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21 Essential Strategies in Teaching Math

Even veteran teachers need to read these.

We all want our kids to succeed in math. In most districts, standardized tests measure students’ understanding, yet nobody wants to teach to the test. Over-reliance on test prep materials and “drill and kill” worksheets steal instructional time while also harming learning and motivation. But sound instruction and good test scores aren’t mutually exclusive. Being intentional and using creative approaches to your instruction can get students excited about math. These essential strategies in teaching mathematics can make this your class’s best math year ever!

1. Raise the bar for all

WeAreTeachers

For math strategies to be effective, teachers must first get students to believe that they can be great mathematicians. Holding high expectations for all students encourages growth. As early as second grade, girls have internalized the idea that math is not for them . It can be a challenge to overcome the socially acceptable thought, I’m not good at math , says Sarah Bax, a math teacher at Hardy Middle School in Washington, D.C.

Rather than success being a function of how much math talent they’re born with, kids need to hear from teachers that anyone who works hard can succeed. “It’s about helping kids have a growth mindset ,” says Bax. “Practice and persistence make you good at math.” Build math equity and tell students about the power and importance of math with enthusiasm and high expectations.

(Psst … you can snag our growth mindset posters for your math classroom here. )

2. Don’t wait—act now!

Look ahead to the specific concepts students need to master for annual end-of-year tests, and pace instruction accordingly. Think about foundational skills they will need in the year ahead.

“You don’t want to be caught off guard come March thinking that students need to know X for the tests the next month,” says Skip Fennell, project director of Elementary Mathematics Specialists and Teacher Leaders Project and professor emeritus at McDaniel College in Westminster, Maryland. Know the specific standards and back-map your teaching from the fall so students are ready, and plan to use effective math strategies accordingly.

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3. Create a testing pathway

You may not even see the results of standardized tests until next school year, but you have to prepare students for it now. Use formative assessments to ensure that students understand the concepts. What you learn can guide your instruction and determine the next steps, says Fennell. “I changed the wording because I didn’t want to suggest that we are in favor of ‘teaching to the test.'”

Testing is not something separate from your instruction. It should be integrated into your planning. Instead of a quick exit question or card, give a five-minute quiz, an open-ended question, or a meaningful homework assignment to confirm students have mastered the math skill covered in the day’s lesson. Additionally, asking students to explain their thinking orally or in writing is a great way to determine their level of understanding. A capable digital resource, designed to monitor your students in real-time, can also be an invaluable tool, providing actionable data to inform your instruction along the way.

4. Observe, modify, and reevaluate

Sometimes we get stuck in a mindset of “a lesson a day” in order to get through the content. However, we should keep our pacing flexible, or kids can fall behind. Walk through your classroom as students work on problems and observe the dynamics. Talk with students individually and include “hinge questions” in your lesson plans to gauge understanding before continuing, suggests Fennell. In response, make decisions to go faster or slower or put students in groups.

5. Read, read, read!

Although we don’t often think of reading as a math strategy, there’s almost nothing better to get students ready to learn a new concept than a great read-aloud. Kids love to be read to, and the more we show students how math is connected to the world around us, the more invested they become. Reading books with math connections helps children see how abstract concepts connect to their lives.

6. Personalize and offer choice

When students are given the opportunity to choose how they learn and demonstrate their understanding of a concept, their buy-in and motivation increase. It gives them the chance to understand their preferred learning style, provides agency over their own learning, and allows for the space to practice different strategies to solve math problems. Give students a variety of options, such as timed exercises, projects, or different materials , to show that they’ve mastered foundational skills. As students show what they’ve learned, teachers can track understanding, figure out where students need additional scaffolding or other assistance, and tailor lessons accordingly.

7. Plant the seeds!

Leave no child inside! A school garden is a great way to apply math concepts in a fun way while instilling a sense of purpose in your students. Measurement, geometry, and data analysis are obvious topics that can be addressed through garden activities, but also consider using the garden to teach operations, fractions, and decimals. Additionally, garden activities can help promote character education goals like cooperation, respect for the earth, and, if the crops are donated to organizations that serve those in need, the value of giving to others.

8. Add apps appropriately

The number of apps (interactive software used on touch-screen devices) available to support math instruction has increased rapidly in recent years. Kids who are reluctant to practice math facts with traditional pencil-and-paper resources will gladly do essentially the same work as long as it’s done on a touch screen. Many apps focus on practice via games, but there are some that encourage children to explore the content at a conceptual level.

9. Encourage math talk

Communicating about math helps students process new learning and build on their thinking. Engage students during conversations and have them describe why they solved a problem in a certain way. “My goal is to get information about what students are thinking and use that to guide my instruction, as opposed to just telling them information and asking them to parrot things back,” says Delise Andrews, who taught math (K–8) and is now a grade 3–5 math coordinator in the Lincoln Public Schools in Nebraska.

Instead of seeking a specific answer, Andrews wants to have deeper discussions to figure out what a student knows and understands. “True learning happens a lot around talking and doing math—not just drilling,” she says. Of course, this math strategy not only requires students to feel comfortable expressing their mathematical thinking, but also assumes that they have been trained to listen respectfully to the reasoning of their classmates.

Learn more: Free Let’s Talk Math Poster

10. The art of math

Almost all kids love art, and visual learners need a math strategy that works for them too, so consider integrating art and math instruction for one of the easiest strategies in teaching mathematics. Many concepts in geometry, such as shapes, symmetry, and transformations (slides, flips, and turns), can be applied in a fun art project. Also consider using art projects to teach concepts like measurement, ratios, and arrays (multiplication/division).

11. Seek to develop understanding

Meaningful math education goes beyond memorizing formulas and procedures. Memorization does not foster understanding. Set high goals, create space for exploration, and work with the students to develop a strong foundation. “Treat the kids like mathematicians,” says Andrews. Present a broad topic, review various strategies for solving a problem, and then elicit a formula or idea from the kids rather than starting with the formula. This creates a stronger conceptual understanding and mental connections with the material for the student.

12. Give students time to reflect

Sometimes teachers get so caught up in meeting the demands of the curriculum and the pressure to “get it all done” that they don’t give students the time to reflect on their learning. Students can be asked to reflect in writing at the end of an assignment or lesson, via class or small group discussion, or in interviews with the teacher. It’s important to give students the time to think about and articulate the meaning of what they’ve learned, what they still don’t understand, and what they want to learn more about. This provides useful information for the teacher and helps the student monitor their own progress and think strategically about how they approach mathematics.

13. Allow for productive struggle

When giving students an authentic problem, ask a big question and let them struggle to figure out several ways to solve it, suggests Andrews. “Your job as a teacher is to make it engaging by asking the right questions at the right time. So you don’t take away their thinking, but you help them move forward to a solution,” she says.

Provide as little information as possible but enough so students can be productive. Effective math teaching supports students as they grapple with mathematical ideas and relationships. Allow them to discover what works and experience setbacks along the way as they adopt a growth mindset about mathematics.

14. Emphasize hands-on learning

WeAreTeachers; Teacher Created Resources

In math, there’s so much that’s abstract. Hands-on learning is a strategy that helps make the conceptual concrete. Consider incorporating math manipulatives whenever possible. For example, you can use LEGO bricks to teach a variety of math skills, including finding area and perimeter and understanding multiplication.

15. Build excitement by rewarding progress

Students—especially those who haven’t experienced success—can have negative attitudes about math. Consider having students earn points and receive certificates, stickers, badges, or trophies as they progress. Weekly announcements and assemblies that celebrate the top players and teams can be really inspiring for students. “Having that recognition and moment is powerful,” says Bax. “Through repeated practice, they get better, and they are motivated.” Through building excitement, this allows for one of the best strategies in teaching mathematics to come to fruition.

16. Choose meaningful tasks

Kids get excited about math when they have to  solve real-life problems. For instance, when teaching sixth graders how to determine area, present tasks related to a house redesign, suggests Fennell. Provide them with the dimensions of the walls and the size of the windows and have them determine how much space is left for the wallpaper. Or ask them to consider how many tiles they would need to fill a deck. You can absolutely introduce problem-based learning, even in a virtual world.

17. Play math games

Life Between Summers/Shape Guess Who via lifebetweensummers.com; 123 Homeschool 4 Me/Tic-Tac-Toe Math Game via 123homeschool4me.com; WeAreTeachers

Student engagement and participation can be a challenge, especially if you’re relying heavily on worksheets. Games, like these first grade math games , are an excellent way to make the learning more fun while simultaneously promoting strategic mathematical thinking, computational fluency , and understanding of operations. Games are especially good for kinesthetic learners and foster a home-school connection when they’re sent home for extra practice.

18. Set up effective math routines

Students generally feel confident and competent in the classroom when they know what to do and why they’re doing it. Establishing routines in your math class and training kids to use them can make math class efficient, effective, and fun! For example, consider starting your class with a number sense routine . Rich, productive small group math discussions don’t happen by themselves, so make sure your students know the “rules of the road” for contributing their ideas and respectfully critiquing the ideas of others.

19. Encourage teacher teamwork and reflection

You can’t teach in a vacuum. Collaborate with other teachers to improve your math instruction skills. Start by discussing the goal for the math lesson and what it will look like, and plan as a team to use the most effective math strategies. “Together, think through the tasks and possible student responses you might encounter,” says Andrews. Reflect on what did and didn’t work to improve your practice.

Learn With Play at Home/Plastic Bottle Number Bowling via learnwithplayathome.com; Math Geek Mama/Skip-Counting Hopscotch via mathgeekmama.com; WeAreTeachers

Adding movement and physical activity to your instruction might seem counterintuitive as a math strategy, but asking kids to get out of their seats can increase their motivation and interest. For example, you could ask students to:

• Make angles with their arms
• Create a square dance that demonstrates different types of patterns
• Complete a shape scavenger hunt in the classroom
• Run or complete other exercises periodically and graph the results

The possibilities of these strategies in teaching mathematics are limited only by your imagination and the math concepts you need to cover. Check out these active math games .

21. Be a lifelong learner

Generally, students will become excited about a subject if their teacher is excited about it. However, it’s hard to be excited about teaching math if your understanding hasn’t changed since you learned it in elementary school. For example, if you teach how to divide fractions by fractions and your understanding is limited to following the “invert and multiply” rule, take the time to understand why the rule works and how it applies to the real world. When you have confidence in your own mathematical expertise, then you can teach math confidently and joyfully to best apply strategies in teaching mathematics.

What do you feel are the most important strategies in teaching mathematics? Share in the comments below.

Want more articles like this be sure to subscribe to our newsletters ., learn why it’s important to honor all math strategies in teaching math . plus, check out the best math websites for teachers ..

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24 Creative Ways to Use Math Manipulatives in Your Classroom

Students learn better when they’re engaged, and manipulatives in the classroom make it easy for kids to get excited. We Continue Reading

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22 Fun Math Activities for Your Classroom

Written by Marcus Guido

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Research showed that Prodigy Math helped drive a significant, positive shift in students' opinion towards math in just a few months.

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• 1. Prodigy Math

2. Read a Math Book

3. create mnemonic devices, 4. deliver a daily starter, 5. visit the national library of virtual manipulatives, 6. run a round of initials, 7. play math baseball, 8. start a game of around the block, 9. play math tic-tac-toe, 10. modify a classic card game, 11. share teachertube videos, 12. co-ordinate live video, 13. research the leaning tower, 14. party on pi day, 15. hold a scavenger hunt, 16. play one-metre dash, 17. put a twist on gym class, 18. run think-pair-share exercises.

• 19. Hold a Game of Jeopardy

20. Take on a Challenge from Get The Math

When students think “fun,” memories of math class likely won’t be the first to pop into their heads. But that doesn’t have to be the case.

There are approaches and exercises, with and without computers, that can  enliven your math lessons .

You’ll likely find that the reward justifies the work of preparing and introducing them. After all, according many studies from  as early as the 1960s , engaged students pay more attention and perform higher than disengaged ones.W

But while making math fun for students is definitely effective, it's not always easy for busy educators to plan, prepare and deliver them on the spot.

That's why we've put together a list of 22 fun math activities for students.  Use these fun activity ideas to engage your students and help them build a lifelong love for learning math.

1. Play Prodigy Math

Grade levels: 1st-8th

Best for: Reinforcing lesson content, differentiating instruction and engaging students through game-based learning.

Use Prodigy — the standards-aligned math platform used by millions of students and teachers worldwide — to  engage your class while  reinforcing lesson content  and teaching essential skills.

Prodigy uses elements from students’ favorite video games as they compete in math duels against in-game characters. To win, they must answer sets of questions. You can customize these questions to supplement class material,  deliver assessments ,  prepare for tests  and more.

And the best bit? Educators and schools get full access to teaching tools at zero cost!

Here's a sneak peak of Prodigy in action! 👇

Grade levels: K-6th

Best for: Introducing new concepts, reinforcing learned skills, and encouraging independent learning.

Show your students that  reading engaging stories isn’t exclusive to language arts class.

There are many age-appropriate  math books  that effectively explain skills and techniques while providing exercises to help students understand content.

For example, the  Life of Fred  series introduces and teaches essential math skills aligned with most elementary school curricula.

The four books, each containing 19 lessons, present content through stories about cats, ice cream and other child-friendly subjects. With full answer keys, the series lends itself to practicing, reviewing or learning entire skills.

You can find age- and topic-specific math books through a few Amazon searches or a brief bookstore visit.

Grade levels: 3rd-8th

Best for: Helping learners remember math facts, equations and sequences.

Dedicate time for students to create mnemonic devices — cues such as rhymes and acronyms — to  help  recall math facts .

A popular example is “I need to be 16 years-old to drive a 4×4 pickup truck.” Such cues should be rhymes or quick stories that distill larger chunks of information, always using tangible objects or scenarios to make them memorable.

Although you can think of mnemonic devices yourself and share them with students, it’s beneficial to run an activity that gets them to make their own. They’ll likely find it easier to remember ones they create.

Grade levels: PreK-8th

Best for: Kicking off the day, focusing students' attention and warming up brains for math learning.

Drop by Scholastic’s  Daily Starters page  each morning to  find entry tickets suited to solo and group work. This includes skills and topics like mental math, place value and number sense.

Content levels range from pre-kindergarten to 8th grade, including problems from subjects other than math. Many teachers either print the questions or project them onto a whiteboard.

Aside from entry tickets, there are different ways to use Daily Starters — such as including them in learning stations or wrapping up a lesson with them.

Grade levels: K-12

Best for: Interactive learning and engagement, especially for visual learners.

Have students visit the online  National Library of Virtual Manipulatives  to  access activities that involve digital objects such as coins and blocks.

Created by Utah State University, the online library aims to engage students. To do so, there are manipulation tasks for students at every grade level.

For example, a 6th grade geometry activity involves using geo-boards to illustrate area, perimeter and rational number concepts. Ideal for classes with one-to-one device use, the website can also act as a learning station.

Grade levels : 4th-8th

Best for: Content reviews and encouraging students to work in teams.

Add a  game-like spin to content reviews  by playing Initials.

Hand a unique sheet to each student that has problems aligned with a common skill or topic. Instead of focusing on their own sheets, students walk around the room to solve questions on their classmates’.

Here’s the catch: A student can only complete one question per sheet, signing his or her initials beside the answer. The exercise continues until all questions on each sheet have answers,  encouraging students to build trust and teamwork .

Grade levels: 2nd-6th

Best for: Creating a competitive environment and reinforcing a variety of math concepts.

Divide your class into two teams to play math baseball —  an activity that gives you full control of the questions students answer.

One team will start “at bat,” scoring runs by choosing questions worth one, two or three bases. You’ll “pitch” the questions, which range in difficulty depending on how many bases they’re worth. If the at-bat team answers incorrectly, the defending team can correctly respond to earn an out. After three outs, switch sides.

Play until one team hits 10 runs, or five for a shorter entry or exit ticket.

Grade levels: 2nd-5th

Best for: Practicing any math skill in a fun, dynamic way.

Play Around the Block as a minds-on activity, using only a ball to  practice almost any math skill.

First, compile questions related to a distinct skill. Second, have students stand in a circle. Finally, give one student the ball and read aloud a question from your list.

Students must pass the ball clockwise around the circle, and the one who started with it must answer the question before receiving again.

If the student incorrectly answers, pass the ball to a classmate for the next question. If the student correctly answers, he or she chooses the next contestant.

Grade levels: 3rd-6th

Best for: Practicing a range of abilities and offering a familiar but math-focused game.

Pair students to  compete against one another while building different math skills  in this take on tic-tac-toe.

To prepare, divide a sheet into squares — three vertical by three horizontal. Fill these squares with questions that collectively test a range of abilities. The first student to link three Xs or Os — by correctly answering questions — wins.

This game can be a  learning station , refreshing prerequisite skills in preparation for new content.

Best for: Reinforcing basic math operations in a competitive and enjoyable manner.

Put a mathematical twist on a traditional card game  by having students play this version of  War .

Students should pair together, with each pair grabbing two decks of cards. Cards have the following values:

• Two to 10 — Face value

The rules of the game will depend on the grade you teach and the skills you’re building. Each student will always play two cards at a time, but younger kids must subtract the lower number from the higher.

Older students can multiply the numbers, designating a certain suit as having negative integers. Whoever has the highest hand wins all four cards.

Best for: Visual learners and supplementing lessons with video content.

Cover core skills by visiting  TeacherTube  — an education-only version of YouTube.

By searching for a specific topic or browsing by category, you can quickly find videos to supplement a lesson or act as a learning station.

For example, searching for “middle school algebra” will load a results page containing study guides, specific lessons and exam reviews.

Students and parents can also visit TeacherTube on their own time, as some videos explicitly apply to them.

Grade levels: 6th-12th

Best for: Enhancing class content with expert insights and diversifying teaching approaches.

Don’t limit yourself to pre-recorded videos — straightforward conferencing technology can allow  subject matter experts to deliver live lessons to your class.

Whether it’s a contact from another school or a seasoned lecturer you reach out to, bringing an expert into your classroom will expose your students to new ideas and can lighten your workload.

Add the person on Skype or Google Hangouts, delivering the lesson through the program. Skype even has a list of guest speakers who will voluntarily speak about their topics of expertise.

Grade levels: 4th-8th

Best for: Interdisciplinary learning, integrating math with real-world structures and events.

Delve into the Leaning Tower of Pisa, one of Italy’s famous landmarks, by running this  popular  interdisciplinary activity .

Although the exercise traditionally spans across subjects through guided research, you can focus on math by requiring students to:

• Develop an itinerary, complete with a budget, for a trip to Pisa
• Calculate measurements such as the tower’s area and volume
• Investigate the tower’s structure, determining if or when it’ll fall

For younger students, you can divide the activity into distinct exercises and allow them to work in groups. Older students should tackle it as an in-class or take-home project.

You can easily adjust the skill complexity to your students' needs, starting off with key math skills like subtraction, addition, multiplication, division and advancing into more complex areas like percentages, fractions and averages. Students at higher grades can even explore graphing and data analysis.

Best for: Making math fun, celebrating a mathematical constant and promoting math-themed camaraderie.

Celebrate Pi Day on March 14 each year by  dedicating an entire period, or more, to the mathematical constant.

Although specific activities depend on your students, you can start the lesson by giving a historical and conceptual overview of pi — from Archimedes to how modern mathematicians use it. After, delve into exercises.

For younger students, get construction paper and choose a colour to represent each digit. Red can be one, blue is two, green can represent three and so on. Their task is to arrange and order the paper to represent as much of pi’s value as possible.

For older students, run learning stations that allow them to complete questions, process content and play  math games  related to pi. For a fun finish, serve students pizza or another kind of pie.

Best for: Integrating technology, promoting research skills, and teaching new math concepts in a fun way.

Send your students on an Internet scavenger hunt, a potential addition to Pi Day fun, allowing them to  build research skills while processing new math concepts.

The exercise starts by providing a sheet of terms to define or questions to solve, which students can complete by using Google or a list of recommended websites. Regardless, the terms and questions should all fall under an overarching topic.

For example, “Find the definition of a negative integer” and “If you multiply a positive integer with a negative integer, will the product be positive or negative? What about multiplying two negative integers together?”

More than engaging, educational hunts introduce your students to resources they can regularly refer to.

Best for: Teaching students about estimation and measurement; hands-on learners; kinetic learners; group work.

Start this quick game to  build students’ perception and understanding of measurement.

Grouping students in small teams, give them metre sticks. They then look around the room for two to four items they think add up to a metre in length. In a few minutes, the groups measure the items and record how close their estimates were.

Want more of a challenge? Give them a centimetre-mark to hit instead of a metre. You can then ask them to convert results to micrometres, millimetres and more.

Best for: Linking physical activity and mathematics; catering to active learners; cross-curricular education; students who enjoy physical activities.

Fuse math and physical education by  delivering ongoing lessons that explain and explore certain motions.

It’s time to practice long jumps. But first, students can estimate how far they’ll jump. After, they can see how close they were.

Such activities can also supplement lessons about lifting, throwing and other actions — potentially interesting students who don’t normally enjoy gym or math.

Grade levels: All grades

Best for: Encouraging discussion and cooperation; fostering critical thinking; aiding understanding and retention; catering to a range of learning styles.

Launch a think-pair-share exercise to  expose students to three lesson-processing experiences in quick succession.

As the strategy’s name implies, start by asking students to individually  think  about a given topic or answer a specific question. Next,  pair  students together to discuss their results and findings. Finally, have each pair  share  their ideas with the rest of the class, and open the floor for further discussion.

The three parts of this exercise vary in length, giving you flexibility when lesson planning.

And because it allows your students to process content individually, in a small group and in a large group, it caters to your classroom’s range of learning and personality types

19. Hold a Game of Math Jeopardy

Grade levels: 3rd-12th

Best for: Reviewing multiple topics; competitive learners; group work; interactive class reviews.

Transform this famous game show to focus on your latest skill or unit,  preparing students for a quiz or test.

Setup involves attaching pockets to a bristol board, dividing them into columns and rows. Each column should focus on a topic, whereas each row should have a point value — 200, 400, 600, 800 and 1,000.

A team can ask for a question from any pocket, but other teams can answer first by solving the problem and raising their hands.

Once the class answers all questions, the team with the highest point total claims your prize. But each student wins in terms of engagement and practicing  peer support .

Looking for more fun math games? Check out this list of 23 classroom math games for kids .

Best for: Applying math to real-world scenarios; career-focused learning; students interested in how math is used in the professional world.

Teach your students about  how math is used in different careers and real-world situations  by visiting  Get the Math .

The website, aimed at middle and high school students, features videos of young professionals who explain how they use algebra. They then pose job-related questions to two teams of students in the video.

Your class can also participate, learning how to apply algebraic concepts in different scenarios. It’s a straightforward way to vary and contextualize your lesson content.

21. Virtual Math Escape Room

Grade levels: 4th-12th

Best for: Group activities where teamwork and problem-solving skills are essential. This activity is excellent for tech-savvy students andfor situations where you want to increase engagement through interactive digital tools.

Escape rooms have been a rising trend in recent years. Take advantage of their popularity by setting up a virtual math escape room. Develop a series of math puzzles that students must solve to "escape." Use a digital platform that allows you to hide clues and puzzles in an online environment. The time pressure and narrative can make solving math problems an exciting adventure!

22. Break the Code (Cryptography)

Grade levels: 5th-9th

Best for: Students who enjoy solving mysteries and puzzles. It's an engaging way to introduce abstract mathematical concepts like modular arithmetic and to showcase the practical applications of mathematics.

Awaken the budding mathematicians and detectives in your students with cryptography. Introduce simple encryption techniques and provide coded messages for your students to decipher. This activity can be tailored to different complexity levels, right from elementary to high school level.

Here's a simple version of math code breaking activity below!

Need further support? Check out these math worksheets!

As well as using fun math activities to deliver math content, you can also use worksheets. Ideal as part of a station rotation, these quick exercises can help students tackle math problems so you can gauge their understanding.

Here are some free printable worksheets to get you started:

• 1st Grade Math Worksheets
• 2nd Grade Math Worksheets
• 3rd Grade Math Worksheets
• 4th Grade Math Worksheets

Final Thoughts

Each of these exercises can inject engagement into your lessons, helping students process content and demonstrate understanding.

What’s more,  they’re versatile.  You can use many of the above activities to introduce concepts or reinforce lessons, and as minds-on exercises or exit tickets. Useful for you, fun for students.

Who says math can’t be engaging?

👉 Try Prodigy today — the standards-aligned, game-based learning platform that delivers fun math activities based on the student’s unique strengths and skill deficits. It’s used by more than 700,000 teachers and millions of students around the world.

Introduction to Sets

Forget everything you know about numbers.

In fact, forget you even know what a number is.

This is where mathematics starts.

Instead of math with numbers, we will now think about math with "things".

What is a set? Well, simply put, it's a collection .

First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property.

For example, the items you wear: hat, shirt, jacket, pants, and so on.

I'm sure you could come up with at least a hundred.

This is known as a set .

So it is just things grouped together with a certain property in common.

There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

The curly brackets { } are sometimes called "set brackets" or "braces".

This is the notation for the two previous examples:

{socks, shoes, watches, shirts, ...} {index, middle, ring, pinky}

Notice how the first example has the "..." (three dots together).

The three dots ... are called an ellipsis, and mean "continue on".

So that means the first example continues on ... for infinity.

(OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)

• The first set {socks, shoes, watches, shirts, ...} we call an infinite set ,
• the second set {index, middle, ring, pinky} we call a finite set .

But sometimes the "..." can be used in the middle to save writing long lists:

Example: the set of letters:

{a, b, c, ..., x, y, z}

In this case it is a finite set (there are only 26 letters, right?)

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

And so on. We can come up with all different types of sets.

We can also define a set by its properties, such as {x|x>0} which means "the set of all x's, such that x is greater than 0", see Set-Builder Notation to learn more.

And we can have sets of numbers that have no common property, they are just defined that way. For example:

Are all sets that I just randomly banged on my keyboard to produce.

Why are Sets Important?

Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.

Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets .

Universal Set

Some more notation.

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?

Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!

Example: Are A and B equal where:

• A is the set whose members are the first four positive whole numbers
• B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal!

And the equals sign (=) is used to show equality, so we write:

Example: Are these sets equal?

• A is {1, 2, 3}
• B is {3, 1, 2}

Yes, they are equal!

They both contain exactly the members 1, 2 and 3.

It doesn't matter where each member appears, so long as it is there.

When we define a set, if we take pieces of that set, we can form what is called a subset .

Example: the set {1, 2, 3, 4, 5}

A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.

But {1, 6} is not a subset, since it has an element (6) which is not in the parent set.

In general:

A is a subset of B if and only if every element of A is in B.

So let's use this definition in some examples.

Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 is in A, and 1 is in B as well. So far so good.

3 is in A and 3 is also in B.

4 is in A, and 4 is in B.

That's all the elements of A, and every single one is in B, so we're done.

Yes, A is a subset of B

Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

Let's try a harder example.

Example: Let A be all multiples of 4 and B be all multiples of 2 . Is A a subset of B? And is B a subset of A?

Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.

The sets are:

• A = {..., −8, −4, 0, 4, 8, ...}
• B = {..., −8, −6, −4, −2, 0, 2, 4, 6, 8, ...}

By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:

A is a subset of B, but B is not a subset of A

Proper Subsets

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.

Let A be a set. Is every element of A in A ?

Well, umm, yes of course , right?

So that means that A is a subset of A . It is a subset of itself!

This doesn't seem very proper , does it? If we want our subsets to be proper we introduce (what else but) proper subsets :

A is a proper subset of B if and only if every element of A is also in B, and there exists at least one element in B that is not in A.

This little piece at the end is there to make sure that A is not a proper subset of itself: we say that B must have at least one extra element.

{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

Notice that when A is a proper subset of B then it is also a subset of B.

Even More Notation

When we say that A is a subset of B, we write A ⊆ B

Or we can say that A is not a subset of B by A ⊈ B

When we talk about proper subsets, we take out the line underneath and so it becomes A ⊂ B or if we want to say the opposite A ⊄ B

Empty (or Null) Set

This is probably the weirdest thing about sets.

As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!"

And right you are. It is a set with no elements .

This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.

It is represented by ∅

Or by {} (a set with no elements)

Some other examples of the empty set are the set of countries south of the south pole .

So what's so weird about the empty set? Well, that part comes next.

Empty Set and Subsets

So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A?

Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A . But what if we have no elements?

It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.

A good way to think about it is: we can't find any elements in the empty set that aren't in A , so it must be that all elements in the empty set are in A.

So the answer to the posed question is a resounding yes .

The empty set is a subset of every set, including the empty set itself.

No, not the order of the elements. In sets it does not matter what order the elements are in .

Example: {1,2,3,4} is the same set as {3,1,4,2}

When we say order in sets we mean the size of the set .

Another (better) name for this is cardinality .

A finite set has finite order (or cardinality). An infinite set has infinite order (or cardinality).

For finite sets the order (or cardinality) is the number of elements .

Example: {10, 20, 30, 40} has an order of 4.

For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.

Arg! Not more notation!

Nah, just kidding. No more notation.

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COMPLETE MATH

George mason university, 2. making assumptions.

Make assumptions and focus the problem with information and mathematics.

• What information do we already know/need to know to make a model?
• What assumptions do we need to make to build a model?
• What are some important quantities in your situation?
• What mathematical tools could
• you use in your model?

                                            KWI CHART

        Gathering Information

If i knew______, i could figure out_______.

(For example, If I knew the cost of a school bus for a field trip,  then I could figure out the cost of the transportation.

If I knew how many students and chaperones were coming on the trip , then I could figure out the total number of school buses we would need. )

Standard for Mathematical Practices:Reason abstractly and quantitatively.

• Master Geometry Formulas: A Comprehensive Guide for Academic Success

Geometry Formulas Unraveled: Your Go-To Resource for Assignments

Geometry, a captivating branch of mathematics, ventures into the nuanced exploration of shapes, sizes, and dimensions within the space we occupy. The intricate world of geometry presents students with a labyrinth of concepts, demanding a profound comprehension of formulas. These formulas are not mere mathematical abstractions; they are the key to unlocking the secrets hidden within angles, polygons, circles, and three-dimensional spaces. Whether engrossed in challenging assignments, gearing up for examinations, or simply striving to grasp the fundamental principles, students stand to gain significantly from a comprehensive resource. This blog serves as a beacon, illuminating the path toward mastery in this mathematical domain by providing practical insights, real-world applications, and a holistic approach that transcends rote memorization. For those seeking assistance with your Geometry assignment , this guide aims to be an invaluable tool in achieving success and mastering the fascinating world of geometry.

As we conclude this exploration, it becomes evident that geometry is not just a theoretical construct but a dynamic tool. Armed with a profound understanding of these formulas, students can navigate the intricate terrain of mathematical problem-solving with confidence. Whether unraveling the mysteries of angles, exploring the symmetry of shapes, or delving into the realms of trigonometry, this blog encapsulates the essence of geometry, offering a comprehensive resource that transcends the boundaries of a traditional academic guide. In the tapestry of mathematical knowledge, geometry unfolds as a vibrant thread, weaving together the abstract and the concrete, and in this journey, we hope to be your steadfast companion, illuminating the path to geometric proficiency in your academic endeavors.

The Basics of Geometry

In laying the groundwork for understanding the intricate realm of geometry, it's imperative to revisit the fundamental concepts that underpin this mathematical discipline. At its core, geometry explores the properties and relationships of various elements in the space we inhabit. Beginning with the basic building blocks such as points, lines, and angles, students gain a foundational understanding that serves as a prerequisite for more advanced geometric principles. Euclidean geometry, with its axioms and postulates, forms the bedrock upon which many geometric theorems rest. Delving into the terminology and classifications of shapes, including triangles, quadrilaterals, and polygons, provides the scaffolding for the application of formulas governing perimeter, area, and other characteristics of these fundamental geometric entities. This section serves as an essential refresher, emphasizing the importance of mastering these basic concepts before venturing into the more complex and nuanced realms of geometric reasoning. As students grasp the basics, they not only develop the necessary skills for problem-solving but also cultivate a solid foundation upon which the edifice of geometric knowledge can be constructed, paving the way for a deeper exploration of the geometrical intricacies that lie ahead.

Properties of Shapes

We delve into the multifaceted realm of shape properties, unraveling the fundamental formulas that underpin geometric understanding. From the simplicity of triangles to the intricacies of polygons, each shape brings a unique set of properties and challenges. Exploring the concept of perimeter, we uncover the formulaic essentials for calculating the total length of a shape's boundary, shedding light on its significance in practical applications. Moving on to the realm of area, we navigate the intricacies of surface measurement, illustrating how these formulas are not just mathematical abstractions but crucial tools for quantifying space within shapes. Through a comprehensive examination of shapes like rectangles, circles, and irregular polygons, we decode the formulas for area, shedding light on the interplay between shape dimensions and space occupancy. Additionally, we unveil the importance of understanding the Pythagorean theorem in the context of right-angled triangles, highlighting its role in calculating sides and angles. As we unravel the properties of shapes, we emphasize the practical relevance of these formulas, showcasing how they extend beyond the classroom into real-world scenarios, from calculating material requirements in construction to determining land areas in urban planning. This section serves as a foundational guide, empowering students and enthusiasts alike with the tools needed to navigate the dynamic landscape of shape properties and apply them adeptly in problem-solving and analysis.

Circle Formulas and Applications

In the realm of geometry, circles stand out as ubiquitous and intriguing entities, playing a pivotal role in numerous facets of our daily lives. Circle formulas, encapsulating essential parameters like circumference, area, and sector calculations, serve as the mathematical backbone for understanding and manipulating these geometric wonders. The circumference, representing the boundary of a circle, is determined by the formula C = 2πr, where r is the radius. This seemingly simple formula finds applications in a myriad of fields, from calculating the length of a bicycle tire to determining the orbits of celestial bodies. The area of a circle, expressed as A = πr², elucidates the extent of space enclosed by the circular boundary, proving indispensable in fields such as land surveying and urban planning. Sector calculations, offering insights into portions of a circle, involve formulas like the area of a sector (A_sector = 0.5r²θ) and the arc length (L = rθ), where θ represents the central angle. These formulas find resonance in diverse areas, ranging from pie-chart constructions to the assessment of angles of elevation in trigonometry. As we unravel the applications of circle formulas, their ubiquity becomes apparent, resonating not only in mathematical problem-solving but also in the intricate tapestry of our tangible, real-world experiences. Whether in the precision of scientific measurements or the aesthetics of design, the understanding and application of circle formulas serve as a cornerstone in comprehending the geometrical symphony that surrounds us.

Angle Relationships and Trigonometry

Angle relationships and trigonometry form a crucial aspect of geometry, offering a deeper understanding of the relationships between angles and their applications in real-world scenarios. In the realm of angle relationships, the study involves exploring the fundamental theorems governing angles, such as the vertical angle theorem, complementary and supplementary angles, and the transversal angle theorems. These principles lay the groundwork for solving intricate geometric problems, providing a systematic approach to angle-related inquiries. Trigonometry, on the other hand, introduces the study of the ratios and functions of angles within right-angled triangles, including sine, cosine, and tangent. These trigonometric functions serve as powerful tools in measuring distances, heights, and angles, extending their applicability to fields ranging from physics and engineering to astronomy. Understanding the relationships between angles and the application of trigonometric functions empowers individuals to solve complex problems involving triangles and circular motion. Moreover, trigonometry serves as a bridge between geometry and algebra, showcasing the interconnectedness of mathematical concepts. As students delve into angle relationships and trigonometry, they not only gain proficiency in handling geometric challenges but also acquire valuable problem-solving skills with broader implications across various disciplines.

Three-Dimensional Geometry

Three-dimensional geometry adds a layer of complexity to our understanding of space, introducing a dynamic realm of shapes and structures. In this multidimensional space, volumes and surfaces take center stage, challenging us to explore formulas that go beyond the confines of traditional two-dimensional geometry. From calculating the volume of prisms and cylinders to determining the surface areas of spheres and cones, three-dimensional geometry provides a rich tapestry of mathematical concepts. The formulas associated with these shapes offer practical insights into real-world applications, influencing fields ranging from architecture to physics. Understanding spatial relationships becomes crucial as we navigate the intricacies of three-dimensional space, requiring a keen awareness of how shapes interact and transform. As we delve into this dimension, the significance of these formulas becomes evident in their role as problem-solving tools for architects designing structures, engineers conceptualizing complex systems, and scientists modeling the behavior of physical objects. Three-dimensional geometry not only expands our mathematical toolkit but also deepens our appreciation for the spatial dimensions that shape the world around us, fostering a connection between abstract mathematical concepts and their tangible manifestations in our physical reality. In this realm of mathematical exploration, students and professionals alike find themselves unraveling the complexities of volumes, surfaces, and spatial relationships, opening doors to new dimensions of understanding and problem-solving.

Transformational Geometry

Transformational geometry is a captivating branch that takes geometry beyond static shapes and explores the dynamic world of spatial transformations. In this realm, geometric figures undergo changes in position, size, and orientation, opening up a rich array of mathematical possibilities. Transformations include translations, where figures move parallel to themselves; rotations, where figures pivot around a fixed point; reflections, which involve flipping figures across a line; and dilations, where figures scale up or down. Each transformation has its set of formulas and rules, serving as tools to manipulate and analyze shapes in various contexts. Understanding transformational geometry is not only vital for solving intricate mathematical problems but also has practical applications in fields like computer graphics, art, and design. In computer-aided design (CAD), for instance, transformations play a crucial role in modeling and rendering three-dimensional objects. Furthermore, in art and animations, transformations bring life to characters and scenes by seamlessly altering their appearance. Transformational geometry adds a dynamic layer to the study of shapes, providing a bridge between mathematical abstraction and real-world applications. Mastery of these transformations empowers individuals to explore the fluidity and adaptability inherent in geometric figures, enhancing problem-solving skills and fostering a deeper appreciation for the dynamic nature of our geometric universe.

Real-World Problem Solving

In the realm of geometry, the application of formulas extends far beyond the confines of a classroom. Section 7 unravels the intricate tapestry of real-world problem-solving, demonstrating the practical implications of geometric concepts in various fields. Architects employ geometry to design structurally sound buildings, ensuring that shapes, angles, and dimensions harmonize for both aesthetic appeal and stability. Engineers rely on geometric principles to calculate stress distributions, optimize designs, and construct efficient structures. The application extends to physics, where geometric formulas come into play when determining distances, trajectories, and spatial relationships between objects in motion. From the intricate calculations involved in satellite orbit trajectories to the design of everyday objects like bridges and vehicles, geometry underpins the fabric of our built environment. Moreover, advancements in technology leverage geometric concepts, enabling innovations in computer graphics, 3D modeling, and virtual simulations. Whether it's the precise mapping of geographical landscapes or the development of cutting-edge virtual reality applications, geometry serves as the silent architect shaping the digital and physical worlds. As students immerse themselves in real-world problem-solving scenarios, they not only reinforce their understanding of geometric principles but also recognize the profound impact these formulas have on shaping the world we inhabit. Thus, Section 7 illuminates the transformative role of geometry, bridging the gap between theoretical knowledge and its tangible manifestations in the complex, multidimensional landscapes of architecture, engineering, physics, and technology.

Conclusion:

In conclusion, the journey through the intricacies of geometry formulas has revealed the underlying structure and beauty inherent in the mathematical world. From foundational concepts to advanced three-dimensional applications, this comprehensive guide serves as a valuable resource for students grappling with assignments and seeking a deeper understanding of geometric principles. As we've unraveled the mysteries of shapes, angles, circles, and transformations, it's evident that geometry is not merely an abstract discipline but a dynamic tool with practical applications across various fields. The real-world problem-solving section highlights the versatility of geometric formulas in architecture, engineering, and physics, showcasing their relevance beyond the classroom. Armed with this knowledge, students can approach assignments with confidence, recognizing the connections between theoretical concepts and their tangible implications. Moreover, the exploration of transformational geometry underscores the dynamic nature of shapes, paving the way for applications in computer graphics and animation. As we navigate the vast landscape of geometric principles, one can appreciate how these formulas are not just academic exercises but essential tools for understanding and shaping the world around us. With this go-to resource in hand, students are equipped to conquer the challenges of geometry, unveiling the elegance and precision that underlie our geometric reality.

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Why Movement Matters in Math

These strategies for building controlled movement into learning can help middle school math students stay focused and engaged.

Make students move: Studies show that movement fosters communication , and thus increases learning. Understanding the middle school brain and development is crucial to creating the activities that will stimulate young minds and foster growth. 

Since many school districts have block scheduling, with classes that are typically about 90 minutes, students need to be motivated, and having them move, with purpose, is a great way to increase engagement. The activities I share here have very little prep and include movement with structure, leaving chaos behind. They work best in classrooms where students can move about freely—i.e., rooms with seating in groups or tables.

Getting students to move

Entry tickets: You can start students moving as soon as they come in with an entry ticket, which can be a useful formative assessment to see what they remember from a prior lesson. Use sticky notes to create a short question that students can answer quickly, and require that they get up to place their answer on the board. (Privately address anyone who answers incorrectly, later.) Students will then be more focused, since they’ve already had a chance to move around.

People search: During the lesson, a “ people search ” is one of my favorite ways to get kids moving while learning. I hand out a worksheet and set the timer for seven to 10 minutes for independent work. When time’s up, the students need to search to get signatures in the squares for the problems answered. 

For example, two students will sign each other’s papers for question one, and then each student must look for another person to sign for question two. No one is allowed to use the same classmate twice, which leads to talking with peers they may not have worked with otherwise. Lastly, students must discuss the answers they don’t agree on. This is the best part: listening to the mathematical conversations and witnessing learning taking place.

Rotating pairs: “Speed dating” is another favorite. The students sit in rows, but paired together. Students on the left get one color index card, and students on the right get another. I use pink and blue for this example. The students rotate, as in actual speed dating, and each person works with every other person. 

The students with the blue cards move back one seat after each round, while the pink card holders stay seated. The student at the back moves to the front of the next row and so forth, until all students have worked together. I have seen some pairs work well together when I would have predicted otherwise.

The timer is set for a couple of minutes, and the time is shortened as the students work faster through each round. Solving equations (each person has a different expression) or slope (each person has a different coordinate point) is a great lesson for this type of activity. The kids love the speed of it and get really proficient at the same time. They record everything on their paper, again discussing errors as they arise, enforcing the learning while moving around.  

Games: Mind aligned strategies are also great for getting students up and out of their seats with a purpose for learning. Students can demonstrate transformations of quadratic functions in algebra by raising their arms up or down, along with taking steps left and right to model the transformation equation being shown. The whole class could act this out as well or use a Simon Says format. Simon Says is great too for learning transformations in geometry, solidifying the rotational directions and degrees for your students.

A true scavenger hunt involves the students actively searching for problems to solve. Teams of three work well, as they search for envelopes hidden around the room. It’s also fun to use plastic eggs when it’s Easter time and put the problems inside the eggs. The envelopes are a lot less conspicuous, so depending on your classroom, you can decide what is best. 

Put several envelopes labeled with a color, and hide them anywhere you like. Leave the envelopes slightly showing, so the students don’t have to dig too much to find them. They then solve what they find collaboratively, using a workmat, and put the envelopes back exactly where they found them so that other teams can find them. This is the workmat I use for systems of equations. It’s a good idea to make two or three envelopes for each color. Students can only hand in work when everyone in the group has the same answers, which encourages conversations among the team.

There are many other activities I have in my toolbox to get students moving with a purpose, such as gallery walks, station work, wipebooks, clock buddies, and digital activities such as Quizlet Live .

The above activities are quick to set up, which is a huge benefit when trying something new. Additionally, as the facilitator, and having students move with a purpose, you’ll undoubtedly see how focused they are with the task at hand. These activities work well for all students, including multilanguage learners and students with special needs. The engagement, discussion, collaboration, and learning that take place are very rewarding and beneficial, so instead of trying to keep the students quiet and seated, let them move. 

‘A Dangerous Assignment’ Director and Reporter Discuss the Risks in Investigating the Powerful in Maduro’s Venezuela

A still from FRONTLINE and Armando.info's documentary "A Dangerous Assignment: Uncovering Corruption in Maduro's Venezuela."

The investigation at the heart of FRONTLINE’s new documentary A Dangerous Assignment: Uncovering Corruption in Maduro’s Venezuela unfolded as Venezuelan journalist Roberto Deniz started looking into complaints about the poor quality of food distributed by a government program.

Venezuela was in the throes of economic collapse in 2016. The value of the country’s oil had fallen, leading to a deficit, and Venezuelans faced high inflation and food shortages. President Nicolás Maduro responded by launching a food program called the Local Committees for Supply and Production (Comité Locales de Abastecimiento y Producción or CLAP).

As Deniz and the Venezuelan independent news site Armando.info where he worked looked into the program, they would help uncover a corruption scheme and the figure at the heart of the scandal: Alex Saab. The documentary, made in collaboration with Armando.info, was directed by Juan Ravell, produced by Jeff Arak and reported by Deniz — who is now living and working in exile.

Deniz and Ravell spoke with FRONTLINE about the risks of reporting on Venezuela, tracing a corruption scandal that reached into the Venezuelan government and spanned continents, and the price that journalists pay for investigating the powerful in Maduro’s government.

This interview has been edited for length and clarity. Some of the responses have been translated from Spanish.

Can you both take me back to how this whole investigation started?

Deniz: This has been a long story for us — “us” being Armando.info, but also for me, as a journalist. My investigation about Alex Saab started in 2016. That was the moment in which I decided to start investigating what was happening behind the CLAP program. But in 2015, the name Alex Saab came up in an investigation about a contract that he got to build buildings for poor people in Venezuela. The moment when I realized that Alex Saab was also behind the CLAP food program, it was a big signal: This is not a simple contractor of the Venezuelan government. He’s a man who is more powerful than we could imagine.

Ravell: I wasn’t there from the beginning, but I did start collaborating with Armando.info around 2019. They were doing short pieces with different styles in their investigation, so I was making shorts for them and little videos. I remember clearly when the Alacran [Scorpion] investigation broke. An investigation by Armando.info found that opposition lawmakers worked secretly to defend some of Alex Saab’s businesses abroad. And I remember listening to the phone call that Roberto had with Venezuelan opposition politician Luis Parra and I was thinking, “This is insane that nobody’s listening to this call and so few people are aware of the job that Roberto is doing.” The way that Roberto behaved — very controlled, pressing but fair — was impressive to me. That’s when I got the initial idea. Then, when Alex Saab was detained in Cape Verde, we were pretty much convinced that this needs to be a documentary.

Alex Saab’s business network was complicated and vast. Juan, how did you decide what aspects of the story to focus on while filming this documentary?

Ravell: Roberto’s investigation led the narrative. We wanted to follow the most important stories Roberto was publishing, and those that had the most impact. The milk investigation is pretty important to Venezuelans and finding out who was behind its import. A chemical analysis requested by Armando.info showed some of the powdered milk in the CLAP boxes was so deficient in calcium and high in sodium that a researcher noted it couldn’t be classified as milk.

We knew Alex Saab before that. There had been some reporting by Armando.info, but when they connect him to the CLAP importing scheme, that’s when this story gets going. So from then on, we’re basically following Roberto through his investigation and his stories. Other journalists were also working on this case like Gerardo Reyes from Univision and Joshua Goodman from The Associated Press.

Roberto, at what point did you realize the scale of Saab’s business network and its connection to so many Venezuelan government projects?

Deniz: Since 2016, when I realized that Alex Saab was behind the CLAP program. For me, it was very clear that Alex Saab was a man that we have to investigate. The idea that he was the man behind this program to provide food to poor people in Venezuela — that Nicolás Maduro gave all this power to these guys — was a very important signal. When I started, I realized that there was a lot of fear to talk about him. Some sources immediately told me, “Well, Roberto, you have to be careful, because this is a powerful man and is very close to Nicolás Maduro.”

Roberto, you say in the film that some of the information about Saab’s dealings was difficult to uncover, and you needed to find alternative sources. Can you share the process you used to vet these sources to make sure that the information that they were providing was legitimate?

Deniz: In a country like Venezuela, there are severe threats and intimidation against the journalists that dare to do this kind of work. Normally, a journalist can access information from public records, and you can access officials and expect some kind of response. But that doesn’t happen in Venezuela. They won’t even want to acknowledge that you have contacted them.

I spoke to many of the sources that I had gathered for many years, whom I thought could have useful information about what was happening with the CLAP program. That was how I started to gain access to information, documents, papers that confirmed and signaled that Alex Saab was behind all this. You have to double-check, check three or even four times, every piece of information.

I also had many off-the-record sources. I think that over time, those sources have seen the determination that I and the team at Armando.info have had regarding this investigation, and that’s the main reason why they have trusted in our rigor and perseverance.

What was the most challenging aspect of telling the story visually?

Ravell: I’d say finding the balance. It’s a lot of documents. It’s a lot of words. It’s a lot of very dry information that we need to present in an interesting way, so I think what we managed to do is just rely on the narrative and try to find the best ways to translate that into a compelling film.

We were present in certain key moments. When Roberto’s house in Venezuela was raided, we had a camera with Roberto and we were able to interview him that night. The day of the prisoner swap — when Saab was returned from Miami to Venezuela — was interesting, because we had a team in Bogotá following Roberto and a team in Miami. So two different teams in two separate cities covering the same thing. It was an interesting experiment. And I think it comes across nicely in the film.

Roberto, you shared how reporting this story has led to you living in exile. How has that affected your ability to tell stories about what’s going on in Venezuela? What kind of challenges do you face now doing the same kind of journalism you used to do from inside the country?

Deniz: Since I had to get out of Venezuela in 2018, the most difficult thing was answering, “How can I do my work now?” It was so difficult. All of my life, since I decided to become a journalist, I was living in Venezuela, working in Venezuela. But ultimately, my exile was a solution for me, because I could keep working.

The most difficult thing, I think, is the personal part, the family. I know that all of these investigations are not easy for my family, all their grief, all the personal costs that I decided to face during all of these years.

People told me, “Wow, Roberto, you are brave,” “You are a strong person.” I am totally convinced that it’s not related to that. It’s related to our duty as journalists, our responsibility as journalists in a country like Venezuela. People don’t have the opportunity to know what is really happening in the country. I think that has pushed me to continue on in this investigation.

Many times I have thought that this is the moment to end the investigation. I cannot continue anymore. But I have to continue on what we have tried to do in Armando.info.

Can you both speak about the government’s reaction to this journalism, and what it says about press freedoms in Venezuela? What impact is the current atmosphere having on reporters still working inside Venezuela?

Ravell: It’s pretty clear from NGOs that research freedom of expression that investigative journalism and free, independent journalism is at risk in Venezuela. If you publish something and you get sued for defamation, that could end up getting you criminal charges and that can put you in jail. What Armando.info decided to do is just go in and report on hard things, subjects like corruption, and report on people who are very connected to the highest reaches of the Venezuelan government. By doing that, the choice they had to make was to leave the country. One of the few ways you can report on Venezuela is by going into exile. Still, in exile, there are risks, as you can see in the film. Roberto’s house in Venezuela was raided right before Alex Saab was extradited. So he’s in exile, and he’s still persecuted.

Deniz: I have been in exile since 2018, and nowadays I don’t feel that I am safe living abroad. I think that shows how powerful the message of an autocratic government is when they decide to oppose the work of independent journalists. If you see all the stories related to the Alex Saab case, the first legal action that I faced was in 2017 when he decided to sue me. I could face jail if I stayed in Venezuela. I’m totally sure about that. But then in 2021, I got a new legal action against me. I think that is a clear message that even if you get out of Venezuela, but you continue with your work, you are going to face all of the power of the Venezuelan government. It’s so sad for us as journalists.

Shortly before the premiere of this film, the Venezuelan government began responding to the documentary. Can you give us your take on their response?

Deniz: The attorney general of Venezuela accused us — Ewald Scharfenberg, editor and founder of Armando.info, and me, as a reporter — of supposedly being part of and benefiting from a “corruption scheme” related to Venezuela’s ex-oil minister, Tarek El Aissami, who was incarcerated some weeks ago and who’s been questioned for more than a year within a corruption investigation in PDVSA, the Venezuelan state-owned oil company.

It’s not a coincidence that this is happening right after we released the documentary’s trailer. For me, it’s more than evident that this accusation is total nonsense, but that doesn’t make it less serious, because this is a criminalization of the journalism that we have been doing in Armando.info. Sometimes I think that if you compare the work of Armando.info with all the power of the Venezuelan government, we’re like a dwarf fighting a giant, a tiny particle against a huge government, but that only shows you the authoritarian nature of this regime. They won’t tolerate, they won’t accept that some people persist and keep investigating.

Watch the full documentary A Dangerous Assignment: Uncovering Corruption in Maduro’s Venezuela :

Max Maldonado , Tow Journalism Fellow, FRONTLINE/Newmark Journalism School Fellowships , FRONTLINE

More stories.

‘It Would Have Been Easier To Look Away’: A Journalist’s Investigation Into Corruption in Maduro’s Venezuela

FRONTLINE's Reporting on Journalism Under Threat

Risks of Handcuffing Someone Facedown Long Known; People Die When Police Training Fails To Keep Up

In Hundreds of Deadly Police Encounters, Officers Broke Multiple Safety Guidelines

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Pirates designate LHP Josh Fleming for assignment one night after a poor relief performance

Pittsburgh Pirates pitcher Josh Fleming throws during the seventh inning of a baseball game against the Milwaukee Brewers Monday, May 13, 2024, in Milwaukee. (AP Photo/Morry Gash)

Pittsburgh Pirates starting pitcher Josh Fleming delivers during the first inning of a baseball game against the Milwaukee Brewers in Pittsburgh, Wednesday, April 24, 2024. (AP Photo/Gene J. Puskar)

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MILWAUKEE (AP) — Left-handed pitcher Josh Fleming was designated for assignment by the Pittsburgh Pirates on Tuesday, one day after he allowed six runs while getting just three outs in a relief appearance.

The Pirates recalled right-hander Ryder Ryan from Triple-A Indianapolis to fill Fleming’s spot.

“We needed a bullpen arm,” Pirates manager Derek Shelton said Tuesday before a game with the Milwaukee Brewers. “We have other guys that are available tonight and we need the innings. Josh became the odd man out.”

Fleming, who turns 28 on Saturday, had a 1-1 record and a 5.68 ERA with one save in 17 appearances with the Pirates this season. He had pitched for the Tampa Bay Rays from 2020-23.

After heading into April 30 with a 1.29 ERA, Fleming had given up 11 runs – 10 earned – over five innings in his last five appearances.

In the Pirates’ 8-6 victory over the Brewers on Monday night, Fleming allowed a grand slam to Jake Bauers and gave up a two-run double to Willy Adames.

He started the seventh inning with the Pirates ahead 5-0 before allowing Adames’ double. After the Pirates extended their lead to 7-2 in the top of the eighth, Fleming faced four batters in the bottom of the inning without retiring any. Blake Perkins and Gary Sánchez opened with singles, and after Joey Ortiz reached on an error by shortstop Oneil Cruz to load the bases, Bauers hit a shot over the right field wall to cut Pittsburgh’s lead to 7-6.

Five of the six runs Fleming allowed were earned.

Ryan, 29, has gone 1-0 with a 3.00 ERA in nine relief appearances with Pittsburgh this season. He also has an 0-0 record and 3.86 ERA in five games with Indianapolis.

“This guy throws a sinker that really does a nice job of getting ground balls when he executes it,” Shelton said.

AP MLB: https://apnews.com/hub/mlb

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Injured Houston Astros Outfielder Continues Rehab Assignment

Matthew postins | 17 hours ago.

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Injured Houston Astros outfielder Chas McCormick continued his injury rehab assignment in the minor leagues, as he joined the Double-A Corpus Christi Hooks on Tuesday.

McCormick batted second and played center field against the San Antonio Missions. He went 0-for-2 at the plate, driving in a run. He left the game after five innings as Rolando Espinosa replaced him defensively.

The Astros intend to have McCormick play the outfield again on Wednesday.

The 29-year-old McCormick started his rehab assignment on Sunday with Triple-A Sugar Land, where he went 2-for-5 as a designated hitter. He reported no issues running the bases.

The right-handed hitting McCormick has been on the 10-day injured list since May 1 with right hamstring discomfort, a move that was retroactive to April 28. He first injured it against the Kansas City Royals .

McCormick is eligible to be activated when he’s ready.

While McCormick is versatile enough to play all three outfield positions, his bat has not been where it was last season. In 16 games he slashed .236/.325/.278/.603 with three doubles and eight RBI. He hasn’t hit a home run this season and has hit at least 14 home runs in each of his first three MLB seasons.

In 2023 he blasted a career-high 22 home runs and 70 RBI in 115 games.

McCormick is one of two Astros on the 10- or 15-day injured list and the only outfielder. Without him, Houston has six listed outfielders — Yordan Alvarez, Trey Cabbage, Mauricio Dubon, Joey Loperfido, Jake Meyers and Kyle Tucker.

Alvarez also plays designated hitter and Loperfido can also play first base.  

MATTHEW POSTINS

Matthew Postins is an award-winning sports journalist who covers the Texas Rangers, Philadelphia Phillies, Chicago Cubs and Houston Astros for Sports Illustrated/FanNation. He also covers he Big 12 for Heartland College Sports.

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College Algebra

Unit 1: linear equations and inequalities, unit 2: graphs and forms of linear equations, unit 3: functions, unit 4: quadratics: multiplying and factoring, unit 5: quadratic functions and equations, unit 6: complex numbers, unit 7: exponents and radicals, unit 8: rational expressions and equations, unit 9: relating algebra and geometry, unit 10: polynomial arithmetic, unit 11: advanced function types, unit 12: transformations of functions, unit 13: rational exponents and radicals, unit 14: logarithms.

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Breaking news, 8th-graders given hitler-themed assignment to rate nazi monster as a ‘solution seeker,’ ‘ethical decision-maker’.

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An Adolf Hitler-themed question-and-answer assignment given to students at a private school in Atlanta has sparked outrage among parents over its suspected antisemitic nature.

Eighth-grade students at the Mount Vernon School in Atlanta were given a series of questions asking them to rate some of the characteristics of Adolf Hitler — the dictator of Nazi Germany from 1933 to 1945, whose antisemitic ideology fueled the Holocaust — as a leader, according to Fox 5 Atlanta . 

One question posed to students asked, “According to the Mount Vernon Mindset rubric, how would you rate Adolf Hitler as a ‘solution seeker’?” 

A second question asked how students would “rate Adolf Hitler as an ethical decision-maker?”

For both questions, the students were given the option of selecting “Lacks Evidence,” “Approaching Expectations,” “Meets Expectations” or “Exceeds Expectations” to describe the ruthless dictator. 

The bizarre questions ignited outrage among parents — many of whom were concerned the queries were antisemitic by nature, according to the outlet. 

Students at the private school also had issues with the questions, with one telling the outlet the assignment was “troubling” and could be seen as glorifying the warmongering totalitarian leader. 

“Obviously, that looks horrible in the current context,” another student told the outlet. “Knowing Mount Vernon, we do things a little odd around here.”

The student added that the school is known to “try to think outside the box” but shared that “oftentimes that doesn’t work.”

Several former students told Fox 5 that those questions weren’t given to them during eighth grade.

While many parents and students were shaken over the assignment, one student believes the school attempted to pose a historically provocative question that required students to use their critical thinking skills. 

“I can definitely see why they’d be upset, but overall, I think it’s important to look at both sides of the coin in every situation, and I think it’s important to be able to compare and contrast everything that’s happened in our world history, whether it’s been good or bad,” said the student.

Upon learning the phrasing of the questions in the assignment, Mount Vernon officials said they had removed it from the school’s curriculum. 

The principal of Mount Vernon, Kristy Lundstrom, wrote in a statement that the assignment was “an exploration of World War II designed to boost student knowledge of factual events and understand the manipulation of fear leveraged by Adolf Hitler in connection to the Treaty of Versailles.” 

“Immediately following this incident, I met with the School’s Chief of Inclusion, Diversity, Equality, and Action, Head of Middle School, and a concerned Rabbi and friend of the School who shared the perspective of some of our families and supported us in a thorough review of the assignment and community impact.”

“Adolf Hitler and the events of the time period are difficult and traumatic to discuss.”

The private school, about 16 miles outside downtown Atlanta, is a “co-educational day school for more than 1200 students in Preschool through Grade 12,” according to the institution’s  website . 

“We are a school of inquiry, innovation, and impact. Grounded in Christian values, we prepare all students to be college ready, globally competitive, and engaged citizen leaders,” its mission statement reads.

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