Resource Overview Quantile® Measure:
630Q
Skill or Concept:
Construct or complete a table of values to solve problems associated with a given relationship. (QT‐A‐180)
Excerpted from:
Gourmet Learning 1937 IH 35 North Suite 105 New Braunfels, TX 78130 www.gourmetlearning.com © Gourmet Learning
This resource may be available in other Quantile utilities. For full access to these free utilities, visit www.quantiles.com/tools.aspx. The Quantile® Framework for Mathematics, developed by educational measurement and research organization MetaMetrics®, comprises more than 500 skills and concepts (called QTaxons) taught from kindergarten through high school. The Quantile Framework depicts the developmental nature of mathematics and the connections between mathematics content across the strands. By matching a student’s Quantile measure with the Quantile measure of a mathematical skill or concept, you can determine if the student is ready to learn that skill, needs to learn supporting concepts first, or has already learned it. For more information and to use free Quantile utilities, visit www.Quantiles.com.
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3rd Grade
Algebraic Thinking
Student Expectation: Students will see how tables help organize lists of information
Unit 2 – Lesson 1 The student uses lists, tables, and charts to express patterns and relationships. The student is expected to generate a table of paired numbers based on a real-life situation such as insects and legs.
Study the TEKS . . .
Prior Knowledge In 2nd grade, the students generate and uses lists of a single set of data. There was no mention of tabular data.
Next Steps
3rd
In 4th grade, the students will fully analyze and describe the relationship between two sets of data in a table.
Grade In third grade . . . This year is more of an “organizational” year. The students have down the concept of generating paired numbers, but in the past, they have only applied the skill using lists. This year, they will take the data from their lists and place it in a table. Then they will begin to analyze patterns—both vertically (down the columns) and between the two sets of data (across the rows). This will be reinforced in 4th grade. Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will see how tables help organize lists of information
Focus Activity Generate Tables of Real-Life Situations
C
An
Teacher note: In this Focus Activity, students will begin to compare lists of data from four different types of cycles and, in doing so, determine the need for an organizational tool (i.e., a table) to collect the data. Group size: whole class divided into 4 groups Materials: cycle cards, page 3; 4 large pieces of butcher paper; markers; tape; scissors Before class: Make a copy of page 3; laminate it, and cut out the four pictures of the cycles. Directions: • Divide the class into 4 groups. Distribute 1 piece of butcher paper, markers, and scissors to each group. • Have the shortest person from each group draw a card with a cycle on it. • Each group will cut a large piece of butcher paper into 8 parts. (These do not have to be uniform. Allow the students to show their creativity.) • On one of the pieces of butcher paper, students will show with pictures and numbers how many wheels there would be if they had one of the cycles pictured on their group’s card. • On the next piece of butcher paper, students will show how many wheels there would be if they had two of the cycles. Students will continue this process until they have shown 8 cycles and the corresponding number of wheels on the last piece of paper. • After all groups are done, the tallest person will tape his/her group’s 8 pieces of butcher paper in a vertical line, to the front board.
Questioning Technique Instructional Strategy Say: Now, I need one volunteer (raise your hands!) from each group to come up to the front of the room to share your group’s work. After the 4 volunteers have shared,— Ask: What do these pages have in common? (Answers will vary, but they should indicate that the more cycles there are, the more wheels there are.) Ask: How do these pages differ? (Answers will vary, but they should indicate that the number of wheels is different on each cycle. So, the numbers increase at different rates.) Ask: Evaluate how easy or difficult it was to compare the data on these 4 pages. (Engage the students in a discussion that it was not easy because there was no organization.) Say: Tables help us organize the data that we collect to make it easier to compare the numbers. Watch as I turn your lists into tables. Create a table of the number of cycles and number of wheels for each type of cycle. Ask: What makes looking at the tables easier than the first sheets you completed? (Answers will vary, but it aligns the numbers and organizes the data.)
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will see how tables help organize lists of information
Focus Activity—Cycle Cards Generate Tables of Real-Life Situations
Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will learn the vocabulary related to tabular data
Initial Instruction—Part I—Vocabulary
K
Generate Tables of Real-Life Situations
Definitions: table: a set of data arranged in rows and columns Examples:
Number of Hands Number of Fingers 1
5
2
10
3
15
4
20
5
25
6
30
Number of Hands
1
2
3
4
5
6
Number of Fingers
5
10
15
20
25
30
tabular: arranged or displayed in a table form paired numbers: a set of two numbers that have a relationship to each other—These can be displayed in the table. (For example, in the table above, the numbers 1 and 5 are paired numbers, as are 2 and 10, 3 and 15, and so on.) ordered pair: a pair of numbers in which order is important, often used to indicate a point on a coordinate plane, graph or map
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Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II Generate Tables of Real-Life Situations
K
C
Teacher note: In the Initial Instruction–Part II, students will learn to analyze given tables, looking for patterns that exist vertically in the table and horizontally between the pairs of numbers. A vertical and horizontal version of each table has been provided, so students understand and can analyze both forms. This is very important as both forms are used throughout the lesson and on the Practice and Application. Group size: pairs Materials: Instructional Strategy, pages 5-7; tables, transparency pages 8-11; overhead markers Before class: Copy tables 2-4, pages 9-11, for each pair; laminate all the tables. Directions: Lead students through the following discussion.
Questioning Technique Instructional Strategy Place transparency page 8 on the overhead. Cover the horizontal table. Ask: What does this table show? (the number of legs on 1 - 10 chairs) Say: Explain how you know that it refers to ten chairs. (The numbers in the first column are under the label “Number of Chairs.”) Ask: Is there a pattern in how these numbers are placed in this table? (Yes, the numbers are each one higher than the last.) (Show this on the overhead.) Ask: Is there a pattern in how the numbers are placed in the 2nd column on this table? (Yes, the numbers are each 4 higher than the last one.) Ask: How does this pattern relate to chairs and legs? (There are 4 legs per chair, so as the number of chairs increases by 1, the number of legs increases by 4.) Ask: How many legs do 4 chairs have? (16) Ask: How does the table show this information? (The first column is the number of chairs. Move your finger down to 4 chairs. The second column is the number of legs. Move your finger across from 4 chairs, and you will find 16 legs.) (Show this on the overhead.) Ask: How many chairs do we have if there are 32 legs? (8) Ask: How does the table show this information? (The second column is the number of legs. Move your finger down to 32 legs. The first column is the number of chairs. Move across from 32 legs, and you will find 8 chairs.) (Show this on the overhead.) Teacher note: The following two questions, while seemingly a repeat of previous questions, include key vocabulary for students to understand the role of multiplication in related data on a table. Ask: Is there a pattern relating the numbers in each row? (Yes, each of the numbers in the 2nd column is 4 times as much as the number in the first column.) Ask: What does that pattern have to do with the tables and legs? (There are 4 legs per chair, so we multiply the number of chairs we have by 4 to get the number of legs.) Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II Generate Tables of Real-Life Situations Questioning Technique Instructional Strategy Uncover the horizontal table. Make sure students know how to analyze the data in this format. They will now have to move their fingers across the first row and then down to the second row. Distribute table 2, page 9, and an overhead marker to each pair. Keep transparency page 8 on the overhead. Say: Compare table 2 to table 1. Using your overhead pen, place a star next to data that you see on table 2 that is the same as that on table 1. (Give the students some time to work.) Ask: What are some of the things that you marked? (the heading of each column: number of chairs, number of legs; “chair x 4 legs” relationship between the numbers in the first column and the numbers in the second column) Say: Compare table 2 to table 1. Using your overhead pen, place a check mark next to the things that you see on table 2 that are different from those on table 1. (Give students some time to work.) Ask: What are some of the things that you marked? (The numbers increase by 2 in the “chair” column and by 8 in the “legs” column.) Ask: Has the number of legs on each chair changed? (No.) Ask: How can you tell? (You still multiply the number of chairs by 4 to get the number of legs.) Ask: Then why are we adding 8 each time down the “legs” column? (Answers will vary, but get the students to see that since the number of chairs increases by 2 each time, there are 8 legs on 2 chairs.) Pick up table 2. Distribute table 3, page 10, to each pair. Place transparency page 10 on the overhead. Say: Look over this table. Using your overhead marker, determine the answers to the three questions. Write your answers on the back of the page. While students are working, circle the class. After most pairs are finished, bring the class back together, and answer the questions as a class. Answers to table 3: 1. This table refers to the relationship between a number of birds and the corresponding number of wings. The heading at the top of each column tells us what the table is referring to. In this case, the first column refers to the number of birds and the second column refers to the number of wings. 2. There are vertical patterns in each column. The number of birds increasesby one each time. The number of wings increases by 2 each time. There are horizontal patterns between the numbers. If you multiply the number of birds by 2, you get the number of wings. 3. Each of these patterns shows that each bird has 2 wings.
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II Generate Tables of Real-Life Situations Questioning Technique Instructional Strategy Pick up table 3. Distribute table 4, page 11, to each pair. Place transparency page 11 on the overhead. Using the same format as the last discussion, give students time to determine the answers to the three questions on the board working with their partners. Then bring them back together, and answer the questions as a class. Answers to table 4: 1. The table refers to the relationship between the number of octopuses and the corresponding number of tentacles. The heading at the top of each column tells us what the table is referring to. In this case, the first column refers to the number of octopuses, and the second column refers to the number of tentacles. 2. There are not vertical patterns. The number of octopuses increases by different numbers each time, and the number of tentacles increases by different numbers each time. There are horizontal patterns between the numbers. If you multiply the number of octopuses by 8, you get the number of tentacles. Teacher note: When discussing the fact there is not a vertical pattern, ask the students why. The answer is the person that is creating the tables decides what values go in each column. Ask if we could have created a table with vertical patterns. The answer is sure! We could have listed every number of octopuses or every other one. 3. The patterns shows that each octopus has 8 tentacles.
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II—Table 1 Generate Tables of Real-Life Situations
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Number of Chairs
Number of Legs
1
4
2
8
3
12
4
16
5
20
6
24
7
28
8
32
9
36
10
40
Number of Chairs
1
2
3
4
5
Number of Legs
4
8
12 16 20 24 28 32 36 40
Gourmet Curriculum Press, Inc.©
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7
8
9
10
Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II—Table 2 Generate Tables of Real-Life Situations
Number of Chairs
Number of Legs
2
8
4
16
6
24
8
32
10
40
12
48
14
56
16
64
18
72
20
80
Number of Chairs
2
4
6
8
10 12 14 16 18 20
Number of Legs
8
6
14 32 40 48 56 64 72 80
Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II—Table 3 Generate Tables of Real-Life Situations
Number of Birds
Number of Wings
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
Number of Birds
1
2
3
4
5
6
7
8
9
Number of Wings
2
4
6
8
10 12 14 16 18 20
1. What is this table referring to? How do you know? 2. What patterns do you see? 3. How are the patterns related to the items referred to in the table? 10 ( T )
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will discover patterns in given tables
Initial Instruction—Part II—Table 4 Generate Tables of Real-Life Situations
Number of Octopuses
Number of Tentacles
1
8
3
24
6
48
7
56
10
80
12
96
16
128
17
136
21
168
23
184
Number of Octopuses
1
3
6
7
10 12 16 17 21 23
Number of Tentacles
8
24 48 56 80 96 128 136 168 184
1. What is this table referring to? How do you know? 2. What patterns do you see? 3. How are the patterns related to the items referred to in the table? Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will generate tables of paired numbers based on reallife situations
Initial Instruction—Part III Generate Tables of Real-Life Situations
K
C
Teacher note: In the third part of the Initial Instruction, the students will take information given to them and make tables. Group size: individual, then pairs Materials: Instructional Strategy, pages 12-13; data information, transparency page 14; paper copies and transparency of blank table, page 15; overhead pens; tape; cardstock (optional); blank paper Before class: Copy the table, page 15, on the front and back of paper or cardstock and laminate for each student. Directions: • Place transparency page 14 on the overhead, and cover the second box. Distribute a page of tables and an overhead pen to each student. • Direct students through the following Instructional Strategy. complete a table during this guided instruction.
Each student will
• Then students will work in pairs to create another table. • Each pair will post its table on the board or wall. The tables should be spread out throughout the room. • Next, distribute a blank piece of paper to each pair. Students will fold the paper vertically (hot dog style). • Pairs will next conduct a “gallery walk” of the work to analyze all the tables. As they walk, pairs will use the paper to list similarities and differences in the tables. Similarities will be written on the left and differences on the right. • Hold a class discussion of the lists using the Instructional Strategy to guide conversation. • Repeat this process with the second box on page 14. • Change up the pairs and the second table that they make, so everyone has multiple chances to create different tables.
Questioning Technique Instructional Strategy Ask: Look at the information in the first box. What is this table going to represent? (the relationship between the number of calculators and the number of batteries) Ask: When we create the table, where will this information go? (at the top of the columns, as headings) Say: Place these headings on your table. (Write these on the overhead table, transparency page 15, while the students copy them on their first table.) Ask: What goes down the first column? (the number of calculators)
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Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will generate tables of paired numbers based on reallife situations
Initial Instruction—Part III Generate Tables of Real-Life Situations Questioning Technique Instructional Strategy Ask: What numbers will I put in here? (The students will probably say 1-10, and that is acceptable for the first table. Other options will be discussed at a later time. If, however, the students reply that any numbers can go there, direct them to start with the easiest table by writing 1-10 in this column.) Ask: What goes down the second column? (the number of batteries) Ask: How do I know what numbers to place in this column? (Answers will vary, but should include the fact there are 4 batteries in each calculator. So, count by 4 down the column, or multiply each of the numbers in column one by 4. Have students determine patterns in the table.) Say: List all the ordered pairs in this table. { (1,4), (2, 8), (3, 12), (4, 16), . . . (10, 40) } Ask: What does the ordered pair (6, 24) represent? (We will need 24 batteries for 6 calculators.) Ask: Which method is easier to use—a table or a list of ordered pairs? (The answers will vary; any reasonable explanation is acceptable.) Say: Pair up with your neighbor. Turn the table over, and create another table together using the same scenario. This time, do not number the first column 1 - 10. Go around the room and ask pairs to create different tables - increasing each number in column 1 by 2, 3, 5, 10, etc. Ask one or two groups to create one without a pattern in the first column. Ask: Which method is easier to use—a table or a list of ordered pairs? (The answers will vary; any reasonable explanation is acceptable.) (After pairs have posted their tables, distribute a piece of paper to each pair. Students will fold it vertically and record similarities and differences in the tables on it during their “gallery walk.”) Use the following questioning when pairs have finished analyzing the tables. Ask: How are these tables the same? (All have 10 entries. All show the relationship between calculators and batteries. In each case, if you take the number in the first column and multiply it by 4, you will get the number in the second column.) Ask: How are these tables different? (All of the first columns are different. Point out the differences, and have the students who created each table verify whether the notes are correct. Some don’t have vertical patterns; point out which ones don’t have vertical patterns.) Uncover the second box on page 14, and repeat this process. (The answers to this one revolve around the relationship that there are 12 eggs in a carton. The ordered pairs in the table will be: { (1, 12), (2, 24), (3, 36), (4, 48), (5, 60), (6, 72), (7, 84), (8, 96), (9, 108), (10, 120) }. Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will generate tables of paired numbers based on reallife situations
Initial Instruction—Part III—Data Information Generate Tables of Real-Life Situations
Each calculator that the teacher bought needs 4 batteries to make it work.
Each carton of eggs contains a dozen eggs.
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will generate tables of paired numbers based on reallife situations
Initial Instruction—Part III—Blank Table Generate Tables of Real-Life Situations
Gourmet Curriculum Press, Inc.©
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will use a recipe to make connections to tabular data
Initial Instruction—Part IV
C
Generate Tables of Real-Life Situations Optional Reading Activity
Ap
Teacher note: Students love to eat and love to make things. The 2 come together in this activity in which students take simple recipes from the book Salad People and More Real Recipes by Mollie Katzen and create “salad people” and tables of related data. Group size: no more than one group for each ingredient listed in the recipe Materials: copy of the book Salad People and More Real Recipes; recipe outline, transparency page 18; ingredients for each student to make a salad person (See page 17 in the book for list.); overhead maker; larger pieces of butcher paper; markers; rulers; sturdy paper plates Before class: Shop for the ingredients. Create a recipe based on the one on page 17 of the book. (See note below for guidance.) Record the recipe on the outline, transparency page 18. Teacher note: The recipe in the book is very generic. In order for students to see relationships in data, you will have to choose how many of each ingredient you want the students to use for the hair, arms, legs, buttons, etc., of their salad people. Read the recipe and decide on the numbers first, and then record the recipe on the outline. Suggestions: 1 pear; 1 marshmallow or 1 scoop of yogurt (for the head); 2 cheese sticks (for the arms); 2 celery sticks (for the legs); 10 Chinese noodles for the hair (instead of cooked pasta); 3 raisins (for the eyes and nose); 5 dried cranberries (one for mouth, 2 for hands, 2 for feet) Directions: • Ask students to explain what a recipe is and how to use one. Invite students to share their favorite recipes with the class from memory – in general, what does the recipe help make? (An extension of this project would be for each student to bring a favorite recipe, type it, and then create a book of all the recipes to send home for a Mother’s Day gift.) • From the book, read “The Critics Rave” reviews on page 16 and the original recipe on page 17 to the students. Then show them the pictures on pages 18-19. • Place your recipe, transparency page 18, on the overhead, and explain that this is what each student will follow to create a basic salad person and to find relationships in data. • Assign each group of students one of the ingredients. For example, Group #1 is assigned celery. If you are using celery for the legs, there are 2 pieces of celery in your recipe for one salad person. One group will be responsible for this relationship. • Hand out a large piece of butcher paper to each group. Direct students to create a table on the paper showing the number of students in the class (from 1 student to the total number of students) and the number of items of the assigned ingredient the students will need to make salad people. For example, in the case of 2 celery sticks with a class of 12, the table will look like the following (the table may be horizontal or vertical). Stu Cel
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1 2
2 4
3 6
4 8
5 10
6 12
7 14
8 16
Gourmet Curriculum Press, Inc.©
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10 20
11 22
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will use a recipe to make connections to tabular data
Initial Instruction—Part IV Generate Tables of Real-Life Situations Optional Reading Activity Directions (continued): • While groups are creating their tables, place the recipe ingredients on paper plates, and arrange the plates on a desk or table for student access. • When groups are finished, tape the tables around the classroom. At this point, students should wash their hands and proper food handling should be discussed. • Then one student from each group will go to the table with the ingredients on it and collect JUST ENOUGH items on a paper plate of the group’s assigned ingredient for the class and return to his/her group. In our example, a student would collect 24 pieces of celery. • Next, give each student a sturdy paper plate, and have students walk from one group’s desk to the next picking up the number of items stated on your recipe to create a salad person. • As the students sit back down to create a salad person, ask whether everyone has enough items. If not, go to the desk with the missing items to determine if there was a mistake in calculations. (It might have been a mistake in counting by the student collecting the items, or by the group gatherer, or a mistake in table data.) • After everyone has created an identical salad person, allow for creativity and individualizing by having students add other items to their salad people suggested in the original recipe or additional items you think might be fun! • Ask the students to share how many of each ingredient they added. • Take digital photos of all the salad people to share on a bulletin board or with the students’ parents before students name and eat their creations. Teacher note: Be sure none of the students have allergies to the items you choose for the salad people. Substitute other items as necessary. Other recipes in the book could be used instead of the salad people or for similar activities at other times during the year.
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will use a recipe to make connections to tabular data
Initial Instruction—Part IV—Recipe Outline Generate Tables of Real-Life Situations Optional Reading Activity
My Recipe for: Salad People Ingredients:
Directions:
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Gourmet Curriculum Press, Inc.©
Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will practice making tables
Initial Instruction—Guided Practice Generate Tables of Real-Life Situations
K
C
Teacher note: Guided Practices are designed to allow the students to practice the skills they have just learned in the Initial Instruction. Although students work the problems individually, you will bring them back together before ending this activity to have a full class discussion regarding the answers and strategies they used. Group size: individual Materials: transparencies, pages 19-20; answer key, page 47 Before class: Gather materials. Directions: • Place pages 18-19, one at a time, on the overhead. • Allow students time to work the problems individually. • Bring the class together at the end for a discussion.
1. A ladder has 7 rungs. Complete the table to show how many rungs are on each of the following numbers of ladders. Number of Ladders
Number of Rungs
1
7
3 5
35
7 9
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Unit 2 – Lesson 1
Algebraic Thinking
Student Expectation: Students will practice making tables
Initial Instruction—Guided Practice Generate Tables of Real-Life Situations
2. Create a table of values that shows the total number of gallons of gas needed to fill up 1-6 cars if each car uses 15 gallons.
3. Create a second table representing the same relationship, but use different numbers of cars.
4. What relationship could this table represent? Write in the headings. 1
2
3
4
5
6
12
18
24
30
5. Extend the table in problem #4 to 10 items in the first row.
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