Grade 5 – CA Common Core Standards and Learning Objectives

5.5.OA Operations and Algebraic Thinking

 

5 Write and interpret numerical expressions.

 

5.5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Perform multiple operations with whole numbers (Fifth grade – O.4)

 

5.5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Write numerical expressions (Fifth grade – O.3)

 

5.5.OA.2.1 Express a whole number in the range 2-50 as a product of its prime factors.

Prime factorization (Fifth grade – F.2)

 

5 Analyze patterns and relationships.

 

5.5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

Complete a table for a two-variable relationship (Fifth grade – U.8)

Complete a table from a graph (Fifth grade – U.9)

Graph a two-variable relationship (Fifth grade – U.10)

5.5.NBT Number and Operations in Base Ten

 

5 Understand the place value system.

 

5.5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

Place values (Fifth grade – A.1)

Convert between place values (Fifth grade – A.2)

Place values in decimal numbers (Fifth grade – G.4)

 

5.5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Scientific notation (Fifth grade – A.11)

Multiplication patterns over increasing place values (Fifth grade – C.3)

Multiply numbers ending in zeroes (Fifth grade – C.4)

Multiply numbers ending in zeroes: word problems (Fifth grade – C.5)

Division patterns over increasing place values (Fifth grade – D.7)

Multiply a decimal by a power of ten (Fifth grade – I.2)

Divide by powers of ten (Fifth grade – J.1)

Decimal division patterns over increasing place values (Fifth grade – J.2)

 

5.5.NBT.3 Read, write, and compare decimals to thousandths.

 

5.5.NBT.3.a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

What decimal number is illustrated? (Fifth grade – G.1)

Model decimals and fractions (Fifth grade – G.2)

Understanding decimals expressed in words (Fifth grade – G.3)

Place values in decimal numbers (Fifth grade – G.4)

Convert decimals between standard and expanded form (Fifth grade – G.5)

Convert decimals between standard and expanded form using fractions (Fifth grade – G.14)

 

5.5.NBT.3.b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Equivalent decimals (Fifth grade – G.6)

Decimal number lines (Fifth grade – G.8)

Compare decimals on number lines (Fifth grade – G.9)

Compare decimal numbers (Fifth grade – G.10)

Put decimal numbers in order (Fifth grade – G.11)

Compare decimals and fractions on number lines (Fifth grade – G.15)

Inequalities with decimal multiplication (Fifth grade – I.10)

 

5.5.NBT.4 Use place value understanding to round decimals to any place.

Round decimals (Fifth grade – G.7)

Estimate sums and differences of decimals (Fifth grade – H.8)

 

5 Perform operations with multi-digit whole numbers and with decimals to hundredths.

 

5.5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

Multiply by 2-digit numbers: complete the missing steps (Fifth grade – C.12)

Multiply 2-digit numbers by 2-digit numbers (Fifth grade – C.13)

Multiply 2-digit numbers by 3-digit numbers (Fifth grade – C.14)

Multiply 2-digit numbers by larger numbers (Fifth grade – C.15)

Multiply by 2-digit numbers: word problems (Fifth grade – C.16)

Multiply three or more numbers up to 2 digits each (Fifth grade – C.17)

Multiply by 3-digit numbers (Fifth grade – C.18)

Multiply three numbers up to 3 digits each (Fifth grade – C.19)

Multiply three or more numbers: word problems (Fifth grade – C.20)

 

5.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Properties of multiplication (Fifth grade – C.6)

Division facts to 12 (Fifth grade – D.1)

Division facts to 12: word problems (Fifth grade – D.2)

Divide multi-digit numbers by 1-digit numbers (Fifth grade – D.3)

Divide multi-digit numbers by 1-digit numbers: word problems (Fifth grade – D.4)

Divide numbers ending in zeroes (Fifth grade – D.8)

Divide numbers ending in zeroes: word problems (Fifth grade – D.9)

Divide 2-digit and 3-digit numbers by 2-digit numbers (Fifth grade – D.10)

Divide 2-digit and 3-digit numbers by 2-digit numbers: word problems (Fifth grade – D.11)

Divide larger numbers by 2-digit numbers (Fifth grade – D.12)

Divide larger numbers by 2-digit numbers: word problems (Fifth grade – D.13)

Choose numbers with a particular quotient (Fifth grade – D.16)

 

5.5.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.

Add and subtract money: word problems (Fifth grade – B.4)

Divide money amounts: word problems (Fifth grade – D.14)

Add decimal numbers (Fifth grade – H.1)

Subtract decimal numbers (Fifth grade – H.2)

Add and subtract decimal numbers (Fifth grade – H.3)

Add and subtract decimals: word problems (Fifth grade – H.4)

Choose decimals with a particular sum or difference (Fifth grade – H.5)

Complete the addition or subtraction sentence (Fifth grade – H.6)

Inequalities with decimal addition and subtraction (Fifth grade – H.7)

Multiply a decimal by a one-digit whole number (Fifth grade – I.3)

Multiply a decimal by a multi-digit whole number (Fifth grade – I.4)

Multiply decimals and whole numbers: word problems (Fifth grade – I.5)

Multiply money amounts: word problems (Fifth grade – I.6)

Multiply three or more numbers, one of which is a decimal (Fifth grade – I.7)

Multiply two decimals using grids (Fifth grade – I.8)

Multiply two decimals (Fifth grade – I.9)

Division with decimal quotients (Fifth grade – J.3)

Division with decimal quotients and rounding (Fifth grade – J.4)

Division with decimal quotients: word problems (Fifth grade – J.5)

Add, subtract, multiply, and divide decimals (Fifth grade – O.5)

Add, subtract, multiply, and divide decimals: word problems (Fifth grade – O.6)

Price lists (Fifth grade – R.1)

Unit prices (Fifth grade – R.2)

5.5.NF Number and Operations-Fractions

 

5 Use equivalent fractions as a strategy to add and subtract fractions.

 

5.5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Equivalent fractions (Fifth grade – K.3)

Reduce fractions to lowest terms (Fifth grade – K.4)

Convert between improper fractions and mixed numbers (Fifth grade – K.5)

Add fractions with unlike denominators using models (Fifth grade – L.6)

Add up to 4 fractions with denominators of 10 and 100 (Fifth grade – L.7)

Add fractions with unlike denominators (Fifth grade – L.8)

Subtract fractions with unlike denominators using models (Fifth grade – L.9)

Subtract fractions with unlike denominators (Fifth grade – L.10)

Add 3 or more fractions with unlike denominators (Fifth grade – L.12)

Complete addition and subtraction sentences with fractions (Fifth grade – L.15)

Inequalities with addition and subtraction of fractions (Fifth grade – L.16)

Add mixed numbers with unlike denominators (Fifth grade – L.18)

Subtract mixed numbers with unlike denominators (Fifth grade – L.19)

Complete addition and subtraction sentences with mixed numbers (Fifth grade – L.22)

Inequalities with addition and subtraction of mixed numbers (Fifth grade – L.23)

 

5.5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

Add and subtract fractions with like denominators: word problems (Fifth grade – L.4)

Add and subtract fractions with unlike denominators: word problems (Fifth grade – L.11)

Add 3 or more fractions: word problems (Fifth grade – L.13)

Compare sums and differences of unit fractions (Fifth grade – L.14)

Add and subtract mixed numbers: word problems (Fifth grade – L.20)

Add and subtract fractions in recipes (Fifth grade – L.21)

 

5 Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

 

5.5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Fractions review (Fifth grade – K.1)

Divide fractions by whole numbers (Fifth grade – N.4)

 

5.5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

 

5.5.NF.4.a Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.

Multiply fractions by whole numbers I (Fifth grade – M.5)

Multiply fractions by whole numbers II (Fifth grade – M.8)

Multiply fractions by whole numbers: input/output tables (Fifth grade – M.11)

 

5.5.NF.4.b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Multiply two unit fractions using models (Fifth grade – M.12)

Multiply two fractions using models: fill in the missing factor (Fifth grade – M.13)

Multiply two fractions using models (Fifth grade – M.14)

Area of squares and rectangles (Fifth grade – Z.16)

Area and perimeter: word problems (Fifth grade – Z.22)

 

5.5.NF.5 Interpret multiplication as scaling (resizing), by:

 

5.5.NF.5.a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Scaling whole numbers by fractions (Fifth grade – M.17)

Scaling fractions by fractions (Fifth grade – M.18)

Scaling mixed numbers by fractions (Fifth grade – M.19)

 

5.5.NF.5.b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

Multiply two fractions using models (Fifth grade – M.14)

 

5.5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Multiply fractions by whole numbers: word problems (Fifth grade – M.10)

Multiply two fractions (Fifth grade – M.15)

Multiply two fractions: word problems (Fifth grade – M.16)

Multiply a mixed number by a whole number (Fifth grade – M.23)

Multiply a mixed number by a fraction (Fifth grade – M.24)

Multiply two mixed numbers (Fifth grade – M.25)

Multiplication with mixed numbers: word problems (Fifth grade – M.27)

Multiply fractions and mixed numbers in recipes (Fifth grade – M.28)

Add, subtract, multiply, and divide fractions and mixed numbers (Fifth grade – O.7)

Add, subtract, multiply, and divide fractions and mixed numbers: word problems (Fifth grade – O.8)

 

5.5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

 

5.5.NF.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

Divide unit fractions by whole numbers (Fifth grade – N.1)

Divide fractions by whole numbers (Fifth grade – N.4)

 

5.5.NF.7.b Interpret division of a whole number by a unit fraction, and compute such quotients.

Divide whole numbers by unit fractions (Fifth grade – N.2)

Divide whole numbers by unit fractions using models (Fifth grade – N.7)

Divide whole numbers by fractions (Fifth grade – N.8)

 

5.5.NF.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

Divide whole numbers and unit fractions (Fifth grade – N.3)

Divide unit fractions by whole numbers: word problems (Fifth grade – N.6)

5.5.MD Measurement and Data

 

5 Convert like measurement units within a given measurement system.

 

5.5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Compare and convert customary units of length (Fifth grade – Y.2)

Compare and convert customary units of weight (Fifth grade – Y.3)

Compare and convert customary units of volume (Fifth grade – Y.4)

Compare and convert customary units (Fifth grade – Y.5)

Conversion tables – customary units (Fifth grade – Y.6)

Compare and convert metric units of length (Fifth grade – Y.8)

Compare and convert metric units of weight (Fifth grade – Y.9)

Compare and convert metric units of volume (Fifth grade – Y.10)

Compare and convert metric units (Fifth grade – Y.11)

Conversion tables – metric units (Fifth grade – Y.12)

Compare customary units by multiplying (Fifth grade – Y.13)

Convert customary units involving fractions (Fifth grade – Y.14)

Convert mixed customary units (Fifth grade – Y.15)

Add and subtract mixed customary units (Fifth grade – Y.16)

 

5 Represent and interpret data.

 

5.5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

Interpret line plots (Fifth grade – V.10)

Create line plots (Fifth grade – V.11)

Create and interpret line plots with fractions (Fifth grade – V.12)

 

5 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

 

5.5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

 

5.5.MD.3.a A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

 

5.5.MD.3.b A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

 

5.5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

 

5.5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

 

5.5.MD.5.a Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

Volume of cubes and rectangular prisms (Fifth grade – Z.25)

 

5.5.MD.5.b Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Volume of cubes and rectangular prisms (Fifth grade – Z.25)

 

5.5.MD.5.c Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Volume of irregular figures made of unit cubes (Fifth grade – Z.24)

5.5.G Geometry

 

5 Graph points on the coordinate plane to solve real-world and mathematical problems.

 

5.5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

Coordinate graphs review – whole numbers only (Fifth grade – T.1)

Coordinate graphs with decimals and negative numbers (Fifth grade – T.2)

Graph points on a coordinate plane (Fifth grade – T.3)

 

5.5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Graph points on a coordinate plane (Fifth grade – T.3)

Coordinate graphs as maps (Fifth grade – T.4)

Relative coordinates: follow directions (Fifth grade – T.5)

 

5 Classify two-dimensional figures into categories based on their properties.

 

5.5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

Classify quadrilaterals (Fifth grade – Z.7)

 

5.5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

Identify 2-dimensional and 3-dimensional shapes (Fifth grade – Z.1)

Types of triangles (Fifth grade – Z.2)

Open and closed shapes and qualities of polygons (Fifth grade – Z.3)

Regular and irregular polygons (Fifth grade – Z.4)

Number of sides in polygons (Fifth grade – Z.5)

Which figure is being described? (Fifth grade – Z.6)

Classify quadrilaterals (Fifth grade – Z.7)

Common Core Arithmetic

Arithmetic is the study of counting, comparison of quantities, number operations and properties of the rational number system.  These studies include applications to daily and professional life.

K.CC   Counting and Cardinality
Know number names and the count sequence.

1. Count to 100 by ones and by tens.

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

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

Count to tell the number of objects.

4. Understand the relationship between numbers and quantities; connect counting to cardinality.

a. 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.

b. 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.

c. Understand that each successive number name refers to a quantity that is one larger.

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.

Compare numbers.

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.1

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

1.NBT  Number and Operations in Base Ten
Extend the counting sequence.

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.

Understand place value.

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

a. 10 can be thought of as a bundle of ten ones—called a “ten.”

b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. Use place value understanding and properties of operations to add and subtract.

4. Add within 100, including adding a two-digit number and a one-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; relate 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.

5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

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 between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

2.NBT  Number and Operations in Base Ten
Understand place value.

1. Understand that the three digits of a 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:

a. 100 can be thought of as a bundle of ten tens—called a “hundred.”

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

3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Use place value understanding and properties of operations to add and subtract.

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

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

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.

7.1 Use estimation strategies to make reasonable estimates in problem solving. CA

8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

9. Explain why addition and subtraction strategies work, using place value and the properties of operations.

3.NBT  Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic.

1. Use place value understanding to round whole numbers to the nearest 10 or 100.

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.

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

3.NF  Number and Operations—Fractions
Develop understanding of fractions as numbers.

1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

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

a. 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.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 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.

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

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

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

d. 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.

4.NBT  Number and Operations in Base Ten
Generalize place value understanding for multi-digit whole numbers.

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. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

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.

3. Use place value understanding to round multi-digit whole numbers to any place.
Use place value understanding and properties of operations to perform multi-digit arithmetic.

4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.2

4.NF  Number and Operations—Fractions
Extend understanding of fraction equivalence and ordering.

1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples:
3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5 (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Understand decimal notation for fractions, and compare decimal fractions.

5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.2

6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram..

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 >, =, or <, and justify the conclusions, e.g., by using the number line or another visual model.

5.NBT  Number and Operations in Base Ten
Understand the place value system.

1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

3. Read, write, and compare decimals to thousandths.

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4. Use place value understanding to round decimals to any place. Perform operations with multi-digit whole numbers and with decimals to hundredths.

5. Fluently multiply multi-digit whole numbers using the standard algorithm.

6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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.

5.NF  Number and Operations—Fractions
Use equivalent fractions as a strategy to add and subtract fractions.

1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

5. Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size o one factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

6. Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

6.NS  The Number System
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.

2. Fluently divide multi-digit numbers using the standard algorithm.

3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9+ 2).

Apply and extend previous understandings of numbers to the system of rational numbers.

5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize red pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7. Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.

c. 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. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.

6.RP  Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems.

1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1

3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

7.NS  The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

 

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.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. 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.

3. Solve real-world and mathematical problems involving the four operations with rational numbers.1

7.RP  Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.

1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.

2. Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed
as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with specialattention to the points (0, 0) and (1, r) where r is the unit rate.

3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

8.NS  The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.

1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g.,π 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

 

Arithmetic Operations Including Division of Fractions – Grade 6 Module 2

OVERVIEW

In Module 1, students used their existing understanding of multiplication and division as they began their study of ratios and rates. In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4).

In Topic A, students extend their previous understanding of multiplication and division to divide fractions by fractions. They construct division stories and solve word problems involving division of fractions (6.NS.1). Through the context of word problems, students understand and use partitive division of fractions to determine how much is in each group. They explore real-life situations that require them to ask, “How much is one share?” and “What part of the unit is that share?”

Students use measurement to determine quotients of fractions. They are presented conceptual problems where they determine that the quotient represents how many of the divisor is in the dividend. For example, students understand that (6 cm)/(3 cm) derives a quotient of 3 because 2 cm divides into 6 centimeters three times. They apply this method to quotients of fractions, understanding 6/7 ÷ 2/7 = (6 sevenths)/(2 sevenths) = 3 because, again, two-sevenths divides into six-sevenths three times. Students look for and uncover patterns while modeling quotients of fractions to ultimately discover the relationship between multiplication and division. Using this relationship, students
create equations and formulas to represent and solve problems.

Later in the module, students learn to and apply the direct correlation of division of fractions to division of decimals. Prior to division of decimals, students will revisit all decimal operations in Topic B. Students have had extensive experience of decimal operations to the hundredths and thousandths (5.NBT.7), which prepares them to easily compute with more decimal places. Students begin by relating the first lesson in this topic to mixed numbers from the last lesson in Topic A. They find that sums and differences of large mixed numbers can sometimes be more efficiently determined by first converting the number to a decimal and then applying the standard algorithms (6.NS.3). They use estimation to justify their answers.

Within decimal multiplication, students begin to practice the distributive property. Students use arrays and partial products to understand and apply the distributive property as they solve multiplication problems involving decimals. By gaining fluency in the distributive property throughout this module and the next, students will be proficient in applying the distributive property in Module 4 (6.EE.3).

Estimation and place value enable students to determine the placement of the decimal point in products and recognize that the size of a product is relative to each factor. Students learn to use connections between fraction multiplication and decimal multiplication. In Grades 4 and 5, students used concrete models, pictorial representations, and properties to divide whole numbers (4.NBT.6, 5.NBT.6). They became efficient in applying the standard algorithm for long division. They broke dividends apart into like base-ten units, applying the distributive property to find quotients place by place.

In Topic C, students connect estimation to place value and determine that the standard algorithm is simply a tally system arranged in place value columns (6.NS.2). Students understand that when they “bring down” the next digit in the algorithm, they are essentially distributing, recording, and shifting to the next place value. They understand that the steps in the algorithm continually provide better approximations to the answer.

Students further their understanding of division as they develop fluency in the use of the standard algorithm to divide multi-digit decimals (6.NS.3). They make connections to division of fractions and rely on mental math strategies to implement the division algorithm when finding the quotients of decimals. In the final topic, students think logically about multiplicative arithmetic.

In Topic D, students apply odd and even number properties and divisibility rules to find factors and multiples. They extend this application to consider common factors and multiples and find greatest common factors and least common multiples. Students explore and discover that Euclid’s Algorithm is a more efficient way to find the greatest common factor of larger numbers and see that Euclid’s Algorithm is based on long division. The module comprises 21 lessons; four days are reserved for administering the Mid-and End-of-Module Assessments, returning the assessments, and re-mediating or providing further applications of the concepts.

The Mid-Module Assessment follows Topic B. The End-of-Module Assessment follows Topic C.

Focus Standards

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc). How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

Foundational Standards

Gain familiarity with factors and multiples.
4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1– 100 is prime or composite.

Understand the place value system.
5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
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.

Apply and extend previous understandings of multiplication and division to multiply and
divide fractions.
5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do
the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷
(1/5) = 20 because 20 × (1/5) = 4.

Focus Standards for Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.

Students use concrete representations when understanding the meaning of division and apply it to the division of fractions. They ask themselves, “What is this problem asking me to find?” For instance, when determining the quotient of fractions, students ask themselves how many sets or groups of the divisor is in the dividend. That quantity is the quotient of the problem. They solve simpler problems to gain insight into the solution. They will confirm, for example, that 10 ÷ 2 can be found determining how many groups of two are in ten. They will apply that strategy to the division of fractions. Students may use pictorial representations such as area models, array models, number lines, and drawings to conceptualize and solve problems.

MP.2 Reason abstractly and quantitatively.

Students make sense of quantities and their relationships in problems. They understand “how many” as it pertains to the divisor in a quotient of fractions problem. They understand and use connections between divisibility and the greatest common factor to apply the distributive property. Students consider units and labels for numbers in contextual problems and consistently refer to what the labels represent to make sense in the problem. Students rely on estimation and properties of operations to justify the reason for their answers when manipulating decimal numbers and their operations. Students reason abstractly when applying place value and fraction sense when determining
the placement of a decimal point.

MP.6 Attend to Precision.

Students use precise language and place value when adding, subtracting, multiplying, and dividing by multi-digit decimal numbers. Students read decimal numbers using place value. For example, 326.31 is read as three hundred twenty-six and thirty-one hundredths. Students calculate sums, differences, products, and quotients of decimal numbers with a degree of precision appropriate to the problem context.

MP.7 Look for and make use of structure.

Students find patterns and connections when multiplying and dividing multi-digit decimals. For instance, they use place value to recognize that the quotient of: 22.5÷0.15, is the same as the quotient of: 2250 ÷ 15. Students recognize that when expressing the sum of two whole numbers using the distributive property, for example: 36 + 48 = 12(3+4), the number 12 represents the greatest common factor of 36 and 48 and that 36 and 48 are both multiples of 12. When dividing fractions, students recognize and make use of a related multiplication problem or create a number line and use skip counting to determine the number of times the divisor is added to obtain the dividend. Students use the familiar structure of long division to find the greatest common factor in another way, specifically the Euclidean Algorithm.

MP.8 Look for and express regularity in repeated reasoning.

Students determine reasonable answers to problems involving operations with decimals. Estimation skills and compatible numbers are used. For instance, when 24.385 is divided by 3.91, students determine that the answer will be close to the quotient of 24 ÷ 4, which equals 6. Students discover, relate, and apply strategies when problem-solving, such as the use of the distributive property to solve a multiplication problem involving fractions and/or decimals (e.g., 350 × 1.8= 350(1+0.8)= 350 + 280 = 630). When dividing fractions, students may use the following reasoning: Since 2/7+2/7+2/7=6/7, then 6/7÷2/7=3; and so I can solve fraction division problems by first getting common denominators and then solving the division problem created by the numerators. Students understand the long-division algorithm and the continual breakdown of the dividend into different place value units. Further, students use those repeated calculations and reasoning to determine the greatest common factor of two numbers using the Euclidean Algorithm.

Terminology ` New or Recently Introduced Terms

 Greatest Common Factor

The largest quantity that factors evenly into two or more integers; the GCF of 24 and 36 is 12 because when all of the factors of 24 and 36 are listed, the largest factor they share is 12.

 Least Common Multiple

The smallest quantity that is divisible by two or more given quantities without a remainder; the LCM of 4 and 6 is 12 because when the multiples of 4 and 6 are listed, the
smallest or first multiple they share is 12.

 Multiplicative Inverses

Two numbers whose product is 1 are multiplicative inverses of one another. For example, 3/4  and 4/3 are multiplicative inverses of one another because  3/4 x 4/3 = 4/3 x 3/4 = 1.

Topic A: Dividing Fractions by Fractions  6.NS.1

In Topic A, students extend their previous understanding of multiplication and division to divide fractions by fractions. Students determine quotients through visual models, such as bar diagrams, tape diagrams, arrays, and number line diagrams. They construct division stories and solve word problems involving division of fractions (6.NS.1). Students understand and apply partitive division of fractions to determine how much is in each group. They explore real-life situations that require them to ask themselves, “How much is one share?” and “What part of the unit is that share?” Students use measurement to determine quotients of fractions. They are presented conceptual problems where they determine that the quotient represents how many of the divisor is in the dividend. Students look for and uncover patterns while modeling quotients of fractions to ultimately discover the relationship between multiplication and division. Later in the module, students will understand and apply the direct correlation of division of fractions to division of decimals.

Lessons 1–2: Interpreting Division of a Whole Number by a Fraction—Visual Models

 Students use visual models such as fraction bars, number lines, and area models to show the quotient of whole numbers and fractions. Students use the models to show the connection between those models and the multiplication of fractions.
 Students divide a fraction by a whole number

 Students understand the difference between a whole number being divided by a fraction and a fraction being divided by a whole number

Lessons 3–4: Interpreting and Computing Division of a Fraction by a Fraction—More Models

 Students use visual models such as fraction bars and area models to show the division of fractions by fractions with common denominators.
 Students make connections to the multiplication of fractions. In addition, students understand that the division of fractions require students to ask, “How many groups of the divisor are in the dividend?” to get the quotient.

 Students use visual models such as fraction bars and area models to divide fractions by fractions with different denominators.
 Students make connections between visual models and multiplication of fractions

Lesson 5: Creating Division Stories

 Students demonstrate further understanding of division of fractions when they create their own word problems.
 Students choose a measurement division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they discover that they must try several situations and units before finding which are realistic with given numbers.

Lesson 6: More Division Stories

 Students demonstrate further understanding of division of fractions when they create their own word problems.
 Students choose a partitive division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they practice trying several situations and units before finding which are realistic with given numbers.

Lesson 7: The Relationship Between Visual Fraction Models and Equations

 Students formally connect models of fractions to multiplication through the use of multiplicative inverses as they are represented in models.

Lesson 8: Dividing Fractions and Mixed Numbers

 Students divide fractions by mixed numbers by first converting the mixed numbers into a fraction with a value larger than one.
 Students use equations to find quotients.

Topic B: Multi‐Digit Decimal Operations—Adding, Subtracting, and Multiplying 6.NS.3

Prior to division of decimals, students will revisit all decimal operations in Topic B. Students have had extensive experience of decimal operations to the hundredths and thousandths (5.NBT.7), which prepares them to easily compute with more decimal places. Students begin by relating the first lesson in this topic to mixed numbers from the last lesson in Topic A. They find that sums and differences of large mixed numbers can be more efficiently determined by first converting to a decimal and then applying the standard algorithms (6.NS.3). Within decimal multiplication, students begin to practice the distributive property. Students use arrays and partial products to understand and apply the distributive property as they solve multiplication problems involving decimals. Place value enables students to determine the placement of the decimal point in products and recognize that the size of a product is relative to each factor. Students discover and use connections between fraction multiplication and decimal multiplication.

Lesson 9: Sums and Differences of Decimals

 Students relate decimals to mixed numbers and round addends, minuends, and subtrahends to whole numbers in order to predict reasonable answers.
 Students use their knowledge of adding and subtracting multi-digit numbers to find the sums and differences of decimals.
 Students understand the importance of place value and solve problems in real-world contexts.

Lesson 10: The Distributive Property and Products of Decimals

 Through the use of arrays and partial products, students strategize and apply the distributive property to find the product of decimals.

Lesson 11: Fraction Multiplication and the Products of Decimals

Students use estimation and place value to determine the placement of the decimal point in products and to determine that the size of the product is relative to each factor.
 Students discover and use connections between fraction multiplication and decimal multiplication.
 Students recognize that the sum of the number of decimal digits in the factors yields the decimal digits in the product.

Topic C: Dividing Whole Numbers and Decimals 6.NS.2, 6.NS.3

In Topic C, students build from their previous learning to fluently divide numbers and decimals. They apply estimation to place value and determine that the standard algorithm is simply a tally system arranged in place value columns (6.NS.2). Students understand that when they “bring down” the next digit in the algorithm, they are distributing, recording, and shifting to the next place value. They understand that the steps in the algorithm continually provide better approximations to the answer. Students further their understanding of division as they develop fluency in the use of the standard algorithm to divide multi-digit decimals (6.NS.3). They make connections to division of fractions and rely on mental math strategies in order to implement the division algorithm when finding the quotients of decimals.

Lesson 12: Estimating Digits in a Quotient

 Students connect estimation with place value in order to determine the standard algorithm for division

Lesson 13: Dividing Multi-Digit Numbers Using the Algorithm

 Students understand that the standard algorithm of division is simply a tally system arranged in place value columns.

Lesson 14: The Division Algorithm—Converting Decimal Division into Whole Number Division Using Fractions

 Students use the algorithm to divide multi-digit numbers with and without remainders. Students compare their answer to estimates to justify reasonable quotients.
 Students understand that when they “bring down” the next digit in the algorithm, they are distributing, recording, and shifting to the next place value.

Lesson 15: The Division Algorithm—Converting Decimal Division into Whole Number Division Using Mental Math

 Students use their knowledge of dividing multi-digit numbers to solve for quotients of multi-digit decimals.
 Students understand the mathematical concept of decimal placement in the divisor and the dividend and its connection to multiplying by powers of 10.

Topic D: Number Theory—Thinking Logically About Multiplicative Arithmetic 6.NS.4

Students have previously developed facility with multiplication and division. They now begin to reason logically about them in Topic D. Students apply odd and even number properties and divisibility rules to find factors and multiples. They extend this application to consider common factors and multiples and find greatest common factors and least common multiples. Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division.

Lesson 16: Even and Odd Numbers

 Students apply odd and even numbers to understand factors and multiples

Lesson 17: Divisibility Tests for 3 and 9

 Students apply divisibility rules, specifically for 3 and 9, to understand factors and multiples

Lesson 18: Least Common Multiple and Greatest Common Factor

 Students find the least common multiple and greatest common factor and apply factors to the Distributive Property

Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm

 Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division.

Core K-5 – Number and Operations in Base Ten – Progression Overview

 

 Review – focussing on the completed thought in this leftmost column.   Test your recall by hiding this column.
1     Students’ work in the ____ ____ is intertwined with their work on counting and cardinality, and with the meanings and properties of addition, subtraction, multiplication, and division. base-ten system
2     Work in the base-ten system relies on these meanings and ____ , but also contributes to deepening students’ understanding of them. properties
3     Position
4    The base-ten system is a remarkably efficient and uniform system for systematically representing ____ ____. all numbers
5    Using only the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, every number can be represented as a ____ ____ ____, where each digit represents a value that depends on its place in the string. string of digits
6    The relationship between ____ ____ by the places in the base-ten system is the same for whole numbers and decimals: the value represented by each place is always 10 times the value represented by the place to its immediate right. values represented
6a   The relationship between values represented by the places in the base-ten system is the same for ____ ____ and decimals: the value represented by each place is always 10 times the value represented by the place to its immediate right. whole numbers
6b   The relationship between values represented by the places in the base-ten system is the same for whole numbers and ____: the value represented by each place is always 10 times the value represented by the place to its immediate right. decimals
6c   The relationship between values represented by the places in the base-ten system is the same for whole numbers and decimals: the value represented by each place is always ____ times the value represented by the place to its immediate right. 10
7    In other words, moving one place to the ____, the value of the place is multiplied by 10. left,
8    In other words, moving one place to the left, the value of the place is ____ by 10. multiplied
9    Because of this uniformity, ____ ____ for computations within the base-ten system for whole numbers extend to decimals standard algorithms
10   Base-ten units 

11    Each ____ of a base-ten numeral represents a base-ten unit: ones, tens, tenths, hundreds, hundredths, etc.

place
12    The ____ in the place represents 0 to 9 of those units. digit
13    Because ____ ____ ____ make a unit of the next highest value, only ten digits are needed to represent any quantity in base ten. ten like units
14    The basic unit is a ____ (represented by the right-most place for whole numbers). one
15    In learning about whole numbers, children learn that ten ones compose a new kind of unit called a ____ . ten
16    They understand ____ ____ numbers as composed of tens and ones, and use this understanding in computations, decomposing 1 ten into 10 ones and composing a ten from 10 ones. two-digit
17    The power of the base-ten system is in repeated bundling by ____: 10 tens make a unit called a “____”. ten, hundred
18    Repeating this process of creating new units by bundling in ____ ____ ____ creates units called thousand, ten thousand, hundred thousand groups of ten
19    In learning about decimals, children ____ a one into 10 equal-sized smaller units, each of which is a tenth. partition
18    Each base-ten unit can be understood in terms of any other ____ ____. base-ten unit
20    For example, ____ ____ can be viewed as a tenth of a thousand, 10 tens, 100 ones, or 1.000 tenths. one hundred
21    ____  for operations in base ten draw on such relationships among the base ten units. Algorithms
22    Computations
23    Standard algorithms for base-ten computations with the four operations rely on ____ numbers written in base-ten notation into base-ten units. decomposing
24    The properties of operations then allow any multi-digit computation to be reduced to a collection of ____  computations. single-digit
25    These single-digit computations sometimes require the ____ ____ ____ of a base-ten unit. composition or decomposition
31    Strategies and algorithms
32    In learning about ____, children partition a one into ____  equal-sized smaller units, each of which is a ____. decimals, 10, tenth
33    Each base-ten unit can be understood in terms of any other ____ ____. base-ten unit
34    For example, ____ hundred can be viewed as a tenth of a thousand,10 tens, 100 ones, or1000 tenths. one
35    Algorithms for operations in base ten draw on such ____  among the base-ten units. relationships
36    Standard algorithms for base-ten computations with the ____ ____  rely on decomposing numbers written in base-ten notation into base-ten units. four operations
37    The properties of operations then allow any ____  computation to be reduced to a collection of single-digit computations. multi-digit
38    These single-digit computations sometimes require the ____ or decomposition of a base-ten unit. composition
39    Beginning in ____, the requisite [computation] abilities develop gradually over the grades. Kindergarten
40    Experience with addition and subtraction made within ____ is a Grade 1 standardand fluency is a Grade 2 standard. 20
41    Computations within 20 that “____ ____,” such as 9 + 8 or 13 – 7 , are especially relevant to NBT because they afford the development of the Level 3 make-a-ten strategies for addition and subtraction described in the OA Progression. cross 10

42    From the NBT perspective, _____ strategies are (implicitly) the first instances of composing or decomposing a base-ten unit.

make-a-ten
43    Such strategies are a  foundation for understanding in Grade 1 that addition may require ____  a tenand in Grade 2 that subtraction may require ____ a ten. composing, decomposing
44    The Standards distinguish ____ from algorithms. strategies
45    For example, students use strategies for addition and subtraction in Grades K-3, but are expected to fluently ____ ____ ____  whole numbers using standard algorithms by the end of Grade 4. add and subtract
46    Use of the ____ ____ can be viewed as the culmination of a long progression of reasoning about quantities, the base-ten system, and the properties of operations. standard algorithms
47    This progression distinguishes between two types of computational strategies: ____ strategies and ____ methods. special, general
48    For ample, a special strategy for computing 398 + 17 is to ____ 17 as 2 + 15 and evaluate (398 + 2) + 15. decompose
49    Special strategies either cannot be ____  to all numbers represented in the base-ten system or require considerable modification in to do so. extended
51    A more readily generalizable method of computing 398 + 17 is to ____ like base-ten units. combine
50    General methods extend to all numbers represented in the base-ten system. A ____ method is not necessarily efficient. general
51    For example, ____ on by ones is a general method that can be easily modified for use with finite decimals. counting
52    General methods based on ____ ____, however, are more efficient and can be viewed as closely connected with standard algorithms. place value
53    Mathematical Practices
51    Both general methods and special strategies are opportunities to ____ competencies relevant to the NBT standards. develop
52    Use and discussion of both types of strategies offer opportunities for developing ____ with place value and properties of operations, and to use these in justifying the correctness of computations (MP 3) fluency
53    Special strategies may be advantageous in situations that require ____ computation, but less so when uniformity is useful. quick
54    Thus, they offer opportunities to raise the topic of ____  appropriate tools strategically (MP 5) using
55    Standard algorithms can be viewed as expressions of regularity in ____ ____ (MP.8) used in general methods based on place value. repeated reasoning
56    ____ ____ and recordings of computations, whether with strategies or standard algorithms, afford opportunities for students to contextualize, probing into the referents for the symbols involved (MP.2). Numerical expressions
57    Representations such as bundled objects or math drawings (e.g., drawings of hundreds, tens, and ones) and diagrams (e.g., simplified renderings of arrays or area models) afford the mathematical practice of explaining ____ among different representations (MP.1). correspondences
58    Drawings, diagrams, and numerical recordings may raise questions related to ____ (MP.6), e.g., does that 1represent 1 one or 1 ten? precision
59    This progression gives examples of ____ that can be used to connect _____ with quantities and to connect numerical representations with combination, composition ,and decomposition of base-ten units as students work towards computational fluency. representations, numerals