Patterns & Functions Final Project – Summer 2008 Grade 6 Unit Aaron Wall Warren-Alvarado-Oslo School [email protected] Pam Bagaason Clearbrook-Gonvick School [email protected] Stewart Wilson Walker-Hackensack-Akeley School [email protected] 1 Executive Summary The 15-day unit that we have prepared is a Grade 6 Algebra unit. It addresses the following MN State Standards: Number and Operation: Convert between equivalent representations of positive rational numbers. 6.1.1.7 For example: Express 10 7 as 77+ 3 = 77 + 73 = 1 73 . Algebra: Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts. Recognize and represent relationships 6.2.1.1 between varying quantities; translate For example: If a student earns $7 an hour in a job, the amount of money from one earned can be represented by a variable and is related to the number of hours representation to worked, which also can be represented by a variable. another; use patterns, tables, graphs and Represent the relationship between two varying quantities with function rules to solve realrules, graphs and tables; translate between any two of these representations. world and mathematical 6.2.1.2 problems. For example: Describe the terms in the sequence of perfect squares t = 1, 4, 9, 16, ... by using the rule 2 t = n2 for n = 1, 2, 3, 4, .... Use properties of arithmetic to generate equivalent numerical expressions and evaluate expressions involving positive rational numbers. Apply the associative, commutative and distributive properties and order of operations to generate equivalent expressions and to solve problems involving positive rational numbers. 6.2.2.1 For example: 32 × 5 = 32×5 = 2×16×5 = 16 × 2 × 5 = 16 15 6 15×6 3×5×3×2 9 2 5 9 . Another example: Use the distributive law to write: ( ) 1 + 1 9 − 15 = 1 + 1 × 9 − 1 × 15 = 1 + 3 − 5 = 2 − 5 = 1 3 2 3 2 8 2 3 2 3 8 2 2 8 8 8 . Represent real-world or mathematical situations using equations and Understand and inequalities involving variables and positive rational numbers. interpret equations and inequalities involving 6.2.3.1 variables and positive rational numbers. Use For example: The number of miles m in a k kilometer race is represented by equations and the equation m = 0.62 k. inequalities to Solve equations involving positive rational numbers using number sense, represent real-world properties of arithmetic and the idea of maintaining equality on both sides of and mathematical the equation. Interpret a solution in the original context and assess the problems; use the idea reasonableness of results. of maintaining 6.2.3.2 equality to solve equations. Interpret solutions in the For example: A cellular phone company charges $0.12 per minute. If the original context. bill was $11.40 in April, how many minutes were used? MCA Connections: Questions from the MN Dept. Ed Sixth Grade Sampler – refer to sample problems at the end of the unit. Our focus is to help students better understand Algebra through the use of models and manipulatives, to connect their learning through the stages of concrete, representational, and abstract thinking where ultimately they are able to explain “why” something is happening…and to help prepare the students for the MCA II testing. ☺ Learning Opportunities: 1) 2) 3) 4) 5) 6) Finding, creating and analyzing patterns Creating tables and charts Understanding functions Creating linear graphs Using Excel and graphing calculators(if available) to organize and graph data Use manipulatives such as Algeblocks to investigate equations 3 Table of Contents Day 1-2 Exploring and Describing Patterns Day 3-4 Patterns and Expression Day 5-6 Patterns and Tables Day 7-8 Table and Chair Activity Day 9-10 Introduction to Functions Day 11-12 Functions and Equality Day 13-14 Functions and Graphing Equations I Day 15 Functions and Graphing Equations II 4 Day 1-2 Exploring and Describing Patterns Lesson Objective: The students will learn to solve problems using patterns. Materials: 2-3 patterned items (teacher) Exploring houses worksheet (Navigating through Algebra Gr. 6-8) ©2001 Mini-Lab worksheet (Glencoe-Mathematics Applications and Connections-Course 1)© 2001 Patterns Worksheet Launch: Bring in 2-3 items with different patterns and ask students to think of something that the items have in common. (A: They all have patterns) (Ex. Checkerboard/quilt/striped shirt etc.) Explore/Activity: Have students observe the classroom to find three patterns and draw them on their paper. Ask for volunteers to share their finding on the over head and see if their classmates can guess what pattern it is. Distribute worksheet (students draw next three patterns). Share: Using manipulatives (shapes) have the students design two patters on their desk. Ask their partner to recognize the next pattern. Continue, trading worksheets with partners. Ask for volunteers to share using document camera/overhead. Assign: “Exploring Houses” worksheets Mini-Lab Activity Summarize: Not only were we looking to see if students could recognize patterns, but that they could analyze them. The next step is to recognize numerical and shape patterns, and describe it using a table/chart. 5 Day One Mini-Lab A “staircase: that is 4 cubes high is shown at the right. Notice that 10 cubes are needed to build the staircase. TRY THIS: 1. Copy the chart below. Then use centimeter cubes to find the number of cubes needed to build each staircase. Height of Staircase 1 Number of Cubes Needed 1 2 3 4 10 6 5 6 7 7 Day 3-4 Patterns and Expressions Lesson Objective/Standard: Students will learn to use tables and record data while exploring patterns, and understand that tables can help represent patterns using symbols. Materials: -“I Spy Patterns” Worksheet (Navigating through Algebra Grades 3-5, pg 82) ©2001 - ©Algeblocks - Centimeter Graph Paper -Tables Worksheet - Document Camera Launch: Display “I Spy Patterns”. Show diamond pattern on overhead and discuss one pattern recognized. Describe numerical pattern. Distribute worksheet to let students explore other design patterns. Label number of diamonds in each row/design to introduce numerical patterns. Explore/Activity: Introduce symbol growth, analyze, and record in a table. Have each student design a letter shape on their desktop using Algeblocks or other manipulatives. Distribute centimeter graph paper and let students draw a letter design. Have them increase the letter design and record on table. Trade papers with three other students to analyze and record results on their tables. Share: Ask for volunteers to come up to document camera and share results of different letters as well as make predictions for what the sixth stage of growth for their letter would be without drawing or use of manipulatives. Summarize: This is the perfect place to discuss the “first difference” column and add vocabulary words “term”, “conjecture”, and “sequence”. Without realizing it, students are recognizing algebraic patterns and learning recursive formulas! We also think it is important that students see tables in both horizontal and vertical format. 8 Patterns Complete the next three patterns in the sequence. 9 Tables Mass (kg) 0 1 2 3 4 5 Length of Spring (cm) 5 9 13 17 21 25 6 7 8 9 Rule: _______________________________________ Mass (kg) Length (cm) 0 4 1 7 2 10 3 4 5 Rule: _______________________________________ Weight in pounds Shipping cost in dollars 1 4.00 2 5.50 3 7.00 4 5 Rule: _______________________________________ Solve with a table. 1. In some states people are refunded 5 cents for every bottle or can returned for recycling. Fill in the table to show the relationship between the number of bottles returned and the amount refunded. Suppose you were given 7 cents for each can returned. Create a graph and a rule for this situation. # of bottles 1 2 3 4 5 6 Amount Refunded $0.05 $0.10 Rule: ________________________________________ 10 Tables 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 11 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 1 2 Rule: ____________________________ 12 13 14 DAY 5-6 Patterns and Tables Lesson Objective: Students will learn to transfer pictoral patterns to numerical tables and charts analyzing the first difference. Also they will begin to see functions and rules. Materials: - Dot Pattern worksheet (Glencoe-Applications and Connections- Course 1) ©2001 - Table and Chair worksheet (Teaching Mathematics, May 2008) Launch: Display a triangular dot pattern on the board. Allow students time to analyze any patterns or relationships. Together, record data on table and discuss first common difference Explore/Activities: Pass out dot pattern worksheet and allow students to explore more dot patterns. Challenge students to create their own dot pattern and exchange with a classmate. Introduce “Table and Chair” activity chart/table to recognize “1st common difference”. Ask if anyone recognizes a rule….? (Introduction to functions) Share: As a group, review previous tables and discuss first common difference, second(if any)… eventually identifying a rule to find “any” difference. Students will create charts representing the relationship between a function table and a drawing. Summarize: Students will be exposed to the algebraic terminology “function” and are now ready to be introduced to the “function machine” and begin to analyze the “rules”. 15 16 Day 7-8 Table and Chair Activity Students investigate properties of perimeter, area, and volume related to various geometric two- and threedimensions shapes. They conjecture, test, discuss, verbalize, and generalize patterns. Through this process they discover the salient features of the pattern, construct understandings of concepts and relationships, develop a language to talk about the pattern, integrate, and discriminate between the pattern and other patterns. When relationships between quantities in a pattern are studied, knowledge about important mathematical relationships and functions emerges. Source: Illuminations, 2008 NCTM Students will: • • compute perimeter, area, and volume of various geometric figures compute maximum and minimum area of geometric figures, given linear dimensions restrictions Material needed: Square tiles A piece of grid paper for each student Tables at a Birthday Party Activity Sheet Investigation: Perimeter and Area LAUNCH Pose the following problem to the students: Tanya Teen started her own summer business - putting on birthday parties for small children. Her neighbors agreed to loan her square card tables to seat the children for refreshments. However, when some of the neighbors were away on vacation, Tanya couldn't use their tables, and she really hated hauling the tables back and forth. Therefore, using as few tables as possible was important to her. Because all the children wanted to sit together, she had to place the card tables together into rectangles. Only one child could sit on each side of a card table. Her first party had eighteen children. How many tables did Tanya need to borrow? Students may use tiles to display a 1 × 2 banquet table on the overhead projector. Ask, How many people can be seated? [6] Alternatively, the problem can be posed as, What is the area of a rectangle with a perimeter of 18 units? 17 EXPLORE Allow small groups of students to explore the problem with the square tiles. Ask, Which rectangles used the fewest square tiles? [1 × 9 or 9 × 1] The most square tiles? [4 × 5 or 5 × 4] How many tables did Tanya need? The language of tables and people sitting at tables is an appropriate story to link the world of students to the world of mathematics. In this example, tables represent area, and people sitting at tables represent perimeter. Translate the language of unit tables and people into area and perimeter. In describing the dimensions of a rectangle, use bottom and side edge in place of length and width. This allows students to include rectangles 4 × 5 and 5 × 4 as different rectangles. Later the terms width and length can be introduced. Arrange all the rectangles with a perimeter of 18 in a table, such as the one shown below. Length (units) Width (units) Perimeter (units) Area (square units) 1 8 18 8 2 7 18 14 3 6 18 18 4 5 18 20 5 4 18 20 6 3 18 18 7 2 18 14 8 1 18 8 A blank table is found on the Tables at a Birthday Party activity sheet. 18 SHARE Ask the students to observe patterns in the chart. As the length increases, the width decreases, and as the length increases, the area increases to a certain point and then starts to decrease. The later pattern is a verbal description of the graph called a parabola, as shown below. This question builds an understanding of the effect of changing one variable, say length, on another variable, such as width or area. If we allow lengths to be real numbers, is there a rectangle with a perimeter of 18 that has a larger area? Students will usually try a rectangle with a side of length 4.5 units. The area is 20.25 square units. Trying sides with lengths larger and smaller than 4.5, say 4.4 or 4.6 or 4.55, will yield areas that are smaller than 20.25. The shape of the rectangle with largest area for a perimeter of 18 is the square rectangle that is closest in shape to a square, 4 × 5 (if we restrict our lengths to whole numbers). Several graphs can be drawn to represent the data in the table above. The graph of the area versus the length (or width) is a parabola, which is the graph of a quadratic function. Area vs. Length 19 The graph of the length versus the width is a linear graph. Length vs. Width Using the graph of the parabola, ask students, What happens between the points (4, 5) and (5, 4)? This is where the maximum point will occur if we extend our dimensions to include real numbers. Ask, What are the dimensions of the rectangle whose area is 16 square units? [6.5 × 2.5] What is the area of the rectangle whose length is 5.5 units? [19.25] The graph visually describes the effects of increasing the length on the area. Can a length be 9? [No, because the width would then be 0.] Ask, What happens if the graph is allowed to cross the horizontal axis? Investigation: Predicting the Maximum Area Have students predict which rectangle has the maximum area for a fixed perimeter. Discussion. Ask the class to predict when the area seems to "turn around." It reaches its maximum area when the shape of the rectangle is as close to a square as possible. If we have real numbers, then the rectangle with largest (maximum) area is a square. Try out this conjecture with perimeters of 20 units and 24 units. Have students guess when the area turns around and then find all the rectangles with integer lengths that have the given perimeter. For a perimeter of 20 units, students may need to list all pairs of whole numbers whose sum is 20/2, or 10, and note that the pair (5,5) is the turn around. Thus the square with sides of length 5 and area of 25 is the rectangle with the largest area for a fixed perimeter of 20 units. 20 For a perimeter of 24 units, the rectangle with the largest area is also a square, with sides of length 6 units and area of 36 square units. For a perimeter of 34, which rectangle has the maximum area? If the dimensions are whole numbers, then the rectangle with dimensions 8 × 9 has the maximum area. If the dimensions are real numbers, then the square whose sides have length 8.5 has the maximum area. Investigation: Generalizing the Process for Finding the Maximum Area Have students generalize the pattern for finding the area of a rectangle given a fixed perimeter. Discussion. Ask the students to describe how they found all the rectangles. Allow the students time to talk about the process and then to translate the process into symbols. Find all pairs of whole numbers whose sum is half the perimeter. Use these pairs of numbers as the lengths and widths of the rectangles and then compute the areas. Or find the pair of numbers that are almost equal; this rectangle has the maximum area. For real numbers, since we know that the rectangle with the maximum area for a fixed perimeter is a square, use the number that is one-fourth the perimeter for the length of the side of the square. The following sets of equations could also be used: Basic Equations 2(length + width) = Perimeter, or length + width = Perimeter ÷ 2 length × width = area Advanced Equations 21 Looking toward Algebra: The equation that describes the area of a rectangle with fixed perimeter is a quadratic equation, which is studied in more detail in algebra. Quadratic equations always have the shape of a parabola (as shown previously), except some open down and have a maximum point and some open up and have a minimum point. The maximum or minimum pint can also be found algebraically. Have students observe the symmetry of the parabola and how this relates to the pairs of rectangles; for example, the rectangles 4 × 5 and 5 × 4 have the same area. They lie on the same horizontal line. The perpendicular line drawn from the maximum point to the horizontal axis is the line of symmetry. This problem provides intuition and understanding for some very important concepts of algebra and calculus. Maximum and minimum points of more general functions are studied in calculus. Geometrically, the maximum or minimum points occur when the tangent line to the graph is a horizontal line or has slope zero. As the students generalize the patterns they have observed and discussed, they are "sneaking" up on the notion of variable in a very natural way. Investigation: Factor Pairs Have students find all the factor pairs for 36. Discussion. Ask students to describe all the rectangles that have an area of 36. Let them use square tiles or grid paper. Some students will recognize that this task is equivalent to finding all the factor pairs of 36. To find the rectangle with the least perimeter, students need to find the rectangle with a shape closest to a square or the factor pair where the factor pairs begin to repeat. For 36, these are the factor pairs: 1 × 36 2 × 18 3 × 12 4×9 6×6 9×4 12 × 3 18 × 2 36 × 1 The factor pairs repeat after the pair of factors (6, 6). The factors in this pair are equal, thus 36 is a square. Let students explore an area of 12, 28, or 64. Ask them to conjecture about when the factor pairs will begin to repeat. Ask, If you were going to find all the factor pairs of a number, how many numbers would you have to check? 22 [1 through the number that when squared is closest to the number; for 12 it is 3; for 28 it is 4; for 64 it is 8.] Ask students if they can predict which numbers are perfect squares (square of a whole number). The number is a square number if it has a factor pair with both factors equal or with an odd number of factors. Assessment options: 1. To evaluate students' understanding of the concept of "maximum area," the teacher can ask them to write about the process of dinging the rectangles with a greatest area, given a fixed perimeter. This question asks the students to reflect on the class exploration and discussion and helps the student (and teacher) assess the level of understanding. 2. Find the perimeter of all rectangles with whole-number dimensions whose area is 72 square units. Make a table and graph your data. Use the table or graph to answer the following questions: Which rectangle has the least perimeter? The greatest perimeter? If we allow the dimensions to be rational numbers (fractions), which rectangle has the least perimeter? The greatest perimeter? NCTM Standards: Algebra 6-8 1. Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships. 2. Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations. 3. Model and solve contextualized problems using various representations, such as graphs, tables, and equations. Geometry 6-8 1. Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. Number & Operations 6-8 1. Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers. Phillips, Elizabeth, et al. Patterns and Functions, 1, 41 - 46. Reston, VA: NCTM, 1991. 23 Name______________________ Tables at a Birthday Party Activity Sheet Work with your group to solve the following problem. You may use square tiles or grid paper to help you. Tanya Teen started her own summer business – putting on birthday parties for small children. Her neighbors agreed to loan her square card tables to seat the children for refreshments. However, when some of the neighbors were away on vacation, Tanya couldn’t use their tables, and she really hated hauling the tables back and forth. Therefore, using as few tables as possible was important to her. Because all the children wanted to sit together, she had to place the card tables together into rectangles. Only one child could sit on each side of a card table. Her first party had eighteen children. How many tables did Tanya need to borrow. Record all of the arrangements in the table below: Length (Units) Width (Units) Perimeter (Units) 24 Area (Square Units) Day 9-10 Introduction to Functions Lesson Objective: Students will use tables to record and describe patterns with expressions. Then, they use expressions to extend their tables. Materials: Input/Output worksheet Launch: A good analogy for “function machine” (rule) would be to make a pitcher of Kool-aid as a demonstration. By just adding water to the pitcher, stirring it up, and pouring it out, students can see that a change has occurred. The same concept can be applied to numbers. If a number is added to a “machine” and an operation is performed a new number will be produced. Explore/Activity: Draw a “function machine” (see below) on the board, and starting with the number one input that number into the left side of the machine and a two on the right side. Continue with this doubling pattern (1,2) (2,4) (3,6) etc. Share: Hand out Input/Output worksheet. Have students work along with teacher at the board, introducing “input” and “output” vocabulary using function tables. Complete the first table together but challenge students giving them less information for each table. Input Output Input Output Input Output Input Output Summarize: Students have now been exposed to the algebraic vocabulary “input”, “output”, and “functions”. Following these exercises students have become aware that when applying different rules or operations to numbers the outcome changes. They are beginning to think of their own rules and are now ready to apply them to tables, graphs and constructed response questions. 25 Input Output Input Output Input Output Input Output Input Output Input Output Input Input Input Input 26 Output Output Output Output Day 11-12 Functions and Equality Lesson Objective: Students will review the concept of equality and be able to describe (verbally and symbolically) and make a connection with the “rule” or function of the pattern with the numbers. Materials: Input/Output worksheet Graph worksheet Classroom computer and projector Computer lab Website: www.nlvm.usu.edu/ Launch: Introduce the www.nlvm.usu.edu/ website under algebra grades 6-8 using the balance scale activity and function machine activity with classroom computer and projector. Explore/Activity: Take students to the computer lab. Using their “input/output” worksheets have them record results of the function machine activities. Share: Bring “input/output” worksheet back to classroom and discuss how to record data on graph paper. In doing so, we will analyze rules that occurred in the computer lab. These graphs will be constructed as an in class activity setting up for tomorrow’s lesson. Summarize: Students should have a good sense of what the “1st difference” is and be able to recognize it quickly. This activity has helped them to understand the rule, or “function” that occurred within each input and output. They are now prepared to use variables and begin graphing equations. 27 Input Output Input Output Input Output Input Output Input Output Input Output Input Input Input Input 28 Output Output Output Output 29 Day 13-14 Functions and Graphing Equations Lesson Objective: Students will take yesterday’s input/output tables and begin transferring data to create linear graphs. Materials: Graph Paper Computer and projector Launch: Review a function machine activity, create a table, and as a group create a graph. Explore: Review yesterday’s function machine activity and have students create multiple linear graphs. Share: Have students share their individual work using the document camera or overhead and discuss the similarities between the various graphs. Summarize: By now, the students should be quite familiar with the four points of algebra: patterns, functions, tables, and graphs. It’s a perfect lead-in to introducing equations with emphasis on different variables using a variety of operations. Day 15 Functions and Graphing II Lesson Objective: Use tables from previous lessons the students will create graphs with their information Materials: Graph/ worksheets from previous lessons Classroom computer and projector Computer lab access Launch: Show examples of how tables can be graphed using ©Microsoft Excel spreadsheet. Explore/Activity: Take students to computer lab to graph their equations using ©Microsoft Excel. Begin doing a few as a class then allow students to use what they have learned to explore on their own. Share: After returning from the computer lab, allow students to show some of the graphs that they have created. Summarize: Following this lesson students will have an understanding of how graphs are applied in the real world. 30 MCA II Sample Questions: Type of Food Lettuce Potatoes Burger Patties Delivered Every 3 days Every 6 days Every 15 days The Burger Hut receives regular deliveries of food. All the types of food listed in the table were delivered today. In how many days will the Burger Hut receive another delivery of all 3 items on the same day? A. 15 B. 30 C. 45 D. 60 Please write your response to question 20 on page 3 of your answer book. Juice boxes are packaged two different ways. • A package of 24 boxes costs $12.98. • A package of 4 boxes costs $2.59. You need 56 juice boxes. Part A Find 2 different possible ways you could purchase the 56 juice boxes and the total cost for each way. Show or explain your work. Part B Using your answer in part A, what is the price per juice box if you purchase the lower priced combination? Show or explain your work. 31

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