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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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Integrated math 2

Welcome to integrated math 2, unit 1: absolute value & piecewise functions, unit 2: quadratics: multiplying & factoring, unit 3: quadratic functions & equations, unit 4: irrational numbers, unit 5: complex numbers, unit 6: rational exponents and radicals, unit 7: exponential models, unit 8: similarity, unit 9: right triangles & trigonometry, unit 10: solid geometry, unit 11: circles, unit 12: conic sections, unit 13: probability.

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Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to formulate queries.

• find roots to quadratic x^2-7x+12
• plot inequality x^2-7x+12<=0
• solve {3x-5y==2,x+2y==-1}
• plot inequality 3x-5y>=2 and x+2y<=-1
• solve 3x^2-y^2==2 and x+2y^2==5
• plot 3x^2-y^2>=2 and x+2y^2<=5
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About solving equations

A value is said to be a root of a polynomial if ..

The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.

How Wolfram|Alpha solves equations

For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase speed and reliability. Other operations rely on theorems and algorithms from number theory, abstract algebra and other advanced fields to compute results. These methods are carefully designed and chosen to enable Wolfram|Alpha to solve the greatest variety of problems while also minimizing computation time.

Although such methods are useful for direct solutions, it is also important for the system to understand how a human would solve the same problem. As a result, Wolfram|Alpha also has separate algorithms to show algebraic operations step by step using classic techniques that are easy for humans to recognize and follow. This includes elimination, substitution, the quadratic formula, Cramer's rule and many more.

Also known as "College Algebra"

OK. So what are you going to learn here?

You will learn about Numbers, Polynomials, Inequalities, Sequences and Sums, many types of Functions, and how to solve them.

You will also gain a deeper insight into Mathematics, get to practice using your new skills with lots of examples and questions, and generally improve your mind.

With your new skills you will be able to put together mathematical models so you can find good quality solutions to many tricky real world situations.

Near the end of most pages is a "Your Turn" section ... do these! You need to balance your reading with doing . Answering questions helps you sort things out in your mind. And don't guess the answer: use pen and paper and try your best before seeing the solution.

So what is this thing called Mathematics? And how do you go about learning it?

• Welcome to Mathematics
• Learning Mathematics
• The Language of Mathematics
• Symbols in Algebra

Next, we need to think about mathematics in terms of sets .

• Introduction to Sets

Now we know what a set is, let us look at different sets of numbers that are useful:

• The Evolution of Numbers
• Prime and Composite Numbers
• Fundamental Theorem of Arithmetic
• Whole Numbers and Integers
• Rational Numbers
• Using Rational Numbers
• Irrational Numbers
• 0.999... = 1
• Real Numbers
• Imaginary Numbers
• Complex Numbers
• Multiplying Complex Numbers
• The Complex Plane
• Common Number Sets

Inequalities

"Equal To" is nice but not always available. Maybe we only know that something is less than, or greater than. So let's learn about in equalities.

• Introduction to Inequalities
• Properties of Inequalities
• Solving Inequalities
• Solving Inequality Word Questions

We will be using exponents a lot, so let's get to know them well.

• Variables with Exponents
• Using Exponents in Algebra
• Squares and Square Roots
• Squares and Square Roots in Algebra
• Fractional Exponents
• Laws of Exponents
• Exponents of Negative Numbers

Polynomials

Polynomials were some of the first things ever studied in Algebra. They are simple, yet powerful in their ability to model real world situations.

• What is a Polynomial?
• Adding And Subtracting Polynomials
• Multiplying Polynomials
• Polynomials - Long Multiplication
• Dividing Polynomials
• Polynomials - Long Division
• Degree (of an Expression)
• Special Binomial Products
• Difference of Two Cubes
• Factoring in Algebra
• Solving Polynomials
• Roots of Polynomials: Sums and Products
• Rational Expressions
• Using Rational Expressions
• Fundamental Theorem of Algebra
• Remainder Theorem and Factor Theorem
• General Form of a Polynomial

Graphing Polynomials

• How Polynomials Behave
• Polynomials: The Rule of Signs
• Polynomials: Bounds on Zeros

And, of course, we need to know about equations ... and how to solve them.

• Equations and Formulas
• Solving Equations
• Solving Word Questions
• Zero Product Property
• Implication and Iff
• Theorems, Corollaries, Lemmas

Graphs can save us! They are a great way to see what is going on and can help us solve many things. But we need to be careful, as they sometimes don't give the full story.

• Cartesian Coordinates
• Pythagoras' Theorem
• Distance Between 2 Points
• Graph of an Equation
• Finding Intercepts From an Equation
• Symmetry in Equations
• Linear Equations

They are just equations for lines. But they come in many forms.

• Equation of a Straight Line
• Point-Slope Equation of a Line
• General Form of Equation of a Line
• Equation of a Line from 2 Points
• Midpoint of a Line Segment
• Parallel and Perpendicular Lines

A function relates an input to an output. But from that simple foundation many useful things can be built.

• What is a Function?
• Domain, Range and Codomain
• Evaluating Functions
• Increasing and Decreasing Functions
• Maxima and Minima of Functions
• Even and Odd Functions
• Set-Builder Notation

Common Functions Reference :

• Square Function
• Square Root Function
• Cube Function
• Reciprocal Function
• Absolute Value Function
• Floor and Ceiling Function
• Function Transformations
• Equation Grapher
• Operations with Functions
• Composition of Functions
• Inverse Functions

Equations of Second Degree

"Second degree" just means the variable has an exponent of 2, like x 2 . It is the next major step after linear equations (where the exponent is 1, like x).

• Quadratic Equations
• Factoring Quadratics
• Completing the Square
• Derivation of Quadratic Formula
• Graphing Quadratic Equations
• Quadratic Equations in the Real World
• Circle Equations

We already have experience in solving, but now we can learn more!

• Mathematical Models and Mathematical Models 2
• Approximate Solutions
• Intermediate Value Theorem
• Solving Radical Equations
• Change of Variables
• Algebra Mistakes

We learned about inequalities above, now let's learn how to solve them.

• Graphing Linear Inequalities
• Inequality Graphing Tool
• Solving Quadratic Inequalities
• Solving Rational Inequalities
• Absolute Value in Algebra

Exponents and Logarithms

We already know about exponents ... well logarithms just go the other way. And together they can be very powerful.

• Introduction to Logarithms
• Exponents, Roots and Logarithms
• Working with Exponents and Logarithms
• Exponential Function
• Logarithmic Function
• Exponential Growth and Decay
• Systems of Linear Equations

What happens when we have two or more linear equations that work together? They can often be solved! It isn't very hard but can take a lot of calculations.

• Types of Matrix
• How to Multiply Matrices
• Determinant of a Matrix
• Inverse of a Matrix:
• Using Elementary Row Operations (Gauss-Jordan)
• Using Minors, Cofactors and Adjugate
• Scalar, Vector, Matrix and Vectors
• Matrix Calculator
• More at Matrix Index
• Solving Systems of Linear Equations Using Matrices
• Systems of Linear and Quadratic Equations
• Probability

Is it likely? You be the judge!

• The Basic Counting Principle
• Combinations and Permutations

Sequences, Series and Partial Sums

A Sequence is a set of things (usually numbers) that are in order. We can also sum up a series, where Sigma Notation is very useful.

• Sequences - Finding A Rule
• Sigma Notation
• Partial Sums
• Arithmetic Sequences and Sums
• Geometric Sequences and Sums

These last few subjects use what we have learned above.

• Partial Fractions
• Mathematical Induction
• Pascal's Triangle
• Binomial Theorem

And that is all!

But there are many other interesting algebra topics such as:

• Euler's Formula for Complex Numbers
• Taylor Series (needs a basic understanding of derivatives )

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Math Word Problem Solver

Math calculators and online math solver apps aren't built to handle math word problems. We've fixed that! Our solver can interpret math word problems and determine what mathematical operations needs to be used to solve the problem.

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Frequently Asked Questions

Can the math solver help me improve my grades (yes).

Now, we can't guarantee you'll get better grades, since the AI Math Solver is only half of the success equation, but the majority of students who use the application report full letter grade improvements in their grades. Most see improvements in their homework grades immediately.

How long does it take to get an answer to my math problem?

Our application is available 24/7 and will start working on a solutions immediately after you send it. The time it takes to solve each problem is dependant on the complexity of the problem. The application will post a step by step solution.

Can I really ask unlimited questions?

How is this app different from your tutoring service.

The AI Math Solver is FREE to try and powered by our proprietary mathematical computation engine and AI while our Tutoring service enables students to chat with professional (human) math tutors.

Can it answer Physics questions?

Yes, the AI has been trainined on physics concepts and problems. However if it can't answer a question, then we recommend subscribing to our Tutoring service to connect with a physics tutor.

Will it work on my phone?

Yes, the application is optimized for mobile and tablet, no need to download another app onto your phone. An internet connection is required.

Can I submit a picture of my math problem?

Yes, simply tap or click the carmera icon next to the Solve button in the application then select the image or if you're on your phone open your camera to immediately take a picture of your math problem.

The application can interpret most handwritten or typed math problems.

Is it free to use?

Yes, it's free to try! You can ask 3 questions for free before reaching the free question limit.

Which math subjects are covered?

The AI LLM has been trained on a large array of mathematical subjects including, but not limited to Basic Algebra, Advanced Algebra, Geometry, Trigonometry, Calculus, Advanced Calculus, Physics and much more.

Can it Solve Basic Math Problems?

Yep! The AI LLM has been trained on a large array of mathematical subjects including, but not limited to Basic Algebra, Advanced Algebra, Geometry, Trigonometry, Calculus, Advanced Calculus, Physics and much more.

How do I cancel my subscription?

You can cancel your subscription very easily in several ways.

1) Login, go to the Manage Account tab, then click the Cancel Plan button.

2) Click the cancel plan link in your sign up email.

3) If you can't find it or no longer have it, just send us an email at [email protected] and we'd be happy to assist you.

Can I change my subscription?

Yes you certainly can! You can change your subscription by accessing the Manage Account page within the application or using the link provided in your sign up email. If you can't find it or no longer have it, just send us an email at [email protected] or ask your tutor for assistance they will be happy to assist.

How much does it cost?

You can view the price of our subscription packages here . We're also providing freebies and exclusive deals from our large network of high quality partners. We obsess over the details to ensure you get 10x more value than the price tag!

Can it Solve Calculus Math Problems?

Can it solve geometry math problems.

P.S. We're currently working to enable in app graphing to accompany solutions.

Can it solve Trigonometric Functions?

Can it solve simple equations math problems.

Yep! The AI LLM has been trained on a large array of mathematical problems and subjects including, but not limited to Basic Algebra, Advanced Algebra, Geometry, Trigonometry, Calculus, Advanced Calculus, Physics and much more.

Can it solve any math problems and equations?

How do i get math solutions.

Simply upload an image of your math problem or enter it and click Solve. Our AI math problem solving calculator will provide a step by step solution to the problem with the answer.

Can it solve word problems?

Yes! Our AI math word problem solver will provide a step by step solution and answer to any word problem.

How do you find the answers to math word problems?

Step by step solutions and answers to any word problem or equations can be found in the chat window indicated in blue text bubbles.

Reading & Math for K-5

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2nd Grade Math Word Problems Worksheets

Math word problem worksheets for grade 2.

These word problem worksheets place 2nd grade math concepts in contexts that grade 2 students can relate to.  We provide math word problems for addition, subtraction, multiplication, time, money and fractions.

We encourage students to read and think about the problems carefully, and not just recognize an answer pattern.  We facilitate this by:

• providing a number of mixed word problem worksheets (e.g. subtraction questions mixed in with addition questions)
• sometimes including irrelevant data within word problems so students must understand the context before applying a solution

Addition word problems for grade 2

Simple addition (1-2 digits)

Slightly harder addition (1-3 digits)

Subtraction word problems for grade 2

Simple subtraction (1 and 2 digit numbers)

Subtracting 1-3 digit numbers

Mixed addition and subtraction word problems

Mixed addition and subtraction within 20

Mixed addition and subtraction (two digits)

More mixed addition and subtraction word problems

Multiplication word problems

Multiplication within 25

Measurement word problems

Length word problems

Time word problems

Time and elapsed time (1/2 hour intervals)

Time and elapsed time (5 minute intervals)

Money word problems

Counting money (coins and bills)

Fraction word problems

Understanding fractions

Write and compare fractions from a story

Mixed word problems

Mix of all above types of grade 2 word problems

Mix of just addition / subtraction / multiplication word problems  

Sample Grade 2 Word Problem Worksheet

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• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
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• Pre Algebra
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Number line.

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• (x+5)(x-5)\gt 0
• 10^{1-x}=10^4
• \sqrt{3+x}=-2
• 6+11x+6x^2+x^3=0
• factor\:x^{2}-5x+6
• simplify\:\frac{2}{3}-\frac{3}{2}+\frac{1}{4}
• x+2y=2x-5,\:x-y=3
• How do you solve algebraic expressions?
• To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true.
• What are the basics of algebra?
• The basics of algebra are the commutative, associative, and distributive laws.
• What are the 3 rules of algebra?
• The basic rules of algebra are the commutative, associative, and distributive laws.
• What is the golden rule of algebra?
• The golden rule of algebra states Do unto one side of the equation what you do to others. Meaning, whatever operation is being used on one side of equation, the same will be used on the other side too.
• What are the 5 basic laws of algebra?
• The basic laws of algebra are the Commutative Law For Addition, Commutative Law For Multiplication, Associative Law For Addition, Associative Law For Multiplication, and the Distributive Law.

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Free Math Worksheets — Over 100k free practice problems on Khan Academy

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That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

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High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

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1.2: Problem Solving and Estimating

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• Page ID 74279

• Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
• Rio Hondo College

1.2 Learning Objectives

• Apply the problems solving process to real life problems
• Use dimensional analysis to solve problems
• Use estimates to solve real life problems
• Round decimals

In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

Problem Solving Process

• Identify the question you’re trying to answer.
• Work backwards, identifying the information you will need and the relationships you will use to answer that question.
• Continue working backwards, creating a solution pathway.
• If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
• Solve the problem, following your solution pathway.
• Check your answer. Does the answer make sense?

In most problems we work, we will be estimating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to estimate the solution using basic knowledge from our lives.

Definition: Estimation

Estimation is the process of arriving at an approximate answer to a question. Usually by rounding answers to a place value that is easier to express in the situation.

Depending on the situation, you may be asked to round your answers to a specific place value. The tables below list more common values, but there are many more. 

Place values of a whole number:

Place values of a decimal number:

Name and Write Decimals

Notice that “ten thousand” is a number larger than one, but “one ten-thousand th ” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.

When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values.

We summarize the steps needed to name a decimal below.

• Name the number to the left of the decimal point.
• Write “and” for the decimal point.
• Name the “number” part to the right of the decimal point as if it were a whole number.
• Name the decimal place of the last digit.

Name the decimal \(4.3\).

Try It Now 1

Name the decimal \(6.7\).

six and seven tenths

Try It Now 2

Name the decimal: \(−15.571\).

When we write or read a check we interpret both the numerals and the name of the number. Let’s see how to write the decimal from the name.

We summarize the steps to writing a decimal .

• Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
• If there is no “and,” write a “0” with a decimal point to its right.
• Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
• Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
• Fill in zeros for place holders as needed.

Write “fourteen and twenty-four thousandths” as a decimal.

Try It Now 3

Write as a decimal: thirteen and sixty-eight thousandths.

Round Decimals

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

We summarize the steps for rounding a decimal here.

• Locate the given place value and mark it with an arrow.
• Underline the digit to the right of the place value.
• Yes—add 1 to the digit in the given place value.
• No—do not change the digit in the given place value.
• Rewrite the number, deleting all digits to the right of the rounding digit.

Round 18.379 to the nearest hundredth.

Try It Now 4

Round 6.582 to the nearest

• whole number.

We can use these estimation techniques to answer questions that involve some problem solving. Bringing the two concepts from above together, we apply them to several examples below.

How many times does your heart beat in a year?

This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

Suppose you count 80 beats in a minute. To convert this beats per year:

\(\dfrac{80 \text { beats }}{1 \cancel {\text { minute }}} \cdot \dfrac{60 \cancel {\text { minutes }}}{1 \cancel {\text { hour }}} \cdot \dfrac{24 \cancel {\text { hours }}}{1 \cancel {\text { day }}} \cdot \dfrac{365 \cancel{ \text { days }}}{1 \text { year }}=42,048,000\) beats per year

How thick is a single sheet of paper? How much does it weigh?

While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

\(\dfrac{2 \text { inches }}{ \cancel{\text { ream }}} \cdot \dfrac{1 \cancel{ \text { ream }}}{500 \text { pages }}=0.004\) inches per sheet

\(\dfrac{5 \text { pounds }}{ \cancel {\text { ream }}} \cdot \dfrac{1 \cancel {\text { ream }}}{500 \text { pages }}=0.01\) pounds per sheet, or 0.16 ounces per sheet.

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

There are several possible solution pathways to answer this question. We will explore one.

To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

We can now execute our plan:

\(12 \cancel{\text { muffins}} \cdot \dfrac{250 \text { calories }}{ \cancel {\text { muffin }}}=3000\) calories for the whole recipe

\(\dfrac{3000 \text { calories }}{20 \text { mini-muffins }}\) gives 150 calories per mini-muffin

\(4 \cancel {\text { mini-muffins }} \cdot \dfrac{150 \text { calories }}{ \cancel {\text { mini-muffin }}}\) totals 600 calories consumed.

You need to replace the boards on your deck. About how much will the materials cost?

There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

Suppose that measuring the deck, it is rectangular, measuring 16 ft by 24 ft, for a total area of \(384 \mathrm{ft}^{2}\).

From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

\(8 \text { feet } \cdot 4 \cancel { \text{inches } }\cdot \dfrac{1 \text { foot }}{12 \cancel {\text{ inches} }}=2.667 \mathrm{ft}^{2}\).

The cost per square foot is then

\(\dfrac{\$ 7.50}{2.667 \mathrm{ft}^{2}}=\$ 2.8125 \text { per } \mathrm{ ft}^{2}\).

This will allow us to estimate the material cost for the whole \(384 \mathrm{ ft}^{2}\) deck

\(\ 384 \cancel {\mathrm{ ft}^{2} }\cdot \dfrac{\$ 2.8125}{\cancel{\mathrm{ ft}^{2}}}=\$ 1080\) total cost.

Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

We can then find the number of gallons each car would require for the year.

\(9000 \cancel {\text { city miles } }\cdot \dfrac{1 \text { gallon }}{24 \cancel{ \text { city miles }}}+3000 \cancel{ \text { hightway miles}}. \dfrac{1 \text { gallon }}{35 \cancel{ \text { highway miles }}}=460.7\text{ gallons}\)

\(9000 \cancel {\text { city miles }}\cdot \dfrac{1 \text { gallon }}{35 \cancel{ \text { city miles }}}+3000 \cancel {\text { highway miles}}. \dfrac{1 \text { gallon }}{40 \cancel {\text { highway miles }}}=332.1\text{ gallons}\)

If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

Sonata: \(460.7 \text { gallons } \cdot \dfrac{\$ 3.50}{\text { gallon }}=\$ 1612.45\)

Hybrid: \(\text { 332.1 gallons } \cdot \dfrac{\$ 3.50}{\text { gallon }}=\$ 1162.35\)

The hybrid will save $450.10 a year. The gas costs for the hybrid are about \(\dfrac{\$ 450.10}{\$ 1612.45} = 0.279 = 27.9\%\) lower than the costs for the standard Sonata.

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

Try it Now 5

If traveling from Seattle, WA to Spokane WA for a three-day conference, does it make more sense to drive or fly?

There is not enough information provided to answer the question, so we will have to make some assumptions, and look up some values.

Assumptions:

a) We own a car. Suppose it gets 24 miles to the gallon. We will only consider gas cost.

b) We will not need to rent a car in Spokane, but will need to get a taxi from the airport to the conference hotel downtown and back.

c) We can get someone to drop us off at the airport, so we don’t need to consider airport parking.

d) We will not consider whether we will lose money by having to take time off work to drive.

Values looked up (your values may be different)

a) Flight cost: \(\$184\)

b) Taxi cost: \(\$25\) each way (estimate, according to hotel website)

c) Driving distance: \(280\) miles each way

d) Gas cost: \(\$3.79\) a gallon

Cost for flying: \(\$184\text{ flight cost }+ \$50\text{ in taxi fares }= \$234\).

Cost for driving: \(560\) miles round trip will require 23.3 gallons of gas, costing \(\$88.31\).

Based on these assumptions, driving is cheaper. However, our assumption that we only include gas cost may not be a good one. Tax law allows you deduct \(\$0.55\) (in 2012) for each mile driven, a value that accounts for gas as well as a portion of the car cost, insurance, maintenance, etc. Based on this number, the cost of driving would be \(\$319\).

Home » Teacher & Parents Tips » Math Made Easy: Helping Grade 2 Students Thrive in Problem Solving

Math Made Easy: Helping Grade 2 Students Thrive in Problem Solving

Mathematics is an essential subject that students cannot afford to neglect. As early as grade 2, students are introduced to the basics of mathematical concepts such as numbers, addition, subtraction, multiplication, and division. An essential aspect of mathematics is problem-solving. While many students find it challenging, it can be made easy with the right approach. This article will discuss how parents and educators can turn problem-solving woes into victories and equip their grade 2 students with the skills needed to excel in math.

Table of Contents

The Importance of Problem Solving in Grade 2 Math

Problem-solving is an essential skill that students need to master to excel in math. It involves critical thinking, analysis, and decision-making. It also helps students develop their reasoning abilities and enables them to apply logic in real-life situations. In grade 2 math, problem-solving skills are essential as they help students understand mathematical concepts better. Students who can solve problems are more confident and enjoy math more than those who struggle.

Turning Problem-Solving Woes into Victories

Many students, especially in grade 2, find problem-solving challenging. However, with the right approach, it can be turned into a victory. One way to turn problem-solving woes into victories is by breaking down the problem into smaller parts. Encourage your child to read the problem carefully and identify the key elements. Once they have identified these elements, they can begin to solve the problem step by step. This approach helps students understand the problem better and reduces anxiety.

Another way to turn problem-solving woes into victories is by practicing regularly. The more students practice, the more comfortable they become with problem-solving. Encourage your child to practice regularly and provide them with different types of problems to solve. As they solve more problems, their confidence will increase, and they will become more efficient problem-solvers.

Strategies for Effective Problem Solving in Math

There are several strategies that students can use to solve problems effectively. One of the most effective strategies is the use of visual aids. Encourage your child to draw diagrams or pictures to help them understand the problem better. This approach helps students visualize the problem and enables them to make better decisions.

Another strategy is to use real-life situations to solve problems. This approach helps students understand how math can be applied in real-life situations. For example, if you want to teach your child about fractions, you can use pizza slices to help them understand the concept better.

Lastly, encourage your child to work with a partner or in a group. Group work helps students learn from one another and can be an effective way to solve problems. It also helps students develop their social skills and enables them to work collaboratively.

Fun and Engaging Math Activities for Grade 2 Students

Learning math can be fun and engaging. There are several math activities that parents and educators can use to help grade 2 students develop their problem-solving skills. One such activity is math games. Games such as Sudoku, Math Bingo, and Math Jeopardy can be used to teach students math concepts while making learning fun.

Another activity is math puzzles. Puzzles such as crosswords, word searches, and logic puzzles can be used to develop critical thinking and problem-solving skills. Additionally, math stories can be used to teach math concepts while making learning fun. Math stories can be found in storybooks or online, and they provide an interactive way to teach math concepts.

Developing Critical Thinking Skills through Math

Mathematics is an excellent way to develop critical thinking skills. Critical thinking involves analyzing, interpreting, and evaluating information to make informed decisions. In math, critical thinking is essential as it helps students understand mathematical concepts and apply them in real-life situations.

Encourage your child to think critically when solving problems. Teach them to ask questions and to consider different solutions to a problem. Additionally, encourage them to explain their reasoning and to justify their solutions. These skills are essential in problem-solving and can be applied in other areas of their lives.

Providing Support and Encouragement for your Child

Providing support and encouragement is essential when helping your grade 2 child excel in math. Encourage your child to ask questions and to seek help when they need it. Additionally, provide them with a positive learning environment and praise their efforts and progress. Celebrate their successes and encourage them to keep learning and practicing.

Equipping Your Child for Math Success in Grade 2

Mathematics is an essential subject that requires problem-solving skills. By turning problem-solving woes into victories, providing support and encouragement, and using effective strategies and fun activities, parents and educators can help grade 2 students excel in math. With the right approach, learning math can be fun and engaging while developing critical thinking skills that are essential in all areas of life.

2nd Grade Fast Math Success Workbook

2nd Grade Fast Math Success Workbook: Math Worksheets Grade 2: Numeration, Addition, Subtraction, Telling Time and More with Answers

• Comparing Numbers Within 200
• Ordering Numbers Within 200
• Skip Counting: Count By 1s and 2s
• Place Value: Ones, Tens, and Hundreds
• Addition: 1 through 100
• Subtraction: 1 through 100
• Addition: Missing Number – 1 through 100
• Subtraction: Missing Number – 1 through 100
• Ordering Numbers – 1 through 1000
• Comparing Numbers – 1 through 1000
• Counting: Count by 4 to 6
• Addition and Subtraction: Double Digit
• Addition and Subtraction: Missing Number – Double Digit
• Addition: Triple Addend – 1 through 100
• Mixed Operations: 1 through 100
• Write the Numbers Before, After, and Between
• Addition and Subtraction: 1 through 1000
• Telling Time
• Time Passages
• Addition and Subtraction Games
• Final Review

MathBear: Math Workbook Grade 2

MathBear: Math Workbook Grade 2: 2nd Grade Math Workbook: Addition, Subtraction, Multiplication, and Numeration with Answers

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• Comparing Numbers
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• Basic Multiplication

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All Seasons Math Tests Grade 2 (Student's Edition): 100 Math Practice Pages Grade 2: Timed Math Tests: For Classroom and Homeschool

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Kids Math Book Ages 6-8: Math Practice workbook Grade 1-3: Addition, Subtraction, Place Value, Telling Time

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A two-step iteration method for solving vertical nonlinear complementarity problems

• Wenxiu Guo 1,2 , 
• Xiaoping Lu 1 ,  ,  , 
• Hua Zheng 2
• 1. School of Computer Science and Engineering, Macau University of Science and Technology, Macao, China
• 2. School of Mathematics and Statistics, Shaoguan University, Shaoguan, China
• Received: 21 February 2024 Revised: 06 April 2024 Accepted: 17 April 2024 Published: 19 April 2024

MSC : 65F10, 90C33

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In this paper, for vertical nonlinear complementarity problems, a two-step modulus-based matrix splitting iteration method is established by applying the two-step splitting technique to the modulus-based matrix splitting iteration method. The convergence theorems of the proposed method are given when the number of system matrices is larger than 2. Numerical results show that the convergence rate of the proposed method can be accelerated compared to the existing modulus-based matrix splitting iteration method.

• vertical nonlinear complementarity problem ,
• modulus-based method ,
• $ H $-splitting

Citation: Wenxiu Guo, Xiaoping Lu, Hua Zheng. A two-step iteration method for solving vertical nonlinear complementarity problems[J]. AIMS Mathematics, 2024, 9(6): 14358-14375. doi: 10.3934/math.2024698

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• This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ -->

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The Opening Days of Trump’s First Criminal Trial

Here’s what has happened so far in the unprecedented proceedings against a former u.s. president..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

It’s the first day of the Trump trial and just walking out the door in my house. It’s a beautiful day, 6:11 AM. The thing that keeps running through my head is it’s kind of amazing that hundreds of jurors are going to show up at the Manhattan courthouse. And some of them are going to know what they’re there for — probably talking to their friends, their relatives about it.

Some of them are going to learn this morning talking to other jurors in line, asking what all the fuss is about. But I really do imagine that there’s going to be at least one potential juror who, headphones on, getting into court. Here they’re going to be there for the first criminal trial of Donald J. Trump. And just, I mean, how would you react?

[MUSIC PLAYING]

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.” Today, what it’s been like inside the lower Manhattan courtroom, where political and legal history are being made? My colleague, Jonah Bromwich, on the opening days of the first criminal trial of a US President. It’s Thursday, April 18.

Is that his mic? Hi, there.

Hello. How are you?

I’m doing good.

OK. Thank you for coming in, Jonah —

Thank you for having me.

— in the middle of a trial. Can you just explain why you’re able to even be here?

Sure. So we happen to be off on Wednesdays during trial, so.

We being not “The New York Times,” but the courts.

That’s right.

Which is why we’re taping with you. And because we now have two full court days of this history-making trial now under our belts. And the thing about this trial that’s so interesting is that there are no cameras in the courtroom for the wider world.

There’s no audio recordings. So all we really have is and your eyes and your notebook, maybe your laptop. And so we’re hoping you can reconstruct for us the scene of the first two days of this trial and really the highlights.

Yeah, I’d be happy to. So on Monday morning, I left the subway. It’s before 7:00 AM. The sun is just rising over these grandiose court buildings in lower Manhattan.

I’m about to turn left onto Center Street. I’m right in front of the big municipal building.

And I turn onto Center Street. That’s where the courthouses are.

I’m crossing.

And I expected to see a big crowd. And it was even bigger than I had anticipated.

Here we go. Here we go. Here we go. Now, I finally see the crowd.

You have camera banks. You have reporters. You have the beginnings of what will eventually become a protest. And you have this most New York thing, which is just a big crowd of people.

[CHUCKLES]: Who just know something is going on.

That’s right. And what they know is going on is, of course, the first trial of an American president.

All right, I’m passing the camera, folks. Camera, camera, camera, camera. Here we go.

Let’s start with Sharon Crowley live outside the courthouse in Lower Manhattan.

I want to get right to ABC’S Aaron Katersky who’s outside of the courthouse.

Robert Costa is following it outside the courthouse in Lower Manhattan. Bob, I saw the satellite trucks lined up all in a row. Good morning.

Talk to us how we got here exactly.

So this is the case that was brought by the Manhattan district attorney. So prosecutors have accused Donald Trump of covering up the actions of his former fixer, Michael Cohen, after Cohen paid hush money to Stormy Daniels. Stormy Daniels had a story about having had sex with Donald Trump, which Trump has always denied.

Cohen paid her money, and then Trump reimbursed Cohen. And prosecutors say that Trump essentially defrauded the American people because he hid this information that could have been very important for the election from those people when he reimbursed Cohen.

Right. And as I remember it, he also misrepresented what that reimbursement was. Claimed it was a legal fee when, in fact, it was just reimbursing Michael Cohen for a hush money payment.

Exactly, yeah. He definitely didn’t say reimbursement for hush money payment to Stormy Daniels. It’s a cover up case. It’s a case about hiding information you don’t want people to see.

Right. And of course, the context of all this is that it is in the middle of a presidential election. It’s 2016. Trump wants to keep this secret, prosecutors allege, so that the American public doesn’t know about it and potentially hold it against him.

Right. And prosecutors are telling a story about election interference. They’re saying that Trump interfered with an election. And Trump himself is also using the phrase “election interference.” But he’s painting the trial itself as election interference as he now runs again in 2024.

Fascinating.

And because we’re in Manhattan, and because the jury pool is going to be largely Democratic, and the judge is a Democrat, and the district attorney is a Democrat, Trump keeps claiming he cannot get a fair shake. This is democrat central. And in democrat central, Trump doesn’t have a chance.

OK. So, what happens once you actually enter the courthouse?

Outside, there’s all this fanfare. But inside, it’s a little bit business as usual. So I go up to the 15th floor, and I walk into the courtroom, and I sit down, and it’s the same old courtroom. And we’re sitting and waiting for the former president.

Around 9:30, Trump walks in. He looks thin. He looks a little tired, kind of slumping forward, as if to say with his body like let’s get this over with. Here we go.

The judge walks in a little bit after that. And we think we’re all set for the trial to start, but that’s not what happens here. And in fact, there are a series of legal arguments about what the trial is going to look like and what evidence is going to be allowed in.

So, for example, prosecutors ask that they be allowed to admit into evidence headlines from “The National Enquirer” that were attacks on Trump’s 2016 opponents — on Ted Cruz, on Marco Rubio, on Ben Carson.

Because prosecutors are in some sense putting Trump’s 2016 campaign on trial. These headlines are a big part of that because what prosecutors say they show is that Trump had this ongoing deal with “The National Enquirer.” And the publisher would promote him, and it would publish damaging stories about his opponents. And then crucially, it would protect Trump from negative stories. And that’s exactly what prosecutors say happened with Stormy Daniels. That “The National Enquirer” tipped Cohen off about Stormy Daniels trying to sell her story of having had sex with Donald Trump, which he denies. And that led to the hush money payment to her. So what prosecutors are doing overall with these headlines is establishing a pattern of conduct. And that conduct, they say, was an attempt to influence the election in Trump’s favor.

And the judge agrees. He’s going to admit this evidence. And this is a pretty big win for the prosecution. But even though they win that one, they’re not winning everything.

They lose some important arguments here. One of them was that after the Access Hollywood tape came out, there were allegations of sexual assault against Donald Trump. And you know this, Michael, because you reported two of them — two of the three in question at this very trial.

Prosecutors had hoped to talk about those during trial in front of the jury to show the jurors that the Trump campaign was really, really focused on pushing back against bad press in the wake of the Access Hollywood tape in which Trump seemed to describe sexual assault. That was a big problem for the campaign. Campaign did everything it could to push back, including against these allegations that surfaced in the wake of the tape.

But the judge, saying that the allegations are hearsay — that they’re based on the women’s stories — says absolutely not. That is incredibly prejudicial to the defendant.

Interesting.

And that Donald Trump would actually not get a fair trial were those allegations to be mentioned. And so he will not let those in. The jurors will not hear about them.

So this is a setback, of course, for the prosecution, a victory for Trump’s legal team.

It’s a setback. And it also just shows you how these pre-trial motions shape the context of the trial. Think of the trial as a venue like a theater or an athletic contest of some sort. And these pre-trial motions are about what gets led into the arena and what stays out. The sexual assault allegations — out. “The National Enquirer” headlines — in.

OK. And how is Trump sitting there at the defense table reacting to these pre-trial motion rulings from the judge?

Well, as I’ve just said, this is very important stuff for his trial.

Right. Hugely important.

But it’s all happening in legal language, and I’m decoding it for you. But if you were sitting there listening to it, you might get a little lost, and you might get a little bored. And Trump, who is not involved in these arguments, seems to fall asleep.

Seems to fall asleep — you’re seeing this with your own eyes.

What we’re seeing, overall, including our colleague, Maggie Haberman, who’s in the overflow room and has a direct view of Trump’s face — I’m sitting behind him in the courtroom, so I can’t see his face that well.

You guys are double teaming this.

That’s right. I’m sitting behind him, but Maggie is sitting in front of him. And what she sees is not only that his eyes are closed. That wouldn’t get you to he is asleep.

And we have to be really careful about reporting that he’s asleep, even if it seems like a frivolous thing. But what happens is that his head is dropping down to his chest, and then it’s snapping back up. So you’ve seen that, when a student —

I’ve done that.

(CHUCKLES) Yeah. We all kind of know that feeling of snapping awake suddenly. And we see the head motion, and it happens several times.

Lawyers kind of bothering him, not quite shaking him, but certainly trying to get his attention. And that head snapping motion, we felt confident enough to report that Trump fell asleep.

During his own criminal trial’s opening day.

Does someone eventually wake him up?

He wakes up. He wakes up. And in fact, in the afternoon, he’s much more animated. It’s almost as if he wants to be seen being very much awake.

Right. So once these pre-trial motions are ruled on and Trump is snapped back to attention, what happens?

Well, what happens in the courtroom is that the trial begins. The first trial of an American president is now in session. And what marks that beginning is jurors walking into the room one by one — many of them kind of craning their necks over at Donald Trump, giggling, raising their eyebrows at each other, filing into the room, and being sworn in by the judge. And that swearing in marks the official beginning of the trial.

The beginning is jury selection, and it’s often overlooked. It’s not dramatized in our kind of courtroom dramas in the same way. But it’s so important. It’s one of the most important parts of the case. Because whoever sits on the jury, these are the 12 people who are going to decide whether Trump is guilty or whether Trump is innocent.

So how does jury selection actually look and feel and go?

So, jury selection is a winnowing process. And in order to do that, you have to have these people go through a bunch of different hurdles. So the first hurdle is, after the judge describes the case, he asks the group — and there are just short of 100 of them — whether they can be fair and impartial. And says that if they can’t, they should leave. And more than half the group is instantly gone.

So after we do this big mass excusal, we’re left with the smaller group. And so now, jurors are getting called in smaller groups to the jury box. And what they’re going to do there is they’re going to answer this questionnaire.

And this part of the process is really conducted by the judge. The lawyers are involved. They’re listening, but they’re not yet asking questions of the jurors themselves.

And what’s on the questionnaire?

Well, it’s 42 questions. And the questions include, their education, their professional histories, their hobbies, what they like to do whether you’re a member of QAnon or Antifa.

Whether you’re far left or far right.

That’s right. Whether you’ve read “The Art of the Deal,” Trump’s book, which some prospective jurors had.

Right. It was a bestseller in its time.

That’s right. And some of it can be answered in yes/no questions, but some of it can be answered more at length. So some of the prospective jurors are going very, very fast. Yes, no, no, no, yes.

Right. Because this is an oral questionnaire.

That’s right. But some of them are taking their time. They’re expanding on their hobbies. So the potential juror in seat 3, for example, is talking about her hobbies. And she says some running, hiking. And then she said, I like to go to the club, and it got a huge laugh. And you get that kind of thing in jury selection, which is one of the reasons it’s so fun. It’s the height of normality in this situation that is anything but normal.

Right. The most banal answer possible delivered in front of the former president And current Republican nominee for president.

Well, that’s one of the fascinating parts about all this, right? is that they’re answering in front of Trump. And they’re answering questions about Trump in front of Trump. He doesn’t react all that much. But whenever someone says they’ve read “The Art of the Deal —” and there are a few of those — he kind of nods appreciatively, smiles. He likes that. It’s very clear. But because there are so many questions, this is taking forever, especially when people are choosing to answer and elaborate and digress.

This is when you fall asleep.

This Is. When I would have fallen asleep if I were a normal person.

And by the end of the day. Where does jury selection stand?

Well, the questionnaire is another device for shrinking that jury pool. And so the questionnaire has almost these little obstacles or roadblocks, including, in fact, a question that jurors have seen before — whether they would have any problem being fair and impartial?

Hmm. And they ask it again.

They’re asked it again. And they’re asked in this more individualized way. The judge is questioning them. They’re responding.

So, remember that woman who said she liked to go to the club got a big laugh. She reaches question 34. And question 34 reads, “Do you have any strong opinions or firmly-held beliefs about former President Donald Trump or the fact that he is a current candidate for president that would interfere with your ability to be a fair and impartial juror?” She said, yes, she does have an opinion that would prevent her from being fair and impartial. And she, too, is excused.

So that’s how it works. People answer the questionnaire, and they get excused in that way, or they have a scheduling conflict once they reach the jury box. And so to answer your question, Michael. At the end of day one, given all these problems with the questionnaire and the length of time it’s taken to respond to and people getting dismissed based on their answers, there is not a single juror seated for this trial.

And it’s starting to look like this is going to be a really hard case for which to find an impartial jury.

That’s the feeling in the room, yeah.

We’ll be right back.

So Jonah, let’s turn to day 2. What does jury selection look like on Tuesday?

So when the day begins, it looks almost exactly like it looked when the day ended on Monday. We’re still with the questionnaire, getting some interesting answers. But even though it feels like we’re going slow, we are going.

And so we’ve gone from about 100 people to now there’s about 24 the room there’s 18 the jury box. And by the time we hit lunch, all those people have answered all those questions, and we are ready for the next step in the process.

Voir dire. And what it is the heart of jury selection. This is the point where the lawyers themselves finally get to interview the jurors. And we get so much information from this moment because the lawyers ask questions based on what they want out of the jurors.

So the prosecution is asking all these different kinds of questions. The first round of wajir is done by a guy named Joshua Steinglass, a very experienced trial lawyer with the Manhattan District Attorney’s Office. And he’s providing all these hypotheticals. I’ll give you one example because I found this one really, really interesting. He provides a hypothetical about a man who wants his wife killed and essentially hires a hitman to do it. And what he asked the jurors is, if that case were before you, would you be able to see that the man who hired the hitman was a part of this crime?

And of course, what he’s really getting at is, can you accept that even though Michael Cohen, Trump’s fixer, made this payment, Trump is the guy who hired him to do it?

That’s right. If there are other people involved, will jurors still be able to see Donald Trump’s hands behind it all?

Fascinating. And what were some of the responses?

People mostly said, yes, we accept that. So that’s how the prosecution did it.

But the defense had a totally different method of voir dire. They were very focused on their client and people’s opinions about their client.

So what kind of questions do we get from them?

So the lawyer, Todd Blanche, is asking people, what do you make of President Trump? What do you think of President Trump?

And what are some of the responses to that?

Well, there’s this incredible exchange with one of the jurors who absolutely refuses to give his opinion of Donald Trump. They go back and forth and back and forth. And the juror keeps insisting you don’t need to know my opinion of him. All you need to know is that I’m going to be fair and impartial, like I said. And Blanch pushes, and the guy pushes back. And the only way the guy budges is he finally kind of confesses almost at the end that, yes, I am a Democrat, and that’s all we get.

And what ends up happening to this potential juror?

Believe it or not, he got dismissed.

[LAUGHS]: I can believe it. And of course, it’s worth saying that this guy and everybody else is being asked that question just feet from Trump himself.

That’s right. And you might think you were going to get a really kind of spicy, like, popcorn emoji-type exchange from that. But because these are now jurors who have said they can be fair and impartial, who, to some extent, want to be on this jury or at least wouldn’t mind being on this jury, they’re being very restrained.

Mostly, what they are emphasizing — much like that guy just described dis — is that they can be fair. They can be impartial. There’s one woman who gives this really remarkable answer.

She says, I thought about this last night. I stayed up all night. I couldn’t sleep, thinking about whether I could be fair. It’s really important to me, and I can.

What ends up happening to that particular juror?

She’s also dismissed. And she’s dismissed without any reason at all. The defense decides it doesn’t like her. It doesn’t want her on the jury. And they have a certain number of chances to just get rid of jurors — no questions asked.

Other jurors are getting dismissed for cause — I’m doing air quotes with my hands — which means that the lawyers have argued they actually revealed themselves through their answers or through old social media posts, which are brought up in the courtroom, to be either non-credible, meaning they’ve said they can be fair and they can’t, or somehow too biased to be on the jury.

Wait, can I just dial into that for a second? Are lawyers researching the jurors in real time going online and saying — I’m making this up — but Jonah Bromwich is a potential juror, and I’m going to go off into my little corner of the courtroom and Google everything you’ve ever said? Is that what’s happening in the room?

Yeah, there’s a whole profession dedicated to that. It’s called jury consultant, and they’re very good at finding information on people in a hurry. And it certainly looked as if they were in play.

Did a social media post end up getting anybody kicked off this jury?

Yes, there were posts from 2016 era internet. You’ll remember that time as a very heated one on the internet, Facebook memes are a big thing. And so there’s all kinds of lock him up type memes and rhetoric. And some of the potential jurors here have used those. And those jurors are dismissed for a reason.

So we have these two types of dismissals, right? We have these peremptory dismissals — no reason at all given. And we have for cause dismissals.

And the process is called jury selection. But you don’t actually get selected for a jury. The thing is to make it through all these obstacles.

You’re left over.

Right. And so when certain jurors are not dismissed, and they’ve made it through all these stages, by the end of the day, we have gone from zero juror seated to seven jurors who will be participating in Donald Trump’s trial.

Got it. And without going through all seven, just give us a little bit of a sketch of who so far is on this jury. What stands out?

Well, not that much stands out. So we’ve got four men. We’ve got three women. One lives on the Upper East Side. One lives in Chelsea. Obviously, they’re from all over Manhattan.

They have these kind of very normal hobbies like spending time with family and friends. They have somewhat anonymous jobs. We’ve got two lawyers. We’ve got someone who’s worked in sales.

So there’s not that much identifying information. And that’s not an accident . One of the things that often happens with jury selection, whether it be for Donald Trump or for anyone else, is the most interesting jurors — the jurors that kind of catch your attention during the process — they get picked off because they are being so interesting that they interest one or the other side in a negative way. And soon they’re excused. So most of the jurors who are actually seated —

Are not memorable.

Are not that memorable, save one particular juror.

OK. All right, I’ll bite. What do I need to know about that one particular juror?

So let me tell you about a prospective juror who we knew as 374, who will now be juror number five. She’s a middle school teacher from Harlem. And she said that she has friends who have really strong opinions about Trump, but she herself does not. And she insisted several times, I am not a political person.

And then she said this thing that made me quite surprised that the prosecution was fine with having her on the jury. She said, quote, “President Trump speaks his mind, and I’d rather that than someone who’s in office who you don’t know what they’re thinking.”

Hmm. So she expressed approval of President Trump.

Yeah, it was mild approval. But the thing is, especially for the defense in this trial, all you need is one juror. One juror can tie up deliberations in knots, and you can end with a hung jury. And this is actually something that I saw firsthand. In 2019, I was the foreperson on a jury.

How you like that?

Yeah. And the trial was really complicated, but I had thought while we were doing the trial, oh, this is going to be a really easy decision. I thought the defendant in that case was guilty. So we get into deliberations, but there’s this one juror who keeps gumming up the works every time we seem to be making progress, getting a conversation started.

This juror proverbially throws up his hands and says, I am not convicting. This man is innocent. And we talked and we talked. And as the foreperson, I was trying to use all my skills to mediate.

But any time we made any progress, this guy would blow it up. And long story short, hung jury — big victory for the defense lawyer. And we come out of the room. And she points at this juror. The guy —

The defense lawyer.

The defense lawyer points at this juror who blew everything up. And she said, I knew it. I knew I had my guy.

OK. I don’t want to read too much into what you said about that one juror. But should I read between the lines to think that if there’s a hung jury, you wonder if it might be that juror?

That’s what everyone in the courtroom is wondering not just about this juror, but about every single person who was selected. Is this the person who swings the case for me? Is this the person who swings the case against me?

These juries are so complex. It’s 12 people who don’t know each other at the start of the trial and, by the end of the trial, have seen each other every morning and are experiencing the same things, but are not allowed to have talked about the case until deliberations start. In that moment when deliberations start —

You’re going to learn a whole lot about each other.

That’s right. There’s this alchemical moment where suddenly, it all matters. Every personality selected matters. And that’s why jury selection is so important. And that’s why these last two days are actually one of the most important parts of this trial.

OK. So by my math, this trial will require five more jurors to get to 12. I know also they’re going to need to be alternates. But from what you’re saying what looked like a really uphill battle to get an impartial jury or a jury that said it could be impartial — and Trump was very doubtful one could be found — has turned out to not be so hard to find.

That’s right. And in fact, we went from thinking, oh, boy, this is going awfully slowly, to the judge himself saying we could be doing opening arguments as soon as Monday morning. And I think that highlights something that’s really fascinating both about this trial and about the jury selection process overall.

One of the things that lawyers have been arguing about is whether or not it’s important to figure out what jurors’ opinions about Donald Trump are. And the prosecution and, I think, the judge have really said, no, that’s not the key issue here. The key issue is not whether or not people have opinions about Donald Trump.

Right. Who doesn’t have an opinion about Donald Trump?

Exactly. They’re going to. Automatically, they’re going to. The question is whether or not they can be fair and impartial. And the seven people we already have seated, and presumably the five people that we’re going to get over the next few days and however many alternates — we expect six — are all going to have answered that question, not I hate Trump; I love Trump, but I can weigh in on the former president’s innocence or guilt, and I can do it as fairly as humanly possible.

Now, Trump is not happy about this. He said after court yesterday, quote, We have a highly conflicted judge, and he’s rushing this trial.” And I think that he is going to see these beats of the system the criminal justice system as it works on him as he is experiencing it as unfair. That is typically how he talks about it and how he views it.

But what he’s getting is what defendants get. This is the system in New York, in the United States. This is its answer to how do you pick a fair jury? Well, you ask people can you be fair? And you put them through this process, and the outcome is 12 people.

And so I think we’re going to see this over and over again in this trial. We’re going to see Trump experience the criminal justice system.

And its routines.

Yeah, openings, witnesses, evidence, closings. He’s going to go through all of it. And I think, at every turn, it makes sense to expect him to say, well, this is not fair. Well, the judge is doing something wrong. Well, the prosecutors are doing something wrong. Well, the jury is doing something wrong.

But at the end of the day, he’s going to be a defendant, and he’s going to sit, mostly silently if his lawyers can make him do that, and watch this process play itself out. So the system is going to try and treat him like any other defendant, even though, of course —

— he’s not. And he is going to fight back like no other defendant would, like no other defendant could. And that tension, him pushing against the criminal justice system as it strives to treat him, as it would anyone else, is going to be a defining quality of this trial.

Well, Jonah, thank you very much. We appreciate it.

Of course. Thanks so much for having me. [MUSIC PLAYING]

PS, have you ever fallen asleep in a trial?

I have not.

[CHUCKLES]:

Here’s what else you need to know today.

It’s clear the Israelis are making a decision to act. We hope they do so in a way that does as little to escalate this as possible and in a way that, as I said —

During a visit to Jerusalem on Wednesday, Britain’s foreign Secretary left little doubt that Israel would retaliate against Iran for last weekend’s aerial attack, despite pressure from the United States and Britain to stand down. The question now is what form that retaliation will take? “The Times” reports that Israel is weighing several options, including a direct strike on Iran, a cyber attack, or targeted assassinations. And —

Look, history judges us for what we do. This is a critical time right now, critical time on the world stage.

In a plan that could threaten his job, Republican House Speaker Mike Johnson will put a series of foreign aid bills up for a vote this weekend. The bills, especially for aid to Ukraine, are strongly opposed by far-right House Republicans, at least two of whom have threatened to try to oust Johnson over the plan.

I can make a selfish decision and do something that’s different, but I’m doing here what I believe to be the right thing. I think providing lethal aid to Ukraine right now is critically important. I really do. I really — [MUSIC PLAYING]

Today’s episode was produced by Rikki Novetsky, Will Reid, Lynsea Garrison, and Rob Zubko. It was edited by Paige Cowett, contains original music by Marion Lozano, Elisheba Ittoop, and Dan Powell, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly Lake.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

• April 19, 2024   •   30:42 The Supreme Court Takes Up Homelessness
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• April 16, 2024   •   29:29 A.I.’s Original Sin
• April 15, 2024   •   24:07 Iran’s Unprecedented Attack on Israel
• April 14, 2024   •   46:17 The Sunday Read: ‘What I Saw Working at The National Enquirer During Donald Trump’s Rise’
• April 12, 2024   •   34:23 How One Family Lost $900,000 in a Timeshare Scam
• April 11, 2024   •   28:39 The Staggering Success of Trump’s Trial Delay Tactics
• April 10, 2024   •   22:49 Trump’s Abortion Dilemma
• April 9, 2024   •   30:48 How Tesla Planted the Seeds for Its Own Potential Downfall
• April 8, 2024   •   30:28 The Eclipse Chaser
• April 7, 2024 The Sunday Read: ‘What Deathbed Visions Teach Us About Living’

Hosted by Michael Barbaro

Featuring Jonah E. Bromwich

Produced by Rikki Novetsky ,  Will Reid ,  Lynsea Garrison and Rob Szypko

Edited by Paige Cowett

Original music by Dan Powell ,  Marion Lozano and Elisheba Ittoop

Engineered by Chris Wood

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music

Political and legal history are being made in a Lower Manhattan courtroom as Donald J. Trump becomes the first former U.S. president to undergo a criminal trial.

Jonah Bromwich, who covers criminal justice in New York, explains what happened during the opening days of the trial, which is tied to Mr. Trump’s role in a hush-money payment to a porn star.

On today’s episode

Jonah E. Bromwich , who covers criminal justice in New York for The New York Times.

Background reading

Here’s a recap of the courtroom proceedings so far.

Mr. Trump’s trial enters its third day with seven jurors chosen.

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The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

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Jonah E. Bromwich covers criminal justice in New York, with a focus on the Manhattan district attorney’s office and state criminal courts in Manhattan. More about Jonah E. Bromwich

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