5th/6th - Math

Fifth grade mathematics is about

- Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions)

- Extending division to two-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations

- Developing understanding of volume.

Sixth grade mathematics is about

- Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems

- Completing understanding of division of fractions and extending the notion of number to the system of rational numbers

- Writing, interpreting, and using expressions and equations

- Developing understanding of statistical thinking.

5/6 Pacing Guide SY 2022-2023

Module 1: Place Value and Operations of Decimal Fractions

(27 Hours)

Standards

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.

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.

5.NBT.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

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

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.

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

5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

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

Overview

Students’ understanding of the patterns in the base ten system are extended. Students deepen knowledge utilizing place value charts to apply new understandings as they reason about and perform decimal operations through the hundredths place.

Students move from whole numbers to multiplication with decimals, again using place value as a guide to reason and make estimations about products. Multiplication is explored as a method for expressing equivalent measures in both whole number and decimal forms. A similar sequence for division begins concretely with number disks as an introduction to division with multi-digit divisors and leads student to divide multi-digit whole number and decimal dividends by two-digit divisors using a vertical written method. In addition, students evaluate and write expressions, recording their calculations using the associative property and parentheses.

This is a combination of two FIFTH GRADE units. Sixth graders should have increased automaticity and gain mastery earlier.

Unit Vocab

Exponent (how many times a number is to be used in a multiplication sentence)

Millimeter (a metric unit of length equal to one-thousandth of a meter)

Thousandths (related to place value)

Conversion factor (the factor in a multiplication sentence that renames one measurement unit as another equivalent unit, e.g., 14 x (1 in) = 14 x ( 1/12 ft); 1 in and 1/12 ft are the conversion factors.)

Decimal Fraction (a proper fraction whose denominator is a power of 10)

Multiplier (a quantity by which a given number—a multiplicand—is to be multiplied)

Parentheses (the symbols used to relate order of operations)

Topic A: Decimal Fractions and Place Value Patterns

Standards

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

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Overview

Students name and compare decimal fractions in various forms. Students will begin naming of decimal fraction numbers in expanded, unit (e.g., 4.23 = 4 ones 2 tenths 3 hundredths), and word forms and concludes with using like units to compare decimal fractions.

Lessons

Topic Assessment

Topic B: Place Value and Rounding Decimal Fractions

Standards

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

Overview

In Grade 5, vertical number lines again provide support for students to make use of patterns in the base ten system, allowing knowledge of whole-number rounding (4.NBT.3) to be easily applied to rounding decimal values (5.NBT.4).

The vertical number line is used initially to find more than or less than halfway between multiples of decimal units. In these lessons, students are encouraged to reason more abstractly as they use place value understanding to approximate by using nearest multiples. Naming those nearest multiples is an application of flexibly naming decimals using like place value units

Lessons

Topic Assessment

Topic C: Adding and Subtracting Decimals

Standards

5.NBT.2 Explain patterns in the number of zeros of the product when m Add, subtract, multiply and divide decimals to hundredths, using concrete models or drawings.

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

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

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.

Overview

Students begin to use base ten understanding of adjacent units and whole-number algorithms to reason about and perform decimal fraction operations—addition and subtraction

Lessons

Topic Assessment

Topic D: Multiplying Decimals

Standards

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.

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

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

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.

Overview

A focus on reasoning about the multiplication of a decimal fraction by a one-digit whole number in Topic E provides the link that connects Grade 4 multiplication work and Grade 5 fluency with multi-digit multiplication. Place value understanding of whole-number multiplication coupled with an area model of the distributive property is used to help students build direct parallels between whole-number products and the products of one-digit multipliers and decimals.

Lessons

Topic Quiz

Topic E: Dividing Decimals

Standards

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

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

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.

Overview

This topic uses a similar exploration of division of decimal numbers by one-digit whole-number divisors. Students solidify their skills with an understanding of the algorithm before moving on to long division involving two-digit divisors in Module 2.

Lessons

Topic Assessment

Topic F: The Standard Algorithm for Multi-Digit Whole Number Multiplication

Standards

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

5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

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

Overview

The algorithm is built over a period of days, increasing in complexity as the number of digits in both factors increases. Reasoning about zeros in the multiplier, along with considerations about the reasonableness of products, also provides opportunities to deepen understanding of the standard algorithm. Although word problems provide context throughout Topic F, the final lesson offers a concentration of multistep problems that allows students to apply this new knowledge.

Lessons

Topic Assessment

Topic G: Decimal Multi-Digit Multiplication

Standards

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.

Overview

Students make connections between what they know of whole number multiplication to its parallel role in multiplication with decimals by using place value to reason and make estimations about products (5.NBT.7). Knowledge of multiplicative patterns from Grade 4 experiences, as well as those provided in Grade 5 Module 1, provide support for converting decimal multiplication to whole number multiplication. Students reason about how products of such converted cases must be adjusted through division, giving rise to explanations about how the decimal must be placed.

Lessons

Topic Assessment

Topic H: Partial Quotients and Multi-Digit Decimal Division

Standards

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.

Overview

The series of lessons in this topic lead students to divide multi-digit dividends by two-digit divisors using the written vertical method. Each lesson moves to a new level of difficulty with a sequence beginning with divisors that are multiples of 10 to non-multiples of 10. Two instructional days are devoted to single-digit quotients with and without remainders before progressing to two- and three-digit quotients (5.NBT.6).

Lessons

Topic Assessment

Module 2: Unit Rates and Ratios

(25 Hours)

Standards

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

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

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

a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Overview

Students investigate and relate ratios, rates, and percents, applying reasoning when solving problems in real world contexts using various tools.

This is traditionally a SIXTH GRADE unit. Ratios becomes an area of emphasis in math as students move on to Middle School. This unit has a heavy focus on real world problem solving which will benefit fifth grade students with additional practice.

Unit Vocab

Ratio (A pair of nonnegative numbers, 𝐴:𝐵, where both are not zero, and that are used to indicate that there is a relationship between two quantities such that when there are 𝐴 units of one quantity, there are 𝐵 units of the second quantity.)

Rate (A rate indicates, for a proportional relationship between two quantities, how many units of one quantity there are for every 1 unit of the second quantity. For a ratio of 𝐴: 𝐵 between two quantities, the rate is 𝐴/𝐵 units of the first quantity per unit of the second quantity.)

Unit Rate (The numeric value of the rate, e.g., in the rate 2.5 mph, the unit rate is 2.5.)

Value of a Ratio (For the ratio 𝐴: 𝐵, the value of the ratio is the quotient 𝐴/𝐵.)

Equivalent Ratios (Ratios that have the same value.)

Percent (Percent of a quantity is a rate per 100.)

Associated Ratios (e.g., if a popular shade of purple is made by mixing 2 cups of blue paint for every 3 cups of red paint, not only can we say that the ratio of blue paint to red paint in the mixture is 2: 3, but we can discuss associated ratios such as the ratio of cups of red paint to cups of blue paint, the ratio of cups of blue paint to total cups of purple paint, the ratio of cups of red paint to total cups of purple paint, etc.)

Double Number Line (See example under Suggested Tools and Representations.)

Ratio Table (A table listing pairs of numbers that form equivalent ratios; see example under Suggested Tools and Representations.)

Topic A: Representing and Reasoning About Ratios

Standards

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

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

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

Overview

Students' previous experience solving problems involving multiplicative comparisons serves as the conceptual foundation for understanding ratios as a multiplicative comparison of two or more numbers used in quantities or measurements.

Students develop fluidity in using multiple forms of ratio language and ratio notation. They construct viable arguments and communicate reasoning about ratio equivalence as they solve ratio problems in real world contexts.

As the first topic comes to a close, students develop a precise definition of the value of a ratio a:b, where b ≠ 0 as the value a/b, applying previous understanding of fraction as division. They can then formalize their understanding of equivalent ratios as ratios having the same value.

Lessons

Topic Assessment

Topic B: Collection of Equivalent Ratios

Standards

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

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

Overview

With the concept of ratio equivalence formally defined, students explore collections of equivalent ratios in real world contexts. They build ratio tables and study their additive and multiplicative structure.

Students continue to apply reasoning to solve ratio problems while they explore representations of collections of equivalent ratios and relate those representations to the ratio table. Building on their experience with number lines, students represent collections of equivalent ratios with a double number line model. They relate ratio tables to equations using the value of a ratio defined in Topic A.

Finally, students expand their experience with the coordinate plane as they represent collections of equivalent ratios by plotting the pairs of values on the coordinate plane.

Lessons

Topic Assessment

Topic C: Unit Rates

Standards

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

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

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Overview

Students build further on their understanding of ratios and the value of a ratio. Students solve unit rate problems involving unit pricing, constant speed, and constant rates of work. They apply their understanding of rates to situations in the real world.

Students determine unit prices and use measurement conversions to comparison shop, and decontextualize constant speed and work situations to determine outcomes. Students combine their new understanding of rate to connect and revisit concepts of converting among different-sized standard measurement units. They then expand upon this background as they learn to manipulate and transform units when multiplying and dividing quantities.

Topic C culminates as students interpret and model real-world scenarios through the use of unit rates and conversions.

Lessons

Topic Assessment

Topic D: Percent

Standards

6.RP.A.3c Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Overview

Students are introduced to percent and find percent of a quantity as a rate per 100. Students understand that N percent of a quantity has the same value as N/100 of that quantity. Students express a fraction as a percent, and find a percent of a quantity in real-world contexts. Students learn to express a ratio using the language of percent and to solve percent problems by selecting from familiar representations, such as tape diagrams and double number lines, or a combination of both

Lessons

Module 3: Addition, Subtraction, Multiplication, and Division of Decimals and Fractions

(35 Hours)

Standards

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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

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

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

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

5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

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.

Overview

Students' understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. This module marks a significant shift away from the elementary grades' centrality of base ten units to the study and use of the full set of fractional units from Grade 5 forward, especially as applied to algebra.

Then, the work extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions. Work proceeds from interpretation of line plots, which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication. The module proceeds to fraction-by-fraction multiplication in both fraction and decimal forms.

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 student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients.

This is a combination of 2 FIFTH GRADE units. Working fluently with fractions and performing all four operations is important for all students. This unit allows for all students to repeatedly practice a traditionally hard skill.

Unit Vocab

Benchmark fraction (e.g., 1/ 2 is a benchmark fraction when comparing 1/ 3 and 3/ 5 )

Like denominators (e.g., 1/8 and 5/8 )

Unlike denominators (e.g., 1/8 and 1/7 )

Decimal divisor (the number that divides the whole and has units of tenths, hundredths, thousandths, etc.)

Simplify (using the largest fractional unit possible to express an equivalent fraction)

Topic A: Equivalent Fractions

Standards

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

Overview

In Topic A, students revisit the foundational Grade 4 standards addressing equivalence. When equivalent, fractions can be represented by the same amount of area of a rectangle as well as the same point on a number line. Students subdivide areas and divide number line lengths to model this equivalence.

Lessons

Topic Assessment

Topic B: Making Like Units Pictorially

Standards

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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.).

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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Overview

Students make like units for all addends or both minuend and subtrahend. First, they draw a wide rectangle and partition it with vertical lines as they would a tape diagram, representing the first fraction with a bracket and shading. They then partition a second congruent rectangle with horizontal lines to show the second fraction. Next, they partition both rectangles with matching lines to create like units.

Lessons

Topic Assessment

Topic C: Making Like Units Numerically

Standards

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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Overview

In Topic C, students use the number line when adding and subtracting fractions greater than or equal to 1. The number line helps students see that fractions are analogous to whole numbers. The number line makes it clear that numbers on the left are smaller than numbers on the right, which leads to an understanding of integers in Grade 6. Using this tool, students recognize and manipulate fractions in relation to larger whole numbers and to each other.

Lessons

Topic Assessment

Topic D: Fractions as Division

Standards

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. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50‐ pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Overview

This topic focuses on interpreting fractions as division. Equal sharing with area models (both concrete and pictorial) provides students with an opportunity to understand the division of whole numbers with answers in the form of fractions or mixed numbers (e.g., seven brownies shared by three girls, three pizzas shared by four people)

Lessons

Topic Assessment

Topic E: Multiplication of a Whole number by a Fraction

Standards

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

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

Overview

Students interpret finding a fraction of a set (34 of 24) as multiplication of a whole number by a fraction (34 × 24) and use tape diagrams to support their understandings. This, in turn, leads students to see division by a whole number as being equivalent to multiplication by its reciprocal. That is, division by 2, for example, is the same as multiplication by 12.

Lessons

Topic Assessment

Topic F: Multiplication of a Fraction by a Fraction

Standards

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.

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

5.NF.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.

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.

Overview

This topic introduces students to multiplication of fractions by fractions—both in fraction and decimal form (5.NF.4a, 5.NBT.7). The topic starts with multiplying a unit fraction by a unit fraction and progresses to multiplying two non-unit fractions.

Students use area models, rectangular arrays, and tape diagrams to model the multiplication. These familiar models help students draw parallels between whole number and fraction multiplication, as well as solve word problems. This intensive work with fractions positions students to extend their previous work with decimal-by-whole number multiplication to decimal-by-decimal multiplication.

Lessons

topic Assessment

Topic G: Division of Fractions and Decimal Fractions

Standards

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

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.

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

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Overview

This topic's major work is division with fractions—both fractions and decimal fractions. Students use tape diagrams and number lines to reason about the division of a whole number by a unit fraction and a unit fraction by a whole number.

Lessons

Topic Assessment

Module 4: Arithmetic Operations Including Dividing by a Fraction

(15 Hours)

Standards

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

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

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

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

Overview

Students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals.

This is a SIXTH GRADE unit. However, this unit will benefit both grade levels as it allows for more practice and instruction in working with fractions. At the end of this module, students should master addition, subtraction, multiplication and division of fractions.

Unit Vocab

Greatest Common Factor (The largest positive integer that divides into two or more integers without a remainder; the GCF of 24 and 36 is 12 because when all of the factors of 24 and 36 are listed, the largest factor they share is 12.)

Least Common Multiple (The smallest positive integer that is divisible by two or more given integers without a remainder; the LCM of 4 and 6 is 12 because when the multiples of 4 and 6 are listed, the smallest or first multiple they share is 12.)

Multiplicative Inverses (Two numbers whose product is 1 are multiplicative inverses of one another. For example, 3 /4 and 4/ 3 are multiplicative inverses of one another because 3/ 4 × 4/ 3 = 4/ 3 × 3/ 4 = 1. Multiplicative inverses do not always have to be the reciprocal. For example 1/ 5 and 10/ 2 both have a product of 1, which makes them multiplicative inverses.)

Topic A: Dividing Fractions by Fractions

Standards

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

Overview

Students extend their previous understanding of multiplication and division to divide fractions by fractions. They construct division stories and solve word problems involving division of fractions. Through the context of word problems, students understand and use partitive division of fractions to determine how much is in each group.

Students use measurement to determine quotients of fractions. They are presented conceptual problems where they determine that the quotient represents how many of the divisor is in the dividend.

Students look for and uncover patterns while modeling quotients of fractions to ultimately discover the relationship between multiplication and division. Using this relationship, students create equations and formulas to represent and solve problems.

Lessons

Topic B: Multi-Digit Decimal Operations - Adding, Subtracting, and Multiplying

Standards

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

Overview

Students have had extensive experience of decimal operations to the hundredths and thousandths, which prepares them to easily compute with more decimal places. Students begin by relating the first lesson in this topic to mixed numbers from the last lesson in Topic A. They find that sums and differences of large mixed numbers can sometimes be more efficiently determined by first converting the number to a decimal and then applying the standard algorithms. They use estimation to justify their answers.

Within decimal multiplication, students begin to practice the distributive property. Students use arrays and partial products to understand and apply the distributive property as they solve multiplication problems involving decimals. Estimation and place value enable students to determine the placement of the decimal point in products and recognize that the size of a product is relative to each factor. Students learn to use connections between fraction multiplication and decimal multiplication.

Lessons

Topic C: Dividing Whole Numbers and Decimals

Standards

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

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

Overview

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

Students further their understanding of division as they develop fluency in the use of the standard algorithm to divide multi-digit decimals. They make connections to division of fractions and rely on mental math strategies to implement the division algorithm when finding the quotients of decimals.

Lessons

Module 5: Rational Numbers, Expressions, and Equations

(26 Hours)

Standards

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

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

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., −(−3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.C.7 Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret −3 > −7 as a statement that −3 is located to the right of −7 on a number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write −3°C > −7°C to express the fact that −3°C is warmer than −7°C.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real‐ world situation. For example, for an account balance of −30 dollars, write | − 30| = 30 to describe the size of the debt in dollars
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than −30 dollars represents a debt greater than 30 dollars. 6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

6.EE.A.1 Write and evaluate numeric expressions involving whole-number exponents.

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.

a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract 𝑦 from 5” as 5 − 𝑦.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems.

6.EE.A.4 Identify when two expressions are equivalent

6.EE.B.5 Understand solving an equation or inequality as a process of answering a question; which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations in the form 𝑥 + 𝑝 = 𝑞 and 𝑝𝑥 = 𝑞 for cases in which 𝑝, 𝑞 and 𝑥 are all nonnegative rational numbers.

6.EE.B.8 Write an inequality of the form 𝑥 > 𝑐 or 𝑥 < 𝑐 to represent a constraint or condition in a real-world mathematical problem. Recognize that inequalities of the form 𝑥 > 𝑐 or 𝑥 < 𝑐 have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Overview

Students understand rational numbers as points on the number line. Students extend the number line to include the opposites of whole numbers, using it to relate numbers to statements of order in real-world contexts.

Students learn to use letters to represent numbers, ultimately extending arithmetic properties from manipulating numbers to manipulating expressions and solving equations.

This is a SIXTH GRADE unit. Fifth graders will benefit from the increase in real world word problems. However, they may need additional scaffolds. They should still gain mathematical knowledge by using variables to write expressions and solve equations. Fifth grade can also work on generating patterns in the problems.

Unit Vocab

Absolute Value (The absolute value of a number is the distance between the number and zero on the number line. For example, |3| = 3, | − 4| = 4, etc.)

Charge (A charge is the amount of money a person must pay, as in a charge to an account, or a fee charged.)

Credit (A credit is a decrease in an expense, as in money credited to an account. For instance, when a deposit is made into a checking account, the money is credited to the account. A credit is the opposite of a debit.)

Debit (A debit is an increase in an expense or money paid out of an account. For instance, using a debit card to make a purchase will result in an expense, and money will be deducted from the related bank account.)

Deposit (A deposit is the act of putting money into a bank account.) Elevation (Elevation is the height of a person, place, or thing above a certain reference level.)

Integers (The numbers ... ,−3, −2, −1, 0, 1, 2, 3, … are integers on the number line.)

Magnitude (The magnitude is the absolute value of a measurement, given the measurement of a positive or negative quantity.)

Negative Number (A negative number is a number less than zero.)

Opposite (In a position on the other side; for example, negative numbers are the opposite direction from zero as positive numbers.)

Positive Number (A positive number is a number greater than zero.)

Quadrants (The four sections of the coordinate plane formed by the intersection of the axes are called quadrants.)

Rational Number (A rational number is a fraction or the opposite of a fraction on the number line.)

Withdraw (To withdraw is to take away; for example, to take money out of a bank account.)

Withdrawal (A withdrawal is the act of taking money out of a bank account.)

Equation (An equation is a statement of equality between two expressions.)

Equivalent Expressions (Two simple expressions are equivalent if both evaluate to the same number for every substitution of numbers into all the letters in both expressions.)

Linear Expression (A linear expression is a product of two simple expressions where only one of the simple expressions has letters and only one letter in each term of that expression or sums and/or differences of such products.)

Simple Expression (A simple expression is a number, a letter that represents a number, a product whose factors are either numbers or letters involving whole number exponents, or sums and/or differences of such products. Each product in a simple expression is called a term, and the evaluation of the numbers in the product is called the coefficient of the term.)

Truth Values of a Number Sentence (A number sentence is said to be true if both numerical expressions are equivalent; it is said to be false otherwise. True and false are called truth values.)

Topic A: Understanding Positive and Negative Numbers on a Number Line

Standards

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

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

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane

Overview

Students use positive integers to locate negative integers, understanding that a number and its opposite will be on opposite sides of zero and that both lie the same distance from zero. Students represent the opposite of a positive number as a negative number and vice-versa. Students realize that zero is its own opposite and that the opposite of the opposite of a number is actually the number itself. They use positive and negative numbers to represent real-world quantities.

Topic A concludes with students furthering their understanding of signed numbers to include the rational numbers. Students recognize that finding the opposite of any rational number is the same as finding an integer’s opposite and that two rational numbers that lie on the same side of zero will have the same sign, while those that lie on opposites sides of zero will have opposite signs.

Lessons

Topic B: Order and Absolute Value

Standards

6.NS.C.6c Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.C.7 Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real‐world contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real‐world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

Overview

Students apply their understanding of a rational number’s position on the number line to order rational numbers. Students understand that when using a conventional horizontal number line, the numbers increase as you move along the line to the right and decrease as you move to the left. Students compare rational numbers using inequality symbols and words to state the relationship between two or more rational numbers. They describe the relationship between rational numbers in real-world situations and with respect to numbers’ positions on the number line.

Students use the concept of absolute value and its notation to show a number’s distance from zero on the number line and recognize that opposite numbers have the same absolute value. In a real-world scenario, students interpret absolute value as magnitude for a positive or negative quantity. They apply their understanding of order and absolute value.

Lessons

Topic C: Relationships of the Operations

Standards

6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + 𝑥𝑥) to produce the equivalent expression 6 + 3𝑥𝑥; apply the distributive property to the expression 24𝑥𝑥 + 18𝑦𝑦 to produce the equivalent expression 6(4𝑥𝑥 + 3𝑦𝑦); apply properties of operations to 𝑦𝑦 + 𝑦𝑦 + 𝑦𝑦 to produce the equivalent expression 3y

Overview

Students understand the relationships of operations and use them to generate equivalent expressions. By this time, students have had ample experience with the four operations since they have worked with them from kindergarten through Grade 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 divided by b times b = a

a times b divided by b = a (when b ≠ 0)

3x = x + x + x

If 12 divided x = 4

Then 12 - x - x - x - x = 0.

Lessons

Topic D: Replacing Letters and Numbers

Standards

6.EE.A.2c Write, read, and evaluate expressions in which letters stand for numbers. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas 𝑉𝑉 = 𝑠𝑠3 and 𝐴𝐴 = 6𝑠𝑠2 to find the volume and surface area of a cube with sides of length 𝑠𝑠 = 1 2 .

6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions 𝑦𝑦 + 𝑦𝑦 + 𝑦𝑦 and 3𝑦𝑦 are equivalent because they name the same number regardless of which number 𝑦𝑦 stands for.

Overview

Students represent letters with numbers and numbers with letters in Topic C. 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 times b = b times a

a + b = b + a

g times 1 = g

g + 0 = g

g divided by 1 = g

g divided by g = 1

1 divided by 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.

Lessons

Topic E: Expanding, Factoring, and Distributing Expressions

Standards

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract 𝑦𝑦 from 5” as 5 − 𝑦𝑦. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + 𝑥𝑥) to produce the equivalent expression 6 + 3𝑥𝑥; apply the distributive property to the expression 24𝑥𝑥 + 18𝑦𝑦 to produce the equivalent expression 6(4𝑥𝑥 + 3𝑦𝑦); apply properties of operations to 𝑦𝑦 + 𝑦𝑦 + 𝑦𝑦 to produce the equivalent expression 3𝑦𝑦.

6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions 𝑦𝑦 + 𝑦𝑦 + 𝑦𝑦 and 3𝑦𝑦 are equivalent because they name the same number regardless of which number 𝑦𝑦 stands for

Overview

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. 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

a ÷ b = a/b

Lessons

Topic F: Expressing Operations in Algebraic Form

Standards

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.

a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract 𝑦𝑦 from 5” as 5 − 𝑦𝑦.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms

Overview

Students express operations in algebraic form. They read and write expressions in which letters stand for and represent numbers. They also learn to use the correct terminology for operation symbols when reading expressions.

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”

Lessons

Module 6: Volume, Area, and Surface Area

(27 Hours)

Standards

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

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

5.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.

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

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.
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.

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

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

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

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Overview

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.

Students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. Extending skills from Module 3 where they used coordinates and absolute value to find distances between points on a coordinate plane, students determine distance, perimeter, and area on the coordinate plane in real-world contexts.

Students apply volume formulas and use their previous experience with solving equations to find missing volumes and missing dimensions.

The final topic includes deconstructing the faces of solid figures to determine surface area. Students apply the surface area formula to real-life contexts and distinguish between the need to find surface area or volume within contextual situations.

This is a combination of a FIFTH and SIXTH grade unit. Everyone needs to find area and volume. Sixth graders are also expected to find surface area.

Unit Vocab

Base (one face of a three-dimensional solid—often thought of as the surface on which the solid rests)

Bisect (divide into two equal parts)

Cubic units (cubes of the same size used for measuring volume)

Height (adjacent layers of the base that form a rectangular prism) Hierarchy (series of ordered groupings of shapes)

Unit cube (cube whose sides all measure 1 unit; cubes of the same size used for measuring volume)

Volume of a solid (measurement of space or capacity)

Altitude and Base of a Triangle (An altitude of a triangle is a perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. For every triangle, there are three choices for the altitude, and hence there are three base-altitude pairs. The height of a triangle is the length of the altitude. The length of the base is called either the base length or, more commonly, the base. Usually, context makes it clear whether the base refers to a number or a segment. These terms can mislead students: base suggests the bottom, while height usually refers to vertical distances. Do not reinforce these impressions by consistently displaying all triangles with horizontal bases.)

Cube (A cube is a right rectangular prism all of whose edges are of equal length.)

Topic A: Concepts of Volume

Standards

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

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.
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.

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

Overview

In Topic A, students extend their spatial structuring to three dimensions through an exploration of volume. They come to see volume as an attribute of solid figures and understand that cubic units are used to measure it (5.MD.3). Using unit cubes, both customary and metric, students build three-dimensional shapes, including right rectangular prisms, and count to find the volume (5.MD.4). By developing a systematic approach to counting the unit cubes, they make connections between area and volume.

Lessons

Topic B: Volume and the Operations of Multiplication and Addition

Standards

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

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.
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.

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

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

Overview

Students come to see that multiplying the edge lengths or multiplying the height by the area of the base yields an equivalent volume to that found by packing and counting unit cubes. Next, students solidify the connection of volume as packing with volume as filling by comparing the amount of liquid that fills a container to the number of cubes that can be packed into it. This connection is formalized as students see that 1 cubic centimeter is equal to 1 milliliter. Complexity increases as students use their knowledge that volume is additive to partition and calculate the total volume of solid figures composed of non-overlapping rectangular prisms.

Lessons

Topic C: Areas of Triangles, Quadrilaterals, and Polygons

Standards

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Overview

Students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. Students learn through exploration that the area of a triangle is exactly half of the area of its corresponding rectangle. They extend their previous knowledge about the area formula for rectangles to evaluate the area of the rectangle using A = bh and discover through manipulation that the area of a right triangle is exactly half that of its corresponding rectangle.

Students discover that triangles have altitude, which is the length of the height of the triangle. The altitude is the perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. Students understand that any side of the triangle can be a base, but the altitude always determines the base. They move from recognizing right triangles as categories to determining that right triangles are constructed when altitudes are perpendicular and meet the base at one side.

Acute triangles are constructed when the altitude is perpendicular and meets within the length of the base, and obtuse triangles are constructed when the altitude is perpendicular and lies outside the length of the base. Students use this information to cut triangular pieces and rearrange them to fit exactly within one half of the corresponding rectangle to determine that the area formula for any triangle can be determined using A = 1/2(bh).

Lessons

Topic D: Polygons on the Coordinate Plane

Standards

6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

Overview

Previously, students used coordinates and absolute value to find distances between points on a coordinate plane. In this Topic, students extend this learning to find edge lengths of polygons (the distance between two vertices using absolute value) and draw polygons given coordinates. From these drawings, students determine the area of polygons on the coordinate plane by composing and decomposing into polygons with known area formulas.

Students further investigate and calculate the area of polygons on the coordinate plane and also calculate the perimeter. They note that finding perimeter is simply finding the sum of the polygon’s edge lengths (or finding the sum of the distances between vertices). This Topic concludes with students determining distance, perimeter, and area on the coordinate plane in real-world contexts.

Lessons

Topic E: Nets and Surface Area

Standards

Overview

Module 6 concludes with deconstructing the faces of solid figures to determine surface area. Students note the difference between finding the volume of right rectangular prisms and finding the surface area of such prisms.

Students build solid figures using nets. They note which nets compose specific solid figures and also understand when nets cannot compose a solid figure. From this knowledge, students deconstruct solid figures into nets to identify the measurement of the solids’ face edges.

Students use nets to determine the surface area of solid figures. They find that adding the areas of each face of the solid will result in a combined surface area. Students find that each right rectangular prism has a front, a back, a top, a bottom, and two sides. They determine that surface area is obtained by adding the areas of all the faces. They understand that the front and back of the prism have the same surface area, the top and bottom have the same surface area, and the sides have the same surface area. Thus, students develop the formula SA = 2lw + 2lh + 2wh.

To wrap up the module, students apply the surface area formula to real-life contexts and distinguish between the need to find surface area or volume within contextual situations

Lessons