The Reading Level provides a supplemental measurement that considers the individual child. With this information, parents and teachers are able to help choose well-fitting books that give the reader just the right amount of challenge.
How to Write a Book Report (Middle and High School level)
Book reports and book reviews are similar. Book reports tend to be a little more descriptive (What is this book about?) and book reviews are usually more persuasive (Why a reader should or shouldn’t read this book). Both offer a combination of summary and commentary.
They are a way to think more deeply about a book you’ve read and to demonstrate your understanding.
Here you want to provide basic information about the book, and a sense of what your report will be about. You should include:
1. Title (underlined)/Author
2. Publication Information: Publisher, year, number of pages
4. A brief (1-2 sentences) introduction to the book and the report/review.
There are two main sections for this part. The first is an explanation of what the book is about. The second is your opinions about the book and how successful it is. There are some differences between reports on fiction or other imaginative writing and reports on non-fiction books.
But for both, a good place to start is to explain the author’s purpose and/or the main themes of the book. Then you can summarize.
For fiction or other creative writing:
Provide brief descriptions of the setting, the point of view (who tells the story), the protagonist , and other major characters. If there is a distinct mood or tone, discuss that as well.
Give a concise plot summary. Along with the sequence of major events, you may want to discuss the book’s climax and resolution, and/or literary devices such as foreshadowing. But, if you are writing a review, be careful not to give away important plot details or the ending.
Provide a general overview of the author’s topic, main points, and argument. What is the thesis? What are the important conclusions?
Don’t try to summarize each chapter or every angle. Choose the ones that are most significant and interesting to you.
Analysis and Evaluation
In this section you analyze or critique the book. You can write about your own opinions; just be sure that you explain and support them with examples. Some questions you might want to consider:
• Did the author achieve his or her purpose?
• Is the writing effective, powerful, difficult, beautiful?
• What are the strengths and weaknesss of the book?
• For non-fiction, what are the author’s qualifications to write about the subject? Do you
agree with the author’s arguments and conclusions?
• What is your overall response to the book? Did you find it interesting, moving, dull?
• Would you recommend it to others? Why or why not?
Briefly conclude by pulling your thoughts together. You may want to say what impression the book left you with, or emphasize what you want your reader to know about it.
Writing a book report can be a lot of fun. It gives you a chance to read a new book and then tell your teacher and friends what you thought about it.
How to Write a Book Report
Here are some of the things you need to include in your book report:
Where did the story take place? Was it in a city or on a farm? Was it a made-up place or somewhere in outer space? Give a good description of the place with as much detail as possible.
Who was the story about? Was there just one main character or were there a few? When you write about the characters, include their names and what they look like.
What happened in the book? Was there a problem the characters were trying to solve? Were the characters on some sort of an adventure? Describe what happened in the beginning, the middle and the end of the book.
Did you like the book? Write a little bit about why you liked or didn’t like the book. Talk about how the book made you feel – happy, sad, excited. Would you tell your friends to read this book?
When you’re finished writing the report, read it over carefully to make sure everything is spelled correctly. You can ask a grown up to help look for spelling mistakes too.
Below are a number of resources for parents to further supplement lessons with online practice questions, videos, printable worksheets, and fun games to play at home.
Extra Math Practice
Unlike traditional workbooks and exercises, IXL offers hours of intrigue for your child. If you purchase a membership, you can also set goals, track your child’s progress, and monitor the usage information.
This site features an extensive library of instructional videos and practice exercises covering a variety of subjects. Each problem is randomly generated, so your child will never run out of practice material.
You design your own games for your child!
Make the Sun rise and set with this interactive clock while learning to tell time.
This site offers interactive lessons and games for your child from kindergarten through the eighth grade.
Simple Ways to Help Your Child with Math
- Encourage your child to separate toys or other objects into groups. For example, you could ask your child to separate blue toy cars from red cars.
- Weigh and measure your child, and tell him/her the results. Talk about how some things are taller or shorter and some things are lighter or heavier than your child is. Ask your child to find things in your home that are both tall and heavy or short and light.
- Talk to your child about numbers. Teachers would like children to be able to recognize the numbers 1 to 10 and to be able to count to 20 when they start school.
- Ask your child to explain what he/she learned in math class today. Letting children take the teacher role gives them the chance to practice new skills and to clarify their thinking on a lesson.
- Talk to your child about how adults use math in their everyday lives–grocery shopping, budgeting, balancing a checkbook, and checking clothing sizes, for example.
- Schedule a time and provide a quiet environment for your child to work only on math homework.
- Cook up some calculations. Get your child to help you measure ingredients while you cook. Ask the child how he/she would convert a recipe or 4 into a dish for 2 or a banquet for 20.
- Discuss unfamiliar words in your child’s math homework. Don’t be afraid to admit you don’t know or don’t remember some of the definitions. Look them up together.
- Teach your child math by teaching him/her about money. It is recommended that children between the ages of 11 and 13 be able to set up a savings plan and a savings account. They also should understand the importance of giving to worthy causes, and they should know how to shop wisely.
- If you have a daughter, you may need to work extra hard to keep her interested in math. Studies show that young girls enjoy math, science, and technology as much as boys do, but by the time they become eighth graders, twice as many boys as girls say they’re interested in math, science, and engineering careers.
- Encourage your teen to read the business section of the newspaper. Articles in this section often have a lot of numbers in them.
- Continually show an interest in your teen’s math education. Every few weeks, you might want to ask, “What are you learning now?” Ask your teen to explain the lesson to you if it’s not a math concept you already know.
- Point out that even if your teen does not plan to pursue a career in which math is important, learning math is still important because it teaches how to think in a disciplined way.
- Be positive about math. If math was a challenge for you, be careful how you express that to your child. Parent’s often say ‘I was never any good at math.’ Unfortunately, the children may begin to believe that they inherited the same inability. Perseverance is more important than heredity.
Family Math Fun
- Play math-related games with your child. Dominoes, Yahtzee, Uno, Monopoly, and many other games require math skills. Point this out to your child as you play, and talk about the ways that people use math every day.
- Family projects, such as remodeling a room or hanging wall paper, can teach your child an incredible amount of math.
- For children interested in sports statistics, you might want to show them how to set up a graph on paper or a spreadsheet on a computer that they can use to track the numbers.
- Let your child help you map a family trip. Ask him/her to help calculate the number of miles between stops.
- Play math games in the car when your family travels. Ask everyone to look at signs and billboards in order to find a series of numbers in order. For example, a sign with a 1 on it gives you the number 1. Then look for a different sign with a 2 on it. See how high you can go.
Having to speak in front of an audience scares most people, but if your children learn crucial skills in their childhood, they can avoid being ever anxious about speaking in public. There are many other benefits of teaching children how to speak in front of people. It helps to build their communication skills and confidence. They learn how to capture the audience’s attention, develop charisma, and write their own speech. They also discover their own potential.
Learn by practice:
Find interesting examples:
Use favorite subject:
To develop important writing skills, it is important for children to start early. Frustration or lack of interest may shut them down to learning in the future. This is where parents can help at home. Follow these tips to help your children build writing skills that they will use throughout their educational career.
Provide a place for your child to write:
The writing area should be quiet and well lit. Stock this area with supplies such as paper, pencils, and crayons. You can also gather family photos and magazines and place them in the area so they can be used as story starters.
Read, read, read:
The best activity to improve writing is reading. If your child reads good books, he/she will be a better writer. Reading exposes students to general vocabulary, word study, and content-specific vocabulary. Through reading, students see a variety of authors’ techniques that they can use in their own writing.
Provide authentic writing opportunities for your child:
Look for opportunities for purposeful writing at home, and encourage your child to read and write letters to family, grocery lists, messages, postcards, thank-you notes, and party invitations.
Be a writing role model:
Make sure that your child sees you reading and writing. It is recommended that you also write when your child writes. You can schedule a day of the week that you will turn off the television and/or other distractions and share your writing.
Always ask your child questions when he/she writes. Ask specific questions about your child’s writing: How did that happen? How did that make you feel? Can you tell me more about …? What are some other words you could use to describe …?
Encourage your child to embrace the concept of revision:
Sit down with your child and read through his/her writing together. Make your child circle grammatical mistakes. Before making corrections, have your child tell you what he/she should have done differently.
It is helpful to point out errors now and then, but if your child thinks you always look for what’s wrong, he/she will not want to share his/her writing with you.
Publish your child’s writing:
Share your child writing with others, place it on the refrigerator, or encourage your child to write for kids’ magazines. When your child’s writing is published in a children’s book, he/she will be on the way to becoming a lifelong writer and author. (Refer to Stone Soup for publishing children’s work.)
Grade K~1 (Leveled Reading Phase)
- Help the student to read aloud once a day for a minimum of three days.
- Help the student read by pointing to each word with his/her finger.
- What if the student encounters an unfamiliar word? Ask him or her to guess the word from a picture that illustrates it or gives a hint about its meaning. Also, have the student read the word by identifying the beginning sound of the word and by using other phonics methods.
- Ask the student to talk about the story. Do not ask the student to identify the main idea unless he/she is in first grade or above. Praise the student for any content that he/she remembers correctly, even if he/she may not remember everything from the book.
- Ask the student which part of the book was the most interesting.
- Have the student write a book report. Instead of requiring them to write a detailed summary, have them write about the main character and what the main character does. In the beginning, it is important to allow the student to use sentences directly from the book and to refrain from correcting spelling errors too vigorously.
Grade 2~3 (Independent Reading Phase)
- Help the student to read silently once a day for a minimum of three days.
- Picture books can help the student develop an active imagination.
- The time from second to third grade is a crucial period during which reading habits are established. It is important to have the student create a reading log so that he/she can feel enjoyment and a sense of accomplishment.
- What if the student encounters an unfamiliar word? The student should try to guess its meaning before consulting the dictionary, especially if the word is a verb.
- Ask the student to talk about the story.
- Ask the student which part of the book was the most interesting.
- Ask the student about the characters and setting.
- Ask the student what he/she learned from the book.
- Have the student write a book report. By second grade, most students can understand the physical setting and time in which the story takes place. In addition, they are able to summarize the contents of the book using their own sentences instead of sentences taken directly from the book.
Grade 4 & Above (Targeted Reading Phase)
- Chapter book reading
- Choosing a favorite author
- Recognizing the book’s structure and themes
- Approaching the story from a literary perspective: understanding genre, point of view, story structure, mood, and symbolism
- Informative reading about diverse topics in society, science, history, and current events (Reading about themes that are directly related to their school studies can help students improve their performance in school.)
- Reading the classics
- Character Study – Have the student interpret the characters’ personalities and actions and write about them.
- Have the student write a summary of each chapter and a summary of the entire book.
- Have the student analyze and compare various books by the same author.
- Have the student write a book report. The student must be able to express his/her personal views about the conflicts that occur in the story and the characters who try to resolve them. The student should also be able to summarize the contents of nonfiction books in addition to those of storybooks.
Why Three Times?
Must Students Read Aloud?
Guide to Public Forum Debate
Public Forum Debate (PFD) is a team event that advocates or rejects a position posed by the monthly resolution topic (announced online at www.nflonline.org). The clash of ideas must be communicated in a manner persuasive to the non-specialist or “citizen judge”, i.e. a member of the American jury. The debate should:
Display solid logic, lucid reasoning, and depth of analysis
Utilize evidence without being driven by it
Present a clash of ideas by countering/refuting arguments of the opposing team (rebuttal)
Communicate ideas with clarity, organization, eloquence, and professional decorum
The Topic ~ Topics are worded as resolutions, meaning they advocate solving a problem by establishing a position. Teams must understand the meaning of terminology in a consistent manner so debates have a clash of ideas. If the topic were “Resolved: Free trade benefits all nations,” it would be vital to understand the concept of free trade. An expert definition from an economics or legal dictionary or encyclopedia would be preferable to a standard dictionary. If the topic, “Resolved: NATO countries should act together on international matters,” the more common terms ‘act’ and ‘together’ could be appropriately defined by a standard dictionary. Given the limited time of a round, debate should not center on obscure claims of minutia.
Case Development & Evidence
A team must develop both a pro and con case, persuasively supported by evidence and reasoning. Given the short nature of a Public Forum round, cases should center on a few quality arguments. A team, however, should research several arguments on both sides of the issue, so it can adapt its case to the opposing team’s claims as necessary. Having arguments in direct contradiction with each other will enhance clash in rebuttals. Organization of speeches through effective communication and clear outlines is important so both judges and the opposing team can follow each of the arguments and their supporting evidence. Effective persuasion requires credible, unbiased, quality supporting evidence, which may include a mix of facts, statistics, expert quotations, studies, polls; but it may also be real-life examples, anecdotes, analogies, and personal experience. Since topics are based on current events, research should be accessible through periodicals, Web search engines and think tanks. Teams should not overwhelm their case with evidence; rather, they should select the best evidence to represent their claims.
The Coin Flip ~ The round starts with a
coin toss; the winning team selects either:
The side (pro or con) they will argue ! The speaker order (begin the debate or give the last speech).
The team that loses the toss will then decide their preference from the option
not selected by the winner (i.e., if the winning team decides to speak last, then the losing team may decide which side they will argue). The debate, therefore may begin with the con side, arguing against the topic. Teams might consider: Is one side of the topic more acceptable to citizen judges? On which side is the team stronger? On which side of the topic are the opponents stronger? Is the first speaker position critical to “sell” the case by making a good first impression? Is the final focus speech critical for the last word to the judge(s)? Are the opponents so effective in either the first or last speaker position that our team needs to select speaker position rather than side? The first team sits to the judge’s left.
Speeches and Time Limits
Speaker 1 (Team A, 1st speaker )…………………….4 min.
Speaker 2 (Team B, 1st speaker)………………………4 min.
Crossfire (between speakers 1 & 2)……………..3 min.
Speaker 3 (Team A, 2nd speaker ) …………………..4 min.
Speaker 4 (Team B, 2nd speaker )……………………4 min.
Crossfire (between speakers 3 & 4)……………..3 min.
Speaker 1 Summary…………………………………………..2 min.
Speaker 2 Summary…………………………………………..2 min.
Grand Crossfire (all speakers) ……………………3 min. Speaker 3
Final Focus………………………………………..2 min.
Speaker 4 Final Focus………………………………………..2 min.
Each team may use up to two minutes of prep time.
First Pro Speech ~ This speech constructs arguments advocating the resolution’s worthiness. The key analysis will be to present major reasons why there is a problem. An underlying concept will always be the risk of change versus the risk of not changing. This speech should have a brief introduction to frame the team’s case for the judge. If a definition is important to understanding the case, it should be presented from the most appropriate source. A few reasons for adopting the topic should be presented with accompanying evidence. Each reason should be an independent reason to vote for the resolution, and should explain why it is pertinent. The speech should conclude with a summary of the arguments covered.
First Con Speech ~ This speech constructs arguments showing disadvantages of the resolution and why it should not be adopted. If the pro speech has the advantage of a changing future, the con speech has a track record of experience (status quo) and why change is ill-advised The rest of the speech elements will be the same as the pro speech.
Strategies for the Second Team ~ If the team feels that the opponent’s case is based on a faulty or unfair interpretation of the resolution, they should provide counter definitions and convincingly explain why their perspective is more appropriate. Whichever side speaks second may also choose to drop a reason from the prepared speech and spend time instead refuting claims presented by the other team. This strategy should be employed when one of the arguments directly clashes with the other team’s or when the team believes one of the opponent’s arguments is based on a false definition or assumption.
Third & Fourth Constructive Speeches
Both of these debaters have the primary burden of refuting the other team’s arguments by analyzing and explaining flaws in the opponent’s position. The debater should identify the opposition’s key arguments and attack their legitimacy by: turning the analysis to the other side; presenting evidence that destroys or reduces the opposing position; presenting alternate causes that are not accounted for by the opposition argument; exposing argument inconsistencies between the speakers or between the opponents and their statements during crossfire. To best accomplish refutation, both members of a team should have a consistent approach and a unified view of what is important and less important. An argument format could be an introduction that links the team’s second speech to the first speech, followed by an overview of the issue, which is frequently the opponent’s argument, followed by reasons/evidence why the opponent is wrong, followed by what this argument clash now means for your side in the debate. In addition, some time in either of these speeches should be allocated to rebuilding the original case. It is important to have clarity that is seldom attained by an intricate outline. Speeches should conclude with a summary.
Summary Speeches ~ These are complicated speeches because each debater has to find a way to explain issues in the light of all that has happened so far – in just two minutes – without speaking too rapidly. New evidence, but not new arguments may be presented, except responses (refutation). This means that a limited number of issues can be addressed. For example, perhaps develop one to two issues from the debater’s side on the resolution and one from the opponent’s side of the
resolution. The speech should have a brief overview. On each key argument, try to add a short original quotation, anecdote, or fact. Wrap up each argument by stressing its importance in arriving at a fair decision.
The Final Focus ~ This frames, with clarity, why your team has won the debate. Again, no new arguments may be presented, however, new evidence may be introduced to support an argument made earlier in the debate. Before the final focus, ask, “If I were judging this round, what would I be voting on?” Strategies may include: ! Choose the most important argument you are winning, and summarize the analysis and evidence that make it so important.
- Turn a major argument from your opponent into the winning analysis and evidence of one of your important arguments; this technique clinches two arguments.
- Answer the most important argument you may be losing by summarizing the analysis and evidence that you believe takes out the opponent’s argument.
- Choose an argument that you believe the community judge will most likely vote on.
Expose a major inconsistency made by your opponent—two arguments that contradict each other—at least one of which the opponent is focusing on to win the debate.
- Art of Argumentation
The quantity of arguments is less important than the quality of arguments, just as the quantity of
evidence is less important than the quality of evidence. Thus we come to
three important components of an argument: claim, evidence, and warrant. A claim is a major argument made on either side of the resolution. On the resolution, “Resolved that NATO countries should have acted together in Iraq,” a claim could be that animosities would be reduced because one nation would not bear the brunt of the responsibility for the invasion. To prove this to be true, a debate must provide evidence, proving that the claim is valid. The debater chooses at least one type of evidence that will support the claim even when challenged. In the above example, much credible evidence exists that resistance is high because the United States for the most part acted alone. Perhaps the most crucial component of argumentation is the warrant. Warrants connect the claim and its support, sometime obviously, sometime subtly. Warrants emerge from the total sum of our experiences and personal observations. Thus it is entirely possible that the debater and the judge have a different set of experiences. The warrant for the claim used in the NATO example should connect the judge to the thesis, perhaps by making anecdotal comments about how everyone is much better satisfied when cooperation exists, whether among people or nations. On the other hand, the opposing team can counter that forcing nations to cooperate with each other when that is not their wish alienates allies and ruins alliances. Turn the evidence against the team and make the logical warrant that such a NATO policy for Iraq would have destroyed NATO, would have kept us operating in Iraq by ourselves, and would have destroyed the unity for future NATO missions. Warrants provide believable reasons why a claim and evidence are true. That is why evidence without analysis can result in an assertion without substance and an argument lost. Arguments and evidence without warrants are seldom persuasive.
Guide to Public Forum Debate © 2009 • National Forensic League
Crossfire ~ Questioning periods give debate interactivity and a change to build clash. In crossfire, both debaters have equal access to the floor, but the first question must be asked to the debater who just finished speaking by a debater from the other team. After the initial question and answer, either debater may question or answer. A debater who attempts to dominate or be rude to his opponent will lose points.
Good questions are brief and good answers must meet the question. In the first two crossfires, only the corresponding speakers may participate, and they stand next to each other.
Grand Crossfire ~ Seated, all debaters interact with one another. The first question is asked to the team that just ended its summary by the other team. After the initial question and answer, any debater may question or answer, and all should participate. The same guidelines for rudeness and stalling apply to the grand crossfire. Resist rushing questions or answers, or trying to do too much in crossfire; desperation is not persuasive.
Prep Time ~ Each team has two minutes of prep time. For very practical reasons, a team should not use prep time until their summary speech or final focus speech. Being prepared on the arguments is the best way to avoid using prep time until it is vital to select the key arguments and issues.
Delivery ~ Effective delivery is critical to impact the arguments for a citizen judge. Practice delivery in front of ordinary people: teachers, parents, relatives, friends, non- debate classmates. Heed their advice. If they tell you to slow down, slow down; if they tell you to quit repeating yourself, start your sentences with the subject and avoid compound complex sentences; if they tell you to enunciate more clearly, practice with a pencil in your mouth; if they tell you to look up, make sure you remember everything about the person to whom you are talking; if they tell you to speak with variety, practice emphasizing key words, especially action verbs; if they tell you to speak louder, practice with cotton in your ears. In other words, do everything before a debate to cultivate a good delivery.
Working Knowledge ~ The more a debater knows about a topic, both arguments and evidence, both pro and con, the more one will be able to practice delivery and hence become truly skilled in the communication of arguments, evidence and analysis.
Evaluation & Judging ~ The judge is the chairperson of the round (facilitating the coin flip and giving time signals if requested), and may halt any crossfire lacking civility. S/he may not interact in the crossfire.
Judges evaluate teams on the quality of the arguments actually made, not on their own personal beliefs, and not on issues they think a particular side should have covered. Judges should assess the bearing of each argument on the truth or falsehood of the assigned resolution. The pro should prove that the resolution is true, and the con should prove that the resolution in not true. When deciding the round, judges should ask, “If I had no prior beliefs about this resolution, would the round as a whole have made me more likely to believe the resolution was true or not true?” Teams should strive to provide a straightforward perspective on the resolution; judges should discount unfair, obscure interpretations that only serve to confuse the opposing team. Plans (formalized, comprehensive proposals for implementation), counterplans and kritiks (off-topic arguments) are not allowed. Generalized, practical solutions should support a position of advocacy.
Quality, well-explained arguments should trump a mere quantity thereof. Debaters should use quoted evidence to support their claims, and well-chosen, relevant evidence may strengthen – but not replace – arguments.
Clear communication is a major consideration. Judges weigh arguments only to the extent that they are clearly explained, and they will discount arguments that are too fast, too garbled, or too jargon-laden to be understood by an intelligent high school student or a well-informed citizen. A team should not be penalized for failing to understand his or her opponent’s unclear arguments.
In short, Public Forum Debate stresses that speakers must appeal to the widest possible audience through sound reasoning, succinct organization, credible evidence, and clear delivery. Points provide a mechanism for evaluating the relative “quality of debating.”
This guide may be freely reproduced for instructional purposes. Revised July 2009 • NFL Programs/Education Office
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM (National Council of Teachers of Mathematics) process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
Standards for Mathematical Practices
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is.
Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.
Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Standards for Mathematical Content
The Standards for Mathematics are described in grade bands, and they are almost identical in domain, but are increasingly difficult as for upper grades and are expanded in coverage.
In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and operating on whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.
In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and composing and decomposing geometric shapes.
In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.
In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.
In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.
In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
To build a foundation for college and career readiness, students must read widely and deeply from among a broad range of high-quality, increasingly challenging literary and informational texts. Through extensive reading of stories, dramas, poems, and myths from diverse cultures and different time periods, students gain literary and cultural knowledge as well as familiarity with various text structures and elements. By reading texts in history/social studies, science, and other disciplines, students build a foundation of knowledge in these fields that will also give them the background to be better readers in all content areas. Students can only gain this foundation when the curriculum is intentionally and coherently structured to develop rich content knowledge within and across grades. Students also acquire the habits of reading independently and closely, which are essential to their future success.
They must also be able to determine or clarify the meaning of grade-appropriate words encountered through listening, reading, and media use; come to appreciate that words have nonliteral meanings, shadings of meaning, and relationships to other words; and expand their vocabulary in the course of studying content.
Through reading a diverse array of classic and contemporary literature as well as challenging informational texts in a range of subjects, students are expected to build knowledge, gain insights, explore possibilities, and broaden their perspective.
The standards establish a “staircase” of increasing complexity in what students must be able to read so that all students are ready for the demands of college- and career-level reading no later than the end of high school. The standards also require the progressive development of reading comprehension so that students advancing through the grades are able to gain more from whatever they read.
The standards mandate certain critical types of content for all students, including classic myths and stories from around the world, foundational U.S. documents, seminal works of American literature, and the writings of Shakespeare.
Whatever they are reading, students must also show a steadily growing ability to discern more from and make fuller use of text, including making an increasing number of connections among ideas and between texts, considering a wider range of textual evidence, and becoming more sensitive to inconsistencies, ambiguities, and poor reasoning in texts.
Key Ideas and Details
Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text.
Determine central ideas or themes of a text and analyze their development; summarize the key supporting details and ideas.
Analyze how and why individuals, events, and ideas develop and interact over the course of a text.
Craft and Structure
Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone.
Analyze the structure of texts, including how specific sentences, paragraphs, and larger portions of the text (e.g., a section, chapter, scene, stanza) relate to each other and the whole.
Assess how point of view or purpose shapes the content and style of a text.
Integration of Knowledge and Ideas
Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words.
Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence.
Analyze how two or more texts address similar themes or topics in order to build knowledge or to compare the approaches the authors take.
Range of Reading and Level of Text Complexity
Read and comprehend complex literary and informational texts independently and proficiently.
The standards expect that students will grow their vocabularies through a mix of conversations, direct instruction, and reading. The standards will help students determine word meanings, appreciate the nuances of words, and steadily expand their repertoire of words and phrases. The use of vocabulary extends across reading, writing, speaking, and listening. The vocabulary standards focus on understanding words and phrases, their relationships, and their nuances and on acquiring new vocabulary, particularly general academic and domain-specific words and phrases.
Determine or clarify the meaning of unknown and multiple-meaning words and phrases by using context clues, analyzing meaningful word parts, and consulting general and specialized reference materials, as appropriate.
Demonstrate understanding of figurative language, word relationships, and nuances in word meanings.
Acquire and use accurately a range of general academic and domain-specific words and phrases sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when encountering an unknown term important to comprehension or expression.