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Copy of Patterns, Relationships, and Algebraic Thinking - 6th grade
Transcript of Copy of Patterns, Relationships, and Algebraic Thinking - 6th grade
Standards Grade Level 111.22. 3-5 Mathematics, Grade 6.
(3) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships. The student is expected to:
(A) use ratios to describe proportional situations;
(B) represent ratios and percents with concrete models, fractions, and decimals; and
(C) use ratios to make predictions in proportional situations.
(4) Patterns, relationships, and algebraic thinking. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes. The student is expected to:
(A) use tables and symbols to represent and describe proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change), perimeter and area; and
(B) use tables of data to generate formulas representing relationships involving perimeter, area, volume of a rectangular prism, etc.
(5) Patterns, relationships, and algebraic thinking. The student uses letters to represent an unknown in an equation. The student is expected to formulate equations from problem situations described by linear relationships. Sixth Grade Math Understand patterns, relations, and functions:
Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;Relate and compare different forms of representation for a relationship;Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.
Represent and analyze mathematical situations and structures using algebraic symbols
Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations Activities Anno's Magic Seeds Anno's Magic Seeds:
In Anno’s activity, students read the story Anno’s Magic Seeds, by Mitsumasa Anno. Students will use the information in from different sections of story to complete three different tables regarding the information about number of seeds produced compared to numbers of seeds eaten and the number of seeds buried over a period of time. After the students complete the tables, they will use the results to answer several questions that focus on describing the observed patterns.
Austin, R. and Thompson, D. (2004). "Anno's magic seeds. exploring algebraic patterns through literature".Virginia: National Council of Teachers of Mathematics, Inc. Bouncing
Tennis Balls Bouncing Tennis Balls:
Bouncing Tennis Balls is an activity in which students can practice collecting and recording data. Students will collect data and discuss the relationship between the number of times the tennis ball is bounced within a given amount of time. Through this activity, students will discover that patterns that are linked to the algebraic content and see that algebra is more than solving equations. This problem solving activity will evoke discussion and connections to real- world situations of algebra connections of patterns.
Friel, S., Rachlin S., and Doyle D. (2010). National Council of Teachers of Mathematics (NCTM). “Bouncing tennis balls.” Retrieved from:http://illuminations.nctm.org/LessonDetail.aspx?ID=L246 Traffic Jam Traffic Jam:
Traffic Jam focuses on a kinesthetic activity of problem solving, reasoning and proof through communication and discussion. The object is for the students to rearrange themselves by discover a process pattern. Traffic Jam states the challenge as “every must move so that the people originally standing on the right-hand stepping stones are on the left-hand stones and so that those originally standing on the left-hand stepping stones are on the right-hand stones, with the center stone again unoccupied” (Traffic Jam CITE). The students on the left can only move right and vise versa. Students may jump another person if the “stone” is unoccupied. However, a student can only jump one stone at a time. Through this activity, the students can expand on their knowledge by looking for patterns. The activity includes questions that promote pattern reflection.
Conway, J. (2009). National Council of Teachers of Mathematics (NCTM). "Traffic jams" Retrieved from
www.nctm.org/eresources/view_article.asp?add=Y&article_id=2072&page=11 Stacking Cups Stacking Cups:
The goal of this activity is for students to understand, by graphing, the positive linear relationship between the number of cups stacked and the cups’ height in centimeters. The objective of this activity is to find the measurement of a given cup and then, through algebraic processing and patterning, the students will predict the height of a given amount of stacked cups. Once the students collect the real the measurements, the students will then graph the data and compare the results of different cup sizes.
The National Council of Teachers of Mathematics, Inc. (2001). Navigating through algebra in grades 6-8. Reston, Virginia: Library of Congress Cataloging Exploring -Houses Journals Spreadsheets, Patterns,
and Algebraic Thinking
Building the Bridge
to Algebraic Thinking Experiences
with Patterning Patterns as Tools
Reasoning Developing Algebraic Thinking through Pattern Exploration Spreadsheets, Patterns, and Algebraic Thinking
By: Don Ploger, Lee Klingler, and Michael Rooney
In this article, a group of 5th graders explore spreadsheets and their many uses. At first, they learn how to use the spreadsheet and do simple problems. Then their curiosity takes over and they start to do more complex problems. After a little practice, the students were using spreadsheets to generate odd and even numbers, make multiplication tables, generate squares and square roots, and explore exponential growth. They were recognizing the patterns within each problem and applying them to real life situations. They then learned how to create Fibonacci numbers. The students were finding the patterns that result from the Fibonacci sequence and creating their own problems with similar patterns. By learning about spreadsheets and how to use them the students learned how to generate a pattern, create a running total, create a pattern using a formula, and how to pose problems and create explorations of their own.
Ploger, D. & Klingler L. & Rooney, M. (1997). Spreadsheets, patterns, and algebraic thinking. Teaching Children Mathematics, 3(6), 330. Experiences with Patterning
By: Joan Ferrini-Mundy, Glenda Lappan and Elizabeth Phillips
This article offers a way for students to experiment with geometric shapes and to illustrate how mathematical ideas can be developed. The children in this article, who studied problems, saw how algebra emerged. There were able to recognize a pattern and estimate the next few figures because they generalized the ideas from a pattern. There are guiding questions to help the children and teacher organize a classroom discussion.
Ferrini-Mundy J., Lappan G. and Phillips E. (1997). Experiences with patterning. National Council of Teachers of Mathematics. Teaching Children Mathematics, 3(6), 282. Developing Algebraic Thinking through Pattern Exploration
By: Lesley Lee and Viktor Freiman
This article makes the connection between patterns and algebraic thinking in children by posing the question, “How can teachers exploit pattern work to further algebraic thinking and introduce the formal study of algebra in middle school” (Freiman & Lee, 2006). In order for children to be able to understand the study of algebra in patterns, the patterns should be visual. Some of the time, teachers find that there is an avoidance of algebra when working with patterns so this article offers guiding questions that encourage students to explore algebra when figuring out a pattern.
Freiman, V., & Lee, L. (2006). Developing algebraic thinking through pattern exploration. Mathematics Teaching in the Middle School, 11(9), 428. Exploring Houses:
The goals for this activity are for the students to understand how patterns develop and to make predictions about and discover the development of a certain patterns. The students work individually in this activity and begin by copying the four images the figure provides, which is a sequence of houses/geometric patterns. The students will determine how many of the pattern blocks are needed in total to make each of the four houses. Through this activity, students will discuss questions such as, how many square pattern blocks are needed, how many triangular pattern blocks are needed, how does the next figure in the sequence differ from the previous? The students then will predict describe what the fifth house in the sequence will look like. Lastly, the students will predict how many blocks would be in the fifteenth house, and write a rule, an equation, or an explanation describing how to determine the “nth” figure. The students do not necessarily have to do this activity with houses; instead, the teachers may instruct or have students perform this activity with any sort of pattern they choose.
The National Council of Teachers of Mathematics, Inc. (2001). Navigating through algebra in grades 6-8. Reston, Virginia: Library of Congress Cataloging. Manipulatives Geoboards Function
Puzzle Counting Chips Colored Blocks:
These blocks are multi-colored single unit blocks that can be used to show a pattern and as well as, providing a visual image for how a pattern can advance. Students can create many different models/patterns using these colored blocks. Teachers may use these blocks for the students to do an open-ended activity, to build patterns, to create and solve critical thinking problems. The blocks can also be used to help students parallel patterns to algebraic equations. Students may use these blocks to aid their understanding in sequencing activities such as “Exploring Houses” because this visual block image can help the students understand how to find the nth figure in a sequence.
Found at Lakeshore Learning
The function machine is a virtual manipulative that consists of a machine in the middle of the screen where the numbers go “in and out”. Inside the machined, the function is performed on that number. On the right-hand side of the screen, there is a chart with two columns. The left hand column has the numbers that go “in the machine” and on the right hand column are the numbers that come “out of the machine”. The goal is that the student figures out what function is being performed and he/she has to fill out the rest of the chart with the correct numbers. Though the students are not studying functions, through observations of the data, the students will be able to evaluate the patterns contained in the chart.
This manipulative may be used virtually or physically. It has many uses including exploring area, perimeter, fractions, graphing, and other geometry concepts. The manipulative consists of a board with vertical pegs. The students’ use rubber bands to place around the pegs to form shapes, solve various mathematical puzzles and even represent graphed figures. This manipulative may be used for multiple mathematical topics and provides students with a tactile activity.
Found at Lakeshore Learning Patterns as Tools for Algebraic Reasoning
By: Kristen Herbert and Rebecca H. Brown
Brown and Herbert’s article, Patterns as Tools for Algebraic Reasoning, discuss the important ways in which, students explore algebraic concepts in an informal way to build a foundation later advanced studies of algebra. Herbert and Brown emphasize, “a broad view of algebraic thinking is taken to show students the real-life uses and relevance of algebra” (Brown & Herbert, 1997, 340). Though algebraic reasoning may be commonly associated with formulas, the main ideas of algebraic reasoning is to use patterns by understanding the information, representing the information and interpreting the data for a further meaning. This journal highlights three parts of the investigation process: pattern seeking, pattern recognition and generalization. During pattern seeking, teachers should guide students through the use of counters or manupulatives. Herbert and Brown suggest that the counters will help guide students to solve the task independently and not be “spoon fed” by the teacher. While some students may recognize a pattern with the manipulatives, other students may need to try additional cases by substituting different numbers into the problem in order to understand a pattern to calculate an answer. Lastly, the step of “generalization.” During this step, students develop a general method how to not only figure out this specific word problem, but a generalized rule in which the student can solve the same problem with any numerical value. The importance of generalization is that students are using their own words to describe how a method relates to a “physical situation.” Brown and Herbert contend that once students can create an understanding of finding and applying patterns, students have grasped the meaning of modeling a concrete situation to algebraic thinking. Brown and Herbert conclude the article with the statement along the lines that the investigative process provides sixth-grade students a solid foundation to build algebraic and formula thinking throughout the middle grades.
Herbert, K. and Brown, R. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 3(6), 340. TEKS Content Area Patterns, Relationships &
Algebraic Thinking Literature Building the Bridge to Algebraic Thinking
By: Roger Day and Graham A. Jones
During problem solving, students solve and may not make an algebraic connection. However, once the teacher recasts the problem, the students are able to explore new routes that may lead to algebraic thinking. These extended problems go beyond a single situation and response. Teachers give students an opportunity in a problem to demonstrate simple solutions and by revising the problem, students may identify, describe and extend a pattern presented in the problem. This exercise will guide students to understand algebraic thinking. By having ongoing discussions and interactions with problem solving problems, a teacher is able to take advantage of the opportunity to aid students in expanding their thinking beyond general arithmetic functions. As Day and Jones suggests, by having the students describe their thinking process, teachers are “building the bridge” from problem solving to algebraic thinking and relationships. Even though students may not at first catch on to using the “algebraic terminology” of “x’s and y’s” the important aspect is the discussion of the students’ thinking process. The development of the thinking process will enable students to move forward to solve more advanced problems.
Roger, D. and Graham, J. (1997). Building Bridges to algebraic thinking. Mathematics Teaching in the Middles School, 2(4), 208. Peg Puzzle:
The manipulative involves a range of pegs from 2 pegs to 8 pegs. There is one more hole than pegs and the pegs are divided evenly. Red on the right, empty hole, and blue on the left. Every peg must move so that the red pegs are on the right hand side and blue pegs move to the left hand side. Each peg must stay in a hole and cannot be removed from the board. Also, pegs may only jump one other peg and may not move backwards. (Back to the side the peg originally started on). This manipulative is an excellent aid for the “traffic jam” activity because it provides a visual representation of the problem. Also, the peg puzzle includes various levels so a teacher can fluctuate the problem according to his or her students’ ability. Since this manipulative illustrates the process of problem solving, students will be able to grasp the concept at hand.
http://nlvm.usu.edu/en/nav/category_g_3_t_2.html Anno's Magic
Seeds Anno’s Magic Seeds
By: Mitsumasa Anno
Anno’s Magic Seeds is a children’s book that allows the reader to get involved in a fun playful story. The reader is asked to do a series of mathematical equations to solve problems presented throughout the book. In this short story a wizard gives Jack, the main character, 2 magic seeds. He was told to eat one of the seeds, which would make him full for 1 year, and to plant the other seed and care for it so it grows and produces 2 more seeds. He repeats this process for many years but then decides to do something different. He planted both of the seeds and when 4 seeds were produced he ate one and buried the other 3. He kept doing this and the amount of seeds kept increasing. The reader is asked to predict and calculate the amount of seeds produced on each page. They are also asked to find the pattern within the book and do many mathematical operations dealing with it.
APA Citation:Anno, M. (1995). Anno’s magic seeds. New York, NY: Philomel Books. Counting Chips:
This manipulative is used for counting negative and positive numbers. The red side represents the negative numbers and the yellow side represents the positive numbers. By placing a yellow chip on a red chip you get zero, because -1 + 1 = 0. We chose to use this manipulative to represent Anno’s Magic Seeds because in the book he starts off with two seeds, or two yellow chips. He then eats one, so the student turns it to red and place it in a separate pile, and then he plants one so the student keeps it yellow and put it in a pile. Then students add two more yellow chips to the yellow pile but takes one away because it turned into those two seeds. The student is then back to where he or she started from. The students continue the process, that is, until Jack changes what he does with the seeds. Both teachers and students have to play around with the chips in order to keep up with what Jack does to the seeds. By using these chips, the students will be able to easily follow along with the concept of adding and taking away of seeds in the story.
Found at Lakeshore Learning THE END!!
Elizabeth, Lindsay, Rachel and Stephanie Footprints in the Snow
This book discusses the pattern of counting by twos. It talks about this pattern by way of footprints. The book starts with two footprints and adds by two each time you turn the page. At the corner of the page on the bottom it has a flashcard that shows the reader the pattern change. The last page of the book contains different resources to expand students’ knowledge. There are fun facts related to the environment in which this book takes place. They also have a web resource that students can visit and find other websites related to this book that will expand upon the knowledge discovered in this book. Even though these patterns seem immature for sixth graders it is important for them to have a good foundation of pattern relationships before advancing to algebraic expressions.
Dahl, M. (2005). “Footprints in the snow.” Minneapolis, Minnesota: Picture Window Books. Footprints in
the Snow Counting:
Mighty Math "Counting" by Sara Pistoia presents mathematical information that is geared towards a younger age. This book includes descriptions about the reasons counting and pattern images are used. Though “Counting” is intended for a younger age, it can be advanced for sixth graders by studying and discussing the relationship of the patterns. Through this discussion, students will verbally communicate their words and thoughts, which in turn will expand their thinking beyond general arithmetic functions.
Pistoia, S. (2003). Counting. Chanhassen, MN: The Child's World. Counting Hands down:
The same author as “Footprints In The Snow” writes this book. “Hands down” displays illustration of counting by fives each time the page is turned. This book also seems geared for a younger age, but on page 14 this book reveals a very important concept that is applicable to sixth graders who are thinking about algebraic patterns. It talks about thirty fingers can create kissing turkeys. Each turkey has is represented with five fingers and there are two turkeys kissing. The picture illustrates six turkeys, each represented by five fingers, which totals 30 fingers. Since there are six turkeys, there are a total of three pairs of “kissing turkeys.” This concept can be expanded for sixth graders by asking the students to find a pattern in which there will be even number of five fingered turkeys occurring. This book is a great way to visually represent the pattern of counting by fives in pictures.
Dahl, M. (2005). Hands down counting by fives” Minneapolis Minnesota: Picture Window Books. Hands
Down Math For All Seasons
Math For All Seasons is a children's book that allows the reader to engage in mathematics through "mind-stretching math riddles" about the different seasons. This book teaches children about problem-solving skills, and supports children to utilize a variety of ways to solve the problem posed in the riddles. Math for All Seasons is primarily aimed towards children practicing skills in addition, subtraction, and counting; however, on each page, there is a type of pattern in the illustration. The illustrations in this book can be advanced for a sixth grade level through discussion and analyzing of patterns. Calculating and determining the nth figure of the various patterns represented in this book is a way to extend the study of this particular book. Another aspect of “Math for All Season” is the variety of patterns the students can evaluate through multiple colors, different amounts of designs, a range of shapes, and different numbers of objects.
Tang, G. (2002). Math for all seasons. New York, NY: Scholastic, Inc. Math For