Standard 4.3 Patterns and Algebra
All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

Big Idea Algebra provides language through which we communicate the patterns in mathematics.

4.3 A. Patterns

Essential Questions

Enduring Understandings

- How can change be best represented mathematically? (4.5C1; 4.5F1; 4.5F2; 4.5F3; 4.5F4)

- How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations? (4.5C1)

- The symbolic language of algebra is used to communicate and generalize the patterns in mathematics.

- Algebraic representation can be used to generalize patterns and relationships.

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

1.    Recognize, describe, extend, and create patterns.

·        Using concrete materials (manipulatives), pictures, rhythms, & whole numbers

·        Descriptions using words and symbols (e.g., “add two” or “+ 2”)

·        Repeating patterns

·        Whole number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward)

1.         Recognize, describe, extend, and create patterns.

·        Descriptions using words and number sentences/expressions

·        Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)

 1.         Recognize, describe, extend, and create patterns.

·        Descriptions using words, number sentences/expressions, graphs, tables, variables (e.g., shape, blank, or letter)

·        Sequences that stop or that continue infinitely

·        Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11, . . . or 800, 400, 200, . . .)

·        Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4, . . . could be 8, or 7, or … )

1.         Recognize, describe, extend, and create patterns involving whole numbers.

·        Descriptions using tables, verbal rules, simple equations, and graphs

Sample MC Item: Last year, the cafeteria at Kyle's school recycled 100 pounds of the trash that was collected. This year was the second year of recycling, and the cafeteria recycled twice as much. If the amount of trash the cafeteria recycles doubles each year, how much will be recycled in the fourth year?
a. 1600 pounds

* b. 800 pounds

c. 600 pounds

d. 400 pounds

 1.         Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.

·        Descriptions using tables, verbal rules, simple equations, and graphs

·        Formal iterative formulas (e.g., NEXT = NOW * 3)

·        Recursive patterns, including Pascal’s Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1,  1,  2,  3,  5,  8, . . .    (where NEXT = NOW + PREVIOUS)

Sample MC Item: Which equation fits this pattern? 2, 6, 18, 24, . . a. NEXT = NOW + 4

b. NEXT = NOW + 3 *

c. NEXT = 3 * NOW

d. NEXT = NOW / 3

 1.         Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

·        Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

·        Finite and infinite sequences

·        Generating sequences by using calculators to repeatedly apply a formula

 1.         Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.

·        Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

·        Finite and infinite sequences

·        Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

·        Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)

·        Generating sequences by using calculators to repeatedly apply a formula

1.         Use models and algebraic formulas to represent and analyze sequences and series.

·        Explicit formulas for n^th terms

·        Sums of finite arithmetic series

·        Sums of finite and infinite geometric series

3.       Use inductive reasoning to form generalizations.

Essential Questions

Enduring Understandings

- How are patterns of change related to the behavior of functions? (4.5F1; 4.5F2; 4.5F3; 4.5F4)

- Patterns and relationships can be represented graphically, numerically, symbolically, or verbally. (4.5E1)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

 1.         Use concrete and pictorial models to explore the basic concept of a function.

·        Input/output tables, T-charts

 1.         Use concrete and pictorial models to explore the basic concept of a function.

·        Input/output tables, T-charts

·        Combining two function machines

·        Reversing a function machine

Sample Classroom Performance Task:
Students are split into groups. Each group receives a large bottle of water and 10 smaller water bottles. Each group also receives a stack of paper or plastic cups—one group receives 2 oz. cups; one group receives 3 oz. cups; one group receives 4 oz. cups; etc.

The students in each group first fill up the cups from the large bottle only. Students record the number of cups filled. The students then fill up additional cups from one small water bottle, recording the total number of cups filled with water. In each group, the process is repeated until the 10 small water bottles have been emptied.

Each group will then prepare a graph of the number of cups filled vs. the number of bottles emptied. Each group then shares the results with the rest of the class. The class compares and contrasts the graphs.
Individual students then attempt, for each size cup, to describe the relationship between the number of bottles and the number of cups that could be filled.

 1.         Graph functions, and understand and describe their general behavior.

·        Equations involving two variables

 1.       Graph functions, and understand and describe their general behavior.

·        Equations involving two variables

·        Rates of change (informal notion of slope)

 2.        Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.

2.       Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

·        Slope of a line or curve

·        Domain and range

·        Intercepts

·        Continuity

·        Maximum/minimum

·        Estimating roots of equations

·        Intersecting points as solutions of systems of equations

·        Rates of change

3.       Understand and perform transformations on commonly-used functions.

·        Translations, reflections, dilations

·        Effects on linear and quadratic graphs of parameter changes in equations

·        Using graphing calculators or computers for more complex functions

4.       Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.

·        Linear vs. non-linear

·        Symmetry

·        Increasing/decreasing on an interval

Essential Questions

Enduring Understandings

- How are mathematical models used to describe physical relationships? (4.5E2)

- How are physical models used to clarify mathematical relationships? (4.5E3)

- Mathematical models can be used to describe and quantify physical relationships. (4.5E2)

- Physical models can be used to clarify mathematical relationships. (4.5E3)

Areas of Focus/Cumulative Progress Indicators

Comments and Examples

2.    Construct and solve simple open sentences involving addition or subtraction.

·        Result unknown (e.g., 6 – 2  = __  or  n = 3 + 5)

·        Part unknown (e.g., 3 + ÿ = 8)

1.         Recognize and describe change in quantities.

·        Graphs representing change over time (e.g., temperature, height)

2.         Construct and solve simple open sentences involving addition or subtraction (e.g., 3 + 6 = __,  n = 15 – 3,  3 + __ = 3,  16 – c = 7).

Sample Assessment Items:
• MC: Kamala bought a box of crayons for 29¢. She also bought a coloring book for 65¢. Which number sentence shows how much money Kamala spent on the crayons and coloring book?
a. 65¢ - 29¢ = ___

b. ___ + 29¢ = 65¢

* c. 29¢ + 65¢ = ___

d. 65¢ - ___ = 29¢

• MC: What does the p equal in 3 + p = 15 ?
a. 3

b. 5

* c. 12

d. 18

1.         Recognize and describe change in quantities.`

·        Graphs representing change over time (e.g., temperature, height)

·        How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration)

  2.         Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = __, n = 15 ¸ 3,  3 x __ = 0,  16 – c = 7).

Sample Assessment Items:
• MC: If 􀂈 x 8 = 96, what is the value of 􀂈?
* a. 12

b. 88

c. 104

d. 768

• MC: If 84 ÷ 􀂈 = 7, then what is the value of 􀂈?
a. 91

b. 77

* c. 12

d. 7

• MC: Some very old books do not have the pages numbered. Mrs. Jensen is a librarian and has developed a rule for estimating the number of pages (P), given the weight of the book. Which of the following is most likely the rule Mrs. Jensen uses for a book weighing two pounds?
* a. P = 100 x 2

b. P = 100 - 2

c. P = 100 + 2

d. P = 100 ÷ 2

1.         Use number sentences to model situations.

·        Using variables to represent unknown quantities

·        Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations

2.         Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

·        Changes over time

·        Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

1.        Use patterns, relations, and linear functions to model situations.

·        Using variables to represent unknown quantities

·        Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

Sample MC Item: Dorothy has $3.00 on September 1. Each week she earns $5.00. Which number sentence shows how much money she will have in 10 weeks?

a. D = 3 + 5(10)

b. D = 5(10)

* c. D = 3(5) + 10

d. D = 3(10) + 5

2.         Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

·        Changes over time

·        Relations between quantities

·        Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

2.        Use patterns, relations, symbolic algebra, and linear functions to model situations.

·        Using manipulatives, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

·        Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

 2.         Use patterns, relations, symbolic algebra, and linear functions to model situations.

·        Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalities

·        Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

1.       Use functions to model real-world phenomena and solve problems that involve varying quantities.

·        Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)

·        Direct and inverse variation

·        Absolute value

·        Expressions, equations and inequalities

·        Same function can model variety of phenomena

·        Growth/decay and change in the natural world

·        Applications in mathematics, biology, and economics (including compound interest)

3.       Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).

Essential Questions

Enduring Understandings

- What makes an algebraic algorithm both effective and efficient? (4.5D1)

-  Algebraic and numeric proced