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Adolescent girl doing homework.

What’s the Right Amount of Homework?

Decades of research show that homework has some benefits, especially for students in middle and high school—but there are risks to assigning too much.

Many teachers and parents believe that homework helps students build study skills and review concepts learned in class. Others see homework as disruptive and unnecessary, leading to burnout and turning kids off to school. Decades of research show that the issue is more nuanced and complex than most people think: Homework is beneficial, but only to a degree. Students in high school gain the most, while younger kids benefit much less.

The National PTA and the National Education Association support the “ 10-minute homework guideline ”—a nightly 10 minutes of homework per grade level. But many teachers and parents are quick to point out that what matters is the quality of the homework assigned and how well it meets students’ needs, not the amount of time spent on it.

The guideline doesn’t account for students who may need to spend more—or less—time on assignments. In class, teachers can make adjustments to support struggling students, but at home, an assignment that takes one student 30 minutes to complete may take another twice as much time—often for reasons beyond their control. And homework can widen the achievement gap, putting students from low-income households and students with learning disabilities at a disadvantage.

However, the 10-minute guideline is useful in setting a limit: When kids spend too much time on homework, there are real consequences to consider.

Small Benefits for Elementary Students

As young children begin school, the focus should be on cultivating a love of learning, and assigning too much homework can undermine that goal. And young students often don’t have the study skills to benefit fully from homework, so it may be a poor use of time (Cooper, 1989 ; Cooper et al., 2006 ; Marzano & Pickering, 2007 ). A more effective activity may be nightly reading, especially if parents are involved. The benefits of reading are clear: If students aren’t proficient readers by the end of third grade, they’re less likely to succeed academically and graduate from high school (Fiester, 2013 ).

For second-grade teacher Jacqueline Fiorentino, the minor benefits of homework did not outweigh the potential drawback of turning young children against school at an early age, so she experimented with dropping mandatory homework. “Something surprising happened: They started doing more work at home,” Fiorentino writes . “This inspiring group of 8-year-olds used their newfound free time to explore subjects and topics of interest to them.” She encouraged her students to read at home and offered optional homework to extend classroom lessons and help them review material.

Moderate Benefits for Middle School Students

As students mature and develop the study skills necessary to delve deeply into a topic—and to retain what they learn—they also benefit more from homework. Nightly assignments can help prepare them for scholarly work, and research shows that homework can have moderate benefits for middle school students (Cooper et al., 2006 ). Recent research also shows that online math homework, which can be designed to adapt to students’ levels of understanding, can significantly boost test scores (Roschelle et al., 2016 ).

There are risks to assigning too much, however: A 2015 study found that when middle school students were assigned more than 90 to 100 minutes of daily homework, their math and science test scores began to decline (Fernández-Alonso, Suárez-Álvarez, & Muñiz, 2015 ). Crossing that upper limit can drain student motivation and focus. The researchers recommend that “homework should present a certain level of challenge or difficulty, without being so challenging that it discourages effort.” Teachers should avoid low-effort, repetitive assignments, and assign homework “with the aim of instilling work habits and promoting autonomous, self-directed learning.”

In other words, it’s the quality of homework that matters, not the quantity. Brian Sztabnik, a veteran middle and high school English teacher, suggests that teachers take a step back and ask themselves these five questions :

  • How long will it take to complete?
  • Have all learners been considered?
  • Will an assignment encourage future success?
  • Will an assignment place material in a context the classroom cannot?
  • Does an assignment offer support when a teacher is not there?

More Benefits for High School Students, but Risks as Well

By the time they reach high school, students should be well on their way to becoming independent learners, so homework does provide a boost to learning at this age, as long as it isn’t overwhelming (Cooper et al., 2006 ; Marzano & Pickering, 2007 ). When students spend too much time on homework—more than two hours each night—it takes up valuable time to rest and spend time with family and friends. A 2013 study found that high school students can experience serious mental and physical health problems, from higher stress levels to sleep deprivation, when assigned too much homework (Galloway, Conner, & Pope, 2013 ).

Homework in high school should always relate to the lesson and be doable without any assistance, and feedback should be clear and explicit.

Teachers should also keep in mind that not all students have equal opportunities to finish their homework at home, so incomplete homework may not be a true reflection of their learning—it may be more a result of issues they face outside of school. They may be hindered by issues such as lack of a quiet space at home, resources such as a computer or broadband connectivity, or parental support (OECD, 2014 ). In such cases, giving low homework scores may be unfair.

Since the quantities of time discussed here are totals, teachers in middle and high school should be aware of how much homework other teachers are assigning. It may seem reasonable to assign 30 minutes of daily homework, but across six subjects, that’s three hours—far above a reasonable amount even for a high school senior. Psychologist Maurice Elias sees this as a common mistake: Individual teachers create homework policies that in aggregate can overwhelm students. He suggests that teachers work together to develop a school-wide homework policy and make it a key topic of back-to-school night and the first parent-teacher conferences of the school year.

Parents Play a Key Role

Homework can be a powerful tool to help parents become more involved in their child’s learning (Walker et al., 2004 ). It can provide insights into a child’s strengths and interests, and can also encourage conversations about a child’s life at school. If a parent has positive attitudes toward homework, their children are more likely to share those same values, promoting academic success.

But it’s also possible for parents to be overbearing, putting too much emphasis on test scores or grades, which can be disruptive for children (Madjar, Shklar, & Moshe, 2015 ). Parents should avoid being overly intrusive or controlling—students report feeling less motivated to learn when they don’t have enough space and autonomy to do their homework (Orkin, May, & Wolf, 2017 ; Patall, Cooper, & Robinson, 2008 ; Silinskas & Kikas, 2017 ). So while homework can encourage parents to be more involved with their kids, it’s important to not make it a source of conflict.

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Unit 5: Rational number arithmetic

Lesson 1: interpreting negative numbers.

  • Missing numbers on the number line examples (Opens a modal)
  • Comparing rational numbers (Opens a modal)
  • Missing numbers on the number line Get 3 of 4 questions to level up!
  • Order rational numbers Get 3 of 4 questions to level up!

Lesson 2: Changing temperatures

  • Signs of sums on a number line (Opens a modal)
  • Adding negative numbers review (Opens a modal)
  • Adding negative numbers Get 5 of 7 questions to level up!

Lesson 3: Changing elevation

  • Adding negative numbers on the number line (Opens a modal)
  • Number equations & number lines (Opens a modal)
  • Adding fractions with different signs (Opens a modal)
  • Signs of sums Get 5 of 7 questions to level up!
  • Adding negative numbers on the number line Get 5 of 7 questions to level up!
  • Number equations & number lines Get 3 of 4 questions to level up!

Lesson 4: Money and debts

  • Rational number word problem: checking account (Opens a modal)

Lesson 5: Representing subtraction

  • Adding the opposite with number lines (Opens a modal)
  • Subtracting negative numbers review (Opens a modal)
  • Understand subtraction as adding the opposite Get 3 of 4 questions to level up!
  • Subtracting negative numbers Get 5 of 7 questions to level up!

Lesson 6: Subtracting rational numbers

  • Graphing negative number addition and subtraction expressions (Opens a modal)
  • Interpreting numeric expressions example (Opens a modal)
  • Absolute value as distance between numbers (Opens a modal)
  • Interpret negative number addition and subtraction expressions Get 3 of 4 questions to level up!
  • Adding & subtracting negative fractions Get 5 of 7 questions to level up!
  • Absolute value to find distance Get 5 of 7 questions to level up!
  • Adding & subtracting rational numbers Get 3 of 4 questions to level up!

Lessons 1-6: Extra practice

  • Adding integers: find the missing value (Opens a modal)
  • Subtracting integers: find the missing value (Opens a modal)
  • Associative and commutative properties of addition with negatives (Opens a modal)
  • Equivalent expressions with negative numbers (Opens a modal)
  • Adding & subtracting negative numbers Get 5 of 7 questions to level up!
  • Addition & subtraction: find the missing value Get 3 of 4 questions to level up!
  • Commutative and associative properties of addition with integers Get 3 of 4 questions to level up!
  • Equivalent expressions with negative numbers Get 3 of 4 questions to level up!

Lesson 7: Adding and subtracting to solve problems

  • Interpreting negative number statements (Opens a modal)
  • Negative number word problem: Alaska (Opens a modal)
  • Interpreting negative number statements Get 3 of 4 questions to level up!
  • Negative number addition and subtraction: word problems Get 3 of 4 questions to level up!

Lesson 9: Multiplying rational numbers

  • Multiplying a positive and a negative number (Opens a modal)
  • Multiplying two negative numbers (Opens a modal)
  • Why a negative times a negative makes sense (Opens a modal)
  • Why a negative times a negative is a positive (Opens a modal)
  • Multiplying negative numbers review (Opens a modal)
  • Multiplying negative numbers Get 3 of 4 questions to level up!

Lesson 10: Multiply!

  • Multiplying positive and negative fractions (Opens a modal)
  • Multiplying positive and negative fractions Get 3 of 4 questions to level up!

Lesson 11: Dividing rational numbers

  • Dividing positive and negative numbers (Opens a modal)
  • Dividing negative numbers review (Opens a modal)
  • Dividing negative fractions (Opens a modal)
  • Dividing negative numbers Get 3 of 4 questions to level up!
  • Dividing positive and negative fractions Get 3 of 4 questions to level up!
  • Dividing mixed numbers with negatives Get 3 of 4 questions to level up!

Lesson 12: Negative rates

  • Interpreting multiplication & division of negative numbers (Opens a modal)
  • Multiplying & dividing negative numbers word problems Get 3 of 4 questions to level up!

Lesson 13: Expressions with rational numbers

  • Substitution with negative numbers (Opens a modal)
  • Ordering expressions (Opens a modal)
  • Negative signs in fractions (Opens a modal)
  • Expressions with rational numbers (Opens a modal)
  • Substitution with negative numbers Get 5 of 7 questions to level up!
  • Ordering negative number expressions Get 3 of 4 questions to level up!
  • Negative signs in fractions Get 3 of 4 questions to level up!
  • Signs of expressions Get 5 of 7 questions to level up!

Lesson 14: Solving problems with rational numbers

  • Intro to order of operations (Opens a modal)
  • Order of operations with rational numbers (Opens a modal)
  • Rational number word problem: cab (Opens a modal)
  • Order of operations with negative numbers Get 5 of 7 questions to level up!
  • Rational number word problems Get 3 of 4 questions to level up!

Lesson 15: Solving equations with rational numbers

  • No videos or articles available in this lesson
  • One-step equations with negatives (add & subtract) Get 5 of 7 questions to level up!
  • One-step equations with negatives (multiply & divide) Get 5 of 7 questions to level up!

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  • 1.1 Real Numbers: Algebra Essentials
  • Introduction to Prerequisites
  • 1.2 Exponents and Scientific Notation
  • 1.3 Radicals and Rational Exponents
  • 1.4 Polynomials
  • 1.5 Factoring Polynomials
  • 1.6 Rational Expressions
  • Key Equations
  • Key Concepts
  • Review Exercises
  • Practice Test
  • Introduction to Equations and Inequalities
  • 2.1 The Rectangular Coordinate Systems and Graphs
  • 2.2 Linear Equations in One Variable
  • 2.3 Models and Applications
  • 2.4 Complex Numbers
  • 2.5 Quadratic Equations
  • 2.6 Other Types of Equations
  • 2.7 Linear Inequalities and Absolute Value Inequalities
  • Introduction to Functions
  • 3.1 Functions and Function Notation
  • 3.2 Domain and Range
  • 3.3 Rates of Change and Behavior of Graphs
  • 3.4 Composition of Functions
  • 3.5 Transformation of Functions
  • 3.6 Absolute Value Functions
  • 3.7 Inverse Functions
  • Introduction to Linear Functions
  • 4.1 Linear Functions
  • 4.2 Modeling with Linear Functions
  • 4.3 Fitting Linear Models to Data
  • Introduction to Polynomial and Rational Functions
  • 5.1 Quadratic Functions
  • 5.2 Power Functions and Polynomial Functions
  • 5.3 Graphs of Polynomial Functions
  • 5.4 Dividing Polynomials
  • 5.5 Zeros of Polynomial Functions
  • 5.6 Rational Functions
  • 5.7 Inverses and Radical Functions
  • 5.8 Modeling Using Variation
  • Introduction to Exponential and Logarithmic Functions
  • 6.1 Exponential Functions
  • 6.2 Graphs of Exponential Functions
  • 6.3 Logarithmic Functions
  • 6.4 Graphs of Logarithmic Functions
  • 6.5 Logarithmic Properties
  • 6.6 Exponential and Logarithmic Equations
  • 6.7 Exponential and Logarithmic Models
  • 6.8 Fitting Exponential Models to Data
  • Introduction to Systems of Equations and Inequalities
  • 7.1 Systems of Linear Equations: Two Variables
  • 7.2 Systems of Linear Equations: Three Variables
  • 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
  • 7.4 Partial Fractions
  • 7.5 Matrices and Matrix Operations
  • 7.6 Solving Systems with Gaussian Elimination
  • 7.7 Solving Systems with Inverses
  • 7.8 Solving Systems with Cramer's Rule
  • Introduction to Analytic Geometry
  • 8.1 The Ellipse
  • 8.2 The Hyperbola
  • 8.3 The Parabola
  • 8.4 Rotation of Axes
  • 8.5 Conic Sections in Polar Coordinates
  • Introduction to Sequences, Probability and Counting Theory
  • 9.1 Sequences and Their Notations
  • 9.2 Arithmetic Sequences
  • 9.3 Geometric Sequences
  • 9.4 Series and Their Notations
  • 9.5 Counting Principles
  • 9.6 Binomial Theorem
  • 9.7 Probability

Learning Objectives

In this section, you will:

  • Classify a real number as a natural, whole, integer, rational, or irrational number.
  • Perform calculations using order of operations.
  • Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
  • Evaluate algebraic expressions.
  • Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. We describe them in set notation as { 1 , 2 , 3 , ... } { 1 , 2 , 3 , ... } where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the opposites of the natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:

  • ⓐ a terminating decimal: 15 8 = 1.875 , 15 8 = 1.875 , or
  • ⓑ a repeating decimal: 4 11 = 0.36363636 … = 0. 36 ¯ 4 11 = 0.36363636 … = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

  • ⓐ 7 = 7 1 7 = 7 1
  • ⓑ 0 = 0 1 0 = 0 1
  • ⓒ −8 = − 8 1 −8 = − 8 1

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

  • ⓐ − 5 7 − 5 7
  • ⓑ 15 5 15 5
  • ⓒ 13 25 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

  • ⓐ − 5 7 = −0. 714285 ——— , − 5 7 = −0. 714285 ——— , a repeating decimal
  • ⓑ 15 5 = 3 15 5 = 3 (or 3.0), a terminating decimal
  • ⓒ 13 25 = 0.52 , 13 25 = 0.52 , a terminating decimal
  • ⓐ 68 17 68 17
  • ⓑ 8 13 8 13
  • ⓒ − 17 20 − 17 20

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

  • ⓑ 33 9 33 9
  • ⓓ 17 34 17 34
  • ⓔ 0.3033033303333 … 0.3033033303333 …
  • ⓐ 25 : 25 : This can be simplified as 25 = 5. 25 = 5. Therefore, 25 25 is rational.

So, 33 9 33 9 is rational and a repeating decimal.

  • ⓒ 11 : 11 11 : 11 is irrational because 11 is not a perfect square and 11 11 cannot be expressed as a fraction.

So, 17 34 17 34 is rational and a terminating decimal.

  • ⓔ 0.3033033303333 … 0.3033033303333 … is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
  • ⓐ 7 77 7 77
  • ⓒ 4.27027002700027 … 4.27027002700027 …
  • ⓓ 91 13 91 13

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1 .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

  • ⓐ − 10 3 − 10 3
  • ⓒ − 289 − 289
  • ⓓ −6 π −6 π
  • ⓔ 0.615384615384 … 0.615384615384 …
  • ⓐ − 10 3 − 10 3 is negative and rational. It lies to the left of 0 on the number line.
  • ⓑ 5 5 is positive and irrational. It lies to the right of 0.
  • ⓒ − 289 = − 17 2 = −17 − 289 = − 17 2 = −17 is negative and rational. It lies to the left of 0.
  • ⓓ −6 π −6 π is negative and irrational. It lies to the left of 0.
  • ⓔ 0.615384615384 … 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.
  • ⓑ −11.411411411 … −11.411411411 …
  • ⓒ 47 19 47 19
  • ⓓ − 5 2 − 5 2
  • ⓔ 6.210735 6.210735

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2 .

Sets of Numbers

The set of natural numbers includes the numbers used for counting: { 1 , 2 , 3 , ... } . { 1 , 2 , 3 , ... } .

The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the negative natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } .

The set of rational numbers includes fractions written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } .

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: { h | h is not a rational number } . { h | h is not a rational number } .

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

  • ⓔ 3.2121121112 … 3.2121121112 …
  • ⓐ − 35 7 − 35 7
  • ⓔ 4.763763763 … 4.763763763 …

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 ⋅ 4 = 16. 4 2 = 4 ⋅ 4 = 16. We can raise any number to any power. In general, the exponential notation a n a n means that the number or variable a a is used as a factor n n times.

In this notation, a n a n is read as the n th power of a , a , or a a to the n n where a a is called the base and n n is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 ⋅ 2 3 − 4 2 24 + 6 ⋅ 2 3 − 4 2 is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 4 2 as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, 24 + 6 ⋅ 2 3 − 4 2 = 12. 24 + 6 ⋅ 2 3 − 4 2 = 12.

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses) E (xponents) M (ultiplication) and D (ivision) A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

  • Step 1. Simplify any expressions within grouping symbols.
  • Step 2. Simplify any expressions containing exponents or radicals.
  • Step 3. Perform any multiplication and division in order, from left to right.
  • Step 4. Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

  • ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 )
  • ⓑ 5 2 − 4 7 − 11 − 2 5 2 − 4 7 − 11 − 2
  • ⓒ 6 − | 5 − 8 | + 3 ( 4 − 1 ) 6 − | 5 − 8 | + 3 ( 4 − 1 )
  • ⓓ 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2
  • ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1
  • ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction.

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

  • ⓒ 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition. 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition.

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

  • ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add. 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.
  • ⓐ 5 2 − 4 2 + 7 ( 5 − 4 ) 2 5 2 − 4 2 + 7 ( 5 − 4 ) 2
  • ⓑ 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6
  • ⓒ | 1.8 − 4.3 | + 0.4 15 + 10 | 1.8 − 4.3 | + 0.4 15 + 10
  • ⓓ 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2
  • ⓔ [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 ) [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 )

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 17 − 5 is not the same as 5 − 17. 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. 20 ÷ 5 ≠ 5 ÷ 20.

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference 12 − ( 5 + 3 ) . 12 − ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , ( 5 + 3 ) , we add the opposite.

Now, distribute −1 −1 and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have ( −6 ) + 0 = −6 ( −6 ) + 0 = −6 and 23 ⋅ 1 = 23. 23 ⋅ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted by (− a ), that, when added to the original number, results in the additive identity, 0.

For example, if a = −8 , a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0. ( −8 ) + 8 = 0.

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if a = − 2 3 , a = − 2 3 , the reciprocal, denoted 1 a , 1 a , is − 3 2 − 3 2 because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

  • ⓐ 3 ⋅ 6 + 3 ⋅ 4 3 ⋅ 6 + 3 ⋅ 4
  • ⓑ ( 5 + 8 ) + ( −8 ) ( 5 + 8 ) + ( −8 )
  • ⓒ 6 − ( 15 + 9 ) 6 − ( 15 + 9 )
  • ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) 4 7 ⋅ ( 2 3 ⋅ 7 4 )
  • ⓔ 100 ⋅ [ 0.75 + ( −2.38 ) ] 100 ⋅ [ 0.75 + ( −2.38 ) ]
  • ⓐ 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify. 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify.
  • ⓑ ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition. ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition.
  • ⓒ 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify. 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify.
  • ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication. 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication.
  • ⓔ 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify. 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify.
  • ⓐ ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ] ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ]
  • ⓑ 5 ⋅ ( 6.2 + 0.4 ) 5 ⋅ ( 6.2 + 0.4 )
  • ⓒ 18 − ( 7 −15 ) 18 − ( 7 −15 )
  • ⓓ 17 18 + [ 4 9 + ( − 17 18 ) ] 17 18 + [ 4 9 + ( − 17 18 ) ]
  • ⓔ 6 ⋅ ( −3 ) + 6 ⋅ 3 6 ⋅ ( −3 ) + 6 ⋅ 3

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x + 5 , 4 3 π r 3 , x + 5 , 4 3 π r 3 , or 2 m 3 n 2 . 2 m 3 n 2 . In the expression x + 5 , x + 5 , 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

  • ⓑ 4 3 π r 3 4 3 π r 3
  • ⓒ 2 m 3 n 2 2 m 3 n 2
  • ⓐ 2 π r ( r + h ) 2 π r ( r + h )
  • ⓑ 2( L + W )
  • ⓒ 4 y 3 + y 4 y 3 + y

Evaluating an Algebraic Expression at Different Values

Evaluate the expression 2 x − 7 2 x − 7 for each value for x.

  • ⓐ x = 0 x = 0
  • ⓑ x = 1 x = 1
  • ⓒ x = 1 2 x = 1 2
  • ⓓ x = −4 x = −4
  • ⓐ Substitute 0 for x . x . 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7
  • ⓑ Substitute 1 for x . x . 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5
  • ⓒ Substitute 1 2 1 2 for x . x . 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6
  • ⓓ Substitute −4 −4 for x . x . 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15

Evaluate the expression 11 − 3 y 11 − 3 y for each value for y.

  • ⓐ y = 2 y = 2
  • ⓑ y = 0 y = 0
  • ⓒ y = 2 3 y = 2 3
  • ⓓ y = −5 y = −5

Evaluate each expression for the given values.

  • ⓐ x + 5 x + 5 for x = −5 x = −5
  • ⓑ t 2 t −1 t 2 t −1 for t = 10 t = 10
  • ⓒ 4 3 π r 3 4 3 π r 3 for r = 5 r = 5
  • ⓓ a + a b + b a + a b + b for a = 11 , b = −8 a = 11 , b = −8
  • ⓔ 2 m 3 n 2 2 m 3 n 2 for m = 2 , n = 3 m = 2 , n = 3
  • ⓐ Substitute −5 −5 for x . x . x + 5 = ( −5 ) + 5 = 0 x + 5 = ( −5 ) + 5 = 0
  • ⓑ Substitute 10 for t . t . t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19 t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19
  • ⓒ Substitute 5 for r . r . 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π
  • ⓓ Substitute 11 for a a and –8 for b . b . a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85 a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85
  • ⓔ Substitute 2 for m m and 3 for n . n . 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12
  • ⓐ y + 3 y − 3 y + 3 y − 3 for y = 5 y = 5
  • ⓑ 7 − 2 t 7 − 2 t for t = −2 t = −2
  • ⓒ 1 3 π r 2 1 3 π r 2 for r = 11 r = 11
  • ⓓ ( p 2 q ) 3 ( p 2 q ) 3 for p = −2 , q = 3 p = −2 , q = 3
  • ⓔ 4 ( m − n ) − 5 ( n − m ) 4 ( m − n ) − 5 ( n − m ) for m = 2 3 , n = 1 3 m = 2 3 , n = 1 3

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2 x + 1 = 7 2 x + 1 = 7 has the solution of 3 because when we substitute 3 for x x in the equation, we obtain the true statement 2 ( 3 ) + 1 = 7. 2 ( 3 ) + 1 = 7.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A A of a circle in terms of the radius r r of the circle: A = π r 2 . A = π r 2 . For any value of r , r , the area A A can be found by evaluating the expression π r 2 . π r 2 .

Using a Formula

A right circular cylinder with radius r r and height h h has the surface area S S (in square units) given by the formula S = 2 π r ( r + h ) . S = 2 π r ( r + h ) . See Figure 3 . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π . π .

Evaluate the expression 2 π r ( r + h ) 2 π r ( r + h ) for r = 6 r = 6 and h = 9. h = 9.

The surface area is 180 π 180 π square inches.

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm 2 ) is found to be A = ( L + 16 ) ( W + 16 ) − L ⋅ W . A = ( L + 16 ) ( W + 16 ) − L ⋅ W . See Figure 4 . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

  • ⓐ 3 x − 2 y + x − 3 y − 7 3 x − 2 y + x − 3 y − 7
  • ⓑ 2 r − 5 ( 3 − r ) + 4 2 r − 5 ( 3 − r ) + 4
  • ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) ( 4 t − 5 4 s ) − ( 2 3 t + 2 s )
  • ⓓ 2 m n − 5 m + 3 m n + n 2 m n − 5 m + 3 m n + n
  • ⓐ 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify. 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify.
  • ⓑ 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify. 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify.
  • ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify. ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify.
  • ⓓ 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify. 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify.
  • ⓐ 2 3 y − 2 ( 4 3 y + z ) 2 3 y − 2 ( 4 3 y + z )
  • ⓑ 5 t − 2 − 3 t + 1 5 t − 2 − 3 t + 1
  • ⓒ 4 p ( q − 1 ) + q ( 1 − p ) 4 p ( q − 1 ) + q ( 1 − p )
  • ⓓ 9 r − ( s + 2 r ) + ( 6 − s ) 9 r − ( s + 2 r ) + ( 6 − s )

Simplifying a Formula

A rectangle with length L L and width W W has a perimeter P P given by P = L + W + L + W . P = L + W + L + W . Simplify this expression.

If the amount P P is deposited into an account paying simple interest r r for time t , t , the total value of the deposit A A is given by A = P + P r t . A = P + P r t . Simplify the expression. (This formula will be explored in more detail later in the course.)

Access these online resources for additional instruction and practice with real numbers.

  • Simplify an Expression.
  • Evaluate an Expression 1.
  • Evaluate an Expression 2.

1.1 Section Exercises

Is 2 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

For the following exercises, simplify the given expression.

10 + 2 × ( 5 − 3 ) 10 + 2 × ( 5 − 3 )

6 ÷ 2 − ( 81 ÷ 3 2 ) 6 ÷ 2 − ( 81 ÷ 3 2 )

18 + ( 6 − 8 ) 3 18 + ( 6 − 8 ) 3

−2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2 −2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2

4 − 6 + 2 × 7 4 − 6 + 2 × 7

3 ( 5 − 8 ) 3 ( 5 − 8 )

4 + 6 − 10 ÷ 2 4 + 6 − 10 ÷ 2

12 ÷ ( 36 ÷ 9 ) + 6 12 ÷ ( 36 ÷ 9 ) + 6

( 4 + 5 ) 2 ÷ 3 ( 4 + 5 ) 2 ÷ 3

3 − 12 × 2 + 19 3 − 12 × 2 + 19

2 + 8 × 7 ÷ 4 2 + 8 × 7 ÷ 4

5 + ( 6 + 4 ) − 11 5 + ( 6 + 4 ) − 11

9 − 18 ÷ 3 2 9 − 18 ÷ 3 2

14 × 3 ÷ 7 − 6 14 × 3 ÷ 7 − 6

9 − ( 3 + 11 ) × 2 9 − ( 3 + 11 ) × 2

6 + 2 × 2 − 1 6 + 2 × 2 − 1

64 ÷ ( 8 + 4 × 2 ) 64 ÷ ( 8 + 4 × 2 )

9 + 4 ( 2 2 ) 9 + 4 ( 2 2 )

( 12 ÷ 3 × 3 ) 2 ( 12 ÷ 3 × 3 ) 2

25 ÷ 5 2 − 7 25 ÷ 5 2 − 7

( 15 − 7 ) × ( 3 − 7 ) ( 15 − 7 ) × ( 3 − 7 )

2 × 4 − 9 ( −1 ) 2 × 4 − 9 ( −1 )

4 2 − 25 × 1 5 4 2 − 25 × 1 5

12 ( 3 − 1 ) ÷ 6 12 ( 3 − 1 ) ÷ 6

For the following exercises, evaluate the expression using the given value of the variable.

8 ( x + 3 ) – 64 8 ( x + 3 ) – 64 for x = 2 x = 2

4 y + 8 – 2 y 4 y + 8 – 2 y for y = 3 y = 3

( 11 a + 3 ) − 18 a + 4 ( 11 a + 3 ) − 18 a + 4 for a = –2 a = –2

4 z − 2 z ( 1 + 4 ) – 36 4 z − 2 z ( 1 + 4 ) – 36 for z = 5 z = 5

4 y ( 7 − 2 ) 2 + 200 4 y ( 7 − 2 ) 2 + 200 for y = –2 y = –2

− ( 2 x ) 2 + 1 + 3 − ( 2 x ) 2 + 1 + 3 for x = 2 x = 2

For the 8 ( 2 + 4 ) − 15 b + b 8 ( 2 + 4 ) − 15 b + b for b = –3 b = –3

2 ( 11 c − 4 ) – 36 2 ( 11 c − 4 ) – 36 for c = 0 c = 0

4 ( 3 − 1 ) x – 4 4 ( 3 − 1 ) x – 4 for x = 10 x = 10

1 4 ( 8 w − 4 2 ) 1 4 ( 8 w − 4 2 ) for w = 1 w = 1

For the following exercises, simplify the expression.

4 x + x ( 13 − 7 ) 4 x + x ( 13 − 7 )

2 y − ( 4 ) 2 y − 11 2 y − ( 4 ) 2 y − 11

a 2 3 ( 64 ) − 12 a ÷ 6 a 2 3 ( 64 ) − 12 a ÷ 6

8 b − 4 b ( 3 ) + 1 8 b − 4 b ( 3 ) + 1

5 l ÷ 3 l × ( 9 − 6 ) 5 l ÷ 3 l × ( 9 − 6 )

7 z − 3 + z × 6 2 7 z − 3 + z × 6 2

4 × 3 + 18 x ÷ 9 − 12 4 × 3 + 18 x ÷ 9 − 12

9 ( y + 8 ) − 27 9 ( y + 8 ) − 27

( 9 6 t − 4 ) 2 ( 9 6 t − 4 ) 2

6 + 12 b − 3 × 6 b 6 + 12 b − 3 × 6 b

18 y − 2 ( 1 + 7 y ) 18 y − 2 ( 1 + 7 y )

( 4 9 ) 2 × 27 x ( 4 9 ) 2 × 27 x

8 ( 3 − m ) + 1 ( − 8 ) 8 ( 3 − m ) + 1 ( − 8 )

9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x ) 9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x )

5 2 − 4 ( 3 x ) 5 2 − 4 ( 3 x )

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π . π . Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of g g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,000 − x = 0. 2,500,000 − x = 0. What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

0.5 ( 12.3 ) 2 − 48 x = 3 5 0.5 ( 12.3 ) 2 − 48 x = 3 5

( 0.25 − 0.75 ) 2 x − 7.2 = 9.9 ( 0.25 − 0.75 ) 2 x − 7.2 = 9.9

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: −18 − 4 ( 5 ) ( −1 ) . −18 − 4 ( 5 ) ( −1 ) .

Determine whether the simplified expression is rational or irrational: −16 + 4 ( 5 ) + 5 . −16 + 4 ( 5 ) + 5 .

The division of two natural numbers will always result in what type of number?

What property of real numbers would simplify the following expression: 4 + 7 ( x − 1 ) ? 4 + 7 ( x − 1 ) ?

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Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
  • Authors: Jay Abramson
  • Publisher/website: OpenStax
  • Book title: College Algebra 2e
  • Publication date: Dec 21, 2021
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
  • Section URL: https://openstax.org/books/college-algebra-2e/pages/1-1-real-numbers-algebra-essentials

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Mathematics LibreTexts

8.4: Homework

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  • Page ID 70331

  • Julie Harland
  • MiraCosta College
  • Submit homework separately from this workbook and staple all pages together. (One staple for the entire submission of all the unit homework)
  • Start a new module on the front side of a new page and write the module number on the top center of the page.
  • Answers without supporting work will receive no credit.
  • Some solutions are given in the solutions manual.
  • You may work with classmates but do your own work.

Find GCF(252, 350) using:

Find GCF(140, 315) using:

Use the Euclidean Algorithm to compute the greatest common factor of the numbers given. Use correct notation, and show each step. Then, show how you check your answer. Also, compute the LCM of the two numbers.

State whether each of the following statements is true or false. If it is false, provide a counterexample. If it is true, provide an example.

a. If (a + b)|c, then a|c and b|c

b. If a|b and a|c, then a|(bc)

c. If a|b and a|(b + c), then a|c

d. If a|bc, then a|b and a|c

e. If a|b and a|c, then a|(b + c)

Write the prime factorization for the following numbers. If it is prime, write "prime" and explain how you know it is prime.

Assume m and n are composite whole numbers in each of the following. Find the following. Then provide an example using numbers for m (and n where used). Remember not to use prime numbers in your example.

a. GCF(m,m) =

b. LCM(m,m) =

c. GCF(m,0) =

d. GCF(m,1) =

e. If GCF(m,n) = 1, then LCM(m,n) =

f. If GCF(m,n) = m, then LCM(m,n) =

g. If LCM(m,n) = mn, then GCF(m,n) =

Find the following sums using methods from this module: Show all work

a. 1 + 2 + 3 + . . . + 313 + 314 + 315 =

b. 111 + 112 + 113 + . . . + 287 + 288 + 289 =

c. 15 + 30 + 45 + . . . + 900 + 915 + 930 =

d. 102 + 105 + 108 + . . . + 300 + 303 + 306 =

On each number line, state all whole number possibilities less than 100 that the man could be standing on.

The factors of a number that are less than the number itself are called proper factors . For instance, the proper factors of 10 are 1, 2 and 5. A number is classified as deficient if the sum of its proper factors is less than the number itself. 10 is a deficient number since 1 + 2 + 5 < 10. A number is classified as abundant if the sum of its proper factors is greater than the number itself. For instance, the proper factors of 18 are 1, 2, 3, 6, and 9. 18 is a deficient number since 1 + 2 + 3 + 6 + 9 > 18. A number is classified as perfect if the sum of its proper factors equals the number itself. For each number, list its proper factors. Then find the sum of its proper factors. Then, classify each number as deficient, abundant or perfect.

Are prime numbers deficient, perfect, or abundant? ________ Explain why.

Answer true or false for each of the following. If it is true, provide an example. If it is false, provide a counterexample.

a. Every prime number is odd.

b. If a number is divisible by 6, then it is divisible by 2 and 3.

c. If a number is divisible by 2 and 6, then it is divisible by 12.

d. If a number is divisible by 3 and 4, then it is divisible by 12.

e. If a \(\neq\) b, then GCF(a, b) < LCM(a, b).

f. If 6 is a factor of mn, then 6 is a factor of m or a factor of n.

g. If 5 is a factor of mn, then 5 is a factor of m or a factor of n.

Can the sum of two odd prime numbers be a prime number? Explain why or why not.

Find the least common multiple of the following sets of numbers:

a. LCM(2, 4, 5, 7, 8, 12, 14, 15)

b. LCM(3, 4, 6, 8, 9, 10, 12, 18)

If GCF(30, x) = 6 and LCM(30, x) = 180, then what is x? (Hint: see page 65)

The theory of biorhythm states that your physical cycle is 23 days long, your emotional cycle is 28 days long, your intellectual cycle is 33 days long. If your cycles all occur on the same day, how many days until your cycles again occur on the same day? About how many years is this?

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Everyone struggles with homework sometimes, but if getting your homework done has become a chronic issue for you, then you may need a little extra help. That’s why we’ve written this article all about how to do homework. Once you’re finished reading it, you’ll know how to do homework (and have tons of new ways to motivate yourself to do homework)! 

We’ve broken this article down into a few major sections. You’ll find: 

  • A diagnostic test to help you figure out why you’re struggling with homework
  • A discussion of the four major homework problems students face, along with expert tips for addressing them 
  • A bonus section with tips for how to do homework fast

By the end of this article, you’ll be prepared to tackle whatever homework assignments your teachers throw at you . 

So let’s get started! 

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How to Do Homework: Figure Out Your Struggles 

Sometimes it feels like everything is standing between you and getting your homework done. But the truth is, most people only have one or two major roadblocks that are keeping them from getting their homework done well and on time. 

The best way to figure out how to get motivated to do homework starts with pinpointing the issues that are affecting your ability to get your assignments done. That’s why we’ve developed a short quiz to help you identify the areas where you’re struggling. 

Take the quiz below and record your answers on your phone or on a scrap piece of paper. Keep in mind there are no wrong answers! 

1. You’ve just been assigned an essay in your English class that’s due at the end of the week. What’s the first thing you do?

A. Keep it in mind, even though you won’t start it until the day before it’s due  B. Open up your planner. You’ve got to figure out when you’ll write your paper since you have band practice, a speech tournament, and your little sister’s dance recital this week, too.  C. Groan out loud. Another essay? You could barely get yourself to write the last one!  D. Start thinking about your essay topic, which makes you think about your art project that’s due the same day, which reminds you that your favorite artist might have just posted to Instagram...so you better check your feed right now. 

2. Your mom asked you to pick up your room before she gets home from work. You’ve just gotten home from school. You decide you’ll tackle your chores: 

A. Five minutes before your mom walks through the front door. As long as it gets done, who cares when you start?  B. As soon as you get home from your shift at the local grocery store.  C. After you give yourself a 15-minute pep talk about how you need to get to work.  D. You won’t get it done. Between texts from your friends, trying to watch your favorite Netflix show, and playing with your dog, you just lost track of time! 

3. You’ve signed up to wash dogs at the Humane Society to help earn money for your senior class trip. You: 

A. Show up ten minutes late. You put off leaving your house until the last minute, then got stuck in unexpected traffic on the way to the shelter.  B. Have to call and cancel at the last minute. You forgot you’d already agreed to babysit your cousin and bake cupcakes for tomorrow’s bake sale.  C. Actually arrive fifteen minutes early with extra brushes and bandanas you picked up at the store. You’re passionate about animals, so you’re excited to help out! D. Show up on time, but only get three dogs washed. You couldn’t help it: you just kept getting distracted by how cute they were!

4. You have an hour of downtime, so you decide you’re going to watch an episode of The Great British Baking Show. You: 

A. Scroll through your social media feeds for twenty minutes before hitting play, which means you’re not able to finish the whole episode. Ugh! You really wanted to see who was sent home!  B. Watch fifteen minutes until you remember you’re supposed to pick up your sister from band practice before heading to your part-time job. No GBBO for you!  C. You finish one episode, then decide to watch another even though you’ve got SAT studying to do. It’s just more fun to watch people make scones.  D. Start the episode, but only catch bits and pieces of it because you’re reading Twitter, cleaning out your backpack, and eating a snack at the same time.

5. Your teacher asks you to stay after class because you’ve missed turning in two homework assignments in a row. When she asks you what’s wrong, you say: 

A. You planned to do your assignments during lunch, but you ran out of time. You decided it would be better to turn in nothing at all than submit unfinished work.  B. You really wanted to get the assignments done, but between your extracurriculars, family commitments, and your part-time job, your homework fell through the cracks.  C. You have a hard time psyching yourself to tackle the assignments. You just can’t seem to find the motivation to work on them once you get home.  D. You tried to do them, but you had a hard time focusing. By the time you realized you hadn’t gotten anything done, it was already time to turn them in. 

Like we said earlier, there are no right or wrong answers to this quiz (though your results will be better if you answered as honestly as possible). Here’s how your answers break down: 

  • If your answers were mostly As, then your biggest struggle with doing homework is procrastination. 
  • If your answers were mostly Bs, then your biggest struggle with doing homework is time management. 
  • If your answers were mostly Cs, then your biggest struggle with doing homework is motivation. 
  • If your answers were mostly Ds, then your biggest struggle with doing homework is getting distracted. 

Now that you’ve identified why you’re having a hard time getting your homework done, we can help you figure out how to fix it! Scroll down to find your core problem area to learn more about how you can start to address it. 

And one more thing: you’re really struggling with homework, it’s a good idea to read through every section below. You may find some additional tips that will help make homework less intimidating. 

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How to Do Homework When You’re a Procrastinator  

Merriam Webster defines “procrastinate” as “to put off intentionally and habitually.” In other words, procrastination is when you choose to do something at the last minute on a regular basis. If you’ve ever found yourself pulling an all-nighter, trying to finish an assignment between periods, or sprinting to turn in a paper minutes before a deadline, you’ve experienced the effects of procrastination. 

If you’re a chronic procrastinator, you’re in good company. In fact, one study found that 70% to 95% of undergraduate students procrastinate when it comes to doing their homework. Unfortunately, procrastination can negatively impact your grades. Researchers have found that procrastination can lower your grade on an assignment by as much as five points ...which might not sound serious until you realize that can mean the difference between a B- and a C+. 

Procrastination can also negatively affect your health by increasing your stress levels , which can lead to other health conditions like insomnia, a weakened immune system, and even heart conditions. Getting a handle on procrastination can not only improve your grades, it can make you feel better, too! 

The big thing to understand about procrastination is that it’s not the result of laziness. Laziness is defined as being “disinclined to activity or exertion.” In other words, being lazy is all about doing nothing. But a s this Psychology Today article explains , procrastinators don’t put things off because they don’t want to work. Instead, procrastinators tend to postpone tasks they don’t want to do in favor of tasks that they perceive as either more important or more fun. Put another way, procrastinators want to do things...as long as it’s not their homework! 

3 Tips f or Conquering Procrastination 

Because putting off doing homework is a common problem, there are lots of good tactics for addressing procrastination. Keep reading for our three expert tips that will get your homework habits back on track in no time. 

#1: Create a Reward System

Like we mentioned earlier, procrastination happens when you prioritize other activities over getting your homework done. Many times, this happens because homework...well, just isn’t enjoyable. But you can add some fun back into the process by rewarding yourself for getting your work done. 

Here’s what we mean: let’s say you decide that every time you get your homework done before the day it’s due, you’ll give yourself a point. For every five points you earn, you’ll treat yourself to your favorite dessert: a chocolate cupcake! Now you have an extra (delicious!) incentive to motivate you to leave procrastination in the dust. 

If you’re not into cupcakes, don’t worry. Your reward can be anything that motivates you . Maybe it’s hanging out with your best friend or an extra ten minutes of video game time. As long as you’re choosing something that makes homework worth doing, you’ll be successful. 

#2: Have a Homework Accountability Partner 

If you’re having trouble getting yourself to start your homework ahead of time, it may be a good idea to call in reinforcements . Find a friend or classmate you can trust and explain to them that you’re trying to change your homework habits. Ask them if they’d be willing to text you to make sure you’re doing your homework and check in with you once a week to see if you’re meeting your anti-procrastination goals. 

Sharing your goals can make them feel more real, and an accountability partner can help hold you responsible for your decisions. For example, let’s say you’re tempted to put off your science lab write-up until the morning before it’s due. But you know that your accountability partner is going to text you about it tomorrow...and you don’t want to fess up that you haven’t started your assignment. A homework accountability partner can give you the extra support and incentive you need to keep your homework habits on track. 

#3: Create Your Own Due Dates 

If you’re a life-long procrastinator, you might find that changing the habit is harder than you expected. In that case, you might try using procrastination to your advantage! If you just can’t seem to stop doing your work at the last minute, try setting your own due dates for assignments that range from a day to a week before the assignment is actually due. 

Here’s what we mean. Let’s say you have a math worksheet that’s been assigned on Tuesday and is due on Friday. In your planner, you can write down the due date as Thursday instead. You may still put off your homework assignment until the last minute...but in this case, the “last minute” is a day before the assignment’s real due date . This little hack can trick your procrastination-addicted brain into planning ahead! 

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If you feel like Kevin Hart in this meme, then our tips for doing homework when you're busy are for you. 

How to Do Homework When You’re too Busy

If you’re aiming to go to a top-tier college , you’re going to have a full plate. Because college admissions is getting more competitive, it’s important that you’re maintaining your grades , studying hard for your standardized tests , and participating in extracurriculars so your application stands out. A packed schedule can get even more hectic once you add family obligations or a part-time job to the mix. 

If you feel like you’re being pulled in a million directions at once, you’re not alone. Recent research has found that stress—and more severe stress-related conditions like anxiety and depression— are a major problem for high school students . In fact, one study from the American Psychological Association found that during the school year, students’ stress levels are higher than those of the adults around them. 

For students, homework is a major contributor to their overall stress levels . Many high schoolers have multiple hours of homework every night , and figuring out how to fit it into an already-packed schedule can seem impossible. 

3 Tips for Fitting Homework Into Your Busy Schedule

While it might feel like you have literally no time left in your schedule, there are still ways to make sure you’re able to get your homework done and meet your other commitments. Here are our expert homework tips for even the busiest of students. 

#1: Make a Prioritized To-Do List 

You probably already have a to-do list to keep yourself on track. The next step is to prioritize the items on your to-do list so you can see what items need your attention right away. 

Here’s how it works: at the beginning of each day, sit down and make a list of all the items you need to get done before you go to bed. This includes your homework, but it should also take into account any practices, chores, events, or job shifts you may have. Once you get everything listed out, it’s time to prioritize them using the labels A, B, and C. Here’s what those labels mean:

  • A Tasks : tasks that have to get done—like showing up at work or turning in an assignment—get an A. 
  • B Tasks : these are tasks that you would like to get done by the end of the day but aren’t as time sensitive. For example, studying for a test you have next week could be a B-level task. It’s still important, but it doesn’t have to be done right away. 
  • C Tasks: these are tasks that aren’t very important and/or have no real consequences if you don’t get them done immediately. For instance, if you’re hoping to clean out your closet but it’s not an assigned chore from your parents, you could label that to-do item with a C. 

Prioritizing your to-do list helps you visualize which items need your immediate attention, and which items you can leave for later. A prioritized to-do list ensures that you’re spending your time efficiently and effectively, which helps you make room in your schedule for homework. So even though you might really want to start making decorations for Homecoming (a B task), you’ll know that finishing your reading log (an A task) is more important. 

#2: Use a Planner With Time Labels 

Your planner is probably packed with notes, events, and assignments already. (And if you’re not using a planner, it’s time to start!) But planners can do more for you than just remind you when an assignment is due. If you’re using a planner with time labels, it can help you visualize how you need to spend your day.

A planner with time labels breaks your day down into chunks, and you assign tasks to each chunk of time. For example, you can make a note of your class schedule with assignments, block out time to study, and make sure you know when you need to be at practice. Once you know which tasks take priority, you can add them to any empty spaces in your day. 

Planning out how you spend your time not only helps you use it wisely, it can help you feel less overwhelmed, too . We’re big fans of planners that include a task list ( like this one ) or have room for notes ( like this one ). 

#3: Set Reminders on Your Phone 

If you need a little extra nudge to make sure you’re getting your homework done on time, it’s a good idea to set some reminders on your phone. You don’t need a fancy app, either. You can use your alarm app to have it go off at specific times throughout the day to remind you to do your homework. This works especially well if you have a set homework time scheduled. So if you’ve decided you’re doing homework at 6:00 pm, you can set an alarm to remind you to bust out your books and get to work. 

If you use your phone as your planner, you may have the option to add alerts, emails, or notifications to scheduled events . Many calendar apps, including the one that comes with your phone, have built-in reminders that you can customize to meet your needs. So if you block off time to do your homework from 4:30 to 6:00 pm, you can set a reminder that will pop up on your phone when it’s time to get started. 

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This dog isn't judging your lack of motivation...but your teacher might. Keep reading for tips to help you motivate yourself to do your homework.

How to Do Homework When You’re Unmotivated 

At first glance, it may seem like procrastination and being unmotivated are the same thing. After all, both of these issues usually result in you putting off your homework until the very last minute. 

But there’s one key difference: many procrastinators are working, they’re just prioritizing work differently. They know they’re going to start their homework...they’re just going to do it later. 

Conversely, people who are unmotivated to do homework just can’t find the willpower to tackle their assignments. Procrastinators know they’ll at least attempt the homework at the last minute, whereas people who are unmotivated struggle with convincing themselves to do it at a ll. For procrastinators, the stress comes from the inevitable time crunch. For unmotivated people, the stress comes from trying to convince themselves to do something they don’t want to do in the first place. 

Here are some common reasons students are unmotivated in doing homework : 

  • Assignments are too easy, too hard, or seemingly pointless 
  • Students aren’t interested in (or passionate about) the subject matter
  • Students are intimidated by the work and/or feels like they don’t understand the assignment 
  • Homework isn’t fun, and students would rather spend their time on things that they enjoy 

To sum it up: people who lack motivation to do their homework are more likely to not do it at all, or to spend more time worrying about doing their homework than...well, actually doing it.

3 Tips for How to Get Motivated to Do Homework

The key to getting homework done when you’re unmotivated is to figure out what does motivate you, then apply those things to homework. It sounds tricky...but it’s pretty simple once you get the hang of it! Here are our three expert tips for motivating yourself to do your homework. 

#1: Use Incremental Incentives

When you’re not motivated, it’s important to give yourself small rewards to stay focused on finishing the task at hand. The trick is to keep the incentives small and to reward yourself often. For example, maybe you’re reading a good book in your free time. For every ten minutes you spend on your homework, you get to read five pages of your book. Like we mentioned earlier, make sure you’re choosing a reward that works for you! 

So why does this technique work? Using small rewards more often allows you to experience small wins for getting your work done. Every time you make it to one of your tiny reward points, you get to celebrate your success, which gives your brain a boost of dopamine . Dopamine helps you stay motivated and also creates a feeling of satisfaction when you complete your homework !  

#2: Form a Homework Group 

If you’re having trouble motivating yourself, it’s okay to turn to others for support. Creating a homework group can help with this. Bring together a group of your friends or classmates, and pick one time a week where you meet and work on homework together. You don’t have to be in the same class, or even taking the same subjects— the goal is to encourage one another to start (and finish!) your assignments. 

Another added benefit of a homework group is that you can help one another if you’re struggling to understand the material covered in your classes. This is especially helpful if your lack of motivation comes from being intimidated by your assignments. Asking your friends for help may feel less scary than talking to your teacher...and once you get a handle on the material, your homework may become less frightening, too. 

#3: Change Up Your Environment 

If you find that you’re totally unmotivated, it may help if you find a new place to do your homework. For example, if you’ve been struggling to get your homework done at home, try spending an extra hour in the library after school instead. The change of scenery can limit your distractions and give you the energy you need to get your work done. 

If you’re stuck doing homework at home, you can still use this tip. For instance, maybe you’ve always done your homework sitting on your bed. Try relocating somewhere else, like your kitchen table, for a few weeks. You may find that setting up a new “homework spot” in your house gives you a motivational lift and helps you get your work done. 

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Social media can be a huge problem when it comes to doing homework. We have advice for helping you unplug and regain focus.

How to Do Homework When You’re Easily Distracted

We live in an always-on world, and there are tons of things clamoring for our attention. From friends and family to pop culture and social media, it seems like there’s always something (or someone!) distracting us from the things we need to do.

The 24/7 world we live in has affected our ability to focus on tasks for prolonged periods of time. Research has shown that over the past decade, an average person’s attention span has gone from 12 seconds to eight seconds . And when we do lose focus, i t takes people a long time to get back on task . One study found that it can take as long as 23 minutes to get back to work once we’ve been distracte d. No wonder it can take hours to get your homework done! 

3 Tips to Improve Your Focus

If you have a hard time focusing when you’re doing your homework, it’s a good idea to try and eliminate as many distractions as possible. Here are three expert tips for blocking out the noise so you can focus on getting your homework done. 

#1: Create a Distraction-Free Environment

Pick a place where you’ll do your homework every day, and make it as distraction-free as possible. Try to find a location where there won’t be tons of noise, and limit your access to screens while you’re doing your homework. Put together a focus-oriented playlist (or choose one on your favorite streaming service), and put your headphones on while you work. 

You may find that other people, like your friends and family, are your biggest distraction. If that’s the case, try setting up some homework boundaries. Let them know when you’ll be working on homework every day, and ask them if they’ll help you keep a quiet environment. They’ll be happy to lend a hand! 

#2: Limit Your Access to Technology 

We know, we know...this tip isn’t fun, but it does work. For homework that doesn’t require a computer, like handouts or worksheets, it’s best to put all your technology away . Turn off your television, put your phone and laptop in your backpack, and silence notifications on any wearable tech you may be sporting. If you listen to music while you work, that’s fine...but make sure you have a playlist set up so you’re not shuffling through songs once you get started on your homework. 

If your homework requires your laptop or tablet, it can be harder to limit your access to distractions. But it’s not impossible! T here are apps you can download that will block certain websites while you’re working so that you’re not tempted to scroll through Twitter or check your Facebook feed. Silence notifications and text messages on your computer, and don’t open your email account unless you absolutely have to. And if you don’t need access to the internet to complete your assignments, turn off your WiFi. Cutting out the online chatter is a great way to make sure you’re getting your homework done. 

#3: Set a Timer (the Pomodoro Technique)

Have you ever heard of the Pomodoro technique ? It’s a productivity hack that uses a timer to help you focus!

Here’s how it works: first, set a timer for 25 minutes. This is going to be your work time. During this 25 minutes, all you can do is work on whatever homework assignment you have in front of you. No email, no text messaging, no phone calls—just homework. When that timer goes off, y ou get to take a 5 minute break. Every time you go through one of these cycles, it’s called a “pomodoro.” For every four pomodoros you complete, you can take a longer break of 15 to 30 minutes. 

The pomodoro technique works through a combination of boundary setting and rewards. First, it gives you a finite amount of time to focus, so you know that you only have to work really hard for 25 minutes. Once you’ve done that, you’re rewarded with a short break where you can do whatever you want. Additionally, tracking how many pomodoros you complete can help you see how long you’re really working on your homework. (Once you start using our focus tips, you may find it doesn’t take as long as you thought!) 

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Two Bonus Tips for How to Do Homework Fast 

Even if you’re doing everything right, there will be times when you just need to get your homework done as fast as possible. (Why do teachers always have projects due in the same week? The world may never know.) 

The problem with speeding through homework is that it’s easy to make mistakes. While turning in an assignment is always better than not submitting anything at all, you want to make sure that you’re not compromising quality for speed. Simply put, the goal is to get your homework done quickly and still make a good grade on the assignment! 

Here are our two bonus tips for getting a decent grade on your homework assignments , even when you’re in a time crunch. 

#1: Do the Easy Parts First 

This is especially true if you’re working on a handout with multiple questions. Before you start working on the assignment, read through all the questions and problems. As you do, make a mark beside the questions you think are “easy” to answer . 

Once you’ve finished going through the whole assignment, you can answer these questions first. Getting the easy questions out of the way as quickly as possible lets you spend more time on the trickier portions of your homework, which will maximize your assignment grade. 

(Quick note: this is also a good strategy to use on timed assignments and tests, like the SAT and the ACT !) 

#2: Pay Attention in Class 

Homework gets a lot easier when you’re actively learning the material. Teachers aren’t giving you homework because they’re mean or trying to ruin your weekend... it’s because they want you to really understand the course material. Homework is designed to reinforce what you’re already learning in class so you’ll be ready to tackle harder concepts later. 

When you pay attention in class, ask questions, and take good notes, you’re absorbing the information you’ll need to succeed on your homework assignments. (You’re stuck in class anyway, so you might as well make the most of it!) Not only will paying attention in class make your homework less confusing, it will also help it go much faster, too. 

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What’s Next? 

If you’re looking to improve your productivity beyond homework, a good place to begin is with time management. After all, we only have so much time in a day...so it’s important to get the most out of it! To get you started, check out this list of the 12 best time management techniques that you can start using today.

You may have read this article because homework struggles have been affecting your GPA. Now that you’re on the path to homework success, it’s time to start being proactive about raising your grades. This article teaches you everything you need to know about raising your GPA so you can

Now you know how to get motivated to do homework...but what about your study habits? Studying is just as critical to getting good grades, and ultimately getting into a good college . We can teach you how to study bette r in high school. (We’ve also got tons of resources to help you study for your ACT and SAT exams , too!) 

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Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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11 Surprising Homework Statistics, Facts & Data

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The age-old question of whether homework is good or bad for students is unanswerable because there are so many “ it depends ” factors.

For example, it depends on the age of the child, the type of homework being assigned, and even the child’s needs.

There are also many conflicting reports on whether homework is good or bad. This is a topic that largely relies on data interpretation for the researcher to come to their conclusions.

To cut through some of the fog, below I’ve outlined some great homework statistics that can help us understand the effects of homework on children.

Homework Statistics List

1. 45% of parents think homework is too easy for their children.

A study by the Center for American Progress found that parents are almost twice as likely to believe their children’s homework is too easy than to disagree with that statement.

Here are the figures for math homework:

  • 46% of parents think their child’s math homework is too easy.
  • 25% of parents think their child’s math homework is not too easy.
  • 29% of parents offered no opinion.

Here are the figures for language arts homework:

  • 44% of parents think their child’s language arts homework is too easy.
  • 28% of parents think their child’s language arts homework is not too easy.
  • 28% of parents offered no opinion.

These findings are based on online surveys of 372 parents of school-aged children conducted in 2018.

2. 93% of Fourth Grade Children Worldwide are Assigned Homework

The prestigious worldwide math assessment Trends in International Maths and Science Study (TIMSS) took a survey of worldwide homework trends in 2007. Their study concluded that 93% of fourth-grade children are regularly assigned homework, while just 7% never or rarely have homework assigned.

3. 17% of Teens Regularly Miss Homework due to Lack of High-Speed Internet Access

A 2018 Pew Research poll of 743 US teens found that 17%, or almost 2 in every 5 students, regularly struggled to complete homework because they didn’t have reliable access to the internet.

This figure rose to 25% of Black American teens and 24% of teens whose families have an income of less than $30,000 per year.

4. Parents Spend 6.7 Hours Per Week on their Children’s Homework

A 2018 study of 27,500 parents around the world found that the average amount of time parents spend on homework with their child is 6.7 hours per week. Furthermore, 25% of parents spend more than 7 hours per week on their child’s homework.

American parents spend slightly below average at 6.2 hours per week, while Indian parents spend 12 hours per week and Japanese parents spend 2.6 hours per week.

5. Students in High-Performing High Schools Spend on Average 3.1 Hours per night Doing Homework

A study by Galloway, Conner & Pope (2013) conducted a sample of 4,317 students from 10 high-performing high schools in upper-middle-class California. 

Across these high-performing schools, students self-reported that they did 3.1 hours per night of homework.

Graduates from those schools also ended up going on to college 93% of the time.

6. One to Two Hours is the Optimal Duration for Homework

A 2012 peer-reviewed study in the High School Journal found that students who conducted between one and two hours achieved higher results in tests than any other group.

However, the authors were quick to highlight that this “t is an oversimplification of a much more complex problem.” I’m inclined to agree. The greater variable is likely the quality of the homework than time spent on it.

Nevertheless, one result was unequivocal: that some homework is better than none at all : “students who complete any amount of homework earn higher test scores than their peers who do not complete homework.”

7. 74% of Teens cite Homework as a Source of Stress

A study by the Better Sleep Council found that homework is a source of stress for 74% of students. Only school grades, at 75%, rated higher in the study.

That figure rises for girls, with 80% of girls citing homework as a source of stress.

Similarly, the study by Galloway, Conner & Pope (2013) found that 56% of students cite homework as a “primary stressor” in their lives.

8. US Teens Spend more than 15 Hours per Week on Homework

The same study by the Better Sleep Council also found that US teens spend over 2 hours per school night on homework, and overall this added up to over 15 hours per week.

Surprisingly, 4% of US teens say they do more than 6 hours of homework per night. That’s almost as much homework as there are hours in the school day.

The only activity that teens self-reported as doing more than homework was engaging in electronics, which included using phones, playing video games, and watching TV.

9. The 10-Minute Rule

The National Education Association (USA) endorses the concept of doing 10 minutes of homework per night per grade.

For example, if you are in 3rd grade, you should do 30 minutes of homework per night. If you are in 4th grade, you should do 40 minutes of homework per night.

However, this ‘rule’ appears not to be based in sound research. Nevertheless, it is true that homework benefits (no matter the quality of the homework) will likely wane after 2 hours (120 minutes) per night, which would be the NEA guidelines’ peak in grade 12.

10. 21.9% of Parents are Too Busy for their Children’s Homework

An online poll of nearly 300 parents found that 21.9% are too busy to review their children’s homework. On top of this, 31.6% of parents do not look at their children’s homework because their children do not want their help. For these parents, their children’s unwillingness to accept their support is a key source of frustration.

11. 46.5% of Parents find Homework too Hard

The same online poll of parents of children from grades 1 to 12 also found that many parents struggle to help their children with homework because parents find it confusing themselves. Unfortunately, the study did not ask the age of the students so more data is required here to get a full picture of the issue.

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Interpreting the Data

Unfortunately, homework is one of those topics that can be interpreted by different people pursuing differing agendas. All studies of homework have a wide range of variables, such as:

  • What age were the children in the study?
  • What was the homework they were assigned?
  • What tools were available to them?
  • What were the cultural attitudes to homework and how did they impact the study?
  • Is the study replicable?

The more questions we ask about the data, the more we realize that it’s hard to come to firm conclusions about the pros and cons of homework .

Furthermore, questions about the opportunity cost of homework remain. Even if homework is good for children’s test scores, is it worthwhile if the children consequently do less exercise or experience more stress?

Thus, this ends up becoming a largely qualitative exercise. If parents and teachers zoom in on an individual child’s needs, they’ll be able to more effectively understand how much homework a child needs as well as the type of homework they should be assigned.

Related: Funny Homework Excuses

The debate over whether homework should be banned will not be resolved with these homework statistics. But, these facts and figures can help you to pursue a position in a school debate on the topic – and with that, I hope your debate goes well and you develop some great debating skills!

Chris

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

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Signing Naturally Units 1-6

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Signing Naturally Units 1-6  is the first part in a series of curricular materials for the instruction of American Sign Language (ASL) as a second language. The goal is to take students with little or no knowledge of ASL and Deaf Culture and provide them with the skills needed to communicate comfortably in a wide variety of situations in the Deaf community. Cultural information taught throughout class allows students to interact with the Deaf community in a way that is respectful and aware.

Signing Naturally Units 1-6  curriculum’s first and foremost goal of language teaching is to bring a person unable to communicate in ASL to a basic level of communicative competency. The curriculum and the lessons are designed to help the class and the program meet the five areas of Communication, Cultures, Connections, Comparisons and Communities outlined by ACTFL. Click here to see how Signing Naturally Units 1-6 match up with the 5 C's!

Units 1-5 comprise several kinds of lessons: conversational (functional), skill building, cultural, and review. Conversation (function) lessons introduce vocabulary and key grammar structures in context of key dialogues. Skill lessons focus on introducing numbers, fingerspelling, spatial elements, and other supporting skills. Cultural lessons focus on behaviors that enable students to act in linguistically and socially acceptable ways. Unit 6 focuses on building narrative skills to prepare students to tell a story from their childhood.

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February 14, 2024

Surreal Numbers Are a Real Thing. Here’s How to Make Them

In the 1970s mathematicians found a simple way to create all numbers, from the infinitely small to infinitely large

By Manon Bischoff

Digitally generated image of two hands reaching out of a purple and red circular spiral touching.

Andriy Onufriyenko/Getty Images

In the early 1970s mathematician Donald Knuth spent a sabbatical with his wife in Norway. The time was meant to be spent relaxing. Yet one night he woke up his partner in a state of agitation. He urgently needed to write a book. Don’t worry, he reassured his spouse, it will only take a week. To concentrate on his writing, he reserved a hotel room just for himself in Oslo.

There he drafted what would become Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness . Though Knuth did not invent the concept of surreal numbers , he was the first to publish a detailed work on the subject and coin the term. To this day, his book is considered the standard work on the subject.

Yet the tome is anything but an ordinary example of nonfiction. It consists of dialogues between two fictional characters, Alice and Bill. It also features the true inventor of surreal numbers, the late mathematician John Horton Conway , who died in 2020.

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“In the beginning, everything was void, and J.H.W.H. Conway began to create numbers,” Knuth wrote. Knuth added extra initials to Conway’s name as an allusion to the Tetragrammaton (the four-letter Hebrew name of God, transliterated as YHWH or Yahweh), he explained in a Numberphile YouTube video .

That’s hardly the only biblical reference; even Knuth’s backstory for writing the book echoes the religious creation story. He put the whole thing to paper in just a week, as he’d promised his wife. “On the sixth day I finished it. On the seventh day I rested,” Knuth told Numberphile.

Surreal numbers are created by adding values between two given preexisting numbers. If you look at 0 and 1, for example, 1 / 2 is in the middle, 1 / 4 is between 0 and 1 / 2 , and so on. This approach makes it possible to resolve the number line ever more precisely.

This idea doesn’t sound very spectacular at first, but instead of simply delivering fractions with ever larger denominators, there is a point at which everything explodes, and suddenly values arise that are not even contained in the real numbers.

Two Axioms Result in an Incredible Universe of Numbers

Conway had established two basic rules from which the immeasurable realm of numbers arises. Every number x is defined by two sets, M L and M R , which contain previously created numbers: x = { M L : M R }.

The M L is the left-hand set and M R is a right-hand set. The first rule is that elements of the left-hand set are always smaller than those of the right-hand set.

The second rule states that 0 is the number that is bounded by two empty sets. These two rules lay the foundation for an incredibly diverse branch of mathematics!

Two circles along a number line represent the number sets surrounding a surreal number x.

Building on those foundations, Knuth’s creation story for surreal numbers takes place over several days. On the zeroth day, 0 is created out of nothing per the second rule, which he represents as 0 = { : }.

On the first day, two more numbers are created: 1 = {0: } and –1 = { :0}. These two numbers arise because they are the next largest and smallest integers (respectively) on the number line.

The second day of surreal creation gets more interesting. Now you can use different numbers for the first time. For example, {0:1} denotes the number that lies between 0 and 1, that is, 1 / 2 . You can also create {1: } = 2, {–1:0} = – 1 / 2 and { :–1} = –2.

Continue this way and on the third day you get 1 / 4 , 3 / 4 , 3, and so on.

Circles loop along a number line of fractions between 0 and 1.

If you keep this up, by the n th day, you will have all integers from – n to n and all fractions with denominators 2, 2 2 , 2 4 , 2 8 ,... to 2 n . Such fractional numbers whose denominator is a multiple of 2 are called dyadic numbers. This makes the surreal numbers seem rather boring: they only consist of integers and dyadic numbers. There is no trace of values such as pi (π) or the square root of two (√2)—nor even a number as trivial as 1 / 3 .

The Dawn of All Real Numbers

The really exciting properties of surreal numbers unfold as soon as you reach the day ω. That omega figure corresponds to a countable infinity . On this day all real numbers that were not there before, that is, all irrational values and all nondyadic fractions, are created in one fell swoop.

For example, √2 results from the following representation: √2 = {1 5 / 4 11 / 8 ... : ... 23 / 16 3 / 2 }. Other irrational values such as π are also obtained in this way; the number is enclosed by two dyadic number sequences.

In fact, the procedure is similar to the established Dedekind method, which is used to construct the real numbers from the rational numbers. In that approach two sets of rational numbers are formed, one of which contains smaller numbers than the other (just like Conway’s first rule). The “intersection” between these two sets then defines a real number.

But in Conway’s curious construction, day ω births new numbers. Suddenly, infinite values also arise—namely, the number ω. To do this, you have to insert all natural numbers into the lefthand set and leave the righthand set empty: ω = {1 2 3 ... : }. This corresponds to the number that is greater than all natural numbers.

And there is something even more unusual. If you enter 0 for the left-hand set and all dyadic fractions for the righthand set, you get an infinitesimal number ε = {0 : ... 1 / 8 1 / 4 1 / 2 1}. This epsilon represents the inverse of infinity: ε is so small that there is no real number that can represent it. And in fact, ε corresponds to the reciprocal of ω: ε = 1 / ω . The infinitesimal number ε appears on the day ω not only on its own but also in combination with all integers and dyadic numbers: 1 / 2 + ε = { 1 / 2 : ... 1 / 8 1 / 4 1 / 2 1}.

Branching nodes depict surreal numbers arising from 0.

On day ω + 1 more surreal numbers arise, such as the numbers ω + 1 and ω – 1, two new infinities. In addition, any real number can now be combined with ε, for example: π + ε = {π : ... 1 / 8 1 / 4 1 / 2 1}. In addition, the number ε / 2 is also created, a value that is half as small as the infinitesimal number.

The Concept of Quantity Collapses

On each subsequent day, new surreal numbers emerge: new infinities and infinitesimals, as well as new values that appear between all the previously generated numbers. Little by little, the variety of numbers continues to grow. In fact, so many objects are created that the surreal numbers can no longer be defined as a set. Instead they form a “class.” As such, they far exceed all other types of numbers: natural, rational and real numbers.

To understand that point, think back to the definition of surreal numbers as the two sets M L and M R , where M L is always the smaller of the two. Assume that the totality of all surreal numbers is a set S . Then you could define a new number x as x = { S : }. This would make x a number that exceeds all values of S —so you would have defined a surreal number that is not contained in S .

This is a contradiction because S contains all surreal numbers by definition. To avoid such paradoxes (which also arise if you want to determine the number of all infinities ), mathematicians have introduced the concept of a class. Because S is a class, it cannot be used to construct surreal numbers.

And surreal numbers hold further surprises. Although there are significantly more surreal numbers than real numbers, they do not form a continuum. A number line made up of surreal numbers is full of holes—unlike the real number line, which has no gaps. The reason for these spaces is that there are always smaller infinitesimals that squeeze in between the previously generated surreal numbers.

For example, the open set [0,1) includes all values that are smaller than 1. The number 1 is therefore an “upper bound.” Such a concept is missing for surreal numbers, however. That’s because you can find a surreal number between the set [0,1) and 1, such as 1 – ε. This number belongs to neither [0,1) nor 1.

A number line with smaller lines parallel to each other that signify breaks or gaps.

This observation has far-reaching consequences. It means that a sequence of numbers such as 1 / n does not have a limit value of 0 if n approaches infinity. Instead the sequence does not converge. It continues to run for all eternity while it takes on the ever smaller values ε, ε / 2 , ..., ε / 100 , and so on. In the universe of surreal numbers, 0.9999... can never equal 1—unlike with real numbers .

With such missing limits, the usual form of analysis that we learn at school in the form of derivatives and integrals also collapses. All fundamental concepts are based on limit value formation and a continuous number space. Nevertheless, experts have succeeded in developing what is called a “ nonstandard analysis ,” which works with surreal numbers.

Even if it all seems very abstract and strange, Knuth is convinced that surreal numbers are just as suitable for describing our world as any other. If “everybody for a hundred years had learned about this in school, [they would have] considered that this is the way numbers are,” he told Numberphile. “There is no reason for us to think that the universe obeys the laws of real numbers.” In fact, physicists have already tried to incorporate surreal numbers into their theories . The effort involved is usually very high, however, and the benefits have so far been marginal.

In mathematics surreal numbers form an interesting structure: an enormous number system that can be used to describe both infinities and infinitesimals. Conway actually came up with this astonishing construction when he was investigating the strategies of the game Go. Surreal numbers have proven themselves in game theory but only in their finite variant as a union of integers and dyadic numbers. As Conway reflected in a lecture in 2016 , the fact that he revealed the infinite expanse of a previously unknown universe of surreal numbers in this way was “the biggest surprise of [his] mathematical life .”

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

Best 2024 Super Bowl commercials: All 59 ranked according to USA TODAY Ad Meter

Super Bowl 58 is over, but the big game's commercials will live forever.

Over 160,000 people registered to vote in the 36th USA TODAY Ad Meter competition, which ranks the Super Bowl commercials each year. The 2024 winner was State Farm's "Like a Good Neighbaaa," starring Arnold Schwarzenegger and Danny DeVito, with a score of 6.68.

Rounding out the this year's top five were ads from Dunkin' , Kia, Uber Eats and the NFL.

Here's a look back at all the 2024 commercials, ranked by their Ad Meter score:

1. "Like a Good Neighbaaa," State Farm – 6.68

2. "the dunkings," dunkin' – 6.52, 3. perfect 10 | the kia ev9, kia – 6.36, 4. "worth remembering," uber eats – 6.26, 5. "born to play," nfl – 6.23, 6. "hard knocks: a dove super bowl film," dove – 6.18, 7. "talkin’ like walken," bmw – 6.10, 8. "old school delivery," budweiser – 6.03, 9. "can’t b broken," verizon – 5.91, 10. "dina & mita," doritos – 5.84, 11. "javier in frame," google – 5.80, 12. "mayo cat," hellmann's – 5.74, 13. "picklebabies," e*trade – 5.72, 14. "an american love story," volkswagen usa – 5.69, 15. michael cerave, cerave – 5.67, 16. "hail patrick," paramount+ – 5.64, 17. 'wicked' trailer, universal – 5.60, 18. "silence," foundation to combat antisemitism – 5.57, 19. "easy night out," bud light – 5.51, 20. "superior beach," michelob ultra – 5.49, 21. "that t-mobile home internet feeling," t-mobile – 5.46, 22. "mullets," kawasaki – 5.44, 23. "yes," reese's – 5.43, 24. "having a blast," mountain dew – 5.41, 25. "the m&m’s almost champions ring of comfort," m&m's – 5.39, 26. "here’s to science," pfizer – 5.34, t-27. "the wait is over," popeyes – 5.31, t-27. "thank you france," etsy – 5.31, 29. "couch potato farms," pluto tv – 5.30, t-30. "mr. p," pringles – 5.27, t-30. "betmgm is for everyone," betmgm – 5.27, 32. "the return of the coors light chill train," coors light – 5.17, 33. "twist on it," oreo – 5.15, 34. "watch me," microsoft – 5.06, 35. "t-mobile auditions," t-mobile – 5.04, 36. "tina fey books whoever she wants to be," booking.com – 5.02, 37. "judge beauty," e.l.f. cosmetics – 4.95, 38. "dareful handle," toyota tacoma – 4.91, 39. "launch," homes.com – 4.90, 40. "mr. t in skechers," skechers – 4.87, 41. "extraterrentrials," apartments.com – 4.84, 42. "nerds big game commercial ft. addison rae," nerds gummy – 4.75, 43. "doordash all the ads," doordash – 4.73, 44. "foot washing," he gets us – 4.65, 45. "the future of soda is now," poppi – 4.63, 46. "hello down there," squarespace – 4.61, 47. "salon," homes.com – 4.60, 48. "life is a ball," lindt usa – 4.59, 49. "shōgun - big game commercial," fx and hulu – 4.53, 50. "doctor on the plane," drumstick – 4.52, 51. "mascot," homes.com – 4.48, 52. "making memories on the water," bass pro shops and cabela’s – 4.42, 53. "the future," crowdstrike – 4.34, 54. "gronk misses the fanduel kick of destiny 2," fanduel – 4.28, 55. "who is my neighbor," he gets us – 4.24, 56. "love triangle," starry – 4.21, 57. "make your moves count," turbotax – 4.20, 58. "less social media. more snapchat," snapchat – 3.84, 59. "american values," robert f. kennedy jr. campaign – 3.41.

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    Number Riddles & Stories page 1 of 2 1 Draw a line to show which number matches each description. The first one is done for you. ex This number has a 2 in the thousands place. 46,305 a This is an even number with a 6 in the hundreds place. 32,617 b This number is equal to 30,000 + 4,000 + 80 + 2. 45,052 c This number is 1,000 less than 46,052 ...

  4. What's the Right Amount of Homework?

    What's the Right Amount of Homework? Decades of research show that homework has some benefits, especially for students in middle and high school—but there are risks to assigning too much. By Youki Terada February 23, 2018 Many teachers and parents believe that homework helps students build study skills and review concepts learned in class.

  5. Rational number arithmetic

    Quiz Unit test Lesson 1: Interpreting negative numbers Learn Missing numbers on the number line examples Comparing rational numbers Practice Up next for you: Missing numbers on the number line Get 3 of 4 questions to level up! Start Not started Order rational numbers Get 3 of 4 questions to level up! Practice Not started Quiz 1

  6. Microsoft Math Solver

    Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

  7. 1.1 Real Numbers: Algebra Essentials

    Classify a real number as a natural, whole, integer, rational, or irrational number. Perform calculations using order of operations. Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity. Evaluate algebraic expressions. Simplify algebraic expressions.

  8. Comparing Numbers Worksheets

    Our comparing numbers worksheets start off by focusing on comparing groups of objects rather than numbers. Later worksheets directly compare or order numbers without representations by pictured objects. Topics include: Kindergarten more than / less than worksheets Which group has more objects (using pictures, not numbers)

  9. 8.4: Homework

    Submit homework separately from this workbook and staple all pages together. (One staple for the entire submission of all the unit homework) ... 2 and 5. A number is classified as deficient if the sum of its proper factors is less than the number itself. 10 is a deficient number since 1 + 2 + 5 < 10. A number is classified as abundant if the ...

  10. Circle the Number

    Circle the Number. Help your little learners build their foundational math skills in this preschool number sense worksheet. In this activity, kids will count the objects and then circle the number that matches the correct amount. This worksheet will give your preschooler practice counting and identifying numbers 0-10 as they demonstrate an ...

  11. Homework

    Mathematics K-2, 3-5, 6-8 Number Theory Homework Problem H1 Prime numbers have exactly two factors. Now find some numbers that have exactly three factors. What do these numbers have in common? That is, how would you categorize these numbers? Look for numbers with three factors, not three prime factors.

  12. Learning Numbers Worksheets for Preschool and Kindergarten

    These free worksheets help kids learn to recognize, read and write numbers from 1-20. Learn the numbers from one to ten Numbers in sequence (tracing 1-10) Numbers in sequence (writing 1-10) Sequences 1-10 with counts Count & color Color by number Print the numbers (1-20) Numbers & number words Match numbers to their words

  13. How to Do Homework: 15 Expert Tips and Tricks

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  14. Preschool Number Worksheets

    Our Preschool Number Worksheets will help your students with building skills in number representation, recognition and formation, names and spelling of numbers, counting and so much more! ... This number nine preschool counting worksheet is great for early finishers and homework packs. Students count, write and trace the number . Preschool ...

  15. 11 Surprising Homework Statistics, Facts & Data (2024)

    A 2018 Pew Research poll of 743 US teens found that 17%, or almost 2 in every 5 students, regularly struggled to complete homework because they didn't have reliable access to the internet. This figure rose to 25% of Black American teens and 24% of teens whose families have an income of less than $30,000 per year. 4.

  16. Homework

    Place Value Homework. Problem H1. Write the base five numbers 1234 five and 1.234 five as base ten numbers. Problem H2. Find the base ten fractions represented by the following: a. a. 0.1 five, 0.2 five, 0.3 five, and 0.4 five. b. b. 0.01 five, 0.02 five, 0.03 five, and 0.04 five.

  17. Homework

    What Is a Number System? Homework Think back to the finite system of units digits that you explored in Part A and answer the following questions about that system. Problem H1 a. The commutative law for multiplication states that (a • b) = (b • a). Does this law hold for the finite number system in your table? Why or why not? b.

  18. Number 5 Worksheets

    Choose a number 5 worksheet. 5 worksheets are great practice for preschool and elementary school kids. Customize your five worksheet by changing the text and font. Try changing the text and creating your own math problems. Grab your crayons and print a 5 worksheet! Categories. Numbers. Counting; Fractions; Number 0; Number 1;

  19. Brainly

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  20. DawnSignPress

    Signing Naturally Units 1-6. Signing Naturally Units 1-6 is the first part in a series of curricular materials for the instruction of American Sign Language (ASL) as a second language.The goal is to take students with little or no knowledge of ASL and Deaf Culture and provide them with the skills needed to communicate comfortably in a wide variety of situations in the Deaf community.

  21. Signing Naturally Unit 1: 1.2 Circle the Number Flashcards

    ASL Homework 3:12 - HOW LONG DOES IT TAKE? Answers. 10 terms. brizzyw5. Preview. ASL 101- 1.2 Write the Number. 11 terms. Emmroz. Preview. Budget 4 series. 6 terms. mame_martin. Preview. Stars3. Teacher 27 terms. Csilla_Kaiser. Preview. ASL Homework 3:10 - STORY CORNER: "THE ELEVATOR INCIDENT" Answers.

  22. Homework Units 1-2

    n/a homework write the number 10 10. homework circle the letter ao se sm ti mi ei on oa os 10. ea homework write the number 11 14 12 10. 11. 10 12. 15 13. 13 14

  23. ASL HW 1.2

    American Sign Language Homework

  24. Surreal Numbers Are a Real Thing. Here's How to Make Them

    On the seventh day I rested," Knuth told Numberphile. Surreal numbers are created by adding values between two given preexisting numbers. If you look at 0 and 1, for example, 1 / 2 is in the ...

  25. Best Super Bowl 2024 commercials: Full USA TODAY Ad Meter rankings

    57. "Make Your Moves Count," TurboTax - 4.20. 58. "Less Social Media. More Snapchat," Snapchat - 3.84. 59. "American Values," Robert F. Kennedy Jr. campaign - 3.41. A look back at all the ...