Deck 15: Thinking Algebraically

Question 1

The pattern below is associated with which function?
1, 2, 4, 8, 16, 32, …

A)2 n
B)2 ^{n + 1}
C)2 ⁿ
D)2 ^{n − 1}

Answer 1

2 ^{n − 1}

Question 2

The pattern below is referred to as what?
1, 8, 27, 64, …

A)square numbers
B)cubic numbers
C)Fibonacci numbers
D)pyramidal numbers

Answer 2

cubic numbers

Question 3

The pattern below is referred to as what?
3, 9, 27, 81, … 

A)the multiples of 3
B)the cubes of 3
C)the factors of 3
D)the powers of 3

Answer 3

the powers of 3

Question 4

Under what circumstance, if any, is the following statement true?
8 + 7 = 3

A)It can never be true.
B)If the equal sign were changed to a less than symbol.
C)If this were related to the number of hours on a clock.
D)If 8 and 7 were both decreased by 5.

Question 5

Variable as a theme in algebra refers to:

A)finding a pattern of numbers.
B)showing the relationship between numbers.
C)a numerical representation having many possible values.
D)two numerical representations having the same value.

Question 6

What is the correct domain and range for the function below?
3 n ² - 2

A)D = {0, 1, 2} R = {−2, 1, 34}
B)D = {3, 5, 7} R = {25, 223, 339}
C)D = {2, 3, 6} R = {10, 25, 106}
D)D = {1, 4, 7} R = {1, 46, 149}

Question 7

The best way to get children to understand equations is to give them:

A)five examples of equations to solve.
B)balance scales to manipulate.
C)familiar formulas such as those for area.
D)nonexamples of equations, such as inequalities.

Question 8

Given an equation such as x + y = 18, children determine that the ordered pair (9, 9)is not possible. The most likely reason for this is:

A)they don't think that x and y can be the same number, since the variables are different.
B)they are incapable of thinking of this pair or any other ordered pair because they are not yet thinking abstractly.
C)they think that the ordered pair (10, 8)is the only possible answer.
D)they don't understand the role of the equal sign.

Question 9

What is the formula for the pattern below?
1, 3, 6, 10, …

A)n + 2 n
B)n ² + 2 n
C)n ² + n
D)

Question 10

What are the next two terms of the pattern below?
2, 1, 4, 3, 6, 5, …

A)8, 7
B)7, 8
C)4, 7
D)7, 4

Question 11

4 y

A)a true open sentence.
B)a false open sentence.
C)an inequality with infinite numbers of values for y .
D)an equation which is true when y

Question 12

Functions can be represented in all of the following ways except :

A)as input/output machines.
B)as tables of values.
C)as rate tables.
D)as inequalities.

Question 13

Research has shown that young children:

A)are incapable of understanding the role that variables play in equation solving.
B)must have extensive practice with variables in order to understand them.
C)must have a teacher who has an extensive background in algebra in order for the teacher to help children understand variables.
D)can use variables to represent unknowns and then solve for the numbers that the variables represent.

Question 14

The following are true statements concerning patterning and functions except for :

A)children who are able to form patterns build a foundation for the algebraic concept of functions.
B)the ability to generalize a relationship from specific data is a hallmark of algebraic reasoning.
C)patterns follow from functions.
D)patterning is the concept of functions that is central to algebra.

Question 15

For younger children, the equal sign in an equation may mean all of the following except for :

A)"The answer is coming."
B)"Get ready to do something."
C)"The left side is balanced with the right side of the equation."
D)"Perform the operation on all of the numbers on the left side of the equal sign and put the result on the right side."

Question 16

Which of the following is not a function?

A){(A, 1), (B, 2), (C, 3)}
B){(A, 1), (B, 1), (C, 3)}
C){(A, 1), (B, 2), (B, 3), (C, 1), (C, 2)}
D){(A, 2), (B, 2), (C, 2)}

Question 17

Algebra in elementary school is best understood as:

A)a simplified version of high school algebra.
B)focusing on the finding and extending of patterns.
C)pre-algebra.
D)a natural extension of other content standards.

Question 18

Young children can best learn about functions in all of the following ways except for :

A)exploring function relationships.
B)working with the definition of functions.
C)solving function problems.
D)applying functions to real-life situations.

Question 19

When teaching patterns to young children, teachers should emphasize:

A)finding, extending, and making different patterns.
B)limiting instruction to one or two examples to decrease confusion.
C)limiting instruction to only those patterns children are likely to be familiar with.
D)the algebraic aspect of patterning.

Question 20

What are the next three numbers in the pattern below?
2, 6, 12, 20, 30, …

A)44, 52, 64
B)40, 50, 60
C)42, 54, 63
D)42, 56, 72

Question 21

Suppose that t = the number of tiles in a problem-solving situation, and a child tells you that 3 t means 3 tiles. How can you explain to him that his reasoning is faulty?

Question 22

What is the inherent difficulty children have when working with an equation such as a + b = 12?

Question 23

Give three specific examples of different types of patterns that a teacher can use to introduce algebraic ideas to young children.

Question 24

PISA: The illustration below shows the footprints of a man walking.

The pace length, p , is the distance between the rear of two consecutive footprints.
For men, the formula n / p = 140 gives an approximate relationship between n and p where
n = number of steps per minute, and
p = pace length in meters
If the formula applies to Heiko's walking and Heiko takes 70 steps per minute, what is Heiko's pace length? Be sure to show your work.

Question 25

Solve for n in the equation below.
4 n + 6 = 3 n + 10 + n - 2

A)n = 2
B)n = 3
C)n = all values
D)no solution

Question 26

NAEP: The terms in this sequence are the squares of consecutive odd numbers:
1, 9, 25, 49, 81, …
The same rule is applied to each number in the given pattern. What is the sixth number in the pattern?

A)40
B)100
C)121
D)144

Question 27

Teacher Licensing Examination Questions
Praxis: If 4 x − 3( x + 1)= 5, what is the value of x ?

A)2
B)4
C)6
D)8

Question 28

The series below is composed of triangular number. What is the next term in the series?
1, 3, 6, 10, …

A)14
B)13
C)15
D)16

Question 29

A newspaper carrier was offered the chance to earn $0.01 on the first day of a month, $0.02 on the second day, and so forth doubling every day. How much did the carrier get paid at the end of the second week?

A)about $160.00
B)about $20.00
C)about $40.00
D)about $80.00

Question 30

Given the value for n , what is the function that produces the following data?

A)2 n
B)2 n + 2
C)n + 1
D)3 n - 1

Question 31

What are some specific ways that a teacher can introduce the idea of variables without actually using letters for variables?

Question 32

What is the major difference in the role that the variable plays in the equation 6 x = 30 and the inequality 6 x > 30? What difficulties do children have with respect to the inequality?

Question 33

Explain the role that patterning plays in helping develop children's algebraic reasoning.

Question 34

In this pattern, what is the value of p when n = 10?

A)15
B)60
C)50
D)55

Question 35

What is the next term in the pattern below?
A, d, g, j…

A)k
B)n
C)l
D)m

Question 36

Using patterns, what is the number of diagonals in an undecagon (11 sides)?

A)40
B)42
C)44
D)46

Question 37

Solve for n in the equation below.
3 n + 6 = n + 8 + 2 n - 2

A)n = 2
B)n = 3
C)n = all values
D)no solution

Question 38

The pattern below is called the Fibonacci sequence. What is the next term in the pattern?
1, 1, 2, 3, 5, 8, …

A)11
B)13
C)12
D)14

Question 39

Using clock arithmetic, what is 6 + 7 = ?

A)11
B)12
C)1
D)2

Question 40

TIMSS: n is a number. When n is multiplied by 7 and then 6 is added, the result is 41. Which of these equations represents this relation?

A)
B)
C)
D)

Question 41

Explain some of the difficulties children have with understanding the role of the equal sign in equations.

Question 42

Suppose you wanted to teach children to solve equations, and you have the following examples:
8 - t = 2
x + 3 = 7
___ + 5 = 6
6 + r = 13
m - 4 = 5
Place the equations in the order you think would be best in order for the children to understand the concept of equations. Explain your decision.

Question 43

A child asks you what a function is. Give several specific examples that you would use to explain this important algebraic concept to her.