Algebra Guide 10 K-2

Patterns, Functions, and Algebra 251 Session 10, K-2

Session 10

Classroom Case Studies, Grades K-2This is the final session of the Patterns, Functions, and Algebra course! In this session, we will examine how thetypes of mathematical tasks involving algebraic thinking from the previous nine sessions might look whenapplied to students in your own classrooms.This session is customized for three grade levels. Select the grade levelmost relevant to your teaching.

The session for grades K-2 begins below. Go to page 273 for grades 3-5 and page 293 for grades 6-8.

Key Terms for This SessionPreviously Introduced

• mathematical thinking tools [Session 1] • algebraic ideas [Session 1]

• representation [Session 1] • variable [Session 2]

• input [Session 2] • output [Session 2]

• recursive description [Session 2] • closed-form description [Session 2]

• function [Session 3] • backtracking [Session 6]

Introduction and ReviewIn the previous sessions, we explored the components of algebraic thinking: fundamental algebraic ideas (con-tent) and mathematical thinking tools (process). You were asked to put yourself in the position of a mathematicslearner, both to analyze your individual approach to solving problems and to get some insights into your own con-ception of algebraic thinking. It may have been difficult for you to separate yourself as a mathematics learner fromyourself as a mathematics teacher. Not surprisingly, this is often the case! In this session, however, we will shift thefocus to your classroom and to the approaches your students might take to mathematical tasks involving alge-braic thinking. [SEE NOTE 1]

Learning ObjectivesIn this session, we will focus on your experiences as a classroom teacher, as you:

• Explore how algebraic thinking is developed at your grade level

• Examine problems for their algebraic content

• Analyze mathematical tasks and their connection to the mathematical themes in this course

• Critique lessons at your grade level for algebraic thinking

NOTE 1. This session focuses on developing algebraic thinking in the K-2 classroom.We’ll consider the mathematics content ofthe previous sessions and the relation of that content to the mathematics you teach in your own classrooms.We’ll explore howalgebraic thinking develops at your grade level by analyzing mathematical tasks appropriate for the K-2 classroom. We’ll alsolook at lessons from existing curriculum materials and critique them in relation to how students are asked to demonstrate theirthinking and to how the mathematics reveals algebraic thinking.

NOTE 1 cont’d. next page

Session 10, K-2 252 Patterns, Functions, and Algebra

To begin the exploration of what algebraic thinking looks like in a classroom at your grade level, watch a videosegment of a teacher who has taken the Patterns, Functions, and Algebra course and has adapted the mathematics to her own teaching situation. When viewing the video, keep the following three questions in mind:[SEE NOTE 2]

a. What fundamental algebraic ideas (content) is the teacher trying to teach? Think back to the big ideas ofthe previous sessions: patterns, functions, linearity, proportional reasoning, nonlinear functions, and alge-braic structure.

b. What mathematical thinking tools (process) does the teacher expect students to demonstrate? Thinkback to the processes you identified in the first seminar: problem-solving skills, representation skills, andreasoning skills.

c. How do students demonstrate their knowledge of the intended content? What does the teacher do toelicit student thinking?

Part A: Classroom Video (30 MINUTES)

NOTE 1, CONT’D.

ReviewBegin the session by reviewing the mathematics content—patterns, functions, linearity, proportional reasoning, nonlinearfunctions, and algebraic structure—in the previous nine sessions. You may want to think about one big idea in each of thesetopics.

Homework ReviewGroups: Discuss any questions about the homework.

NOTE 2. Before examining specific problems at this grade level with an eye toward algebraic thinking, we’ll watch anotherteacher, one who has also taken the course, teaching in her classroom. The purpose is not to be critical of the teacher’s meth-ods or teaching style. Instead, look closely at how the teacher brings out algebraic ideas in teaching the topic at hand, as wellas how the teacher extends the lesson and asks questions that elicit algebraic thinking.

Review the meaning of algebraic ideas (that is, the content of algebra) and mathematical thinking tools (the processes used inanalyzing problems). Keep in mind questions (a), (b), and (c) as you watch the video.

NOTE 3. Groups: Work in small groups on Problems A1-A4. Share answers to Problem A4 especially, because recursive thinkingis not necessarily something that first-grade teachers often consider in their teaching. The vocabulary (recursive) is not theimportant part here.The method of thinking recursively, however, is important for teachers to consider as students make senseof patterns.

VIDEO SEGMENT (approximate times: 18:36-24:06): You can find this segmenton the session video approximately 18 minutes and 36 seconds after theAnnenberg/CPB logo. Zero the counter on your VCR clock when you see theAnnenberg/CPB logo.

In this video segment, Gina Webber asks her students to think about pat-terns. She began the lesson by reading from the book How Many Feet in theBed? Students made a chart of the number of people and feet in the bed.Watch them discuss the relationship between people and feet. [SEE NOTE 3]


Problem A1. Reflect on the questions (a), (b), and (c) on the previous page.

Problem A2. How does Ms. Webber encourage students to use patterns to predict the number of feet and people in the bed?

Problem A3. How does Ms. Webber incorporate the theme of “doing and undoing” we explored in Session 3?

Problem A4. How does Ms. Webber use recursive thinking in this lesson?

Part A, cont’d.


The National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (2000) identifiesalgebra as a strand for grades Pre-K-12. The Standards identify the following concepts that all students shouldcover and be able to do: [SEE NOTE 4]

• Understand patterns, relationships, and functions

• Represent and analyze mathematical situations and structures using algebraic symbols

• Use mathematical models to represent and understand quantitative relationships

• Analyze change in various contexts

For the Pre-K-2 classroom, understanding patterns includes the following expectations:

• Recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numericpatterns, and translate from one representation to another

• Analyze how repeating and growing patterns are both generated

In this part, we’ll look at problems that foster algebraic thinking as it relates to these standards, and we’ll exploreways of asking questions that elicit algebraic thinking. The situations you will be exploring are representative ofthe kinds of problems you would find in some existing texts; in fact, you may recognize some of them! The goal isfor you to examine these problems with the critical eye of someone who has taken this course and is beginningto view algebraic thinking with a different perspective.

Consider the situation below, appropriate for exploration in a K-2 classroom:

Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the water.Tat Minguses blue tiles to represent the water. Around each pool there is a border of white tiles. Here are pictures of thethree smallest square pools that he can design, with blue tiles for the interior and white tiles for the border.[SEE NOTE 5]

Part B: An Example for DevelopingAlgebraic Thinking (25 MINUTES)

NOTE 4. Look at NCTM’s recommendations for content in the algebra strand in the Standards, then look at the problem fordesigning square swimming pools. After reading the problem, work on Problems B1-B5.

NOTE 5. Read the commentary on the swimming pool problem in “Experiences With Patterning,”by Joan Ferrini-Mundy, GlendaLappan, and Elizabeth Phillips, in Teaching Children Mathematics (February 1997), pp. 282-288. The article can be found on thecourse Web site. Go to www.learner.org/learningmath and find Patterns, Functions, and Algebra Session 10, Grades K-2, Part B,Note 5.


Problem B1. What questions would you, as an adult mathematics learner, want to ask about this situation?

Problem B2. How do your questions reflect the algebra content in the situation?

Now focus on the questions you want the students in your classroom to consider. In kindergarten, begin by hav-ing students focus on the relationship between the numbers of blue and white tiles by sorting the tiles into bluetiles for the water and white tiles for the border. In a first-grade class, students might focus on the blue tiles andwhat it means to be square. In a second-grade class, students might begin to organize their data into a table.

Problem B3. What patterns, conjectures, and questions will your students find as they work with this situation?

Problem B4. What questions could you as the teacher pose to elicit and extend student thinking at your gradelevel?

Problem B5. Recall the framework you explored in Session 2 in looking at patterns: finding, describing, explain-ing, and using patterns to predict. Which of these skills will your students use in approaching this problem?

Problem B6. Read the article “Experiences With Patterning” from Teaching Children Mathematics, which can befound on the course Web site. Go to www.learner.org/learningmath and find Patterns, Functions, and Algebra Session 10, Grades K-2, Part B, Problem B6. What ideas mentioned seem appropriate for your classroom?

Part B, cont’d.


Here’s a problem on patterns developed for a kindergarten class:

Shoe PatternsFocus: Extend patterns

Materials: Children’s shoes

Children sit in a circle, and each child removes one shoe. Arrange some of the shoes in a patterned line. (For exam-ple, the pattern might be: vertical, horizontal, vertical, horizontal; this is a nice opportunity to use these terms.) Askif children think they know how to add their shoes to the line, following the pattern. Let several children add theirshoes to either end of the line before you ask them whether they can describe the pattern. Let the rest of the children add their shoes and see how far the pattern reaches.

Problem B7. What questions could you ask to develop students’ skills in describing patterns?

Problem B8. What questions could you ask to develop students’ skills in predicting?

Part B, cont’d.

The swimming pool problem is adapted from Algebra in the K-12 Curriculum: Dilemmas and Possibilities, Final Report to the Boardof Directors, by the NCTM Algebra Working Group (East Lansing, Mich.: Michigan State University, 1995).

The problem and analysis of algebraic thinking is discussed in “Experiences With Patterning,” by Joan Ferrini-Mundy, Glenda Lappan,and Elizabeth Phillips, in Teaching Children Mathematics (February 1997), pp. 282-288.

Problem B6 is taken from Everyday Mathematics, Teacher’s Guide to Activities, Kindergarten, developed by the University ofChicago Math Project (New York: SRA/McGraw-Hill, 2001).


At the K-2 grade level, there are many problems that prepare students for their later work in algebra. Experienceswith these problems can also build bridges between arithmetic and algebra, while increasing the students’chances for success in future mathematics. [SEE NOTE 6]

In this section we’ll analyze problems at the K-2 level for their algebraic content. For each K-2 problem that follows,find a mathematical solution, then answer questions (a) through (f ) listed below.

a. What algebraic content is in the problems?

b. What content does it prepare students for later?

c. How does this content relate to the mathematical ideas in this course?

d. How would your students approach this problem?

e. What are other questions that might extend students’ thinking about the problem?

f. Does your current program in mathematics at your school include problems of this type?

Strips and Balancing BlocksIn Problem C1, the goal is for students to predict and verify the numbers of characters in a pattern. Some ques-tions you might ask your students include:

• What is the pattern?

• How many Xs are in the pattern?

• How many Os are in the pattern?

• What will you draw in the first box? Second box? Third box? Fourth box? Fifth box?

You may think that your students will have difficulty predicting the character in each box without filling in theentire strip. Students should be encouraged to find ways of predicting without having to do the filling first.

Problem C1. Draw two strips with this pattern:

Part C: More Problems That IllustrateAlgebraic Thinking (45 MINUTES)

NOTE 6. It is often difficult for teachers to see how the mathematics content at their grade level builds algebraic thinking.Therefore, it is important to look at many examples of elementary mathematics that build bridges to algebraic thinking.Groups: Work on Problems C1-C9, recording answers to questions (a) through (f ) as you do each pair of problems.


a. How many Xs are in the two strips all together?

b. How many + s are in the two strips all together?

c. Predict how many Os will be in four strips all together.

d. How do you know?

e. Draw the strips to check.

Now draw four strips with this pattern:

a. How many Xs are in the four strips all together?

b. How many Os are in the four strips all together?

c. Predict how many Xs will be in six strips all together.

d. How do you know?

e. Draw the strips to check.

Part C, cont’d.


In Problem C2, the goal is to recognize that a balance scale in essence represents equality. Some questions youmight ask your students are:

• What does Scale B show?

• In Scale A, what blocks are in the left pan? In the right pan?

• In Scale A, which pan is heavier?

Problem C2. For each scale, determine which weighs more—the cube, the cylinder, or the sphere—and explainhow you know.

Which scale has the heavier blocks? Explain how you know.

Shape MachinesIn Problems C3 and C4, the goal is to figure out the function rule from the example, and then to apply the rule todetermine the output when given the input, and the input when given the output. Students are also asked to finda rule that describes what the machine does.

Some questions you might ask your students are:

• What shape went into the first machine?

• What shape came out of the first machine?

• What shape went into the second machine?

• What shape came out of the second machine?

• What is alike about the two figures that came out?

Part C, cont’d.


Problem C3. Notice the pattern in these two machines:

These machines below work in the same way. What goes into the machines? What comes out of them? Describewhat each machine does.



Part C, cont’d.


Number MachinesIn Problems C5-C7, the goals are the same as Problem C3. Some questions you might ask your students are:

• Look at the first example. What number goes in? What number comes out?

• Look at the second example. What number goes in? What number comes out?

• What do you think the machine is doing?

Students may find it useful to act out what is happening in these machines.



Part C, cont’d.




Mystery MachinesProblem C7. Notice the pattern in these two machines:

Part C, cont’d.


For each of the number machines below, describe what goes in or what goes out, then explain how the machineworks.

Part C, cont’d.


Two-SteppersIn Problems C8 and C9, the goal is to apply the function rules to determine the output when given the input, theinput when given the output, and also to describe the rule.

Some questions you might ask your students are:

• What number goes into the stepper first?

• What does the first part of the stepper do to the number?

• What does the second part of the stepper do to that number?

• What is the final result?

Problem C8. Notice the pattern:

What comes out?

What goes in?

Explain what the hexagon does.

Part C, cont’d.


Problem C9. Notice the pattern:

What comes out?

What goes in?

Explain what the hexagon does.

Part C, cont’d.

Problems C1-C9 are taken from Groundworks: Algebraic Thinking, Grade 1 and Grade 2, by Carole Greenes and Carol Findell (NewYork: Creative Publications, Wright Group/McGraw-Hill, 1998). The above materials may not be reproduced without the written per-mission of Creative Publications.


In the K-2 curriculum, students are frequently asked to think about patterns, but often their “pattern-sniffing”skillsend with simply finding the next object. [SEE NOTE 7]

Look at the problem below on patterns.

Problem D1. What algebraic ideas are in this lesson?

Problem D2. How are patterns used in this lesson?

Problem D3. What mathematics do you think students would learn from this lesson?

Part D: Critiquing Student Lessons (20 MINUTES)

NOTE 7. You may be puzzled by this series of subtraction problems, as well you should! After working on Problems D1-D5, youmay notice that in certain problems—where students “see”the pattern and then mindlessly fill in numbers that satisfy the pat-tern—the mathematical thinking and reasoning gets lost.

Problems D1-D5 are taken from Addison Wesley Mathematics, Grade 2 (Menlo Park, Calif.: Scott Foresman-Addison Wesley, 1993).


Problem D4. Are there any misconceptions that students might develop from this lesson?

Problem D5. How would you modify the problem, or what additional questions might you ask, to incorporate theframework for analyzing patterns?

Problem H1. Interview a teacher in the grade level above you. Pick one of the problems above and ask them thefollowing:

a. How does the content of this problem prepare students for algebraic thinking in their grade?

b. Why do they think this content is important?

c. How could this problem be extended for students in their grade?

Problem H2. Look at a problem in your own mathematics program for your grade level that you think illustratesalgebraic thinking. If you were to teach this problem after taking this course, how might you modify or extend itto bring out more of the content of algebraic thinking?

Part D, cont’d.

Homework

Session 10, K-2: Solutions 268 Patterns, Functions, and Algebra

Part A: Classroom VideoProblem A1. Reflection on the three questions should include the ideas described below.

a. The fundamental algebraic ideas (content) in this video are patterns that lead to functions and propor-tional reasoning.

b. The skills that the teacher expects students to demonstrate involve representing patterns with tallies,pictures, or charts to find relationships between numbers of people and numbers of feet or toes.

c. Students demonstrate their knowledge of the intended content by drawing pictures or tallies, or count-ing by twos or fives to answer questions about the relationships.

Problem A2. Ms. Webber encourages students to use patterns to predict when she asks students to look at thetable to answer questions such as the numbers of feet for six or 10 people.

Problem A3. Ms. Webber uses “doing and undoing” when she asks students to work backwards to find the num-ber of people for 70 toes.

Problem A4. Ms. Webber uses recursive thinking when she encourages students to count by twos or fives toanswer the questions.

Part B: An Example for Developing Algebraic ThinkingProblem B1. Answers will vary. One question might ask how to use the pictures to help find the relationshipbetween the pool number and the numbers of white and blue tiles.

Problem B2. Answers will vary. For the question above, you might discuss building each pool. For example, theblue part of Pool 1 is a 1-by-1 square and takes one blue tile, the blue part of Pool 2 is a 2-by-2 square and takesfour blue tiles, etc.

Problem B3. Answers will vary. At this level, many students will see that the blue tiles are always in the shape of asquare, and that there are always an even number of white tiles. Some may recognize that the number of whitetiles is always a multiple of four.

Problem B4. Answers will vary. At this level, teachers should encourage students to describe the shape of the bluetiles. Ask students to tell you how to build the blue part of Pool 4.Then ask them to think about putting white tilesaround that blue pool. How many white tiles do you need for the top? How many for the bottom? Now thinkabout putting tiles around the sides.

Problem B5. Answers will vary. At this level, students should actually build the pools with two different colorsquare tiles, then describe what they built. The process of building will help students put the patterns into words.Many students will use all these skills when solving this problem. For students that are having difficulty, build onlythe blue part first.

Problem B6. Answers will vary. All of the ideas described in the article’s K-2 section are appropriate for this levelstudent. Many students at this level will be able to answer questions posed in the article’s Grades 3-4 section.

Problem B7. Answers will vary. Ask students to describe the directions that the toes of the shoes face, using wordslike up and left.

Problem B8. Answers will vary. Ask students to repeat the pattern with you by saying: Up, left, up, left, up, left,up, ... . Now ask them to continue the pattern.

Solutions

Patterns, Functions, and Algebra 269 Session 10, K-2: Solutions

Part C: More Problems That Illustrate Algebraic Thinking Problem C1.

a. The mathematical content is patterns and proportional reasoning.

b. This prepares students for reasoning with proportions, predicting how many of a particular item will bein a strip of a given length, and using drawings to help solve problems.

c. This content requires the use of proportional reasoning and all the skills related to patterns: finding,describing, explaining, and using patterns to predict.

d. Students could approach these problems in several ways:

• Draw two strips and count the number of each shape in two strips

• Double the number for two strips to get the number for four strips

• Count by ones, twos, or threes to get the number for four or six strips

• Draw two, four, or six strips and count the number of each shape drawn

e. To extend students’ understanding, ask questions such as the following:

• If there are three circles, how many strips are there?

• For the first strip: How many more plus signs than circles in four strips?

• For the second strip: How many more Xs than circles in six strips?

• For the second strip: Could there be 10 Xs in some number of strips?

f. Answers will vary. Very few textbooks contain problems of this type.

Problem C2.

a. The mathematical content is balance, the notion that the objects on the lower pan on a pan balanceweigh more than the objects on the higher pan, and that one object that balances with two objects isthe heaviest of the three objects.

b. This prepares students for understanding equality as balance.

c. This content introduces equality as balance to set the stage for solving equations by manipulating equa-tions while maintaining balance.

d. Students might use the following reasoning to solve this problem:

Since Scale B has both a cube and a sphere on one side, and Scale A has only a sphere on one side, ScaleB must have heavier blocks.

e. To extend students’ understanding, ask students to describe other ways to solve the problem or to makeup weights for each block that would preserve the balance, and ask questions such as the following:

• Are there any blocks that are on both Scale A and Scale B?

• What will happen to the balance if you remove one sphere from both scales?


Solutions, cont’d.

Session 10, K-2: Solutions 270 Patterns, Functions, and Algebra

Problems C3-C4.

a. The mathematical content is function as a machine that takes one input, does something to that input,and returns exactly one output.

b. This prepares students for understanding function.

c. This content gives students an intuitive understanding of function. By using the machine metaphor andchoosing shapes rather than numbers, students realize that you must get exactly one output for everyinput.

d. Students approach these problems by examining and comparing the given inputs and outputs anddescribing what is the same and what is different in each case.

e. To extend students’ understanding, ask them to make their own shape machines.


Problems C5-C7.

a. The mathematical content is function as a machine that takes one input, does something to that input,and returns exactly one output.

b. This prepares students for understanding function.

c. This content gives students an intuitive understanding of function. These machines extend students’understanding of function by using numbers as inputs and outputs.

d. Students approach these problems by examining and comparing the given inputs and outputs anddescribing what is the same and what is different in each case.

e. To extend students’ understanding, ask them to make their own number machines.


Problems C8-C9.

a. The mathematical content is two-step functions depicted as a flow chart showing one input that goesthrough two operations and gives one output.

b. This prepares students for understanding two-step functions.

c. This content gives students an intuitive understanding of multi-step functions. These machines extendstudents’ understanding of function by using more than one operation.

d. Students approach these problems by following the flow chart. Start with an In number, take it throughthe first operation, put that number through the second operation, and then write the Out number.

e. To extend students’ understanding, ask them to make their own two-steppers.



Patterns, Functions, and Algebra 271 Session 10, K-2: Solutions

Part D: Critiquing Student LessonsProblem D1. Answers will vary.

Problem D2. Answers will vary. The patterns shown only relate to the answers. For the first row, the answersincrease by 10, but the problems themselves are not directly related. The second row’s answers are consecutivemultiples of 11, but the problems are not related. The third row’s answers decrease by one with unrelated prob-lems, and the fourth row’s answers decrease by five with unrelated problems.

Problem D3. Answers will vary. Students should strengthen their subtraction with regrouping skills.

Problem D4. Answers will vary. Students could learn to look for patterns in answers without actually looking atthe problems. The answers may not have any pattern.

Problem D5. Answers will vary. One way to modify the problem would be to make the problems relate to the pat-tern of the answers. For example, the first row could show 63 – 55, 63 – 45, 63 – 35, 63 – 25, 63 – 15, then ask whatcomes next.



Notes

Algebra Guide 10 K-2

Documents

Transcript of Algebra Guide 10 K-2