the numbers on the horizontal ais are the number of hours, or the time, that the train travels. Label the horizontal ais, as shown in the completed graph below. ASK: What does the vertical ais show? (the number of miles the train travels, the distance in miles) Label the vertical ais, as shown in the completed graph below. Invite volunteers to plot the points from the table on the graph and join the points with a line. The completed graph should look like this: Distance (mi) y Time (h) ASK: If I etend the line toward the aes, will it pass through the origin? (yes) Have a volunteer check by etending the line. SAY: The graph tells us more than the ratio table does. The graph has a line that joins the points, and any point on that line has the same ratio. Draw a point on the line at (1.5, 105) and ASK: What are the coordinates of this point? ((1.5, 105)) What is the meaning of this point? (the train travels 105 miles in 1.5 hours) How do you know? (the -coordinate is the time in hours and = 1.5, so the train travels for 1.5 hours; the y-coordinate shows the distance in miles and y = 105, so the train travels 105 miles) PROMPTS: What does the -coordinate show? What does the y-coordinate show? ASK: Which point on the graph shows how far the train travels in half an hour? (point (0.5, 35)) Have a volunteer draw the point on the graph and identify the coordinates. ASK: How far does the train travel in half an hour? (35 miles) Which point on the graph shows how long it takes the train to travel 175 miles? (point (2.5, 175)) Again, have a volunteer identify the point on the graph and its coordinates. ASK: How long does it take the train to travel 175 miles? (2.5 hours) Keep the graph on the board for later use. Eercises: Satellite A is moving through space, around Earth. It travels 25,000 km every hour. a) Make a ratio table for the satellite s movement. b) Graph the ratio. Skip count by 12,500 km on the vertical ais. c) Use the graph to tell how far the satellite moves in 2.5 hours. d) Use the graph to tell how long it takes the satellite to travel 37,500 km. F-22 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

22 Answers: a) Time (h) Distance (km) 1 25, , ,000 c) 62,500 km d) 1.5 hours b) Distance (km) 75,000 62,500 50,000 37,500 25,000 12,500 y Time (h) Use graphs to compare ratios for everyday situations. Return to the graph showing the distance a train travels. Label the graph itself (i.e., the line and the points) Train A. SAY: Another train, Train B, travels 105 miles in 2 hours. I want to know which train travels faster. I can plot the ratio for Train B on the same graph. Have students make a ratio table for Train B. Point out that since the graph only shows a distance up to 210 miles, students do not need to create too many rows in the table. When they reach 210 or a number larger than that, they should stop. ASK: How many rows does your table have? (2) Have a volunteer write the ratio table on the board. Have students write the ordered pairs in their notebooks, then have another volunteer plot the ratio for Train B on the same graph as Train A. Have the volunteer use a different color to plot the second ratio. The ratio table and graph should look like this: Train B Time (h) Distance (mi) Distance (mi) y Train A Time (h) Train B ASK: How far does Train A travel in 2 hours? (140 mi) How far does Train B travel in 2 hours? (105 mi) Which train travels farther in 2 hours, Train A or B? (Train A) How can we see that on the graph? (the point that shows the train s position at 2 hours is higher on the grid for Train A than for Train B) SAY: The points closest to the top on the graph are on the same level. ASK: What does this mean? (the trains both travelled 210 mi) How long does it take Train A to travel 210 mi? (3 hours) How long does it take Train B to travel 210 mi? (4 hours) Which train travels faster? (Train A) (MP.4) Eercises: Satellite B travels 15,000 km in half an hour. a) Make a ratio table to show how far Satellite B travels in 1 hour, 1.5 hours, and 2 hours. b) Graph the ratio using the table from part a). Use the same coordinate grid you used in the previous eercises, for Satellite A. c) Compare Satellite A with Satellite B. Which satellite travels farther in 1 hour and in 2 hours? d) Which travels faster, Satellite A or Satellite B? Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-23

23 Answers: a) Time (h) Distance (km) , , , ,000 c) Satellite B d) Satellite B b) Distance (km) 75,000 62,500 50,000 37,500 25,000 12,500 y Satellite B Time (h) Satellite A Ask the class to compare the lines for the slower and the faster objects in the previous eercises and eamples (satellites, trains) on the grids. ASK: How can you see which object goes faster from the graph? (the line for the object that goes faster is steeper) Etensions 1. Randi runs 2 km every 15 minutes. a) Draw a graph to show Randi s running. Skip count by 15s on the -ais and count by 1s on the y-ais. b) At that same pace, how far will Randi run in 30 minutes? Which point on the graph shows the answer? c) How far will she run in 1 hour? Which point on the graph shows the answer? Answers: a) y Time (min) b) Randi will run 4 km in 30 minutes, (30, 4) c) Randi will run 8 km in 1 hour, (60, 8) Distance (km) 2. Ethan earns $17 every 2 hours. a) Draw a graph to show Ethan s earnings. Show intervals of half an hour on the -ais. Skip count by $4.25 on the y-ais. b) How much will Ethan earn in 30 minutes? How does the graph show that? c) How much will he earn in 3 hours? How does the graph show that? d) How long will it take him to earn $42.50? How does the graph show that? F-24 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations

24 Answers: a) y Time (h) b) Ethan earns $4.25 in 30 minutes; the line passes through point (1/2, 4.25). c) Ethan earns $25.50 in 3 hours, because the line passes through point (3, 25.50). d) It will take 5 hours to earn $42.50, because the line passes through point (5, 42.50). Earnings ($) (MP.1) 3. Tony and Lynn are saving money for a gift for their grandfather. They will not share the cost equally. a) Tony saves $15 every 2 weeks. Make a graph to show his savings. b) Lynn saves $10 every week. Graph her savings on the same grid you used in part a). c) How much money will they have together after 2 weeks? After 4 weeks? Show their combined savings on the same grid. d) The gift costs $ Draw a horizontal line showing the price. e) When will Tony and Lynn have enough money together to pay for the gift? f) How much money will Tony pay? How much money will Lynn pay? Answers: parts a), b), c), d): y Savings ($) Time (weeks) Total savings Gift price Lynn s savings Tony s savings c) Tony and Lynn will save $35 after 2 weeks and $70 after 4 weeks e) Tony and Lynn will have enough money after 5 weeks f) Tony will pay $37.50 and Lynn will pay $50 Teacher s Guide for AP Book 8.1 Unit 5 Epressions and Equations F-25