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&&&Jj&&& Lesson 2  Patterns That Grow  Gwen, Stacy, and Phoebe
Prelesson date Oct. 25th, 2006
Research Aim:
Students will grow into persistent and flexible problem solvers.
Broad Content Goal:
Students will communicate their mathematical ideas clearly and respectfully.
Lesson Objectives :
Students will recognize growing patterns.
analyze how growing patterns are created.
extend growing patterns.
communicate clearly to explain a growing pattern.
NCTM Content Standard Algebra 35
Describe, extend, and make generalizations about geometric and numeric patterns.
Represent and analyze patterns and functions, using words, tables, and graphs.
Express mathematical relationships using equations.
Represent the idea of a variable as an unknown quantity using a letter or symbol.
Lesson Overview:
Using several growing patterns, students will work with pictorial, numerical, and verbal representations to analyze, extend, and describe these patterns.
Assumptions about Prior Knowledge:
Students have had experience with repeating patterns (ex AB, AB) and representing patterns visually with manipulatives or pictures.
Accommodations/Modifications/ Extensions:
See notes at end of lesson.Steps
Instructional activitiesAnticipated Student ResponsesRemarks on TeachingIntroduction
Present problem #1
Model communicating ones ideas clearly and respectfully
Ask students if they have played with Legos, discuss what they built, etc.
Present the following problem to the students:
The Staircase Problem
Sara is building a skyscraper with her Legos. She needs to make a staircase for her little Lego people to get to the top. As she builds the staircase, it looks like this:
1 step
2 steps
3 steps
How many Lego pieces will she need for the staircase when there are 10 steps?
Problem Extension: Present this to students who are ready.
How many Legos when there are 25 steps?
Ask students to share their solutions to the problem.
This may be done at the board, on individual whiteboards, on overheads, or on large pieces of paper, chart paper, or posterboard.
.Students will make connections to their own play experiences with Legos by sharing what they have built. Teacher may ask, Have you ever built a very tall building? How would you get your little Lego people to the top? Encourage creative thinking but focus on building a large staircase.
Students (individually or in pairs) may solve the problems and identify the pattern in the following ways:
Strategy A: Building staircases with manipulatives and count to find the answer.
Strategy B: Draw a picture using graph paper or regular paper.
Strategy C: Use a numerical pattern: for a staircase with 10 rows, the total number of Legos is 10+9+8+7+6+5+ 4+3+2+1=55
Strategy D: Make a Tchart or organized list:
Height of Staircase
Number of Legos
1
2
3
4
5
6
7
8
9
10
1
3
6
10
15
21
28
36
45
55
Strategy E: Use a formula: if n = height of staircase the number of Legos= EMBED Equation.3
.
An example of how a student may describe the pattern could be: Each time Sara adds a new step to her staircase, she adds the previous row of Legos plus one more. This is a growing pattern.
Materials Needed: manipulatives (Legos, blocks, tiles, counters, etc.), graph paper
This problem is a modification of the bowling pin problem from the NCTM Illuninations website.
Ask the students to describe what is happening as Sara builds her staircase. Help them to identify this pattern as a growing pattern and talk about its characteristics. What makes it different from a pattern that just repeats?
It is likely that students will solve the problem of finding the total Legos in a 10 step staircase by using manipulatives or drawing a picture (Strategies A and B).
Encourage students to find the growing pattern and then seek efficient ways to add consecutive numbers (Strategy C) by pairing compatible numbers. (Example 9+1, 8+2, 7+3, 6+4 then add the 10 +5).
If students do not come up with it on their own, help them (individually or as a class) think about using a table to organize their data (Strategy D). The teacher may need to ask questions to guide students to this method so that the numeric pattern becomes apparent to them. It is important that students see the value in organizing data in a table.
Students who have had experience with InputOutput Boxes (or a Function Machine) may recognize this chart to be the same idea. What then is the rule for this machine?
While it is highly unlikely that elementaryage students could generate this formula, there are some who could use it with success if they are presented with it.
Solving the Problem Extension will be cumbersome (yet, possible) using any strategy except Strategy E. Present Strategy E to only those students who are ready for it.
The teacher models and encourages appropriate responses to those who are sharing. Use a chart paper or the board to model how to write about this pattern. Begin by drawing the original 3 steps and then draw steps 4 and 5. Then write a clear explanation as a model.Present Problem #2
Sharing solutions
Ask students to solve The I Problem. **
See handout.
Ask the students to share their solutions.
Students (individually or in pairs) may solve the problems and identify the pattern in the following ways:
Strategy A: Building the I with manipulatives and count to find the answer.
Strategy B: Draw a picture using graph paper or regular paper.
Strategy C: Complete the Tchart.
# of Steps
# of Tiles
1
2
3
4
5
6
7
8
9
10
25
7
8
9
10
11
12
13
14
15
16
31
Strategy D: Make a function rule or formula.
Steps + 6 = Tiles or
S + 6 = T or
+ 6 =
An example of how a student may describe the pattern could be:
The number of tiles in this growing pattern increases from one step to another by six tiles. The two horizontal rows remain the same (3 + 3 = 6) and the vertical column increases by one for each step. So the number of steps plus 6 will equal the number of tiles.
This is a similar problem, but this time students have been given a Tchart as a prompt, thereby encouraging them to use this method.
Note again, the similarity with the InOut Boxes or Function Machine.
The S and T are highlighted in the Tchart to facilitate the usage of letters to replace numbers. Also the symbol of a square and a circle were used on the worksheet, again to facilitate the using of symbols to represent numbers in an equation.
It will be much easier for students to discover the formula for this growing pattern. Can the students explain why 6 is added to the number of steps?
Students may need help with the math vocabulary that makes for clear communication. Post on a chart such words as horizontal, vertical, column, row, and increase.
During this sharing time, the teacher will have an opportunity to informally assess how well students understand the content as well as the level of communication. Students have an opportunity to demonstrate how clearly they can communicate their thinking. The class will also demonstrate their level of peer respect. SummaryStudent Assessment
Students complete Growing T (see handout).
Student Reflection
This worksheet is similar to the problems in this lesson and will allow the students to show their thinking.
1 The teacher could choose up to one question from each column Clear Communication, Respectful Communication, Flexible Thinking, and Persistence for students to write about. (See handout: Reflecting on Problem Solving) These questions could be presented as a handout or on an overhead
2. Another way to use the reflection questions from the 4 column chart would be to make 4 small groups and ask each group to discuss all 3 questions from a column and then briefly share with the whole class.
Teacher Reflection
Were students engaged in the activities?
Were they motivated to complete the tasks?
Did they demonstrate persistence?
Were they able to identify and analyze growing patterns?
To what degree were the students able to extend a growing pattern?
Was there evidence that students are ready to use symbols to represent numbers in equations?
To what degree were students able to form a rule for a pattern?
How clearly were they able to communicate their thinking as they described the patterns?
Were students respectful of their classmates?
Were students able to solve the problems in more than one way?
How successfully were the students able to correctly complete Growing T?
What can be learned from the students reflections?Accommodations, Modifications, and Extensions.For all students, there are many childrens books that could be used with this lesson as an integral part or as an extension. See the Literature Resources handout.
For students who need a more gradual introduction to the idea of growing patterns, use such literature selections as The House That Jack Built or There Was an Old Lady Who Swallowed a Fly. See Literature Resources.
For students who need more practice, offer The U Problem ** (see handout).
For the able student there are several extensions that could be offered:
Other letters of the alphabet will provide similar patterns as I. Encourage the student to choose another letter of the alphabet to explore. Ask them to discover the pattern, analyze, and describe it.
Ask students to make up growing patterns for others to explore.
Pascals Triangle offers many patterns to discover, analyze, and describe. Go to
HYPERLINK "http://mathforum.org/workshops/usi/pascal/pascal_middisc.html" mathforum.org/workshops/usi/pascal/
pascal_middisc.html for a good discussion of the triangles properties and ideas for using it with students of many abilities.
**  These problems are adapted from Lessons for Algebraic Thinking Grades 35 by Maryann Wickett, Katharine Kharas, and Marilyn Burns, Math Solutions, Sausalito, CA, 2002
Lesson #2 Patterns That Grow
Literature Resources:
The House That Jack Built There are several versions (Diana Mayo, Simms Taback, Jeanette Winter, etc.) All tell the cumulative tale of cats, dogs, cows, and other creatures involved with the The house that Jack built.
There Was an Old Lady that Swallowed a Fly (Simms Taback or Mary Ann Hoberman and Nadine Bernard Westcott) Another classic tale which can be sung.
Two of Everything by Lily Toy Hong (Morton Grove, IL: Albert Whitman & Company, 1993) One spring morning, Mr. Haktak, a poor farmer, unearths a brass pot in his garden. Placing his coin purse inside for safekeeping, he carries his discovery home to his wife. After she accidently drops her hairpin inside, Mrs. Haktak reaches into the pot and, to her amazement, pulls out two identical hairpins and two matching coin purses. Quickly deducing the magic secret, husband and wife work feverishly to duplicate their few coins, creating enough gold to fill their hut. But their good fortune takes an unexpected turn.
One Grain Of Rice: A Mathematical Folktale by Demi, Scholastic Press, 1997) It's the story of Rani, a clever girl who outsmarts a very selfish raja and saves her village. When offered a reward for a good deed, she asks only for one grain of rice, doubled each day for 30 days. Remember your math? That's lots of rice: enough to feed a village for a good long timeand to teach a greedy raja a lesson.
Annos Mysterious Multiplying Jar by Mitsumasa Anno, (New York: Philomel Books, 1983) In the guise of a tale about a jar, the fatherson authors illustrate the amplitude of multiplication. For more advanced students, an afterword provides a textual explanation of factorials. The book is a stunning visualization of concepts.
Annos Math Games II by Mitsumasa Anno, (New York: Philomel Books, 1989) The Magic Machine is a machine that follows a rule to change the items put inot it into the items that come out. Its possible to change the controls on the machine and therefore change the rule so students can figure out what the machine is doing each time.
Anno's Magic Seeds by Mitsumasa Anno Putnam Juvenile, 1999) An old man gives Jack two golden seeds and a simple formula for becoming selfsufficient. He faithfully follows the directions, eating one of the seeds, which amazingly takes care of his hunger for the year, and planting the other the following spring, which produces two new seeds. He enjoys several years of easy subsistence until he decides to fend for himself one winter and plant both seeds. The next and each successive season begin a geometric progression of harvests?2 sprouts produce 4 seeds (one of which he eats), 3 plants produce 6 seeds, 5 yield 10, etc. In no time at all, he has a bountiful surplus. Even when a hurricane devastates their crops and storehouse, 10 seeds are saved and the family begins anew.
Name ___________________________
The I Problem
step 1 step 2 step 3 step 4
Talk with your partner about what is happening to this pattern?
What would the I look like at step 10? 25?100? (How can you figure this out?)
# of Steps # of Tiles 1
2
3
4
.
.
.
.
.
.
.
.
.
10
.
.
.
.
257
8
9
10Describe the I pattern below.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Name____________________________
The U Problem
1 2 3
Build Us up to at least five steps.
Draw the Us on graph paper.
Make a Tchart.
Figure out the number of tiles for a 10step U.
Write about the pattern. (Remember to describe the pattern and tell about a rule.)
If you can, figure out how many tiles for a 25 step U and a 100 step U.
Name: ____________________ Date: _____________
The Growing T
Tommy is building the letter T with cubes. This is what he started with.
Tommy continues to add another cube to his letter T. The next stage looks like this:
He wanted to add another block so that is letter T is growing longer. The third stage looks like this:
1. What would Tommys T look like if he continues the pattern to the fourth stage?
2. How many blocks would Tommys T have if he continues the pattern to the fifth stage?
3. How many blocks in his pattern at the tenth stage?
4. Describe the pattern in Tommys T.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
5. Create a formula or a rule that fits the pattern.
Reflecting on Problem Solving
Clear CommunicationRespectful CommunicationFlexible Thinking
PersistenceWhat math words could help us share our thinking about this problem? Choose 2 and explain what they mean in your own words.
Did someone else solve the problem in a way you had not thought of? Explain what you learned by listening to a classmate.What other problems or math topics does this remind you of? Explain your connection.What did you do if you got stuck or felt frustrated? What could you use besides words to show how to solve the problem? Explain how this representation would help someone understand.
Did you ask for help or offer to help a classmate? Explain how working together helped solve the problem.Briefly describe at least 2 ways to solve the problem. Which is easier for you?What helped you try your best?
or
What do you need to change so that you can try your best next time?If you needed to make your work easier for someone else to understand, what would you change?What helped you share and listen respectfully when we discussed the problem?
or
What do you need to change so that you can share and listen respectfully next time?What strategies did you use that you think will be helpful again for future problems?Do you feel more or less confident about math after trying this problem? Explain why.
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