Table Test

What is mathematical thinking?

First we address the question: “what is thinking”.

Thinking is a mental process in which we pose a set of directed questions that together with their answers serve a purpose – usually to consider something or to act.

The questions we pose are directed to serve some purpose such as making a plan, defining what we are asked to do, solving a problem we have been asked to solve,  defining a problem we set our selves to solve, or to outline something to be communicated, perhaps then written.

The product of consciousness in the “thinking process” is a directed set of questions with their answers.

When we speak of “mathematical thinking” we mean that the questions and answers address a mathematical purpose – perhaps to pose a mathematical question, to answer that question, consider the justification for the answer or reflect on the process we used to develop answers.

Mathematics is the science of spatial, temporal and logical relationships..

 

This a test for using tables.


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Table of Contents - Number Sense
1997 CA Math Stndards - 2010 CA Core Standards Mathematics
KIndergarten
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Grade 6
Grade 7
Number Sense

KNS Number Sense Top
KNS1.0 Students understand the relationship between numbers and quantities (i.e., that a set of objects has the same number of objects in different situations regardless of its position or arrangement).

K.CC: Know number names and the counting sequence.

K.CC: Count to tell the number of objects. Compare numbers (Cluster Statement)

KNS1.1 Compare two or more sets of objects (up to ten objects in each group) and identify which set is equal to, more than, or less than the other.

K.CC.6: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.*

K.CC.7: Compare two numbers between 1 and 10 presented as written numerals.


KNS1.2 Count, recognize, represent, name, and order a number of objects (up to 30).and by tens.

K.CC.2: Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

K.CC.3: Write numbers from 0–20. Represent a number of objects with written numeral 0 -20 (with 0 representing a count of no objects).

K.CC.5: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects


KNS1.3 Know that the larger numbers describe sets with more objects in them than the smaller numbers have a relationship between numbers and quantities; connect counting to cardinality.

K.CC.4a: When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

K.CC.4b: Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

K.CC.4c: Understand that each successive number name refers to a quantity that is one larger.

K.CC.6: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.*

KNS2.0 Students understand and describe simple additions and subtractions

K.OA: (Cluster Statement) Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

KNS3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones and tens places.

NO

KNS3.1 Recognize when an estimate is reasonable

NO


1NS Number Sense Top

1NS1.0 Students understand and use numbers up to 100.

1.NBT: Extend the counting sequence. (Cluster Statement)

1NS1.1 Count, read, and write whole numbers to 100.

1.NBT.1: Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

1NS1.2 Compare and order whole numbers to 100 by using the symbols for less than, equal to, or greater than (<, =, >).

1.NBT.3: Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

1NS1.3 Represent equivalent forms of the same number through the use of physical models, diagrams, and number expressions (to 20) (e.g., 8 may be represented as 4 + 4, 5 + 3, 2 + 2 + 2 + 2, 10 -2, 11 -3).

1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on: making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one know 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

1NS1.4 Count and group object in ones and tens (e.g., three groups of 10 and 4 equals 34, or 30 + 4).

1.NBT.2: Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

1NS2.0 Students demonstrate the meaning of addition and subtraction and use these operations to solve problems

1.OA: Represent and solve problems involving addition and subtraction. (Cluster Statement)

1NS2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory.

1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.Use strategies such as counting on:making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 –4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one know 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

1NS2.2 Use the inverse relationship between addition and subtraction to solve problems

1.OA.4: Understand subtraction as an unknown-addend problem..

!NS2.3 Identify one more than, one less than, 10 more than, and 10 less than a given number.

1.NBT.5: Given a two-digit number, mentally find 10 more or 10 less that the number, without having to count: explain the reasoning used.

1NS2.4 Count by 2s, 5s, and 10s to 100. 1.OA.5: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.5: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1NS2.5 Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference).

NO

1NS2.5 Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference).

NO
1NS2.6 Solve addition and subtraction problems with one-and two-digit numbers (e.g., 5 + 58 = __).

1.NBT.4: Add within 100, including adding a two-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; related the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

1.NBT.6: Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; related the strategy to a written method and explain the reasoning used.

1NS2.7 Find the sum of three one-digit numbers.

1.OA.2: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1OA.8: Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers.

1NS3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones, tens, and hundreds places.

NO

1NS3.1 Make reasonable estimates when comparing larger or smaller numbers.

NO

2NS Number Sense Top

2NS1.0 Students understand the relationship between numbers, quantities, and place value in whole numbers up to 1,000:

2.NBT: Understand Place Value. Use place value understanding and properties of operations to add and subtract. (Cluster Statement)
2NS1.1* Count, read, and write whole numbers to 1,000 and identify the place value for each digit.

2.NBT.1: Understand that the three-digit number represent amounts of hundreds, tens and ones; e.g. 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

2.NBT.1a: 100 can be thought of as a bundle of ten tens-called a “hundred.”

2.NBT.1b: The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2.NBT.2: Count within 1000; skip-count by 5s, 10s and 100s.

2.NBT.3: Read and write numbers to 1000 using base-ten numerals, number names and expanded form.
2NS1.2 Use words, models, and expanded forms (e.g., 45 = 4 tens + 5) to represent numbers (to 1,000).

2.NBT.3: Read and write numbers to 1000 using base-ten numerals,number names and expanded form.
2NS1.3* Order and compare whole numbers to 1,000 by using the symbols <, =, >.

2.NBT.4: Compare two three-digit numbers based on meanings of the hundreds, tens and ones digits, using >, =, and < symbols to record the results of the comparisons.
2NS2.0 Students estimate, calculate, and solve problems involving addition and subtraction of two- and three-digit numbers:

2.NBT.1: (Cluster Statement) Use place value understanding and properties of operations to add and subtract.
2NS2.1* Understand and use the inverse relationship between addition and subtraction (e.g., an opposite number sentence for 8 + 6 = 14 is 14 – 6 = 8) to solve problems and check solutions.

2.NBT.5: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

2.NBT.7: Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

2.NBT.9: Explain why addition and subtraction strategies work, using place value and the properties of operations.
2NS2.2* Find the sum or difference of two whole numbers up to three digits long.

2.NBT.6: Add up to four two-digit numbers using strategies based on place value and properties of operations.

2.NBT.7: Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2NS2.3 Use mental arithmetic to find the sum or difference of two two-digit numbers. &&&

2.OA.1: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions ,e.g., by using drawing and equations with a symbol for the unknown number to represent the problem.

2.OA.2: Fluently add and subtract within 20 using mental strategies.By the end of Grade 2, know from memory all sums of two one-digit numbers.

2.NBT.5: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

2.NBT.8: Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.
2NS3.0* Students model and solve simple problems involving multiplication and division:

NO
2NS3.1* Use repeated addition, arrays, and counting by multiples to do multiplication.

2.OA.4: Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
2NS3.2* Use repeated subtraction, equal sharing, and forming equal groups with remainders to do division.

NO
2NS3.3* Know the multiplication tables of 2s, 5s, and 10s (to “times 10”) and commit them to memory.

NO
2NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole:

NO
2NS4.1* Recognize, name, and compare unit fractions from 1/12 to 1/2.

NO
2NS4.2* Recognize fractions of a whole and parts of a group (e.g., one-fourth of a pie, two-thirds of 15 balls).

NO
2NS4.3* Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one.

2.G.3: Partition circles and rectangles into two, three, or four equal shares, describe the shares using words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
2NS5.0 Students model and solve problems by representing, adding, and subtracting amounts of money:

2.MD: Work with time and money. (Cluster Statement)
2NS5.1* Solve problems using combinations of coins and bills.

2.MD.8: Solve word problems involving dollar bills, quarters, dimes, nickels and pennies, using dollar and cent symbols appropriately.
2NS5.2* Know and use the decimal notation and the dollar and cent symbols for money.

NO
2NS6.0 Students use estimation strategies in computation and problem solving that involve numbers

NO
2NS6.1 Recognize when an estimate is reasonable in measurements (e.g., closest inch).

NO

3NS Number Sense Top

3NS1.0 Students understand the place value of whole numbers:

NO -- CCS explains in the Grade 4 overview: “Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place.” CCS does not mention what the place value limit in Grade 3.
3NS1.1 Count, read, and write whole numbers to 10,000

NO -- 4.NBT.2: Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

NO -- CCS Grade 4 expectations in this domain are limited to whole numbers
less than or equal to 1,000,000.

3NS1.2 Compare and order whole numbers to 10,000.

NO -- 4.NBT.2: Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi=digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

3NS1.3* Identify the place value for each digit in numbers to 10,000.

NO -- 4.NBT.1: Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. CCS adds the comparison of one place value to the next (x 10).

3NS1.4 Round off numbers to 10,000 to the nearest ten, hundred, and thousand.

Partial -- 4.NBT.3: Use place value understanding to round multi-digit whole numbers to any place.

3NS1.5* Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6).

NO

3NS2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division:

3.OAT (Cluster Statement) Use place value understanding and properties of operations to perform multi-digit arithmetic

3NS2.1* Find the sum or difference of two whole numbers between 0 and 10,000

3.NBT.2: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3NS2.2 Memorize to automaticity the multiplication table for numbers between 1 and 10.

3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one know 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.

3NS2.3* Use the inverse relationship of multiplication and division to compute and check results.

3.OA.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers

3.OA.7: Fluently multiply and divide within 100, using strategies such as the
relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one know 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one digit numbers.

3.OA.6: Understand division as an unknown-factor problem

3NS2.4* Solve simple problems involving multiplication of multi-digit numbers by one-digit numbers (3,671 × 3 = ___).

3.OA.1: Interpret products of whole number, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each.

3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one know 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

3NS2.5 Solve division problems in which a multi-digit number is evenly divided by a one-digit number (135 ÷ 5 = ___).

3NS2.6 Understand the special properties of 0 and 1 in multiplication and division.

3NS2.7 Determine the unit cost when given the total cost and number of units.

3NS2.8 Solve problems that require two or more of the skills mentioned above.

3NS3.0 Students understand the relationship between whole numbers, simple fractions, and decimals:

3.NF: (Cluster Statement) Develop an understanding of fractions as numbers.

CCS does not explicitly describe the relationship between whole numbers,
simple fractions, and decimals.

4.NF: (Cluster Statement) Understand decimal notation for fractions, and compare decimal fractions.

3NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4).

3.NF.3: Explain equivalence of fractions in special cases, and
compare fractions by reasoning about their size

3.NF.3a: Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

3.NF.3b: Recognize and generate simple equivalent fractions, e.g., = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

3.NF.3c: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

3.NF.3d: Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer to the samewhole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model

3NS3.2* Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 is the same as 1/2).

Partial - CCS does not mention adding and subtracting of simple fractions. However, it is implied in statement 3.NF.2b.

3.NF.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram.

3.NF.2a: Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

3.NF.2b: Represent a fraction a/b on a number line diagram by marking off a length of 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

3NS3.3* Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors.

NO -5.NBT.7: Add, subtract, multiply, and
divide decimals to hundredths, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

3NS3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents is 1/2 of a dollar, 75 cents is 3/4 of a dollar).

NO --- 4.NF.6: Use decimal notation for fractions with denominators 10 or 100.

CCS describes the two different representations with denominators that are the powers of 10.

4NS Number Sense Top

4NS1.0 Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts of negative numbers:

4.NBT: Generalize place value understanding for multi-digit whole numbers (Cluster Statement).

4.NF Cluster Statement: Understand decimal notation for fractions, and compare decimal fractions).

4NS1.1* Read and write whole numbers in the millions.

4.NBT.2: Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.
Compare two multi-digit numbers based on meaning of the digits in each place, using >, =, and < symbols to record the results of comparisons.

CCS expectations for Grade 4 are limited to whole numbers less than or equal to 1,000,000.

4NS1.2* Order and compare whole numbers and decimals to two decimal places.

4.NBT.2: Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form.
Compare two multi-digit numbers based on meaning of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NF.7: Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using a visual model

4NS1.3* Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand.

4.NBT.3: Use place value understanding to round multi-digit whole numbers to any place.

4NS1.4 Decide when a rounded solution is called for and explain why such a solution may be appropriate

4.OA.3: Solve multi-step word problems posed with whole numbers and having whole-number answers using the four
operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the
reasonableness of answers using mental computation and estimation strategies including rounding.

4NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions (see Standard 4.0).

NO

4NS1.6 Write tenths and hundredths in decimal and fraction notations, and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50; 7/4 = 1 3/4 = 1.75).

4.NF.6: Use decimal notation for fractions with denominators 10 or 100.

CCS does not specify other fractions than those with denominators of powers of 10.

3.NF.3: Explain equivalence of fractions in special cases, and
compare fractions by reasoning about their size.

3.NF.3a: Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

3.NF.3b: Recognize and generate simple equivalent fractions, e.g.,. = 2/4, 4/6 = 2/3. Explain whey the fractions are equivalent, e.g.,by using a visual fraction model.

3.NF.3c: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

3.NF.3d: Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that
comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model

4NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.

4.NF.5: Express a fraction with
denominator 10 as an equivalent fraction with a denominator 100, and use this
technique to add two fractions with respective denominators 10 and 100. Use decimal notation for fractions with
denominators 10 or 100.

4.NF.7: Compare two decimals to hundredths by reasoning about their size Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using a visual model.

CCS does not ask students to draw a fraction.

CCS does not specify other fractions than those with denominators of powers of 10.

4NS1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in "owing").

CCS states two numbers on a
number line not explicitly two
negative numbers.

6.NS.7: Understand ordering and absolute value of rational numbers.

6.NS.7a: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

6.NS.b7: Write, interpret, and explain statements of order for
rational numbers in real-world contexts.

6.NS.7c: Understand the absolute value of a rational number as its distance from 0 on the number line;interpret absolute value as magnitude for a positive or negative
quantity in a real-world situation.

6.NS.7d: Distinguish comparisons of absolute value from statements about order.

4NS1.9* Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to t