EE.6.EE.1–2. Identify equivalent number sentences. Concept: Number sentences and equations show a relationship and can be written in different ways.
Skills: Recognize equivalent algebraic expressions; represent the unknown in an equation; use properties of operation to generate equivalent expressions involving addition, subtraction, multiplication or division; identify equivalent number sentences; use symbols for equal and not equal.
Big Idea: A number sentence uses numbers and the equal sign to show that two quantities have equal value, whereas a number expression is a math problem that uses numbers and letters to represent variables and an equals sign to show that two quantities have equal value.
Essential Questions: Do the two sides of this problem have equal value? Is this expression true (equal) or false (not equal)?
Initial Precursor • Combine sets • Compare sets
Initial Precursor: Understanding how to evaluate equations and recognize equivalent expressions requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. The educator presents a set, labels it (e.g., two balls, one marker, three CDs), counts the items, labels it again, and encourages students to use numbers to label and count the separate sets. Then, combine the sets, give it a new label, and count the set.
NOTE: Educators can work on the Initial Precursor level using the sets of numbers that students working with.
Distal Precursor • Demonstrate the concept of addition • Demonstrate the concept of subtraction
Distal Precursor: As students begin to understand labeling and counting small sets, they begin to use the number sequence, and students become more adept at tracking individual objects and can recognize when items are added to a set or when items are taken away. Work on this skill using a variety of sets, labeling and counting the set, and moving items in and out of the set labeling and counting the set again.
NOTE: Educators can work on the Distal Precursor level using the sets of numbers that students working at the Target level are working with.
Proximal Precursor • Represent addition with equations • Represent the unknown in an equation • Represent subtraction with equations
Target • Evaluate if equations are true or false • Recognize equivalent algebraic expressions
Successor • Use properties of addition to create an equivalent algebraic expression
EE.6.EE.3. Apply the properties of addition to identify equivalent numerical expressions.
Concept: Number sentences and equations show a relationship and can be written in different ways.
Skills: Recognize equivalent algebraic expressions; represent the unknown in an equation; use properties of operation to generate equivalent expressions involving addition, subtraction, multiplication or division; identify equivalent number sentences; use symbols for equal and not equal.
Big Idea: A number sentence uses numbers and the equal sign to show that two quantities have equal value, whereas a number expression is a math problem that uses numbers and letters to represent variables and an equals sign to show that two quantities have equal value.
Essential Questions: Do the two sides of this problem have equal value? Is this expression true (equal) or false (not equal)?
Initial Precursor • Compare sets • Combine sets
Initial Precursor: Understanding how to evaluate equations and using the properties of addition to create equivalent expressions requires a student to be able to recognize that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. The educator presents a set, labels it (e.g., two balls, one marker, three CDs), counts the items, labels it again, and encourages students to use numbers to label and count the separate sets. Then, combine the sets, give it a new label, and count the set.
NOTE: Educators can work on the Distal Precursor level using the sets of numbers that students working at the Target level are adding and subtracting.
Distal Precursor • Represent the unknown in an equation • Represent subtraction with equations • Represent addition with equations
Distal Precursor: As students begin to understand labeling and counting small sets, they begin to use the number sequence and become more adept at tracking individual objects. Work on this skill using a variety of sets, labeling and counting the sets, and moving items in and out of the sets, labeling and counting the set again. Additionally, the educators will pair those sets with the symbolic representations for addition and subtraction (e.g., 3 + 2 = ?, 3 − 2 = ?).
NOTE: Educators can work on the Distal Precursor level using the sets of numbers that students working at the Target level are adding and subtracting.
Proximal Precursor • Evaluate if equations are true or false • Apply associative property of addition • Apply commutative property of addition
Target • Recognize equivalent algebraic expressions • Use properties of addition to create an equivalent algebraic expression
Successor • Use properties of operations to generate equivalent expressions involving addition • Use properties of operations to generate equivalent expressions involving subtraction
EE.6.EE.5–7. Match an equation to a real-world problem in which variables are used to represent numbers.
Concept: Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations.
Skills: Identify what operation is needed in the real-world problem; identify the known quantities and the unknown variable; identify the structure of the equation; match an equation to a real world-problem.
Big Idea: Letters are used in mathematics to represent generalized properties, unknowns in equations, and relationships between quantities.
Essential Questions: What operation is needed in this problem? What are the known quantities and the unknown variable in the problem? What does the variable represent? Which equations matches this problem?
Initial Precursor • Partition sets • Combine sets
Initial Precursor: The knowledge needed to solve addition and subtraction real world problems links back to an understanding of how to create sets, but it also requires learning to manipulate sets (i.e., combining and separating or partitioning). Provide students many opportunities to take a set of objects (e.g., tiles, linking cubes, buttons) and separate them based on a given characteristic (e.g., shape, color, size) into two distinct sets, and separate them again based on another characteristic. Guide students to notice how the set size changes each time the educator combines or partitions the sets.
Distal Precursor • Represent subtraction with equations • Represent addition with equations
Distal Precursor: As student understanding of combining and partitioning sets increases, educators should take care to use the words “addition” and “subtraction” while defining and demonstrating their meanings and as students combine and partition sets. While students do not need to say the words, they do need to learn the meanings. Educators provide lessons that help students represent addition and subtraction in multiple ways (e.g., using objects, fingers, drawings, sounds, acting out situations, and writing equations). Proximal Precursor • Represent expressions with variables • Represent the unknown in an equation
Target • Represent real-world problems as equations
Successor • Solve real-world problems using equations with nonnegative rational numbers
EE.7.EE.1. Use the properties of operations as strategies to demonstrate that expressions are equivalent.
Concept: Operations create relationships between numbers.
Skills: Apply the properties of operations (i.e., commutative, associative); recognize equivalent expressions (e.g., A + (B x C)= (C x B) + A, and (A+B) – C x (D x E)= (A+B) – (C x D) x E); identify arithmetic sequence with common difference (e.g., 5, 7, 9, 11, 13, 15 common difference of 2).
Big Idea: The commutative and associative properties for addition and multiplication of whole numbers allow computations to be performed flexibly. Subtraction is not commutative or associative for whole numbers. The difference between successive terms in some sequences is constant.
Essential Questions: What is the correct order for performing mathematical operations? How can the properties of operations be used to determine if two equations are equivalent? What is the difference between each of the numbers in this sequence? What is the rule for this sequence?
Initial Precursor • Partition sets • Combine sets
Initial Precursor: In order to use properties of operations, students begin by counting small units, recognizing that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. As educators present a set, they label it (e.g., two balls, one marker, three CDs), count the items, label it again, and encourage students to use numerals to label and count the separate sets. The general goal is to explore how the set changes when items are separated out (partitioned) or combined.
Distal Precursor • Model associativity of multiplication • Model additive commutativity • Model associativity of addition • Model multiplicative commutativity
Distal Precursor: As students continue developing their understanding of how sets change, educators can use manipulatives to create sets that model the associative and associative properties of addition and multiplication.
Proximal Precursor • Apply the associative property of multiplication • Apply commutative property of addition • Apply associative property of addition • Apply the commutative property of multiplication
Target • Use properties of operations to generate equivalent expressions involving subtraction • Use properties of operations to generate equivalent expressions involving addition
Successor • Use equivalent expressions in real-world context
EE.7.EE.2. Identify an arithmetic sequence of whole numbers with a whole number common difference.
Concept: Operations create relationships between numbers.
Skills: Apply the properties of operations (i.e., commutative, associative); recognize equivalent expressions (e.g., A + (B x C)= (C x B) + A, and (A+B) – C x (D x E)= (A+B) – (C x D) x E); identify arithmetic sequence with common difference (e.g., 5, 7, 9, 11, 13, 15 common difference of 2).
Big Idea: The commutative and associative properties for addition and multiplication of whole numbers allow computations to be performed flexibly. Subtraction is not commutative or associative for whole numbers. The difference between successive terms in some sequences is constant.
Essential Questions: What is the correct order for performing mathematical operations? How can the properties of operations be used to determine if two equations are equivalent? What is the difference between each of the numbers in this sequence? What is the rule for this sequence?
Initial Precursor • Classify • Contrast objects • Order objects
Initial Precursor: In order to identify arithmetic sequences, students begin by learning to recognize what is the same and different between familiar items, such as color, shape, quantity, size, texture, and pattern. Educators should take care to use attribute words (e.g., circle/square, more/less/same, rough/smooth, red, green, red, green) while defining and demonstrating their meaning. While students do not need to say these words, they do need to learn the meanings. Educators will also provide activities in which students work on grouping two or more items in the same set based on an attribute and ordering the items by size or shape.
Distal Precursor: As students develop their understanding of attributes and work toward arithmetic sequences, educators provide interactive lessons around patterns using attributes like shape, size, and color. At this level, students are also expected to recognize symbolic (letter and number) patterns. This also requires that students recognize numerals in order. (i.e., 1, 2, 3...). Educators should take care to use number names while defining and demonstrating symbolic sequences. While students do not need to say these words, they do need to learn the meanings and the sequence.
Successor • Recognize the recursive rule for arithmetic sequences
EE.7.EE.4. Use the concept of equality with models to solve one step addition and subtraction equations.
Concept: Equality means that both values on the left and the right side of the equal sign '=' will have the same value.
Skills: Use models to solve one step addition and subtraction equations (e.g., p + 12 = 12 + p, and p + 7 = 12 - 7).
Big Idea: The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality and you can subtract the same value from each side of an equation and maintain the equality.
Essential Questions: What is meant by equality in mathematics? How can I use addition or subtraction to solve one-step equations? What information do we know from the equation? What information is missing? What operation could be used to find the solution? Which representation will I use to help me solve this problem (concrete manipulatives, pictures, words, or equations)?
EE.8.EE.1. Identify the meaning of an exponent (limited to exponents of 2 and 3).
Concept: Numbers have relationships and can be written in different ways.
Skills: Identify the base and exponent; use multiplication strategies to demonstrate the meaning of exponents; solve problems involving exponents of 2 or 3; multiply by the same number each time to get the next term in the geometric sequence (e.g., 3, 6, 12..., the common ratio is 2); compose and decompose whole numbers up to three digits.
Big Idea: Exponents are notations of repeated multiplication. Geometric sequence represent multiplication or division by a common ratio (number). Numbers can be taken apart to create smaller groups or put together to create larger groups.
Essential Questions: Which number is the numbers expressed in scientific exponent? How do I represent multiplication notation, including problems where using exponents? How do I find the pattern of both decimal and scientific notation are a geometric sequence? What is the common used. Use scientific notation, and ratio between this sequence of numbers? How choose units of appropriate size for can I represent the same quantity in different measurements of very large or very ways?
Initial Precursor • Combine • Combine sets • Demonstrate the concept of addition
Initial Precursor: Recognizing exponents requires students to be able to recognize that two or more sets or groups of items exist. Educators can work on this skill using a variety of sets. Help students recognize when items are grouped together into a set and when they are separated out. The educator presents a set, labels it (e.g., two balls, one marker, three CDs), counts the items, labels it again, and encourages students to use numbers to label and count the separate sets. Then, combine the sets, give it a new label, and count the set.
Distal Precursor • Explain repeated addition • Represent repeated addition with a model • Solve repeated addition problems
Distal Precursor: As students' understanding of labeling and counting sets develops, they begin working on adding items to a set and combining sets to create a new set. Additionally, students work on developing an understanding of equal shares by actively participating in one-to-one distribution of objects to person, objects to objects, and objects to available space (e.g., giving each person in the group a pencil; given four counters, they would line up four more counters in front of or on top of the first set; given three chairs at a table, the students would place a cup on the table for each available chair). As students learn to work with sets and connect their understanding of equal shares to sets, educators can provide students experience with combining multiple sets (e.g., 3 sets with 4 counters each) and represent the problem (e.g., 4 + 4 + 4 = ?). Students also learn to represent the problem using a pencil or their communication system (e.g., the students are shown 4 equal sets each with 2 counters. The students count the first set and write a 2 or indicates 2, then write or indicate the plus sign. The students repeat for all 4 sets and then indicate the equal sign and solve the problem.).
Proximal Precursor • Demonstrate the concept of multiplication • Explain multiplication problems • Explain product
Target • Recognize exponents
Successor • Explain product of powers property of exponents • Apply zero exponent property • Explain power of product property of exponents • Explain quotient of powers property of exponents
EE.8.EE.2. Identify a geometric sequence of whole numbers with a whole number common ratio.
Concept: Numbers have relationships and can be written in different ways.
Skills: Identify the base and exponent; use multiplication strategies to demonstrate the meaning of exponents; solve problems involving exponents of 2 or 3; multiply by the same number each time to get the next term in the geometric sequence (e.g., 3, 6, 12..., the common ratio is 2); compose and decompose whole numbers up to three digits.
Big Idea: Exponents are notations of repeated multiplication. Geometric sequence represent multiplication or division by a common ratio (number). Numbers can be taken apart to create smaller groups or put together to create larger groups.
Essential Questions: Which number is the numbers expressed in scientific exponent? How do I represent multiplication notation, including problems where using exponents? How do I find the pattern of both decimal and scientific notation are a geometric sequence? What is the common used. Use scientific notation, and ratio between this sequence of numbers? How choose units of appropriate size for can I represent the same quantity in different measurements of very large or very ways?
Initial Precursor • Classify • Contrast objects • Order objects
Initial Precursor: In order to recognize geometric patterns, students begin by learning to notice what is new. The educator draws the students' attention to new objects or stimuli, labels them (e.g., “this set has all red objects; this set has all blue,” “these fidgets are big; these fidgets are small”) and the student observes, feels, or otherwise interacts with them. Educators encourage students to begin placing like objects together, drawing attention to the characteristics that make an item the same or different.
Distal Precursor: As students develop their awareness of attributes and putting like objects together, educators will draw the students' attention to patterns and sequences in numbers and letters (symbolic patterns) and allow the student to observe, feel, or otherwise interact with the patterns and sequences.
Successor • Recognize the recursive rule for geometric sequences
EE.8.EE.3–4. Compose and decompose whole numbers up to 999.
Concept: Numbers have relationships and can be written in different ways.
Skills: Identify the base and exponent; use multiplication strategies to demonstrate the meaning of exponents; solve problems involving exponents of 2 or 3; multiply by the same number each time to get the next term in the geometric sequence (e.g., 3, 6, 12..., the common ratio is 2); compose and decompose whole numbers up to three digits.
Big Idea: Exponents are notations of repeated multiplication. Geometric sequence represent multiplication or division by a common ratio (number). Numbers can be taken apart to create smaller groups or put together to create larger groups.
Essential Questions: Which number is the numbers expressed in scientific exponent? How do I represent multiplication notation, including problems where using exponents? How do I find the pattern of both decimal and scientific notation are a geometric sequence? What is the common used. Use scientific notation, and ratio between this sequence of numbers? How choose units of appropriate size for can I represent the same quantity in different measurements of very large or very ways?
EE.8.EE.5–6. Graph a simple ratio by connecting the origin to a point representing the ratio in the form of y/x. For example, when given a ratio in standard form (2:1), convert to 2/1, and plot the point (1,2).
Concept: Ratios show a comparison and can be used for mathematical reasoning.
Skills: Identify a coordinate plane and its parts; identify the origin on a coordinate plane; identify the x value and the y value on a coordinate plane; identify that the x values move left and right, and the y value moves up and down; graph the points on the plane; given a ratio, identify which number goes on the x 8.EE.6. Use similar triangles to explain axis, and which number goes on the y axis.
Big Idea: A ratio can be displayed on a graph to show a relationship between horizontal and vertical axis.
Essential Questions: What are the parts of the coordinate plane? Where is the origin? Where is the x value and the y value on a coordinate plane? Which value moves left and right? Which value moves up and down? Where would this ratio be located on the coordinate plane? Given a ratio, which number represents the y value, and which number represents the x value?
EE.8.EE.7. Solve simple algebraic equations with one variable using addition and subtraction.
Concept: Equations express a relationship that can be used to solve an unknown.
Skills: Determine the unknown in an equation; use property of inverse operation (addition/subtraction) to complete the inverse to each side of the equation; isolate the variable to solve; solve algebraic expressions using addition or subtraction.
Big Idea: Variables represent the unknown in an equation.
Essential Questions: What am I trying to figure out in this equation? What do I know about the properties of addition and subtraction that can help me solve this problem?
Initial Precursor • Combine sets • Partition sets
Initial Precursor: Solving linear equations requires a student to count small units, recognizing that two or more sets or groups of items exist. Work on this skill using a variety of sets. Help students recognize when items are grouped together into a set or separated out. The educator presents a set, labels it (e.g., two balls, one marker, three CDs), counts the items, labels it again, and encourages students to use numbers to label and count the separate sets. The general goal is to explore how the set changes when items are separated out (partitioned) or combined.
Distal Precursor • Demonstrate the concept of addition • Demonstrate the concept of subtraction
Distal Precursor: As students begin to understand labeling and counting small sets, they begin to use the number sequence and become more adept at tracking individual objects. They can recognize when items are added to a set or when items are taken away. Work on this skill using a variety of sets, labeling and counting the set, and moving items in and out of the set, labeling and counting the set again.
NOTE: Educators can work on the Distal Precursor level using the sets of numbers that students working at the Target level are working with.
Proximal Precursor • Determine the unknown in an addition equation • Determine the unknown in a subtraction equation
Target • Solve linear equations in one variable
Successor • Solve linear inequalities in 1 variable