1.7: Summary of Key Concepts - Mathematics

Number / Numeral
A number is a concept. A numeral is a symbol that represents a number. It is customary not to distinguish between the two (but we should remain aware of the difference).

Hindu-Arabic Numeration System
In our society, we use the Hindu-Arabic numeration system. It was invented by the Hindus shortly before the third century and popularized by the Arabs about a thousand years later.

The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called digits.

Base Ten Positional System
The Hindu-Arabic numeration system is a positional number system with base ten. Each position has value that is ten times the value of the position to its right.

Commas / Periods
Commas are used to separate digits into groups of three. Each group of three is called a period. Each period has a name. From right to left, they are ones, thou­sands, millions, billions, etc.

Whole Numbers
A whole number is any number that is formed using only the digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Number Line
The number line allows us to visually display the whole numbers.

Graphing a whole number is a term used for visually displaying the whole number. The graph of 4 appears below.

Reading Whole Numbers
To express a whole number as a verbal phrase:

  1. Begin at the right and, working right to left, separate the number into distinct periods by inserting commas every three digits.
  2. Begin at the left, and read each period individually.

Writing Whole Numbers
To rename a number that is expressed in words to a number expressed in digits:

  1. Notice that a number expressed as a verbal phrase will have its periods set off by commas.
  2. Start at the beginning of the sentence, and write each period of numbers individ­ually.
  3. Use commas to separate periods, and combine the periods to form one number.

Rounding is the process of approximating the number of a group of objects by mentally "seeing" the collection as occurring in groups of tens, hundreds, thou­sands, etc.

Addition is the process of combining two or more objects (real or intuitive) to form a new, third object, the total, or sum.

Addends / Sum
In addition, the numbers being added are called addends and the result, or total, the sum.

Subtraction is the process of determining the remainder when part of the total is removed.

Minuend / Subtrahend Difference

Commutative Property of Addition
If two whole numbers are added in either of two orders, the sum will not change.
3 + 5 = 5 + 3

Associative Property of Addition
If three whole numbers are to be added, the sum will be the same if the first two are added and that sum is then added to the third, or if the second two are added and the first is added to that sum.
(3 + 5) + 2 = 3 + (5 + 2)

Parentheses in Addition
Parentheses in addition indicate which numbers are to be added first.

Additive Identity
The whole number 0 is called the additive identity since, when it is added to any particular whole number, the sum is identical to that whole number.
0 + 7 = 7
7 + 0 = 7

Key Shifts in Mathematics

The Common Core State Standards for Mathematics build on the best of existing standards and reflect the skills and knowledge students will need to succeed in college, career, and life. Understanding how the standards differ from previous standards—and the necessary shifts they call for—is essential to implementing them.

The following are the key shifts called for by the Common Core:

Greater focus on fewer topics

The Common Core calls for greater focus in mathematics. Rather than racing to cover many topics in a mile-wide, inch-deep curriculum, the standards ask math teachers to significantly narrow and deepen the way time and energy are spent in the classroom. This means focusing deeply on the major work of each grade as follows:

  • In grades K–2: Concepts, skills, and problem solving related to addition and subtraction
  • In grades 3–5: Concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions
  • In grade 6: Ratios and proportional relationships, and early algebraic expressions and equations
  • In grade 7: Ratios and proportional relationships, and arithmetic of rational numbers
  • In grade 8: Linear algebra and linear functions

This focus will help students gain strong foundations, including a solid understanding of concepts, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the classroom.

Coherence: Linking topics and thinking across grades

Mathematics is not a list of disconnected topics, tricks, or mnemonics it is a coherent body of knowledge made up of interconnected concepts. Therefore, the standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. For example, in 4 th grade, students must “apply and extend previous understandings of multiplication to multiply a fraction by a whole number” (Standard 4.NF.4). This extends to 5 th grade, when students are expected to build on that skill to “apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction” (Standard 5.NF.4). Each standard is not a new event, but an extension of previous learning.

Coherence is also built into the standards in how they reinforce a major topic in a grade by utilizing supporting, complementary topics. For example, instead of presenting the topic of data displays as an end in itself, the topic is used to support grade-level word problems in which students apply mathematical skills to solve problems.

Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity

Rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades. To help students meet the standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skills and fluency, and application.

Conceptual understanding: The standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives in order to see math as more than a set of mnemonics or discrete procedures.

Procedural skills and fluency: The standards call for speed and accuracy in calculation. Students must practice core functions, such as single-digit multiplication, in order to have access to more complex concepts and procedures. Fluency must be addressed in the classroom or through supporting materials, as some students might require more practice than others.

Application: The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.

Summary of the 2010 NAEYC Standards for Initial Early Childhood Professional Preparation Programs

These 2010 Initial Standards are used in NAEYC Accreditation of associate, baccalaureate, and master's degree programs providing degree candidates with their first experience and/or credential in early childhood studies.

Download a printable version of the Initial Standards summary here.


Candidates prepared in early childhood degree programs are grounded in a child development knowledge base. They use their understanding of young children’s characteristics and needs, and of multiple interacting influences on children’s development and learning, to create environments that are healthy, respectful, supportive, and challenging for each child.

Key elements of Standard 1

  • 1a: Knowing and understanding young children’s characteristics and needs, from birth through age 8.
  • 1b: Knowing and understanding the multiple influences on early development and learning
  • 1c: Using developmental knowledge to create healthy, respectful, supportive, and challenging learning environments for young children


Candidates prepared in early childhood degree programs understand that successful early childhood education depends upon partnerships with children’s families and communities. They know about, understand, and value the importance and complex characteristics of children’s families and communities. They use this understanding to create respectful, reciprocal relationships that support and empower families, and to involve all families in their children’s development and learning.

Key elements of Standard 2

  • 2a: Knowing about and understanding diverse family and community characteristics
  • 2b: Supporting and engaging families and communities through respectful, reciprocal relationships
  • 2c: Involving families and communities in young children’s development and learning


Candidates prepared in early childhood degree programs understand that child observation, documentation, and other forms of assessment are central to the practice of all early childhood professionals. They know about and understand the goals, benefits, and uses of assessment. They know about and use systematic observations, documentation, and other effective assessment strategies in a responsible way, in partnership with families and other professionals, to positively influence the development of every child.

Key elements of Standard 3

  • 3a: Understanding the goals, benefits, and uses of assessment – including its use in development of appropriate goals, curriculum, and teaching strategies for young children
  • 3b: Knowing about and using observation, documentation, and other appropriate assessment tools and approaches, including the use of technology in documentation, assessment and data collection.
  • 3c: Understanding and practicing responsible assessment to promote positive outcomes for each child, including the use of assistive technology for children with disabilities.
  • 3d: Knowing about assessment partnerships with families and with professional colleagues to build effective learning environments


Candidates prepared in early childhood degree programs understand that teaching and learning with young children is a complex enterprise, and its details vary depending on children’s ages, characteristics, and the settings within which teaching and learning occur. They understand and use positive relationships and supportive interactions as the foundation for their work with young children and families. Candidates know, understand, and use a wide array of developmentally appropriate approaches, instructional strategies, and tools to connect with children and families and positively influence each child’s development and learning.

Key elements of Standard 4

  • 4a: Understanding positive relationships and supportive interactions as the foundation of their work with young children
  • 4b: Knowing and understanding effective strategies and tools for early education, including appropriate uses of technology
  • 4c: Using a broad repertoire of developmentally appropriate teaching /learning approaches
  • 4d: Reflecting on own practice to promote positive outcomes for each child


Candidates prepared in early childhood degree programs use their knowledge of academic disciplines to design, implement, and evaluate experiences that promote positive development and learning for each and every young child. Candidates understand the importance of developmental domains and academic (or content) disciplines in early childhood curriculum. They know the essential concepts, inquiry tools, and structure of content areas, including academic subjects, and can identify resources to deepen their understanding. Candidates use their own knowledge and other resources to design, implement, and evaluate meaningful, challenging curriculum that promotes comprehensive developmental and learning outcomes for every young child.

Key elements of Standard 5

  • 5a: Understanding content knowledge and resources in academic disciplines: language and literacy the arts – music, creative movement, dance, drama, visual arts mathematics science, physical activity, physical education, health and safety and social studies.
  • 5b: Knowing and using the central concepts, inquiry tools, and structures of content areas or academic disciplines
  • 5c: Using own knowledge, appropriate early learning standards, and other resources to design, implement, and evaluate developmentally meaningful and challenging curriculum for each child.


Candidates prepared in early childhood degree programs identify and conduct themselves as members of the early childhood profession. They know and use ethical guidelines and other professional standards related to early childhood practice. They are continuous, collaborative learners who demonstrate knowledgeable, reflective and critical perspectives on their work, making informed decisions that integrate knowledge from a variety of sources. They are informed advocates for sound educational practices and policies.

Key elements of Standard 6

  • 6a: Identifying and involving oneself with the early childhood field
  • 6b: Knowing about and upholding ethical standards and other early childhood professional guidelines
  • 6c: Engaging in continuous, collaborative learning to inform practice using technology effectively with young children, with peers, and as a professional resource.
  • 6d: Integrating knowledgeable, reflective, and critical perspectives on early education
  • 6e: Engaging in informed advocacy for young children and the early childhood profession


Field experiences and clinical practice are planned and sequenced so that candidates develop the knowledge, skills and professional dispositions necessary to promote the development and learning of young children across the entire developmental period of early childhood – in at least two of the three early childhood age groups (birth – age 3, 3 through 5, 5 through 8 years) and in the variety of settings that offer early education (early school grades, child care centers and homes, Head Start programs).

Algebra and Patterning

In eighth grade, students will analyze and justify the explanations for patterns and their rules at a more complex level. Your students should be able to write algebraic equations and write statements to understand simple formulas.

Students should be able to evaluate a variety of simple linear algebraic expressions at a beginning level by using one variable. Your students should confidently solve and simplify algebraic equations with four operations. And, they should feel comfortable substituting natural numbers for variables when solving algebraic equations.

Dive deeper

Here are examples of what it looks like when people struggle with number sense.

Adding and subtracting. Imagine a pile of seven beads. Then take away two of them. People with poor number sense might not realize that:

The number of beads has shrunk

Subtracting the two beads means the group of seven is now a group of five

Now imagine adding three beads to the pile. If someone struggles with number sense, they might not recognize that:

The group of beads has gotten larger

Adding three beads to the pile of seven makes it a pile of 10

Multiplying and dividing. When people need to combine items from several groups, they might go through the trouble of adding them. They may not grasp that it’s simpler to multiply them.

Likewise, they might not recognize that division is the simplest way to break up groups into their component parts.

Not grasping these concepts makes learning math and using it in everyday life a challenge. Learn more about math challenges in kids .

When kids struggle with math, schools often focus first on reteaching the specific math skills being taught in class. But this approach often doesn’t work for kids who struggle with number sense.

In that case, schools usually turn to an intervention process, where kids typically:

Work with “manipulatives” like blocks and rods to understand the relationships between amounts

Do exercises that involve matching number symbols to quantities

Get a lot of practice estimating

Learn strategies for checking an answer to see whether it’s reasonable

Talk with their teacher about the strategies they use to solve problems

Get help correcting mistakes they make along the way

For many kids with weak number sense, intervention is enough to catch up. But some kids need more support. They may need to be evaluated for special education to get the help they need.

Learn about intervention systems like RTI or MTSS .

It takes time for kids to build number sense skills. But there are many ways to help. Here are some examples:

Practice counting and grouping objects. Then add to, subtract from, or divide the groups into smaller groups to practice operations. Combine groups to show multiplication.

Work on estimating. Build questions into everyday conversations, using phrases like “about how many” or “about how much.”

Talk about relationships between quantities. Ask kids to use words like more and less to compare things.

Build in opportunities to talk about time. Ask kids to keep track of how long it takes to drive or walk to the grocery store. Compare that with how long it takes to get to school. Ask which takes longer.

Key to Fractions

Whether you're introducing fractions for the first time, or find that your students need review and additional practice, Key to Fractions covers all major topics. Written with secondary students in mind, these self-paced workbooks provide the reinforcement needed for fraction mastery. Book 1 teaches fraction concepts, Book 2 teaches multiplying and dividing, Book 3 teaches adding and subtracting, and Book 4 teaches mixed numbers. Each book has a practice test at the end.


Singaporean elementary-school students (N = 299) completed Child Implicit Association Tests (Child IAT) as well as explicit measures of gender identity, math–gender stereotypes, and math self-concepts. Students also completed a standardized math achievement test. Three new findings emerged. First, implicit, but not explicit, math self-concepts (math = me) were positively related to math achievement on a standardized test. Second, as expected, stronger math–gender stereotypes (math = boys) significantly correlated with stronger math self-concepts for boys and weaker math self-concepts for girls, on both implicit and explicit measures. Third, implicit math–gender stereotypes were significantly related to math achievement. These findings show that non-academic factors such as implicit math self-concepts and stereotypes are linked to students' actual math achievement. The findings suggest that measuring individual differences in non-academic factors may be a useful tool for educators in assessing students' academic outcomes.


Two events are independent if one happening has nothing to do with another , like the sun shining and you eating a sandwich for lunch. The sun may shine, and you may eat a sandwich for lunch, but one does not cause or prevent the other.

Two events are mutually exclusive if they cannot both occur , like me wearing a hat and me not wearing a hat. I can’t do both.

For independent, non-mutually exclusive events: P(A and B) = P(A)*P(B), whereas P(A or B) = P(A) + P(B) - P(A and B).

For mutually exclusive events: P(A or B) = P(A) + P(B).

There are other formulas for more complicated scenarios, but these will get you pretty far — they’re all you should need on the SAT.

Let’s take a look at this problem:

Twenty-five people passed the bar exam of these, seven did not take the review course. So, the probability that the interviewed person in question did not take the bar exam is 7/25, or (B).

Math and gambling are closely linked, too. It’s all about that probability.

Math Courses of Study: Grade by Grade

See Below For Grade by Grade Goals
Although mathematics curricula will vary from state to state and country to country, you'll find that this list provides the basic concepts that are addressed and required for each grade. The concepts have been divided by topic and grade for easy navigation. Mastery of the concepts at the previous grade is assumed. Students preparing for each grade will find the listings to be extremely helpful. When you understand the topics and concepts that are required, you will find tutorials to help you prepare under the perspective subjects on the home page. Calculators and computer applications are also required as early as kindgarten. Most curriculum documents request that you are also able to use the corresponding technologies such as software applications, regular calculators, and graphing calculators.

For more specific details regarding the math requirements for each grade, you may want to do a search for the curriculum in your state, province or country. Most boards of education will provide you with the details to access the documents.

A Few More Things About Khan Academy

Important note: If you see a concept in the ETS Math Review that you don't know but can't easily find on the Khan page I've linked to, just type it into the Khan Academy search bar. (BTW, the only topic in the ETS math review that I'm confident will NOT be tested is how to calculate standard deviation. You should still understand standard deviation - just don't worry about calculating it with the formula.)

BUT if you find a concept in Khan that is NOT in the ETS Math Review, don't worry about it*. I'll list anything you don't need in Khan next to each Khan link.

*With that said, ETS does sometimes test a very small number of topics that aren't explicitly covered in the math review. For example, sequences. But don't worry - I got you, bro. Sequences are in the list of topics linked to Khan below.

If you do have a topic for which you're not sure whether it might be tested, please ask about it on Reddit (r/GRE). I'm on there a lot, and I don't want you losing sleep over a random topic if you don't need to.

7th Grade Numbers and Operations Jeopardy

This 7th grade jeopardy game can be used to review important concepts about numbers and operations with rational numbers.

This interactive game has 3 categories: Comparing Rational Numbers, Adding and Subtracting Rational Numbers, and Multiplying and Dividing Rational Numbers. This jeopardy game can be played on computers, iPads, and other tablets. You do not need to install an app to play this game on the iPad. Let the best team win!

The game is based on the following Common Core Math Standards:

CCSS 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers represent addition and subtraction on a horizontal or vertical number line diagram.

CCSS 7.NS.2.a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

CCSS 7.NS.2.b Convert a rational number to a decimal using long division know that the decimal form of a rational number terminates in 0s or eventually repeats.

CCSS 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

Watch the video: Geoboard: Διδασκαλία Μαθηματικών και Γεωμετρικών Εννοιών Εξ Αποστάσεως (November 2021).