Mathematics TEKS Refinement 2006 – K-5

Tarleton State University

Patterns, Relationships, and Algebraic Thinking Activity:

Step’n Out!

TEKS:

(2.6) Patterns, relationships and algebraic thinking. The student uses patterns to describe relationships and make predictions. The student is expected to: (B) identify patterns in a list of related number pairs based on a real-life situation and extend the list; and (C) identify, describe, and extend repeating and additive patterns to make predictions and solve problems.

Overview:

Prerequisite: Students should have a basic understanding of the difference between a repeating pattern and an additive pattern. A repeating pattern can be defined as a pattern in which there is a distinct unit of repetition (Threlfall, 1999). Therefore a repeating “pattern can be generated by the repeated application of a smaller portion of the pattern. This would include patterns such as A, B, A, B, …, the days of the week, or a tessellation” (Liljedahl, 2004, p. 2). The numerical sequence 2, 4, 6, 8, 10,… is considered an additive pattern. Additive patterns are patterns that get larger with each step in the sequence (NCTM, 2000-2006; Oster, 2005). Even 1, 2, 3, 4, 5,… is a simple additive pattern. This lesson allows students to explore the steps used to investigate an additive pattern. This is a 10-step stair pattern that addresses building the pattern concretely, creating a chart, and looking at patterns within a number sentence. The students are also introduced to a graphic organizer that will guide them through the process.

Materials:

One container of 60 one-inch cubes and centimeter cubes for each pair of students Graphic Organizer #1 Graphic Organizer #2 Colored pencils, crayons, or markers Large chart paper Marker

Grouping:

Whole group or small groups with students working in pairs

Time:

Four parts, each 30 to 45 minutes

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 1

Mathematics TEKS Refinement 2006 – K-5

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Lesson: Procedures 1.

Part I Give the students the following problem: Tanya is building a staircase in the pattern shown. The blocks are 1-inch cubes. She wants the last step to be 3 inches tall. How many cubes does she need in order to build the staircase? (You may want to change the name to a student’s name in your class.)

Notes Show a picture of the following staircase to the students, and read the problem at the left.

Have the students tell you what they are being asked to find. Have the students restate the problem in their own words, listing the facts of the problem. Ask them what tools they need to solve the problem. Model writing this information in the top left hand section of Graphic Organizer #1. Have the students also write this information in their graphic organizer.

As part of George Polya’s fourstep process, students are to first “understand the problem” (Motter, 2006). This includes restating the problem in their own words, using Begin by having the students explain how to build the staircase, and teacher demonstrates materials to model the problem, this on the overhead or document camera. As listing given facts or stating the goal and deciding if an exact teacher builds staircase, instruct students to answer is needed. build one at their desks. Give the students time to build their three-step stair. Walk around to help any students that need help forming the three-step stair and to assess their understanding. 2.

Show the students how to record what they have just made. Draw the three steps on the graph paper that is imbedded in the graphic organizer. Instruct students to begin their drawing in the bottom left hand corner of the graph paper.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Walk around to make sure everyone has been able to draw the three steps on their graphic organizer. Give assistance to any student that may need the help. All of the following graphs from graphic organizers begin with the Grade 2 Page 2

Mathematics TEKS Refinement 2006 – K-5

Procedures

Tarleton State University

Notes yellow block. Each time a new set of blocks is added to enlarge the steps, the color is changed. The students do not have to complete the problem in this way. It is shown this way to demonstrate the additive process. Picture

3.

After everyone has successfully drawn the three steps, pose the following question:

Record the students’ estimates on large chart paper as they give them.

“I wonder how many blocks it would take for Tanya to build the stairs to 10 steps?” 4.

After taking estimates, have the students repeat the process of writing what they understand about this problem on Graphic Organizer #2.

5.

Instruct the students to build the stairs to five steps using 1-in. cubes. Have them add the steps to the picture section of Graphic Organizer #2.

Their graphic organizers should look similar to the following: Picture

After they have completed the preceding steps, ask: “Would anyone like to change their guess now that half of the steps are built?” Record any new estimates on the same chart paper. Mark off any estimates that have been surpassed. Have the children explain why they should be marked off.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 3

Mathematics TEKS Refinement 2006 – K-5

Procedures 6.

7.

1.

2.

Instruct the students to continue building to complete the original problem (10 steps). When they are finished, they record the number of blocks used on their graphic organizer.

Tarleton State University

Notes The graphic organizer should look similar to the following: Picture

Take up Graphic Organizer #2 and save for the next day. Part II Hand out Graphic Organizer #2 from the previous day’s lesson. Have students determine how many blocks they used to build the ten steps.

Picture

55 blocks

Have blocks available for the students to rebuild the steps if necessary. Instruct the students to use any method they choose to determine the number of cubes used. Remind them about the two other areas of the graphic organizer – make a table/chart/list or a number sentence. Have the students record their answer in the top right corner of the paper.

3.

Ask for the totals that each of the students recorded in the corner of their papers. As

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Just record the total given by each student; do not worry about correct Grade 2 Page 4

Mathematics TEKS Refinement 2006 – K-5

Tarleton State University

Procedures students give their totals, write the answers on a new piece of chart paper.

Notes answers now.

After having all the possible answers, ask the students to select a number on the estimate chart that can be marked off and justify why they believe the number should be crossed off. (Students might say: The number is more than my total, or it’s less than my total blocks.) Ask the class to show a thumb up if they agree or thumb down if they disagree. Call on students that have their thumbs down to explain why they disagree. Allow students to justify their answers. Mark off numbers as the class comes to a total agreement on each number. Circle numbers that are left. 4.

At the end of the discussion if there is a discrepancy in the total number of blocks, tell the class: “From what you have told me the total is between _________ and _________. (numbers are based on the totals given by the students—the least and greatest number) Record this at the bottom of the chart. Explain that tomorrow they will have a chance to show how they found their answer, and they will have a chance to prove why they believe their answer is the correct one.

1.

Part III The following day allow students to explain how they found their answers. Continue this process until all the students’ methods have been presented.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

All numbers that are not between the two numbers you have given as the totals should be marked out on the estimation chart. If they are not, go back to the chart and ask if the number or numbers come between the two numbers that are used to find the range of possible totals. Continue this process until all numbers that are not between the two totals are eliminated.

If this is the first time you have had the students present their problem solving process, they may be timid about sharing. It takes time to develop a trusting relationship with your students so they feel comfortable sharing their thinking processes (metacognition). Grade 2 Page 5

Mathematics TEKS Refinement 2006 – K-5

Procedures

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Notes If students have wrong answers, do not tell them. Many times when students have to explain and justify their answers, they will find their own mistakes. Allow the other students to ask questions and challenge the processes that are being used. Please remember that this is done with utmost respect for those presenting. Sometimes students will need to demonstrate what they mean because they have a difficult time verbalizing their thought processes. For assessment purposes, take anecdotal notes regarding the processes used by the students.

1.

PART IV Other methods: If students do not use the following three methods, teacher should model them. Do not do this until all the students’ processes have been presented.

Find a Pattern: Show that the answer could also be found by restacking the blocks so each row contains 10 blocks. This is accomplished by taking the blocks from the top and adding them to the bottom steps.

It is important for students to see methods to solve the problem other than counting all the blocks one at a time. There are more solutions than the ones presented so do not exclude any of the students’ work if they can justify how they got their answer. Do this on your copy of the ten steps after everyone has had a chance to show their method of getting the total if no one used the totaling to ten method. The process of finding a pattern by restacking follows. On the original drawing, note the labeling of blocks with letters A thru J.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 6

Mathematics TEKS Refinement 2006 – K-5

Tarleton State University

Procedures

Notes Picture

55 blocks

A B C D E F G H I

J

You move blocks one row at a time from the top of the figure in order to make additional rows of ten.

Step 1: Move block A. Picture

55 blocks

B C D E F G H I

J

A

You now have 2 rows of ten.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 7

Mathematics TEKS Refinement 2006 – K-5

Tarleton State University

Procedures

Notes Step 2: Move blocks B and C. Picture

55 blocks

D E F G H I

J

C B A

You now have 3 rows of ten.

Step 3: Move blocks D, E, and F. Picture

G H I

55 blocks

J

F E D C B A

You now have 4 rows of ten.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 8

Mathematics TEKS Refinement 2006 – K-5

Procedures

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Notes Step 4: Move blocks G, H, I, and J. Picture

55 blocks

J I

H G

F E D C B A

You now have 5 rows of ten with 5 additional blocks. Table/Chart Method: Draw a chart on the bottom left hand side of the graphic organizer. Step Number

Number of Blocks/Steps

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

Total number of blocks

1 1+2=3 3+3=6 6+4=10 10+5=15 15+6=21 21+7=28 28+8=36 36+9=45 45+10=55

Number Sentence: The number sentence can be originally written: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

Patterns, Relationships, and Algebraic Thinking Step’n Out!

All students should learn to use a table/chart. So continue putting together problems that have a growing pattern and can be worked faster using a chart than drawing out a picture of each object. Example: number of wheels on 5 bicycles.

When finding a total, some students will understand looking for patterns by grouping numbers together but others will not.

Grade 2 Page 9

Mathematics TEKS Refinement 2006 – K-5

Tarleton State University

Procedures

Notes

Or 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55

The commutative property of addition allows us to move the numbers around: 1 + 2 = 2 + 1.

Or 10 + (9 + 1) + (8 + 2) + (7 + 3) + (6 + 4) + 5 = 55

The associative property enables us to group the numbers together: (1 + 2) + 3 = 1 + (2 + 3).

Or 10 + 10

+

10

+

10 +

10

+ 5 = 55

In this example, teacher shows the students how to look for a pattern. This process may lead to the answer more quickly than just counting each block.

Some students will continue to count each block because they are not as advanced in their understanding of mathematics. If you continue to demonstrate “looking for a pattern” and “creating a table/chart/list,” students will begin to utilize these more advanced problem-solving skills.

Homework:

Assign other additive patterns as homework and have students show their method of finding what the total number of pieces would be at step 10. As students learn to use a chart, require them to find the answer to the 10th step using a chart only (especially concerning TEKS 2.6A).

Assessment:

Since you have just introduced some other methods to finding the total to a additive pattern, assessment at this stage is still by observation and writing anecdotal notes. Continue doing additive patterns, and use a chart to record answers throughout the year.

Extensions:

Integrate with books from literature that have a additive pattern in them—these would be books that have an item that grows by one or more each time the page is turned. Document the additive pattern on paper as the book is read orally to your students a second time.

Resources:

Liljedahl, P. (Summer 2004). Repeating pattern or number pattern: The distinction is blurred. Focus on Learning Problems in Mathematics. Online at http://www.findarticles.com/p/articles/mi_mONVC/is_3_26/ai_n9505504/pg_1

Motter, A. (September 2006) George Polya: 1887 – 1985. Online at http://www.math.wichita.edu/history/men/polya.html Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 10

Mathematics TEKS Refinement 2006 – K-5

Tarleton State University

National Council of Teachers of Mathematics. (2000-2006). Illuminations Powerful Patterns. Online at: http://illuminations.nctm.org/LessonDetail.aspx?ID=U69. Reston, VA: NCTM. Nugent, Glenda. (1995). Hands-On Math : Manipulative Activities for the 2-3 Classroom Cypress, CA: Creative Teaching Press, Inc. Oster, C. (2005). Growing Patterns. Online at: http://www.mathperspectives.com/pdf_docs/lesson_3.pdf. Bellingham,

WA: MathPerspectives, Teacher Development Service. Threlvall, J. (1999). Repeating patterns in the early primary years. In A. Orton (Ed.), Patterns in the teaching and learning of mathematics (pp. 18-30). London: Cassell. Van de Walle, J.A. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.) Boston, MA: Allyn and Bacon Modifications: Be careful to select peer tutors that can teach and be supportive through the use of caring words so as not to cause struggling students to “shut down”.

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Grade 2 Page 11

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Explain how you derived your answer.

What am I supposed to find?

Picture

Graphic Organizer #1 Write down your thought processes as you solve the following problem.

Mathematics TEKS Refinement 2006 – K-5

Grade 2 Page 12

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Graphic Organizer #2

Patterns, Relationships, and Algebraic Thinking Step’n Out!

Explain how you derived your answer.

Table/Chart/List

Grade 2 Page 13

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Number Sentence

Write down your thought processes as you solve the following problem. Picture What am I supposed to find?

Mathematics TEKS Refinement 2006 – K-5