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Chapter 14: Algebraic Thinking: Generalizations, Patterns, and Functions
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
1) Although algebra has few connections to real-life applications, it is highly intertwined with other areas of
mathematics, and, therefore, is an important topic.
2) Students should begin developing their ability to reason algebraically in elementary school.
3) Children typically have no difficulty understanding the meaning of the = symbol.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
4) Of the options below, the earliest form of algebraic thinking demonstrated by a student would most likely
be a) Graphing linear functions.
b) Recognizing and generalizing patterns.
c) Working with expressions that involve variables.
d) Making conjectures about number properties.
5) Which of the following is NOT one of the five forms of algebraic thinking described by Kaput?
a) Meaningful use of symbols
b) Generalization from arithmetic and from patterns in all of mathematics
c) Number classification
d) Study of patterns and functions
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
6) A ___________________________ is a tool, normally thought of as teaching numeration, that can help
students to connect number concepts and algebraic thinking.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
7) Which of the following is NOT true regarding students’ understanding of the = sign?
a) Because of their early experiences, many students tend to believe the = sign represents “and
the answer is.”
b) It is important, because the = sign is one of the principle methods of representing important
relationships within the number system.
c) The = function can be represented concretely by a number balance scale, which can lead to
deeper conceptual understanding.
d) Failure to understand the = sign does not usually cause difficulties understanding the process
of solving equations.
8) The use of true/false and open sentences
a) Can help clarify for students the meaning of the equal sign.
b) Is only appropriate for students of middle school age and up.
c) Is not an effective method of helping students to develop relational thinking.
d) Should only involve textbook and teacher-created problems.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
9) The ☐ in an open sentence plays the same role as a __________________, which frequently takes the
form
of a letter.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
10) Which of the following is NOT a true statement regarding simplifying expressions and equations?
a) For many students, they have been abstract, meaningless tasks.
b) They are not essential skills to working algebraically.
c) They are specifically mentioned in the NCTM Curriculum Focal Points.
d) Students can gain a better understanding if they examine examples with errors and explain
how to fix them.
11) When helping students to develop a conceptual understanding of number properties, it is helpful to
a) Provide students with a list or properties to examine and memorize.
b) Build them into students’ explorations with number sentences.
c) Encourage students to only use mathematical symbols to express the properties.
d) Discourage students’ conjectures, as they lead to confusion.
12) Which of the following is a true statement about repeating patterns?
a) Children’s literature is not a good source of examples.
b) The core of the pattern is the string of elements that repeats.
c) It is overly confusing to show young children the same pattern constructed with different
materials.
d) It is difficult, but worthwhile, to find examples from the real world.
13) Growing patterns
a) Are frequently functions.
b) Are best represented by numbers only.
c) Should not include fractions and decimals.
d) Have either a recursive formula or an explicit formula.
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
14) Students are more likely to describe the explicit formula for a function verbally before they will be able to
describe it symbolically.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
15) When students create graphs of functions
a) They are representing them in the manner that makes it the hardest to visualize relationships
between patterns.
b) They should place the independent variable (step number) along the vertical axis.
c) It is helpful to provide them with examples within a real-life context.
d) They should always be given specific data, equations, or numbers.
16) A proportional relationship
a) Can only be represented with an equation.
b) Will always have a graph that forms a straight line and passes through the origin.
c) Will always have a positive slope.
d) Is much more challenging for students to generalize than a non-proportional one.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
17) The rate of change of a linear function can be determined from a graph by examining the line’s _________.
18) Functions can be represented in five ways: context, table, verbal description, symbolic equation,
or____________________.
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
19) Describe two different ways you could determine whether a function is linear. Describe how these two
methods relate to one another, and a possible classroom activity that would help students to see this
connection.
20) Describe three different ways algebra can be connected to other areas of the mathematics curriculum.
