1

20

In Module 1, students’ understanding of the patterns in the base ten system are extended to decimals to the thousandths place.  Students deepen their knowledge through a more generalized understanding of the relationships between and among adjacent places on the place value chart, e.g., 1 tenth times any digit on the place value chart moves it one place value to the right.  Toward the module’s end students apply these new understandings as they reason about and perform decimal operations through the hundredths place.

• I can recognize that the value of a digit changes by a factor of 10 when it moves by one place value, having a value 10 times greater when it moves left and 1/10 of the value when it moves to the right. Students do not need to extend this knowledge to moving multiple places values (e.g.knowing that moving a digit two places values to the left increases its value 100 times)

• I can explain patterns in the number of zeros of a product when multiplying a whole number by a power of 10

• I can explain patterns in the placement of the decimal point when multiplying or dividing a decimal by a power of 10.

• I can read and write decimals to thousandths using base-ten numerals (standard form), number names, and expanded form (including both fraction and decimal format), e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

• I can convert numbers between forms.

• I can compare decimals to the thousandths using place value understanding. Use < , >,  and = symbols to record comparisons.

• I can convert between number forms in order to compare decimals.

• I can use place value understanding to round decimals to any place.

• I can add and subtract 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; I can relate these strategies to a written method.

2

35

In Module 2 students apply patterns of the base ten system to mental strategies and a sequential study of multiplication via area diagrams and the distributive property leading to fluency with the standard algorithm.  A similar sequence for division begins concretely with number disks as an introduction to division with multi-digit divisors and leads students to divide multi-digit whole number dividends by two-digit divisors using a vertical written method.  The standard algorithm for division is NOT an expectation in grade 5, that comes in grade 6.  Students apply the work of the module to solve multi-step word problems.  An emphasis on the reasonableness of both products and quotients and an interpretation of remainders. 

• I can explain patterns in the number of zeros of a product when multiplying a whole number by a power of 10.

• I can use the standard algorithm to multiply numbers including 2 digit X 4 digit and 3 digit X 4 digit whole numbers.

• I can find whole-number quotients of whole numbers with up to four-digit dividends and one-digit divisors or two digit divisors  using strategies based on place value and/or the distributive property, models, and strategies based on the relationship between multiplication and division

3

22

In Module 3, students' understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. Students will learn how to model and solve addition and subtraction of fractions problems with any denominators by exploring equivalency of fractions and patterns discovered from this exploration.

• I can explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) using visual fraction models.

• I can recognize and generate equivalent fractions using the principle a / b = (n x a)/(n x b) including fractions greater than 1.

• I can rename 2 fractions with unlike denominators as equivalent fractions with like denominators.

• I can add and subtract 2 proper fractions and/or mixed numbers with unlike denominators by renaming them as equivalent fractions with like denominators.

• I can add and subtract 3 proper fractions or mixed numbers with unlike denominators by renaming them as equivalent fractions with like denominators.

• I can add proper fractions or mixed numbers in problems involving regrouping an additional whole.

• I can subtract mixed numbers in problems involving regrouping from the whole.

4

38

Grade 5’s Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions. Students interpret fractions as division and reason about finding fractions of sets.  Students learn fraction by fraction multiplication in both fraction and decimal forms.  An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa.  Students are introduced to the work of division with fractions and decimal fractions.  Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers.  Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow students to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients.

• I can  interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).

• I can solve whole-number division problems in which the remainder is represented as the numerator and the divisor is represented as the denominator in the answer.

• I can use visual fraction models or equations to represent a whole-number division problem where the answer involves fractions.

• I can I interpret the product (a/b) × q as a parts of a partition of q into b equal parts, where q represents either a whole number or a fraction

• I can create visual models, such as tape diagrams, to demonstrate multiplication of fractions by whole numbers, fractions by fractions, and problems involving mixed numbers.

• I can multiply decimals to hundredths, using concrete models or drawings (such as area models) and strategies based on place value and/or properties of operations; relate these strategies to a written method.

• I can recognize that (a/b) × (c/d) = ac/bd.

• I can interpret multiplication as scaling (resizing) by 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.

• I can  interpret division of a whole number by a unit fraction by creating visual models, story contexts, and equivalent multiplication expressions.

• I can interpret division of a unit fraction by a non-zero whole number by creating visual models, story contexts, and equivalent multiplication expressions.

• I can compute quotients of a unit fraction and a non-zero whole number.

• I can compute quotients of a whole number and a unit fraction.

• I can divide decimals to hundredths, using concrete models or drawings and strategies based on place value and/or properties of operations; relate these strategies to a written method.

5

25

In module 5, students work with two- and three-dimensional figures.  Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms.  The second half of the module turns to extending students’ understanding of two-dimensional figures.  Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths.  They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes.  This module fills a gap between Grade 4’s work with two-dimensional figures and Grade 6’s work with volume and area.

• I can recognize volume as an attribute of solid figures.

• I can identify that a cube with side length 1 unit is called a unit cube and can be used to measure volume. 

• I can know that 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.

• I can figure out the volume of a specific three-dimensional figure by experiment or by counting unit cubes.

• I can find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes

• I can understand and visualize "hidden" cubes and layers.

• I can demonstrate that packing a right rectangular prism with cubes, multiplying the dimension lengths, and multiplying the height by the area of the base all produce equivalent measurements of volume.

• I can represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

• I can apply the formula V = l x w x h to find volumes or right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.

• I can apply the formula V = B x h (where B stands for the area of the base) to find volumes or right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.

• I can recognize volume as additive.

• I can apply the technique of adding the volumes of non-overlapping parts to solve real-world problems.

• I can find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts.

• I can find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths.

• I can show that the area found using the method above is the same as would be found by multiplying the side lengths.

• I can multiply fractional side lengths to find areas of rectangles.

• I can represent fraction products as rectangular areas.

• I can multiply mixed numbers by whole numbers, fractions, and mixed numbers.

• I can use visual models to represent and solve a fraction or mixed-number multiplication problem.

• I can use equations to represent and solve a fraction or mixed-number multiplication problem.

• I can understand that attributes belonging to a category of triangles or quadrilaterals also belong to all subcategories of that category.

• I can create a line of reasoning to explain how attributes of two-dimensional figures spread through categories. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

• I can classify quadrilaterals in a hierarchy based on properties: for example, squares belong to the category of rectangles; rectangles belong to the category of parallelograms; parallelograms belong to the category of quadrilaterals.

• I can classify triangles in a hierarchy based on properties: for example, equilateral triangles belong to the category of isosceles triangles; isosceles triangles belong to the category of triangles.

6

40

In module 6, students develop a coordinate system for the first quadrant of the coordinate plane and use it to solve problems.  Students use the familiar number line as an introduction to the idea of a coordinate, and they construct two perpendicular number lines to create a coordinate system on the plane.  Students see that just as points on the line can be located by their distance from 0, the plane’s coordinate system can be used to locate and plot points using two coordinates.  They then use the coordinate system to explore relationships between points, ordered pairs, patterns, lines and, more abstractly, the rules that generate them.  This study culminates in an exploration of the coordinate plane in real world applications.

• I can 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.

• I can 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).

• I can plot points on the coordinate plane, given an ordered pair of whole-number coordinates.

• I can identify an ordered pair of whole-number coordinates, given a plotted point on a coordinate plane

• I can generate two numerical patterns using two given rules.

• I can identify apparent relationships between corresponding terms.

• I can form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

• I can 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.

• I can 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).

• I can plot points on the coordinate plane, given an ordered pair of whole-number coordinates.

• I can identify an ordered pair of whole-number coordinates, given a plotted point on a coordinate plane.