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)

Expressions and Equations – Grade 6 Module 4

Extend your arithmetic work to include using letters to represent numbers. Understand that letters are simply “stand-ins” for numbers and that arithmetic is carried out exactly as it is with numbers.

Expressions and Equations

Module Overview .

In Module 4, students extend their arithmetic work to include using letters to represent numbers. Students understand that letters are simply “stand-ins” for numbers and that arithmetic is carried out exactly as it is with numbers. Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions because nothing has changed—they are still doing arithmetic with numbers. Students determine that letters are used to represent specific but unknown numbers and are used to make statements or identities that are true for all numbers or a range of numbers. Students understand the importance of specifying units when defining letters.

Students say, “Let K= Karolyn’s weight in pounds” instead of “Let K= Karolyn’s weight” because weight cannot be a specific number until it is associated with a unit, such as pounds, ounces, grams, etc. They also determine that it is inaccurate to define K as Karolyn because Karolyn is not a number. Students conclude that in word problems, each letter (or variable) represents a number and its meaning is clearly stated.

Topic A: Relationships of the Operations (6.EE.A.3)

To begin this module, students are introduced to important identities that will be useful in solving equations and developing proficiency with solving problems algebraically. In Topic A, students understand the relationships of operations and use them to generate equivalent expressions (6.EE.A.3).

By this time, students have had ample experience with the four operations since they have worked with them from kindergarten through Grade 5 (1.OA.B.3, 3.OA.B.5). The topic opens with the opportunity to clarify those relationships, providing students with the knowledge to build and evaluate identities that are important for solving equations.

In this topic, students discover and work with the following identities: w-x+x=w, w+x-x=w , a÷b∙b=a, a∙b÷b=a   (when b≠0), and 3x=x+x +x.  Students will also discover that if 12÷x=4, then 12÷x−x−x−x=0.

Lesson 1: The Relationship of Addition and Subtraction

Students build and clarify the relationship of addition and subtraction by evaluating identities such as w−x+x=w and w+x−x=w.

Lesson 2: The Relationship of Multiplication and Division

Students build and clarify the relationship of multiplication and division by evaluating identities such as a÷b∙b=a and a∙b÷b=a.

Lesson 3: The Relationship of Multiplication and Addition

Students build and clarify the relationship of multiplication and addition by evaluating identities such as 3∙g=g+g+g.

Lesson 4: The Relationship of Division and Subtraction

Students build and clarify the relationship of division and subtraction by determining that 12÷x=4 means 12−x−x−x−x=0

Topic B: Special Notations of Operations (6.EE.A.1, 6.EE.A.2c)

In Topic B, students experience special notations of operations. They determine that 3x=x+x+x is not the same as x~3, which is x∙x∙x.

Applying their prior knowledge from Grade 5, where whole number exponents were used to express powers of ten (5.NBT.A.2), students examine exponents and carry out the order of operations, including exponents.

Students demonstrate the meaning of exponents to write and evaluate numerical expressions with whole number exponents (6.EE.A.1).

Lesson 5: Exponents

 Students discover that 3x=x+x+x is not the same thing as x^3 which is x·x·x.
 Students understand that a base number can be represented with a positive whole number, positive fraction, or positive decimal and that for any number a, we define a to be the product of m factors of a. The number a is the base and m is called the exponent or power of a.

Lesson 6: The Order of Operations

Students evaluate numerical expressions. They recognize that in the absence of parentheses, exponents are evaluated first.

Topic C: Replacing Letters and Numbers (6.EE.A.2c, 6.EE.A.4)

Students represent letters with numbers and numbers with letters in Topic C. In past grades, students discovered properties of operations through example (1.OA.B.3, 3.OA.B.5). Now, they use letters to represent numbers in order to write the properties precisely. Students realize that nothing has changed because the properties still remain statements about numbers. They are not properties of letters, nor are they new rules introduced for the first time.

Now, students can extend arithmetic properties from manipulating numbers to manipulating expressions. In particular, they develop the following identities: a∙b=b∙a, a+b=b+a, gg∙1=g, g+0=g, g÷1=g, g÷g=1, and 1÷g=1/g. Students understand that a letter in an expression represents a number. When that number replaces that letter, the expression can be evaluated to one number.

Similarly, they understand that a letter in an expression can represent a number. When that number is replaced by a letter, an expression is stated (6.EE.A.2).

Lesson 7: Replacing Letters with Numbers

Students understand that a letter represents one number in an expression. When that number replaces the letter, the expression can be evaluated to one number.

Lesson 8: Replacing Numbers with Letters

 Students understand that a letter in an expression or an equation can represent a number. When that number is replaced with a letter, an expression or an equation is stated.
 Students discover the commutative properties of addition and multiplication, the additive identity property of zero, and the multiplicative identity property of one. They determine that g÷1=g, g÷g=1, and 1÷g=1/g.

Topic D: Expanding, Factoring, and Distributing Expressions(6.EE.A.2a, 6.EE.A.2b, 6.EE.A.3, 6.EE.A.4)

In Topic D, students become comfortable with new notations of multiplication and division and recognize their equivalence to the familiar notations of the prior grades. The expression 2 × b is exactly the same as 2∙b and both are exactly the same as 2b. Similarly, 6÷2 is exactly the same as 6/2. These new conventions are practiced to automaticity, both with and without variables.

Students extend their knowledge of GCF and the distributive property from Module 2 to expand, factor, and distribute expressions using new notation (6.NS.B.4). In particular, students are introduced to factoring and distributing as algebraic identities. These include: a+a=2∙a=2a, (a+b)+(a+b)=2∙(a+b)=2(a+b)=2a+2b, and a÷b=a/b.

Lesson 9: Writing Addition and Subtraction Expression

Students write expressions that record addition and subtraction operations with numbers.

Lesson 10: Writing and Expanding Multiplication Expressions

Students identify parts of an expression using mathematical terms for multiplication. They view one or more parts of an expression as a single entity.

Lesson 11: Factoring Expressions

Students model and write equivalent expressions using the distributive property. They move from an expanded form to a factored form of an expression.

Lesson 12: Distributing Expressions

Students model and write equivalent expressions using the distributive property. They move from a factored form to an expanded form of an expression.
Lessons 13–14: Writing Division Expressions

Students write numerical expressions in two forms, dividend÷divisor and dividend/divisor, and note the relationship between the two.

Topic E: Expressing Operations in Algebraic Form (6.EE.A.2b)

In Topic E, students express operations in algebraic form. They read and write expressions in which letters stand for and represent numbers (6.EE.A.2). They also learn to use the correct terminology for operation symbols when reading expressions.For example, the expression 3/(2x−4) is read as “the quotient of three and the difference of twice a number and four.”

Similarly, students write algebraic expressions that record operations with numbers and letters that stand for numbers. Students determine that 3a+b can represent the story “Martina tripled her money and added it to her sister’s money” (6.EE.A.2b).

A Mid-Module Assessment follows Topic E.

Lesson 15: Read Expressions in Which Letters Stand for Numbers

 Students read expressions in which letters stand for numbers. They assign operation terms to operations when reading.
 Students identify parts of an algebraic expression using mathematical terms for all operations.

Lessons 16–17: Write Expressions in Which Letters Stand for Numbers

Students write algebraic expressions that record all operations with numbers and letters standing for the numbers.

Mid-Module Assessment and Rubric

Topics A through E (assessment 1 day, return 1 day, remediation or further applications 3 days)
1 Each lesson is ONE day and ONE day is considered a 45 minute period.

Topic F: Writing and Evaluating Expressions and Formulas (6.EE.A.2, 6.EE.A.2c, 6.EE.B.5)

Students write and evaluate expressions and formulas in Topic F. They use variables to write expressions and evaluate those expressions when given the value of the variable (6.EE.A.2).

From there, students create formulas by setting expressions equal to another variable. For example, if there are 4 bags containing c colored cubes in each bag with 3 additional cubes, students use this information to express the total number of cubes as 4c+3. From this expression, students develop the formula t=4c+3, where t is the total number of cubes. Once provided with a value for the amount of cubes in each bag (c=12 cubes), students can evaluate the formula for t: t=4(12)+3,  Students continue to evaluate given formulas such as the volume of a cube,V=s^3    given the side length, or the volume of a rectangular prism, V=l⋅w⋅ℎ given those dimensions (6.EE.A.2c).

Lesson 18: Writing and Evaluating Expressions—Addition and Subtraction (P)1

 Students use variables to write expressions involving addition and subtraction from real-world problems.
 Students evaluate these expressions when given the value of the variable.

Lesson 19: Substituting to Evaluate Addition and Subtraction Expressions (P)

 Students develop expressions involving addition and subtraction from real-world problems.
 Students evaluate these expressions for given values.

Lesson 20: Writing and Evaluating Expressions—Multiplication and Division (P)

 Students develop expressions involving multiplication and division from real-world problems.
 Students evaluate these expressions for given values.

Lesson 21: Writing and Evaluating Expressions—Multiplication and Addition (P)

 Students develop formulas involving multiplication and addition from real-world problems.
 Students evaluate these formulas for given values.

Lesson 22: Writing and Evaluating Expressions—Exponents (P)

Students evaluate and write formulas involving exponents for given values in real-world problems

Topic G   Solving Equations    6.EE.5, 6.EE.6, 6.EE.7

In Topic G, students are introduced to the fact that equations have a structure similar to some grammatical sentences. Some sentences are true: “George Washington was the first president of the United States.” or “2+3=5.” Some are clearly false: “Benjamin Franklin was a president of the United States.” or “7+3=5.” Sentences that are always true or always false are called closed sentences. Some sentences need additional information to determine whether they are true or false. The sentence “She is 42 years old” can be true or false depending on who “she” is. Similarly, the sentence “x+3=5” can be true or false depending on the value of x. Such sentences are called open sentences. An equation with one or more variables is an open sentence. The beauty of an open sentence with one variable is that if the variable is replaced with a number, then the new sentence is no longer open: it is either clearly true or clearly false. For example, for the open sentence x+3=5:
If x is replaced by 7, the new closed sentence, 7+3=5, is false because 10≠5.
If x is replaced by 2, the new closed sentence, 2+3=5, is true because 5=5.
From here, students conclude that solving an equation is the process of determining the number(s) when substituted for the variable, result in a true sentence (6.EE.B.5). In the previous example, the solution for x+3=5 is obviously 2. The extensive use of bar diagrams in Grades K–5 makes solving equations in Topic G a fun and exciting adventure for students. Students solve many equations twice, once with a bar diagram and once using algebra. They use identities and properties of equality that were introduced earlier in the module to solve one-step, two-step, and multistep equations. Students solve problems finding the measurements of missing angles represented by letters (4.MD.C.7) using what they learned in Grade 4 about the four operations and what they now know about equations.

In Topic G, students move from identifying true and false number sentences to making true number sentences false and false number sentences true. In Lesson 23, students explain what equality and inequality symbols represent. They determine if a number sentence is true or false based on the equality or inequality symbol.

In Lesson 24, students move to identifying a value or a set of values that make number sentences true. They identify values that make a true sentence false. For example, students substitute 4 for the variable in x + 12 = 14 to determine if the sentence is true or false. They note that when 4 is substituted for x, the sum of x + 12 is 16, which makes the sentence false because 16 ≠ 14. They change course in the lesson to find what they can do to make the sentence true. They ask themselves, “What number must we add to 12 to find the sum of 14?” By substituting 2 for x, the sentence becomes true because x + 12 = 14, 2 + 12 = 14, and 14 = 14.

They bridge this discovery to Lesson 25 where students understand that the solution of an equation is the value or values of the variable that makes the equation true. Students begin solving equations in Lesson 26. They use bar models or tape diagrams to depict an equation and apply previously learned properties of equality for addition and subtraction to solve the equation. Students check to determine if their solution makes the equation true.

Given the equation  1 + a = 6  students recognize that the solution can also be found using properties of operations. They make connections to the bar model and determine that 1 +a − 1 =6 − 1 and, ultimately, that a = 5. Students represent two step and multi-step equations involving all operations with bar models or tape diagrams while continuing to apply properties of operations and the order of operations to solve equations in the remaining lessons in this topic.

Lessons 23–24: True and False Number Sentences (P, P)1

Students explain what the equality and inequality symbols including =, <, >, ≤, and ≥ stand for. They determine if a number sentence is true or false based on the given symbol.

 Students identify values for the variable in an equation and inequality that result in true number sentences.
 Students identify values for the variable in an equation and inequality that result in false number sentences.Lesson 25: Finding Solutions to Make Equations True (P)

Students learn the definition of solution in the context of placing a value into a variable to see if it makes the equation true.

Lesson 26: One-Step Equations—Addition and Subtraction (M)

 Students solve one-step equations by relating an equation to a diagram.

 Students check to determine if their solution makes the equation true.

Lesson 27: One-Step Equations—Multiplication and Division (E)

 Students solve one-step equations by relating an equation to a diagram.
 Students check to determine if their solution makes the equation true.

Lesson 28:Two-Step Problems—All Operations (M)

 Students calculate the solution of one-step equations by using their knowledge of order of operations and the properties of equality for addition, subtraction, multiplication, and division. Students employ tape diagrams to determine their answer.
 Students check to determine if their solution makes the equation true.

Lesson 29: Multi-Step Problems—All Operations (P)

 Students use their knowledge of simplifying expressions, order of operations and properties of equality to calculate the solution of multi-step equations. Students use tables to determine their answer.
 Students check to determine if their solution makes the equation true.

Topic H: Applications of Equations (6.EE.B.5, 6.EE.B.6, 6.EE.B.7, 6.EE.B.8, 6.EE.C.9)

In Topic H, students use their prior knowledge from Module 1 to contruct tables of independent and dependent values in order to analyze equations with two variables from real-life contexts. They represent equations by plotting the values from the table on a coordinate grid (5.G.A.1, 5.G.A.2, 6.RP.A.3a, 6.RP.A.3b, 6.EE.C.9). The module concludes with students referring to true and false number sentences in order to move from solving equations to writing inequalities that represent a constraint or condition in real-life or mathematical problems (6.EE.B.5, 6.EE.B.8). Students understand that inequalities have infinitely many solutions and represent those solutions on number line diagrams.

In Topic H, students apply their knowledge from the entire module to solve equations in real-world, contextual problems. In Lesson 30, students use prior knowledge from Grade 4 to solve missing angle problems. Students write and solve one step equations in order to determine a missing angle. Lesson 31 involves students using their prior knowledge from Module 1 to contruct tables of independent and dependent values in order to analyze equations with two variables from real-life contexts. They represent equations by plotting values from the tables on a coordinate grid in Lesson 32. The module concludes with Lessons 33 and 34, where students refer to true and false number sentences in order to move from solving equations to writing inequalities that represent a constraint or condition in real-life or mathematical problems. Students understand that inequalities have infinitely many solutions and represent those solutions on number line diagrams.
An End-of-Module Assessment follows Topic H.

Lesson 30: One-Step Problems in the Real World

Students calculate missing angle measures by writing and solving equations

Lesson 31: Problems in Mathematical Terms

 Students analyze an equation in two variables to choose an independent variable and dependent variable. Students determine whether or not the equation is solved for the second variable in terms of the first variable or vice versa. They then use this information to determine which variable is the independent variable and which is the dependent variable.
 Students create a table by placing the independent variable in the first row or column and the dependent variable in the second row or column. They compute entries in the table by choosing arbitrary values for the independent variable (no constraints) and then determine what the dependent variable must be.

Lesson 32: Multi-Step Problems in the Real World

 Students analyze an equation in two variables, choose an independent variable and a dependent variable, make a table and make a graph for the equation by plotting the points in the table. For the graph, the independent variable is usually represented by the horizontal axis and the dependent variable is usually represented by the vertical axis.

Lesson 33: From Equations to Inequalities

 Students understand that an inequality with numerical expressions is either true or false. It is true if the numbers calculated on each side of the inequality sign result in a correct statement and false otherwise.
 Students understand solving an inequality is answering the question of which values from a specified set, if any, make the inequality true.

Lesson 34: Writing and Graphing Inequalities in Real-World Problems

 Students recognize that inequalities of the form x<c and x>c, where x is a variable and c is a fixed number have infinitely many solutions when the values of x come from a set of rational numbers.

End-of-Module Assessment and Rubric

Topics A through H (assessment 1 day, return 1 day, remediation or further applications 4 day)