Question 1

Use specific values for m and n in $\left( a ^ { m } \right) ^ { n } = a ^ { m n }$ 
to give a basis for justifying that $a ^ { \frac { 1 } {  2}} = \sqrt { a }$ 
.

Accepted Answer

Let $m = \frac { 1 } { 2 } \text { and } n = 2$ Then $\left( a ^ { \frac { 1 } { 2 } } \right) ^ { 2 } = a ^ { \frac { 1 } { 2 } \cdot 2 } = a ^ { 1 } = a$ So $a ^ { \frac { 1 } { 2 } }$ , when squared, gives a. That is, $a ^ { \frac { 1 } { 2 } }$ must be $\sqrt { a }$ 
.

Question 2

Test each algebra statement to see whether they appear to be true in general. If a statement appears always to be true, draw a diagram to justify the statement. Add an explanation if the diagram is not self-explanatory. If a statement is not true in general, give a counterexample.
A) $\frac { a } { b + c } = \frac { a } { b } + \frac { a } { c }$ 
B) $( a + b ) - ( c + d ) = ( a - c ) + ( b - d )$ 
C) $( a + b ) ^ { 2 } = a ^ { 2 } + b ^ { 2 }$ 
D) $( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 }$ 
E) $( a - b ) + c = a - ( b + c )$

Accepted Answer

A) Not true in general; student should have shown a counterexample.
B) True. Sample diagram:   
C) Not true in general; student should have shown a counterexample.
D) Not true in general; student should have shown a counterexample.
E) Not true in general; student should have shown a counterexample.

Question 3

Complete the following and tell how they are alike.
A) $\frac { 2 } { 3 } + \frac { 7 } { 12 }$ 
B) $\frac { 2 } { x + 2 } + \frac { 3 x + 4 } { ( x + 2 ) ( x + 3 ) }$

Accepted Answer

A) $\frac { 2 } { 3 } + \frac { 7 } { 12 } = \frac { 2 \times 4 } { 3 \times 4 } + \frac { 7 } { 12 } = \frac { 2 \times 4 + 7 } { 3 \times 4 } = \frac { 15 } { 12 } = \frac { 5 } { 4 }$

B) $\frac { 2 } { x + 2 } + \frac { 3 x + 4 } { ( x + 2 ) ( x + 3 ) } = \frac { 2 ( x + 3 ) } { ( x + 2 ) ( x + 3 ) } + \frac { 3 x + 4 } { ( x + 2 ) ( x + 3 ) } = \frac { 2 x + 6 + 3 x + 4 } { ( x + 2 ) ( x + 3 ) } = \frac { 5 x + 10 } { ( x + 2 ) ( x + 3 ) } = \frac { 5 ( x + 2 ) } { ( x + 2 ) ( x + 3 ) } = \frac { 5 } { x + 3 }$

The two are alike in finding a common denominator and in simplifying toward the end.

Question 4

Over a four-day period, one Girl Scout troop sold 178 boxes of cookies on the first day, 39 more boxes on the third day than on the second day, and 10 boxes fewer on the fourth day than on the second day. The troop sold 489 boxes in the four days. How many boxes did they sell each day?

Accepted Answer

The answer of Over a four-day period, one Girl Scout...

Question 5

Does each of the following give a function? Include correct reasons for credit.
A) Associate with each whole number n its third power n 3.
B) Assign to each person in the town his/her current last name.
C) Assign to the last names of people in town the first names.
D) The ordered pairs (5, 2), (7, 2), and (4, 9)
E) This dot diagram:

Accepted Answer

Question 6

The "balance" diagram below shows x + 2 = x3. (If it is not true, give a correct equation.)

Accepted Answer

A) True 
 B)False

Question 7

Express symbolically, using variables, the general properties in algebra that are illustrated.
A) $( 400 + 25 ) \times 3 = ( 400 \times 3 ) + ( 25 \times 3 )$ 
B) $( 57 \times 16 ) + ( 43 \times 16 ) = ( 57 + 43 ) \times 16$ 
C) $( 600 + 32 ) \div 4 = ( 600 \div 4 ) + ( 32 \div 4 )$ 
D) $\frac { 75 + 18 } { 3 } = \frac { 75 } { 3 } + \frac { 18 } { 3 }$

Accepted Answer

A) @#LAT-DLM& , or @#LAT-DLM& , or just ac + bc
B)

Question 8

Make a drawing to justify $a ( b + c + d ) = a b + a c + a d$ 
.

Accepted Answer

The answer of Make a drawing to justify \(a (...

Question 9

Take-home question: Here are examples of a shortcut for mentally squaring a number.
Example 1) 762: Go to the closest multiple of 10-here, for 76, plus 4 to 80. Then go the opposite way from 76 by the same amount, minus 4 to 72. Multiply 80 and 72 mentally for 5760. Add the square of the up-down number 4. As a result, 5760 + 16 = 5776 and 762 = 5776.
Example 2) 622: Go to 60 (down 2). Then go up from 62 by 2 to 64. Therefore, 60 × 64 = 3840. Add the square of 2 for 622 = 3844.
Example 3) 572: Go to 60 then 54. Therefore, 60 × 54 = 3240. Add 32 for 572 = 3249.
Example 4) 1982: 200 × 196 = 39,200. Add 22 for 39,204 (= 1982).
A) Use the shortcut to calculate 372 mentally, and then write the mental steps you did.
B) Give a justification that the method works for squaring any n. Label the up-down number x.

Accepted Answer

(Note to the instructor: Cauti

Question 10

The repeating decimal for $\frac { 36350 } { 11111 }$ is 3.2715427154…. What digit is in the 99th decimal place in the repeating decimal? Explain how you know.

Accepted Answer

The 99th digit is 5. The repea

Question 11

Finish each story problem so that it can be described by the given equation.
A) 50 - n = 16: "Jamal had 50 pieces of paper …"
B) 3n + 16.99 = 37: "Jose went shopping and bought a CD for $16.99 …"

Accepted Answer

The answer of Finish each story problem so that it...

Question 12

A child is making "space modules with antennas" from toothpicks.   
The child wonders, "How many toothpicks would it take to make a 100-room module with antennas?!"

Accepted Answer

The answer of A child is making "space modules with...

Question 13

A) Part I: It is correct that 12nine × 32nine = 384nine. How might that inform $( x + 2 ) ( 3 x + 2 )$ ? 
B) Part II: Explain the "might" in part A by considering 4nine × 13nine = 53nine. Give another calculation that would misinform an algebraic expression. Bonus: Why does 4nine × 13nine = 53nine, or your calculation, give an incorrect idea for algebra?

Accepted Answer

Question 14

You find a function rule for a given table of data. Explain why your answer might not be the correct one.

Accepted Answer

Some other rule migh

Question 15

Name the property (or properties) that justifies each of the following.
A) (700 + 60 + 3) + (200 + 30 + 5) = (3 + 5) + (60 + 30) + (700 + 200)
B) 5 × (17 + 3) = (5 × 17) + (5 × 3)
C) 984 + (717 + 563) = 984 + (563 + 717)
D) (56 × 89) × 
 $\frac { 113 } { 113 }$ 
= 56 × 89 
E) ${}^{-} 75 + ( {}^{-} 23 + 75 ) = {}^{-}23$
 
(Hint: More than one property!)
F) $\frac { 1 } { 3 } \times 95 + \frac { 1 } { 3 } \times 4 = \frac { 1 } { 3 } \times ( 95 + 4 )$ 
G) ${}^{-}26 \times \frac { 52 } { 52 } = {}^{-} 26$
 
H) 5 × (17 × 3) = 5 × (3 × 17)

Accepted Answer

The answer of Name the property (or properties) that justifies...

Question 16

Use $\frac { a ^ { m } } { a ^ { n } } = a ^ { m - n }$ as the basis for defining a0 = 1 and a for nonzero values for a.

Accepted Answer

Letting m

Question 17

In each part, use the given, correct algebraic statements to answer the calculations.
A) Part I: 
 $( x + 4 ) ( 2 x + 1 ) = 2 x ^ { 2 } + 9 x + 4$ 
implies that 14seventeen × 21seventeen = _____seventeen. 
B) Part II: 
 $( 2 x + 3 ) ( 2 x + 1 ) = 4 x ^ { 2 } + 8 x + 3$ 
implies that 23eleven × 21eleven = _____eleven. 
C) Bonus: What other bases could be used in parts I and II?

Accepted Answer

A) 294_seventeen (Let x = 17 in the alge

Question 18

Give the 100th and the nth entries for these lists, assuming the patterns continue. A) $12,22,32,42,52 , \ldots \quad$ $\quad $$\quad $100th $\underline{\quad\quad}$ $\quad $ nth$\underline{\quad\quad}$
B) $3,5,7,9,11 , \ldots \quad$ $\quad $$\quad $$\quad $$\quad $100th $\underline{\quad\quad}$ $\quad $nth$\underline{\quad\quad}$
C) $2 \frac { 1 } { 2 } , 4,5 \frac { 1 } { 2 } , 7,8 \frac { 1 } { 2 } , 10 , \ldots \quad$ $\quad $100th $\underline{\quad\quad}$$\quad $$\quad $nth $\underline{\quad\quad}$

Accepted Answer

A) 1002; 1

Question 19

Show your mastery of the conventional order of operations by evaluating each.
A) 6 + 3 × 7 - (2 + -1) 5
B) 10 - 4 ÷ 3 × 2 + 1
C) 10 - 4 ÷ 3 × (2 + 1)
D) $4 x ^ { 2 } - 7 x + {}^{-} 2$, when $x = 3$ 
E) $9 - 3 x - 5 x ^ { 3 }$ , when $x = {}^{-} 2$

Accepted Answer

A) 26
B) @#LAT-DLM&

Question 20

Suppose g(x) = 3x - 2, and h(x) is defined by machine X below. What number is each of the following? Show your work.   
A) g(10) + g(5)
B) h(20)

Accepted Answer

The answer of Suppose g(x) = 3x - 2, and...

Exam 12: What Is Algebra

Use specific values for m and n in (am)n=amn\left( a ^ { m } \right) ^ { n } = a ^ { m n }(am)n=amn to give a basis for justifying that a12=aa ^ { \frac { 1 } { 2}} = \sqrt { a }a21​=a​ .

Complete the following and tell how they are alike. A) 23+712\frac { 2 } { 3 } + \frac { 7 } { 12 }32​+127​ B) 2x+2+3x+4(x+2)(x+3)\frac { 2 } { x + 2 } + \frac { 3 x + 4 } { ( x + 2 ) ( x + 3 ) }x+22​+(x+2)(x+3)3x+4​

Over a four-day period, one Girl Scout troop sold 178 boxes of cookies on the first day, 39 more boxes on the third day than on the second day, and 10 boxes fewer on the fourth day than on the second day. The troop sold 489 boxes in the four days. How many boxes did they sell each day?

The "balance" diagram below shows x + 2 = x3. (If it is not true, give a correct equation.)

Make a drawing to justify a(b+c+d)=ab+ac+ada ( b + c + d ) = a b + a c + a da(b+c+d)=ab+ac+ad .

The repeating decimal for 3635011111\frac { 36350 } { 11111 }1111136350​ is 3.2715427154…. What digit is in the 99th decimal place in the repeating decimal? Explain how you know.

Finish each story problem so that it can be described by the given equation. A) 50 - n = 16: "Jamal had 50 pieces of paper …" B) 3n + 16.99 = 37: "Jose went shopping and bought a CD for $16.99 …"

A child is making "space modules with antennas" from toothpicks. The child wonders, "How many toothpicks would it take to make a 100-room module with antennas?!"

You find a function rule for a given table of data. Explain why your answer might not be the correct one.

Use aman=am−n\frac { a ^ { m } } { a ^ { n } } = a ^ { m - n }anam​=am−n as the basis for defining a0 = 1 and a for nonzero values for a.

Suppose g(x) = 3x - 2, and h(x) is defined by machine X below. What number is each of the following? Show your work. A) g(10) + g(5) B) h(20)

Reasoning About Quantities

Numeration Systems

Understanding Whole Number Operations

Some Conventional Ways of Computing

Using Numbers in Sensible Ways

Meanings for Fractions

Computing With Fractions

Multiplicative Comparisons and Multiplicative Reasoning

Ratios, Rates, Proportions, and Percents

Integers and Other Number Systems

Number Theory

A Quantitative Approach to Algebra and Graphing

Understanding Change: Relationships Among Time, Distance, and Rate

Further Topics in Algebra and Change

Polygons

Polyhedra

Symmetry

Tessellations

Similarity

Curves, Constructions, and Curved Surfaces

Transformation Geometry

Measurement Basics

Area, Surface Area, and Volume

Counting Units Fast: Measurement Formulas

Special Topics in Measurement

Quantifying Uncertainty

Determining More Complicated Probabilities

Introduction to Statistics and Sampling

Representing and Interpreting Data With One Variable

Dealing With Multiple Data Sets or With Multiple Variables

Variability in Samples

Special Topics in Probability

Filters

Use specific values for m and n in $\left( a ^ { m } \right) ^ { n } = a ^ { m n }$ to give a basis for justifying that $a ^ { \frac { 1 } { 2}} = \sqrt { a }$ .

Complete the following and tell how they are alike. A) $\frac { 2 } { 3 } + \frac { 7 } { 12 }$ B) $\frac { 2 } { x + 2 } + \frac { 3 x + 4 } { ( x + 2 ) ( x + 3 ) }$

The "balance" diagram below shows x + 2 = x³. (If it is not true, give a correct equation.)

Make a drawing to justify $a ( b + c + d ) = a b + a c + a d$ .

The repeating decimal for $\frac { 36350 } { 11111 }$ is 3.2715427154…. What digit is in the 99th decimal place in the repeating decimal? Explain how you know.

Use $\frac { a ^ { m } } { a ^ { n } } = a ^ { m - n }$ as the basis for defining a⁰ = 1 and a for nonzero values for a.