Patterns, Patterns, Patterns… - Carnegie Learning · Step 5 and 6: She continues this pattern two more times, alternating between black and green sets of beads. rite the first six
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2.1 Developing Sequences of Numbers from Diagrams and Contexts • 43A
Essential Ideas
•A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term in a sequence is an individual number, figure, or letter in the sequence.
•A diagram can be used to show how each term changes as the sequence progresses.
• There are many different patterns that can generate a sequence of numbers. Some possible patterns are:
• adding or subtracting by the same number each time.
•multiplying or dividing by the same number each time.
• adding by a different number each time, with the numbers being part of a pattern.
• alternating between adding and subtracting.
Common Core State Standards for Mathematics8.FFunctions
Define,evaluate,andcomparefunctions.
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Key Terms sequence
term
ellipsis
Learning GoalsIn this lesson, you will:
Write sequences of numbers generated
from the creation of diagrams and
written contexts.
State varying growth patterns of
sequences.
Patterns, Patterns, Patterns…Developing Sequences of Numbers from Diagrams and Contexts
•How could you determine the next term in Sequence C?
•How could you determine the next term in Sequence D?
•How did you determine the third term in Sequence B?
•How did you determine the twenty-fifth term in Sequence C?
•How did you determine the twelfth term in Sequence D?
•Was it necessary to list all the terms leading up to the term you were trying to determine?
Problem 1Sequence, term in a sequence, and ellipsis are defined. Examples of sequences that involve numbers, figures, and letters are provided. Students determine the next term in each sequence.
Grouping
•Ask a student to read the introduction to Problem 1 aloud. Discuss the definitions and worked example as a class.
•Have students complete Questions 1 through 4 with a partner. Then share the responses as a class.
Discuss Phase, Problem 1
• Explain what a sequence is in your own words.
•Do all sequences have terms? How many terms are in a sequence?
• Is the term in a sequence a number, a figure, or a letter?
•How have you heard the word sequence or term used outside of the math class? How is the usage of these words outside of the math class related to their meaning in math class?
Share Phase, Questions 1 through 4
•How did you determine the next term in Sequence A?
•How could you determine the next term in Sequence B?
NoteIn mathematical usage, the terms sequence and series have different meanings. This chapter addresses sequences, not series; it would be inaccurate to use sequence and series interchangeably in the mathematical sense of the terms.
Problem 2Students write a sequence to represent designing a bead necklace. The terms of the sequence follow the pattern of even numbers. The context helps students make sense of this pattern.
GroupingHave students complete Problems 2 through 5 with a partner. Then share the responses as a class.
Share Phase, Problem 2
• Explain how many black beads and green beads are added to the necklace during each of the first 6 steps.
• Explain how you can determine how many beads would be added to the necklace in the next step without drawing the beads.
•Will there be more green or black beads on the necklace when you are done? Explain?
•How did you determine the values in your sequence?
•Did you draw all the beads on the necklace to complete the sequence? If so, did you count all the beads or just the newly-added beads to determine the next term in the sequence? If not, explain your thinking process.
•How much did the numbers in your sequence grow by each time? What do those numbers represent in the context of the problem? Why are all those numbers even?
Problem 3Students write a sequence to represent crafting toothpick houses. The terms of the sequence increase by a constant value; however, the initial term is one more than the amount of increase between consecutive terms. The context helps students make sense of this discrepancy between the initial value and the amount of increase between consecutive terms.
Share Phase, Problem 3
•How many houses require the use of 6 new toothpicks?
•How many houses require the use of 5 new toothpicks?
•What operation was used to determine the terms in your sequence?
•What do you notice about every other term in your sequence? Is there a pattern?
•Why is every other term in your sequence even?
•Why is every other term in your sequence odd?
•Did you draw all the houses to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process.
• If a house is comprised of six toothpicks, then why doesn’t your sequence grow by six each time?
Problem 4Students write a sequence to represent a card trick. This is the only problem in this lesson where the sequences are decreasing. The first sequence decreases by a constant value of two. The second sequence decreases by a pattern of odd numbers, with the odd numbers being the terms in the first sequence. The context helps students make sense of why the sequences are decreasing and their patterns of decrease.
Share Phase, Problem 4
•How did you determine the terms in the first sequence?
•How did you determine the terms in the second sequence?
•Did you use the diagram of cards to complete the sequence? If so, explain how you wrote the sequence from the diagram. If not, explain.
• If the original diagram has eight columns and six rows, why are only 13 cards removed instead of 14 cards to determine the first term in the first sequence?
•How is the decrease by 2 in the first sequence represented in the diagram?
•How is the decrease in the second sequence represented in the diagram?
•Add the first terms of the two sequences together, the second terms, etc. and use the sum in each instance to create a new sequence. What familiar pattern do you notice in the new sequence?
Problem 5Students write a sequence to represent triangular arrangements of pennies. The terms of the sequence increase by adding consecutive integers. The context helps students make sense of this pattern.
Share Phase, Problem 5
•How did you determine the first eight terms in your sequences?
•How many new rows of pennies are added in each term?
•Did you draw all of the pennies to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process.
•How is the increase by consecutive numbers represented in the diagram?
Problem 6Students write a sequence to represent building stairs. This is the only problem in this lesson which requires students to interpret three-dimensional drawings. The terms of the sequence increase by adding consecutive odd integers. The context helps students make sense of this pattern.
GroupingHave students complete Problems 6 through 11 with a partner. Then share the responses as a class.
MisconceptionStudents may have difficulty drawing the stacked cubes and counting the number of exposed faces. Some students may benefit from using actual cubes to build the models.
Share Phase, Problem 6
•How did you determine the first five terms in your sequence?
•How many additional exposed faces appear between the first and second diagram?
•How many additional exposed faces appear between the second and the third diagram?
•Did you draw or construct the stacked cubes to complete the sequence? If so, explain how you wrote the sequence from your diagram or model. If not, explain your thinking process.
•How were you able to organize your counting process when counting the number of exposed faces of the cubes? What patterns did you notice?
•How much does the increase between the terms grow each time?
• The increase between the terms grows by two every time. How is this two represented in the diagram?
•How is this problem similar to Problem 3 when you crafted toothpick houses?
•How is this problem different from Problem 3 when you crafted toothpick houses?
Problem 7Students write a sequence to represent arranging classroom tables. The terms of the sequence increase by a constant value; however, the initial term is two more than the amount of increase between consecutive terms. The context helps students make sense of this discrepancy between the initial value and the amount of increase between consecutive terms. This problem is similar to Problem 3 because adjacent figures are sharing sides thus reducing the total count for the term number. This problem is different from Problem 3 because the sequence represents the perimeter rather than the total number of sides in the diagram.
Share Phase, Problem 7
•How did you determine first five terms in your sequence?
•Did you draw all the tables to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process.
• If a table seats five students, why doesn’t your sequence grow by five each time?
•How much does each term grow when a table is added?
Problem 8Students write a sequence to represent the numbers of petals in various stages when drawing a flower. Students follow the directions to draw the flower petals and generate the sequence of numbers. This is the only problem in this lesson where the sequence is increasing by a factor of 2. The context helps students make sense of this pattern.
MisconceptionStudents may have difficulty keeping track of what petals were drawn in which stage. If this is the case, have students use a different color of pencil for the petals generated in each stage. Also, encourage students to write the value for each term of the sequence as they complete each stage of the flower drawing, rather than writing all the values after the entire flower has been drawn.
Share Phase, Problem 8
•Without drawing, how could you determine the next term in the sequence?
•What is the growth pattern of this sequence?
•What other mathematical operation could you use to represent the growth of this sequence?
• In what way is the growth of the terms in this sequence different from all other problems?
•Problems 3 and 7 also increase by a constant value. Why is it that the sequence in this problem is a list of multiples, but that is not the case with the sequences in Problems 3 and 7?
Problem 9Students write a sequence to represent babysitting earnings. This is the only problem in this lesson where the sequence alternates between decreasing by one values and increasing by a different value. The context helps students make sense of this pattern.
Share Phase, Problem 9
•What is the growth pattern of this sequence?
•Why does this growth pattern make sense in context of the problem?
Problem 10Students write a sequence to represent recycling. This is the only problem in this lesson where the sequence begins with zero. Because the sequence begins with zero and the terms of the sequence increase by a constant value, the sequence represents a list of multiples.
Problem 11Students write a sequence to represent selling tickets. In this problem, students must do some arithmetic to determine the initial value of the sequence. The sequence increases by a constant value.
Share Phase, Problem 11
•How did you determine the starting value of the sequence?
•What is the growth pattern of this sequence?
• Explain how the growth pattern of this sequence connects to the context of the problem.
Talk the TalkStudents summarize the patterns of each sequence in Problems 2 through 11, and describe the similarities.
GroupingHave students complete the Questions 1 and 2 with a partner. Then share the responses as a class.