Algebra/Chapter 1/Exercises

A set of exercises related to concepts from Chapter 1.

This set contains 55 exercises (including the Conceptual Questions)

Conceptual Questions

Q1.1 (Alien Society) Imagine you came across an alien from a distant planet, where all of its civilians have a solid grasp of English and had three fingers on each hand. Though this civilization is intelligent, they had never learned about what "counting" is or how to do it. In addition to that, they had never learned about what a "number" is, what they're called, or what they looked like. Think about how you might teach this civilian about counting and numbers so that they can go back to their planet to teach their people of this knowledge. What tools might you use to explain the idea? What are some of the concepts you'd want to get across? What are some of the difficulties that might arise from this task?

Q1.2 (What is a Number?) Define what a "number" is in your own words. Define what a "numeral" is in your own words.

Q1.3 (Sign of Zero) Is the number zero positive, negative, or neither? Explain your reasoning.

Q1.4 (Difference of Decimals) What is the difference between "ten" and "one-tenth"?

Q1.5 (Picture Perfect) Suppose the number line actually existed physically. Would you be able to take a photo of the entire number line if you backed away far enough?

Q1.6 (Explaining the Writing of Numbers) Explain in your own words how you write numbers, both in word form and with numerical symbols.

Q1.7 (Largest Number Possible) What is the largest and smallest three-digit number you can write using the digits 0, 8, and 4? Use each digit only once, and explain how you obtained your results. If you wrote these numbers to the right of a decimal point, what is the largest number you can make.

Q1.8 (A Million) A million is one thousand thousands. Explain how this is so.

Q1.9 (Reading it Wrong) Explain what is wrong with reading "50,002" as "fifty-thousand and two". Explain what is wrong with reading "2.203" as "two and two hundred and three thousanths".

Q1.11 (Number Associations) What whole numbers are associated with each word?

a. zilch
b. duo
c. decade
d. a pair
e. naught
f. trio
g. four score
h. century

Q1.12 (Problem with Fractions) Why can't we say that 3/5 of the figure below have been shaded in?

Image: 300 pixels

Q1.13 (Large Numbers) Determine if the following is true: "The more digits a number has, the larger it is".

Q1.14 (Signs) A fast-food menu has the cost of a hamburger listed as .99¢. Explain what is wrong with this.

Q1.15 (Operations on the Number Line) Determine the performed operation that is being represented in each diagram.

Q1.16 (Inverse Operations) What is the inverse operation of “I put my shoes on today, and I walk out of my house”?

Q1.17 (Powers of 1) Find $1^{2}$, $1^{3}$, and $1^{4}$. What can you assume about any power of 1?

Q1.18 (Viral Math Expression) The seemingly simple expression below has stumped many people across the Internet. Some will argue the answer is 9, while others will argue it is 1. However, there is a fundamental issue with the way that the expression is written, leading to these two different answers, can you figure out what it is?

$6/2(1+2)$

Exercises

Section 1.1

1.1 (Locating Numbers) Draw a number line, and then figure out where the following values might be located on it.

${\frac {3}{11}},0,0.0001,5,{\frac {1}{4}},-2.3$

1.2 (Comparing Numbers) For each given pair of numbers, determine which of the two is larger.

a. 4, 100
b. 9, 9.0001
c. -7, -2
d. -5, 0
e. 100, 100

1.3 (Weighing Bull Sharks) A biologist is studying bull shark populations. She records the weights of four sharks, in pounds, that she has caught. Order the bull sharks from lightest to heaviest.

1.4 (Place Values) Find the place value of the number 5 in each of the following numbers.

a. 5,000,000
b. 0.5
c. 105
d. 3572896
e. 123,456,789
f. 0.000005
g. 8051
h. 85,931
i. 800,026

1.5 (Writing Numbers) Translate to mathematical symbols

1.6 (Writing Numbers in Words) Write the following numbers in words

a. 9
b. 10
c. 274
d. 8,322
e. 1,000,000,009
f. 1,343,234,985
g. 0.01

1.7 (Numbers in Expanded Form) In the number 7,893, there are "7 thousands", "8 hundreds", "9 tens", and "3 ones". We therefore say that a number is in expanded form when it is written as follows:

7 thousands + 8 hundreds + 9 tens + 3 ones
or
7000 + 800 + 9 + 3

Write the following numbers in expanded form:

a. 473
b. 6852
c. 73,016
d. 570,003
e. 3,519,803
f. 48,000,061
g. 37.89
h. 124.575
i. 7496.5467
j. 6.40941

1.8 (Fraction Diagrams) Write a fraction to describe what part of the diagrams below are shaded. Write a fraction to describe what part of the diagrams aren't shaded in.

1.9 (Fruit Basket) A basket of fruit holds 5 mangoes, 7 apples, 12 oranges, and 20 pomegranates.
a. What fraction of the fruits in the basket are apples?
b. What fraction of the fruits in the basket are not oranges?
c. What fraction of the fruits in the basket are oranges or pomegranates?

Section 1.2

1.10 (Make 1000 out of 8) Eight digits “8” are written together, like below, and plus signs “+” are inserted in between to get the sum of 1000. Where were the plus signs added?

$8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8$

1.11 (Unknown Sum) In the addition problem below, A, B, and C each represent three different digits. What are the digits?

${\begin{array}{r}AAA\\+\ BBB\\\hline AAAC\end{array}}$

1.12 (Unknown Product) A six-digit number with 1 as its left-most digit is three times bigger when we put the one at the end of the number instead. What number is this?

1.13 (Fractions and Decimals) Use long division to find the decimal expansion of each fraction.

1.14 (Terminating and Repeating Decimals) You may notice from Problem 1.7 that when you convert a fraction to a decimal, you will sometimes get what is called a repeating decimal. Take for example the fraction ${\frac {3}{11}}$.

${\frac {3}{11}}=0.272727272727...$

The decimal form of ${\frac {3}{11}}$ consists of the two digits 2 and 7 in an infinitely repeating sequence. To simplify things, instead of writing the above, we denote it as 0.27.

a. Use this bar notation to write each of the repeating decimals from Problem 1.13.
b. We see that in the fraction above, the fractional part repeats after two digits. We say that this number has a period of 2. Likewise, we say that the number ${\frac {1}{7}}$ has a period of 6, because the number repeats after 6 digits. From the numbers below, which of them has the largest period?

1.15 (Mixed Fractions) Write the following improper fractions as mixed fractions.

1.16 (Sharing Pizza) Billy's family ordered a large pizza. His father had ${\frac {1}{6}}$ of it, and his mother had ${\frac {1}{5}}$ of what remained. Later on, Billy's sister ate some pizza, and then Billy had the remaining pizza when there was exactly a half of what they started with (Billy is a large kid). What fraction of what their parents had left for her did the sister have?

A set of stamps (For Problem 1.17)

1.17 (Stamp Collection) The picture to the right shows stamps, arranged in four groups of four. How many stamps are in that image? While you can count them individually, there is a much faster way of getting the total.

1.18 (Decimal Operations) Explain how addition with decimals is comparable to addition with whole numbers, how are they different? Do the same thing with multiplication with decimals.

1.19 (A Sum and a Difference) The sum of two numbers is 104 and their difference is 32. What is the value of the larger number?

1.20 (On Allison's Street) Allison's house is on the same street as the library, post office, and supermarket, as shown in the diagram below. The distance from Allison's house to each of the three buildings is different. Based on this information, at which point is Allison's house located?

Section 1.4

1.21 (Bank Account) Nick deposits $2 into a bank account on the first day, $4 on the second day, and $8 on the third day. He will continue to double the deposit each day. How much will he deposit on the tenth day?

Section 1.6

1.22 (Using Divisibility Rules) Use divisibility tests to find the remainder of the following quotients:

Reason and Apply

1.23 (Count the 24ths) Without performing division, how many ${\frac {1}{24}}$'s are in ${\frac {2}{3}}?$

1.24 (Negative Negative Negative Negative...)

a. What is $-(-(-2))$?
b. What is $-(-(-(-2)))$?
c. What if there were 20 minus signs in front of the 2?
d. What if there were 75 minus signs in front of the 2?

1.25 (Using Bar Graphs) Look at the diagram below, and use it to answer the following questions.

1.26 (Using Multibar Graphs) Look at the diagram below, and use it to answer the following questions.

1.27 (Using Line Graphs) Look at the diagram below, and use it to answer the following questions.

1.28 (Creating a Bar Graph) Look at the table below, and use it to create a bar graph.

1.29 (Reading Meters) The amount of electricity in a household is measured in kilowatt-hours. Determine the reading on the meter shown below. (When a pointer is between two numbers, use the smaller number).

1.30 (Sky High) The table below shows the altitude each of the cloud types are found at. Graph the numbers on the vertical number line below.

1.31 (Rulers) Look at the diagram of a ruler below.

a. How many tick marks are between 0 and 1?
b. What number is the arrow pointing to?

1.32 (Page Numbers) How many numerals are required to number all of the pages of a book containing 450 pages?

1.33 (Operations with Repeating Decimals) Calculate:

a. 0.55555... + 0.66666...
b. 0.99999... + 0.11111...
c. 1.11111... - 0.22222...
d. 0.33333... * 0.66666...
e. 1.22222... * 0.81818...

Challenge Problems

1.34 (One to Ten) To get yourself thinking about this, try this simple mathematical game:

Take the numbers 1 through 10 on the left side of an equation, and pick a number for the right side.

Example: 1 2 3 4 5 6 7 8 9 10 = 1

Now put operators between those numbers. Only use parentheses when necessary.

Example: 1 + 2 - 3 + 4 - 5 + 6 + 7 + 8 - 9 - 10 = 1

Change the number on the right-hand side. Can you generate an expression for this number? If not, can you prove why not?

Does this change if you change the order of the numbers?

1.35 (Diffy Squares) Draw a square. One each of the corners of that square, write the numbers 7, 5, 9, and 2. Now, draw a second square around the first one so that it it goes through each of the four corners. At each corner of the second square, write the difference of the numbers at the closest corners of the smaller square: 7-5 = 2, 9-5 = 4, 9-2 = 7, and 7-2 = 5.

Repeat this process until you come to a pattern of four numbers that do not change.

a. What is the pattern?
b. Try this same procedure with another set of four starting numbers. Do you end up with the same pattern?
c. Explains what happened.

1.36 (The Binary Number System)

1.37 (The Hexadecimal Number System)