Module 6, Topic E provides an opportunity for students to encounter complex, multi-step problems requiring the application of the concepts and skills mastered throughout the Grade 5 curriculum. Students use all four operations with both whole and fractional numbers in varied contexts. The problems in Topic E are designed to be non-routine problems that require students to persevere to solve them.

While wrestling with complexity is an important part of Topic E, the true strength of this topic is derived from the time allocated for students to construct arguments and critique the reasoning of their classmates. After students have been given adequate time to ponder and solve the problems, two lessons are devoted to the sharing of approaches and solutions. Students partner to justify their conclusions, communicate them to others, and respond to the arguments of their peers.

Content Standard(s):

Mathematics MA2019 (2019) Grade: 5

9. Model and solve real-word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, 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.

Example: Recognize an incorrect result ^{2}/_{5}+ ^{1}/_{2}= ^{3}/_{7}by observing that ^{3}/_{7} < ^{1}/_{2}.

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Evidence Of Student Attainment:

Students:

Model and solve real-world problems involving sums and differences of fractions (including mixed numbers) with unlike denominators.

Use visual models to illustrate the problem situation involving fractions.

Use fraction understanding and estimation strategies to assess the reasonableness of answers.

Teacher Vocabulary:

Fraction

Benchmark fraction

Denominator

Fraction model

Estimate

Reasonableness

Equation

Unlike denominator

Unlike units

Knowledge:

Students know:

The meaning and magnitude of fractions expressed in units of halves, fourths, eighths, thirds, sixths, twelfths, fifths, tenths, and hundredths.

Strategies to find sums of two or more fractions with like denominators.

Strategies to find the difference of two fractions with like denominators.

How to decompose a fraction greater than 1 and express as a mixed number. Example: 7/3 = 3/3 + 3/3 + 1/3 = 2 1/3.

Skills:

Students are able to:

Solve real-word problems involving addition and subtraction of fractions with unlike denominators.

Represent problems using fraction models or equations.

Assess reasonableness of answers using estimation and benchmark fractions.

Understanding:

Students understand that:

solving word problems involving addition and subtraction of fractions with unlike units

Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit.

Can be assessed for reasonableness of answers using estimation strategies.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.9.1: Add and subtract mixed numbers with like denominators.
M.5.9.2: Recognize that comparisons are valid only when the two fractions refer to the same whole.
M.5.9.3: Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
M.5.9.4: Recognize a fraction as a number on the number line; represent fractions on a number line diagram.
M.5.9.5: Recognize key terms to solve word problems.
M.5.9.6: Apply properties of operations for addition and subtraction.
M.5.9.7: Recall basic addition and subtraction facts.

Prior Knowledge Skills:

Use fraction equivalence to add and subtract fractions and mixed numbers with unlike denominators.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.9 Illustrate equivalent fractions using models of wholes, halves, thirds, and fourths to add fractions with like denominators.

Mathematics MA2019 (2019) Grade: 5

11. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

a. Model and interpret a fraction as division of the numerator by the denominator (^{a}/_{b}= a ÷ b)

b. Use visual fraction models, drawings, or equations to represent word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers

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Evidence Of Student Attainment:

Students:

Solve problems involving division of whole numbers leading to quotients of a fraction or mixed number.

Example: Given that 3 cookies are shared equally with 6 people, find what fraction of the cookies each person receives. Each person receives 3/6 of a cookie or 1/2 of a cookie. Example: Given that 3 cookies are shared equally with 2 people, find what fraction of the cookies each person receives. Each person receives 3/2 cookies or 1 1/2 cookies.

Model and interpret a fraction as division.

Use models, drawings, or equations to represent word problems.

Teacher Vocabulary:

Fraction

Numerator

Denominator

Division

Remainder

Dividend

Divisor

Knowledge:

Students know:

Contextual situations for division.

Strategies to equipartition.

Skills:

Students are able to:

Solve word problems involving division of whole numbers leading to quotients with fractions.

Use fraction models, drawings, equations to represent word problems.

Model and interpret a fraction as division.

Understanding:

Students understand that:

a ÷ b is a division expression and can be written as a/b showing division of the numerator by the denominator (including cases where the value of a < b).

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.11.1: Define a mixed number.
M.5.11.2: Generate equivalent fractions.
M.5.11.3: Recognize a fraction as a number on the number line; represent fractions on a number line diagram.

Prior Knowledge Skills:

Find products of a fraction times a whole number and products of a fraction times a fraction.

Use area models, linear models or set models to represent products.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.11 Using vocalization, sign language, augmentative communication or assistive technology, identify models of thirds (e.g., 1/3. 2/3, 3/3) and tenths (e.g., 1/10, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10, 10/10).

Mathematics MA2019 (2019) Grade: 5

14. Model and solve real-world problems involving multiplication of fractions and mixed numbers using visual fraction models, drawings, or equations to represent the problem.

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Evidence Of Student Attainment:

Students:

Use a variety of strategies, including models, pictures, tables, and patterns to solve problems that provide a context for multiplying fractions and mixed numbers.

Teacher Vocabulary:

Fraction

Models

Mixed number

Multiplication

Knowledge:

Students know:

Contextual situations for multiplication.

How to use an area model to illustrate the product of two whole numbers and its relationship to partial products and extend this knowledge to illustrate products involving fractions and mixed numbers.

Skills:

Students are able to:

Solve real-word problems involving multiplication of fractions and mixed numbers.

Write equations to represent the word situation.

Use visual fraction models to represent the problem.

Understanding:

Students understand that:

A variety of strategies are used to model and solve problems that provide a context for multiplying fractions and mixed numbers.

Solutions are interpreted based on the meaning of the quantities and the context of the problem situation.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.14.1: Define improper fraction, mixed number, fraction, equations, numerator, denominator.
M.5.14.2: Multiply proper fractions with common denominators 2-10.
M.5.14.3: Solve problems using whole numbers.
M.5.14.4: Write number sentences for word problems.
M.5.14.5: Identify key terms to solve multiplication word problems. Examples: times, every, at this rate, each, per, equal/equally, in all, total.
M.5.14.6: Recall basic multiplication facts.

Prior Knowledge Skills:

Divide unit fractions by a whole number and whole numbers by unit fractions.

Use visual models to illustrate quotients.

Create story contexts for division.

Use relationship between multiplication and division to explain quotients.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.14 Use a model to solve multiplying a whole number by a unit fraction of 1/2, 1/3, and 1/4.

Mathematics MA2019 (2019) Grade: 5

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

a. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions and illustrate using visual fraction models, drawings, and equations to represent the problem.

b. Create a story context for a unit fraction divided by a whole number, and use a visual fraction model to show the quotient.

c. Create a story context for a whole number divided by a unit fraction, and use a visual fraction model to show the quotient.

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Evidence Of Student Attainment:

Students:

Solve real-world problems involving division of a unit fraction by a non-zero whole number, or division of a whole number by a unit fraction.

Justify solutions using visual models, drawings, and equations to represent the problem context.

Explain quotients using the relationship between multiplication and division.

Create a story context for a unit fraction divided by a whole number and use models to illustrate the quotient.

Create a story context for a whole number divided by a unit fraction and use models to illustrate the quotient.

Teacher Vocabulary:

Unit fraction

Whole number

Division

Dividend

Divisor

Quotient

Equation

Multiplication

Factor

Fraction models

Knowledge:

Students know:

Contextual situations involving division with whole numbers and unit fractions.

Strategies for representing a division problem with a visual model.

Skills:

Students are able to:

Use previous understandings of operations to

Divide unit fractions by a whole number and whole numbers by unit fractions.

Use visual models to illustrate quotients.

Create story contexts for division.

Use the relationship between multiplication and division to explain quotients.

Understanding:

Students understand that:

A variety of contextual situations are represented with division of a whole number by a fraction or a fraction by a whole number.

Quotients resulting from division of a whole number by a fraction or a fraction by a whole number can be illustrated and justified with a visual model.

The relationship between multiplication and division can be used to justify quotients resulting from division of a whole number by a fraction or a fraction by a whole number.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.15.1: Define quotient.
M.5.15.2: Multiply a fraction by a whole number.
M.5.15.3: Recognize key terms to solve word problems. Examples: times, every, at this rate, each, per, equal/equally, in all, total.
M.5.15.4: Recall basic multiplication and division facts.
M.5.15.5: Express whole numbers as fractions.
M.5.15.6: Recognize fractions that are equivalent to whole numbers.
M.5.15.7: Recall basic multiplication and division facts.
M.5.15.8: Solve word problems involving multiplication of a fraction by a whole number.
M.5.15.9: Recognize key terms to solve word problems.
M.5.15.10: Recall basic multiplication and division facts.

Prior Knowledge Skills:

Find products of a fraction times a whole number and products of a fraction times a fraction.

Use area models, linear models or set models to represent products.

Create a story context to represent equations (a/b) × q and (a/b) × (c/d) to interpret products.

Find area of rectangles with fractional side lengths and represent products as rectangular areas.

Find the area of a rectangle by tiling the area of a rectangle with unit squares.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.15 Use a model to solve dividing a whole number by a unit fraction of 1/2.

Mathematics MA2019 (2019) Grade: 5

17. Convert among different-sized standard measurement units within a given measurement system and use these conversions in solving multi-step, real-world problems.

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Evidence Of Student Attainment:

Students:

Convert different-sized measurement units within the same system.

Solve multi-step word problems involving conversion of metric or customary units.

Teacher Vocabulary:

Measurement system

US Customary

Metric

Unit

Conversion

Equivalent measurements

Knowledge:

Students know:

Strategies for converting a larger unit of measure to a smaller unit in the same system.

Relative size of customary and metric units of measure.

Strategies for converting between units of measure in the same system.

Skills:

Students are able to:

Convert measurement units.

Solve multi-step word problems involving measurement conversions.

Understanding:

Students understand that:

the multiplicative relationship between units of measures given in the same measurement system is essential when converting units to a larger or smaller unit.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.17.1: Identify relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec.
M.5.17.2: Express measurements in a larger unit in terms of a smaller unit.
M.5.17.3: Solve two-step word problems.
M.5.17.4: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).
M.5.17.5: Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
M.5.17.6: Recall basic addition, subtraction, multiplication, and division facts.

Prior Knowledge Skills:

Create a line plot with appropriate intervals.

Represent data on a line plot.

Apply strategies for solving problems involving all four operations with the fractional data.

Convert measurement units.

Solve mulit-step word problems involving measurement conversions.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.17 Using vocalization, sign language, augmentative communication, or assistive technology, to tell time using an analog or digital clock to the half or quarter hour.
M.AAS.5.17a Use standard units to measure the weight and length of objects.
M.AAS.5.17b Sort a collection of coin according to their value.

Mathematics MA2019 (2019) Grade: 5

19. Relate volume to the operations of multiplication and addition, and solve real-world and mathematical problems involving volume.

a. Use the associative property of multiplication to find the volume of a right rectangular prism and relate it to packing the prism with unit cubes. Show that the volume can be determined by multiplying the three edge lengths or by multiplying the height by the area of the base.

b. Apply the formulas V = l x w x h and V = B x 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.

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

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Evidence Of Student Attainment:

Students:
Given right rectangular prisms with whole number edge lengths,

Use associative property of multiplication to find volume and relate it to packing a solid with unit cubes.

Apply formula V = l × w × h, where V represents volume and l, w, and h represent the three dimensions of the prism (length, width, height) and relate the formula to a unit cube filled model.

Apply formula V = B × h, where V represents volume, B is the base-area, and h represents the height (number of layers of the base-area) and relate the formula to a unit cube filled model.

Given a solid figure composed of two or more right rectangular prisms in real-world or mathematical contexts, find the total volume by decomposing the figure into non-overlapping rectangular prisms and find the sum of the volumes.

Teacher Vocabulary:

Volume

Unit cube

Rectangular prism

Base

Base-area

Dimensions

Face

Length

Width

Height

Layers

Edge

Equivalent

Conservation of volume

Attribute

Composition

Decomposition

Formula

Knowledge:

Students know:

Measurable attributes of area and how it relates to finding the volume of objects.

Units of measurement for volume, specifically unit cubes.

Skills:

Students are able to:

Solve word problems involving volume.

Use associative property of multiplication to find volume.

Relate operations of multiplication and addition to finding volume.

Apply formulas to find volume of right rectangular prisms.

Find volume of solid figures composed of two rectangular prisms.

Understanding:

Students understand that:

Volume is a derived attribute based on a length unit and can be computed as the product of three length measurements or as the product of one base area and one length measurement.

Volume is an extension of area and can be found as the area of the base being repeated for a given number of layers.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.19.1: Define volume.
M.5.19.2: Recognize angle measure as additive.
M.5.19.3: Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
M.5.19.4: Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
M.5.19.5: Recognize the formula for volume.
M.5.19.6: Recall the attributes of three-dimensional solids.
M.5.19.7: Recall basic multiplication facts.
M.5.19.8: Fluently add.
M.5.19.9: Compare the unit size of volume/capacity in the metric system including milliliters and liters.
M.5.19.10: Measure and estimate liquid volumes.
M.5.19.11: Recall basic multiplication facts.
M.5.19.12: Compare the unit size of volume/capacity in the metric system including milliliters and liters.
M.5.19.13: Recognize the formula for volume.
M.5.19.14: Recall basic multiplication facts.
M.5.19.15: Describe attributes of three-dimensional figures.
M.5.19.16: Describe attributes of two-dimensional figures.
M.5.19.17: Identify solid figures.

Prior Knowledge Skills:

Count unit cubes to find volume.

Demonstrate volume by packing a solid figure with unit cubes.

Convert measurement units.

Solve mulit-step word problems involving measurement conversions.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.19 Determine the volume of a rectangular prism by counting units of measurement (e.g., unit cubes).

Mathematics MA2019 (2019) Grade: 5

20. Graph points in the first quadrant of the coordinate plane, and interpret coordinate values of points to represent real-world and mathematical problems.

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Evidence Of Student Attainment:

Students:

Use the first quadrant in a coordinate plane to identify coordinates of a given point.

Use the first quadrant in a coordinate plane to explain how the location of an ordered pair is determined.

Given a real-world situation involving a relationship between two variables, graph a representation of the situation and interpret coordinate values of the points in the context of the problem.

Given a graph representing a real-world situation, interpret the coordinate values of the points in the context of the situation.

Teacher Vocabulary:

Coordinate system

Coordinate plane

First quadrant

Points

Lines

Perpendicular

X-axis

Y-axis

Origin

Ordered pair

Coordinate plane

Horizontal

Vertical

Intersection of lines

Knowledge:

Students know:

Specific directions and vocabulary to explain ordered pair location.

The first number of an ordered pair 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.

Skills:

Students are able to:

Graph points in the first quadrant.

Interpret coordinate values in context of the problem.

Understanding:

Students understand that:

graphing points on a coordinate plane provides a representation of a mathematical context which aids in visualizing situations and solving problems.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.5.20.1: Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.
M.5.20.2: Label the horizontal axis (x).
M.5.20.3: Label the vertical axis (y).
M.5.20.4: Identify the x- and y- values in ordered pairs.
M.5.20.5: Model writing ordered pairs.

Prior Knowledge Skills:

Graph points in the first quadrant.

Interpret coordinate values in context of the problem.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.5.20 Identify a point on a horizontal number line representing the horizontal x-axis (no greater than 5) and identify a point on a vertical number line representing the y-axis (no greater than 5).