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Gli studenti saranno in grado di comprendere il significato di derivata e integrale in riferimento al comportamento di una funzione.

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About this unit

The derivative of a function describes the function's instantaneous rate of change at a certain point. Another common interpretation is that the derivative gives us the slope of the line tangent to the function's graph at that point. Learn how we define the derivative using limits. Learn about a bunch of very useful rules (like the power, product, and quotient rules) that help us find derivatives quickly.

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Derivatives: definition and basic rules | Khan Academy

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The derivative of a function describes the function's instantaneous rate of change at a certain point. Another common interpretation is that the derivative gives us the slope of the line tangent to the function's graph at that point. Learn how we define the derivative using limits.

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Gli studenti saranno in grado di determinare concavità, punti critici e punti di flesso di una funzione da un grafico.

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Concavity, Inflection Points, Increasing Decreasing, First & Second Derivative - Calculus

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Come possiamo analizzare una funzione per comprenderne completamente il comportamento?

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Q&A for people studying math at any level and professionals in related fields

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Spiegazione dei concetti chiave di studio di funzione, inclusi derivata, integrale, dominio, intervalli di crescita/decrescita, concavità e asintoti.

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An Introduction to Functional Analysis by John Cagnol

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Esempi lavorati insieme in classe sull'analisi completa di una funzione data.

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Calculus I - Functions (Practice Problems)

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Here is a set of practice problems to accompany the Functions Section of the Review chapter of the notes for Paul Dawkins Calculus I course at Lamar University.

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Esercizi di gruppo per analizzare il comportamento della funzione, utilizzando strumenti tecnici o collaborativi come Padlet.

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From the instructionaldesign community on Reddit

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Explore this post and more from the instructionaldesign community

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Ideas for group data analysis activity?
by u/Anonymous-Anglerfish in instructionaldesign

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Esercizi per ogni studente riguardanti lo studio di funzione di equazioni differenti.

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Evaluate functions | Algebra (practice) | Khan Academy

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Materiali per studenti che richiedono più supporto: video spiegazioni e fogli di lavoro di pratica differenziata.

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Math classrooms are mosaics of strengths and experiences. When we have students with diverse backgrounds-with various languages, achievements, and interests-in the same space, everyone learns from each other and broadens their world.

On the flip side, though, teaching math to a broad array of students can be challenging. Do you struggle to reach all of your students? Are you a newer teacher who is looking to improve your practice? The strategies for differentiated instruction provided here might help you out.

What Is Differentiated Math Instruction?

Differentiated math instruction refers to the collection of techniques, strategies, and adaptations you can use to reach your diverse group of learners and make mathematics accessible to every single one. Dr. Timothy Kanold, former president of the National Council of Supervisors of Mathematics (NCSM)-and HMH author- clarifies that differentiation in a math lesson is "differentiation on the entry points into the task for support or the exit point to advance student thinking."

By applying various tools and strategies, such as incorporating technology, assigning hands-on projects, and teaching in math small-group formats, you can help every student meet expectations. We know that there are different schools of thought regarding what differentiation means. When we use the term, we are talking about providing student choice, voice, and agency. Differentiating instruction isn't meant to add more work to your day. Quite the opposite, in fact; it's meant as a teaching approach that will help you to reach more students in terms of accessibility and equity, making your job both easier and more effective in the long run.

Why Is Differentiating Math Instruction Important?

Some people think that math, more than any other subject, is the best fit for differentiation. Even though a 2018 survey by Texas Instruments found that 46% of kids said they really liked math, there are hundreds of books, websites, and memes discussing the difficulty of the subject. From the anxiety caused by there being only one correct answer to the cultural buy-in to the myth of being-or not being-a "math person" to the fear of solving a word problem, many students struggle with math. In addition, many students and educators alike find it hard to make the connection between math and the real world, which only increases disillusionment with the subject. That's why it's especially important to be open to new ways of providing instruction.

The National Council of Teachers of Mathematics (NCTM) promotes differentiating math instruction for differences in learning as well as differences in achievement, interest, and confidence. NCTM advises that the need is greater in middle and high school, as higher-level math relies on more complex reasoning. When you differentiate your math instruction, you support all learners by targeting and addressing specific needs of groups and individual students.

Examples of Differentiated Instruction in Math

Do you need ideas for how to differentiate your teaching to be sure your math students are progressing? Below are seven differentiation strategies for math instruction, along with ways that you can use them in your math classroom. They serve as examples of differentiated instruction in math and may work better for some classrooms and math topics than others. Customize these ideas however you need to serve you and your students.

Strategy 1: Math Centers

For this, you'll need to come up with a few activities your students can rotate through (be sure to browse our library of free activities and resources!), such as watching a video, reading an article, or solving a word problem. We spoke with Kristy McFarlane, an instructional supervisor at Sandshore Elementary School in New Jersey, about differentiation. She says math teachers at her school spend about 10 minutes on a mini-lesson for the whole class and then students spend about 15 minutes at various math centers. "They might meet with the teacher in a small group for extra help, use math software, do a game or project at the hands-on station, or do seat work based on the day's mini-lesson," she says.

Math centers are a powerful way to facilitate independent and small group learning within your classroom. Our program, for example, is known for embedding resources and instructional time to math centers. If a select group of your students are all struggling to, say, add fractions, they may benefit from an activity that has them practice finding least common denominators. Think about ways to customize the groupings and centers so they're perfect for your students' strengths, misconceptions, and interests, and make use of tools that strategically group students and recommend activities for you.

Strategy 2: Activity Cards

Choice is an important part of differentiation, and letting students decide how they want to spend their time is a great way to appeal to various learning preferences. You'll need to come up with math problems, tasks, or questions. As much as possible, use or create cards that span several lessons and offer options to work independently, with a partner, or in a small group. Ask for feedback so you can adjust future learning accordingly. Many of HMH's math programs, including , , and include inquiry-based task and project cards that help teachers differentiate.

Strategy 3: Choice Boards

As we just mentioned, giving students the ability to make decisions about their learning is an important part of differentiation. A choice board is a graphic organizer that gives students activities to choose from. There are different types of choice boards, but they need to focus on specific learning needs, interests, and skills. Choice boards increase student ownership; students pace themselves and get to decide how to engage with information, along with how to demonstrate their learning. Some teachers create different versions of the same choice board; others will color-code options to signify topic, activity type, or expected level of challenge. Check out the choice board we developed for remote learning. This board covers all subjects but also includes a free template to get you started on a math-only version.

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7 Strategies for Differentiated Math Instruction

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These strategies provide differentiated math instruction to K-12 teachers looking for ways to personalize their mathematics teaching.

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Gli studenti saranno in grado di identificare il dominio di una funzione e riconoscere l'esistenza di asintoti verticali e orizzontali.

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Horizontal and Vertical Asymptotes - Slant / Oblique - Holes - Rational Function - Domain & Range

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Problemi più complessi per rafforzare le capacità di pensiero critico e problem solving.

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Home

CHALLENGING MATH PROBLEMS WORTH SOLVING

DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL

WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS?

HERE'S OUR GROWING COLLECTION

WANT TO SHARE OPEN MIDDLE WITH OTHERS?

CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK

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CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS? HERE’S OUR GROWING COLLECTION Get Google Slide Versions WANT TO SHARE OPEN MIDDLE WITH OTHERS? CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK Elementary Version

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Osservazione delle discussioni in classe, revisioni di esercizi e correzioni durante la lezione.

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Formative assessment-discovering what students know while they're still in the process of learning it-can be tricky. Designing just the right assessment can feel high stakes-for teachers, not students-because we're using it to figure out what comes next. Are we ready to move on? Do our students need a different path into the concepts? Or, more likely, which students are ready to move on and which need a different path?

When it comes to figuring out what our students really know, we have to look at more than one kind of information. A single data point-no matter how well designed the quiz, presentation, or problem behind it-isn't enough information to help us plan the next step in our instruction.

Add to that the fact that different learning tasks are best measured in different ways, and we can see why we need a variety of formative assessment tools we can deploy quickly, seamlessly, and in a low-stakes way-all while not creating an unmanageable workload. That's why it's important to keep it simple: Formative assessments generally just need to be checked, not graded, as the point is to get a basic read on the progress of individuals, or the class as a whole.

7 Approaches to Formative Assessment

1. Entry and exit slips: Those marginal minutes at the beginning and end of class can provide some great opportunities to find out what kids remember. Start the class off with a quick question about the previous day's work while students are getting settled-you can ask differentiated questions written out on chart paper or projected on the board, for example.

A quick way to see the big picture if you use paper exit tickets is to sort the papers into three piles: Students got the point; they sort of got it; and they didn't get it. The size of the stacks is your clue about what to do next.

No matter the tool, the key to keeping students engaged in the process of just-walked-in or almost-out-the-door formative assessment is the questions. Ask students to write for one minute on the most meaningful thing they learned. You can try prompts like:

What are three things you learned, two things you're still curious about, and one thing you don't understand?

How would you have done things differently today, if you had the choice?

What I found interesting about this work was...

Right now I'm feeling...

Today was hard because...

Or skip the words completely and have students draw or circle emojis to represent their assessment of their understanding.

2. Low-stakes quizzes and polls: If you want to find out whether your students really know as much as you think they know, polls and quizzes created with Socrative or Quizlet or in-class games and tools like Quizalize, Kahoot, FlipQuiz, Gimkit, Plickers, and Flippity can help you get a better sense of how much they really understand. (Grading quizzes but assigning low point values is a great way to make sure students really try: The quizzes matter, but an individual low score can't kill a student's grade.) Kids in many classes are always logged in to these tools, so formative assessments can be done very quickly. Teachers can see each kid's response, and determine both individually and in aggregate how students are doing.

Because you can design the questions yourself, you determine the level of complexity. Ask questions at the bottom of Bloom's taxonomy and you'll get insight into what facts, vocabulary terms, or processes kids remember. Ask more complicated questions ("What advice do you think Katniss Everdeen would offer Scout Finch if the two of them were talking at the end of chapter 3?"), and you'll get more sophisticated insights.

3. Dipsticks: So-called alternative formative assessments are meant to be as easy and quick as checking the oil in your car, so they're sometimes referred to as dipsticks. These can be things like asking students to:

write a letter explaining a key idea to a friend,

draw a sketch to visually represent new knowledge, or

do a think, pair, share exercise with a partner.

Your own observations of students at work in class can provide valuable data as well, but they can be tricky to keep track of. Taking quick notes on a tablet or smartphone, or using a copy of your roster, is one approach. A focused observation form is more formal and can help you narrow your note-taking focus as you watch students work.

4. Interview assessments: If you want to dig a little deeper into students' understanding of content, try discussion-based assessment methods. Casual chats with students in the classroom can help them feel at ease even as you get a sense of what they know, and you may find that five-minute interview assessments work really well. Five minutes per student would take quite a bit of time, but you don't have to talk to every student about every project or lesson.

You can also shift some of this work to students using a peer-feedback process called TAG feedback (Tell your peer something they did well, Ask a thoughtful question, Give a positive suggestion). When you have students share the feedback they have for a peer, you gain insight into both students' learning.

For more introverted students-or for more private assessments-use Flipgrid, Explain Everything, or Seesaw to have students record their answers to prompts and demonstrate what they can do.

5. Methods that incorporate art: Consider using visual art or photography or videography as an assessment tool. Whether students draw, create a collage, or sculpt, you may find that the assessment helps them synthesize their learning. Or think beyond the visual and have kids act out their understanding of the content. They can create a dance to model cell mitosis or act out stories like Ernest Hemingway's "Hills Like White Elephants" to explore the subtext.

6. Misconceptions and errors: Sometimes it's helpful to see if students understand why something is incorrect or why a concept is hard. Ask students to explain the " muddiest point" in the lesson-the place where things got confusing or particularly difficult or where they still lack clarity. Or do a misconception check: Present students with a common misunderstanding and ask them to apply previous knowledge to correct the mistake, or ask them to decide if a statement contains any mistakes at all, and then discuss their answers.

You can use sticky notes to get a quick insight into what areas your kids think they need to work on. Ask them to pick their own trouble spot from three or four areas where you think the class as a whole needs work, and write those areas in separate columns on a whiteboard. Have you students answer on a sticky note and then put the note in the correct column-you can see the results at a glance.

Several self-assessments let the teacher see what every kid thinks very quickly. For example, you can use colored stacking cups that allow kids to flag that they're all set (green cup), working through some confusion (yellow), or really confused and in need of help (red).

Similar strategies involve using participation cards for discussions (each student has three cards-"I agree," "I disagree," and "I don't know how to respond") and thumbs-up responses (instead of raising a hand, students hold a fist at their belly and put their thumb up when they're ready to contribute). Students can instead use six hand gestures to silently signal that they agree, disagree, have something to add, and more. All of these strategies give teachers an unobtrusive way to see what students are thinking.

No matter which tools you select, make time to do your own reflection to ensure that you're only assessing the content and not getting lost in the assessment fog. If a tool is too complicated, is not reliable or accessible, or takes up a disproportionate amount of time, it's OK to put it aside and try something different.

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7 Smart, Fast Ways to Do Formative Assessment

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Within these methods you'll find close to 40 tools and tricks for finding out what your students know while they're still learning.

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Un quiz con domande aperte e a scelta multipla sullo studio di funzione.

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function analysis 12th Grade Quiz | Quizizz

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function analysis quiz for 12th grade students. Find other quizzes for Mathematics and more on Quizizz for free!

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Ogni studente condivide in breve un aspetto chiave dello studio di funzione che ha trovato utile o interessante.

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Ogni studente deve descrivere un concetto chiave dello studio di funzione e un'applicazione pratica in una frase.

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Math Exit Tickets: What are they and 10 ideas you can use in your classroom today

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Discover 10 functional ‘Math Exit Tickets’ ideas for your classroom on our Knowledgehook blog!

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Foglio di lavoro con esercizi aggiuntivi sullo studio di funzione.

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Math Exercises & Math Problems: Analysis of Functions, Properties of Functions

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Math exercises on analysis of functions & properties of functions. Use the analysis of functions to find the properties of functions at Math-Exercises.com.

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Breve ricerca o relazione che mostra come l'analisi delle funzioni viene utilizzata in un campo della vita reale, come l'ingegneria o l'economia.

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What are the applications of functional analysis?

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I recently had a course on functional analysis. I was thinking of studying the mathematical applications of functional analysis. I came to know it had some applications on calculus of variations. I…