Fifth Grade - MATH
5: Math
Unit 1: Place Value
Lesson 1
Numbers to 1,000,000 Review
Students practice writing numbers in expanded notation (e.g., (3(x)1,000,000) + (4(x)100,000) + ...) in Activity 2 and the foldable in Activity 5, where they place parentheses and write each digit multiplied by its place value. In Activity 3 (Big Number Search) students read and use numerical expressions such as 5 × (100,000 + 10,000 + 100) + (1 × 2 × 100) + (5 × 10) + (4 × 1) to identify the corresponding whole number. The Parent Plan and answer keys show examples of students converting between word form, expanded form, and multiplicative expanded notation.
Lesson 2
Comparing and Ordering Numbers
Students are asked to write numbers in expanded form and in expanded notation (for example, 9,000,000 + 600,000 + 50,000 + ... and (9 × 1,000,000) + (6 × 100,000) + ...). The Unit 1 quiz requires students to produce expanded form and expanded notation for given multi-digit numbers, so students practice writing numerical expressions that represent place-value calculations.
Lesson 3
Digits to the Left and Right
Students practice writing and evaluating multiplication expressions involving place value (e.g., 3 × 10, 30 × 10, and writing values as products such as 2,000 as 2 × 100). They complete fill-in-the-blank statements that interpret phrases like "10 times as much as" and "1/10 of" (e.g., 200 is 10 times as much as 20; 50 is 1/10 of 500). Activities ask students to interpret fractions as division (1/10 = 1 ÷ 10 = 0.1) and to reason about how place value changes when multiplying or dividing by 10.
Lesson 4
Decimals to Thousandths
Students write numbers in expanded notation using multiplication and addition, e.g., 1.875 = (1 × 1) + (8 × 0.1) + (7 × 0.01) + (5 × 0.001). The Try It! page shows expressions such as (9 × 10^2) + (5 × 10^1) + 4 that students convert to number form. Students also convert digits to fractional values (e.g., value of 3 in 24.673 is 3/1000) and match decimals to written names, demonstrating interpretation of numerical expressions without performing full calculations.
Lesson 6
Rounding to Thousandths
Students are asked to write 3.485 in multiple forms including fractional and decimal expanded notation, e.g., (3 × 1) + (4 × 0.1) + (8 × 0.01) + (5 × 0.001). Students are given expressions such as (8 × 1/10) + (3 × 1/100) + (2 × 1/1000) and asked to write them in number form. The answer key explicitly shows expanded notation expressions that record calculations with numbers.
Lesson 7
Powers of 10
Students write repeated multiplication expressions such as 10 × 10, 10 × 10 × 10, and are asked to express 1,000,000 in exponential form (10^6). Students complete matching problems that require multiplying and dividing whole numbers and decimals by powers of ten and describe patterns connecting the number of 10s multiplied to the number of zeros in the product. Students also explain in their own words what a power of ten is and say what number is ten times a given large number.
Lesson 8
Working With Powers of 10
Students repeatedly write and rewrite calculations using exponent notation and scientific notation (for example, tasks to rewrite 3.2 × 1000 as 3.2 × 10^3 and 62 ÷ 1,000 as 62 ÷ 10^3). Tasks ask students to fill in and compute expressions like 5 × 10^3, 12 × 10^2, and to match equivalent expressions such as 20 ÷ 10^2 and 0.02 × 10. Activity directions explicitly require students to rewrite problems using scientific notation and to match cards showing equivalent expressions.
Lesson 9
Roman Numerals
Students write numerical decompositions such as "48 = 40 + 8 = ______" and see examples like "124 = C + XX + IV = CXXIV," which require writing simple additive expressions that record how a number is formed. Students match Arabic numbers with Roman numerals (1–20 and beyond) and play comparison activities where they turn over Roman numeral cards and decide which shows the larger amount.
Lesson 10
Problem Solving
Students work with written numerical expressions that use multiplication, division, and powers of 10 (e.g., 0.005 × 10^6, 100 × 0.08, 40,000 ÷ 10^6) and sort them into "Greater Than 1" or "Less Than 1." Students are prompted to explain how digits and decimal points move when multiplying and dividing by powers of 10. Students are asked to write times in fractional expanded form and word form (e.g., express 7.82 as 7 + 8/10 + 2/100 and "seven and eighty-two hundredths").
Lesson 11
Unit Test
Students write numbers in fraction expanded notation as expressions (for example, (2 × 1) + (4 × 1/10) + (7 × 1/100) for 2.47) and convert decimals to fractional or summed forms. Students complete problems that use exponents and multiplication/division by powers of ten (e.g., 2.98 × 10^3 = 2980, 3.78 × 100 = 378). Students fill in blank expressions such as 12.35 × 10 = ___ and complete tables that show number form, word form, fraction form, and expanded notation. These tasks require students to write and work with expressions that record calculations with digits and powers of ten.
Final Project
Number Trading Cards
Students are required to show each featured number in at least three different forms, and the back-of-card requirements explicitly list "Addition or subtraction problem" as one allowed representation. The requirements also ask students to show numbers using "Powers of ten or scientific notation" and "decimal expanded form," which requires writing numerical expressions (including exponent and expanded-sum forms). Students collect and order their cards from smallest to greatest, which has them compare numerical values and consider relationships among representations.
Unit 2: Four Operations
Lesson 1
Basic Operations with Multi-Digit Numbers
Students sort and match verbal keywords (e.g., "altogether," "how many fewer," "times") into columns for addition, subtraction, multiplication, and division, and they label parts of number sentences (factors, product, dividend, divisor, quotient). Students break multiplication problems into partial products (e.g., 185 × 5 and 185 × 20) and rewrite multi-digit multiplications as sums of those partial products (314×523 → 314×3 + 314×20 + 314×500). Students highlight operation words in word problems and set up the corresponding calculations (for example, totaling ticket sales then multiplying by $25).
Lesson 2
Prime Factorization
Students create factor trees and write prime factorizations such as 36 = 2 × 2 × 3 × 3 and express repeated factors with exponents (e.g., 2^2 × 3^2). Students complete practice pages that require writing products of prime factors for numbers like 12, 16, 24, 30, 45, and 50. Students also record multiplication expressions (e.g., 2 × 3 × 3) as part of finding prime factorizations.
Lesson 3
Dividing Multi-Digit Numbers
Students are asked to write division in different symbolic forms (for example, the prompt to write "172 divided by 4" in three different ways and pages showing 72 ÷ 9, 72/9, and 9)72). Students convert word problems into division equations (e.g., word problems that become 800 ÷ 20 and 3,600 ÷ 30) and use multiplication sentences to check division answers (for example, checking 1,200 ÷ 30 by forming 40 × 30 = 1,200). Student pages also require creating and using multiplication tables to support division, showing students record calculations with number operations.
Lesson 4
Adding Decimals
Students are asked to write and solve a verbal decimal sum: "two and six-tenths plus eight and seven-hundredths (2.6 + 8.07 = 10.67)" and complete many problems that require writing numeric addition expressions (e.g., 1.6 + 2.7, 5.32 + 6.19, 3.46 + 2.75 + 1.27). Student activity pages require stacking decimal addends with aligned decimal points and recording the sums, so students write calculations with numbers in decimal form.
Lesson 5
Subtracting Decimals
Students are asked to write subtraction problems from drawn cards and record them on the 'Subtracting Decimals' sheet, which requires creating numeric expressions such as 8.45 - 6.03. Activities include blank equation items (e.g., ______ - ______ = 2.6) and missing-number problems (e.g., 5.6 - □ = 1.8) that require students to form and manipulate subtraction expressions. The flapbook and practice pages explicitly prompt students to set up decimal subtraction expressions with proper alignment and zeros.
Lesson 6
Decimal Problem Solving
Students are asked to "show two ways to create a SUM of 6.83" and "show two ways to create a DIFFERENCE of 6.83," which requires writing numerical expressions that record calculations (for example, 3.99 + 2.84). Several fill-in-the-blank equations such as "5.72 + □ = 9" and "□ + 6.4 = 13.9" require students to set up and complete numerical expressions. Word problems (e.g., combining ounce amounts or computing change) ask students to represent and compute sums and differences of decimals.
Lesson 7
Multiplying Decimals
Students set up and solve multiplication number sentences with decimals in many activities and word problems (for example, Angelica: 8.25 × 5) and are instructed to write the whole number sentence for word problems. Students are taught the meaning of multiplication as equal groups (e.g., 3 × 12 means 3 groups of 12) when they model problems with base-10 blocks. Students use estimation and matching activities (e.g., choosing which product is reasonable for 5.6 × 7.27 and matching problems to products) to judge numerical expressions without fully calculating them.
Lesson 8
Dividing Decimals
Students set up division calculations in long-division form and rewrite word problems as numerical division expressions (for example, 2.8 ÷ 4, 2.5 ÷ 5). Students represent division with base-10 blocks and record the corresponding numerical expression (2.8 ÷ 4) and check answers by multiplying the quotient by the divisor (4 × 0.7 = 2.8). Students are asked to place decimal points correctly in dividends and quotients and to rewrite problems in numeric form for the standard algorithm.
Lesson 9
Solving Problems Using the Four Operations
Students are asked to "Write number sentence(s) for each of the following problems" and the answer key shows expressions such as 2.9 × 0.90, 2.9 − 1.6, 8.6 + 15.2 + 11.2 + 12.4, 47.4 ÷ 12, 20.6 × 7, and 10.40 × 8 that record calculations with numbers. The Coinstar and Money Matters activities require students to set up multiplication and addition of coin values and wages (e.g., coin count × coin value, hourly pay × hours) and to estimate totals without precise computation in one task. The parent notes explicitly tell students to write each number sentence and to label answers, reinforcing that students must translate word problems into numerical expressions.
Lesson 10
Mathematical Expressions
Students match five word problems about cupcakes to corresponding numerical expressions, requiring them to write or identify expressions that record calculations with numbers (e.g., 36 - 12, 12 ÷ 12, 12 + 24). In the Parentheses and Brackets activity students cut and match verbal descriptions (like "3 times the sum of 6 and 2") with expressions such as (6 + 2) × 3, directly translating phrases to expressions. Activities ask students to write their own cupcake problem and expression and to place parentheses in 5 × 3 + 2 to create a desired result, showing explicit practice in both writing expressions and interpreting their structure without necessarily evaluating large calculations.
Lesson 11
Introducing Order of Operations
Students are asked to write numerical expressions from verbal descriptions on the Unit 2 Quiz 5 (e.g., "6 times the sum of 3 and 6," "7 plus the product of 10 and 20," "10 less than the quotient of 36 and 3"), and the answer key shows the corresponding symbolic expressions. The Parent Plan Skills list explicitly includes "Write simple expressions that record calculations with numbers," and students are instructed to write examples for parentheses, exponents, multiplication, division, addition, and subtraction in their Interactive Notebook.
Lesson 12
Working With PEMDAS
Students are asked to "Write out the mathematical expression for each problem and then solve using PEMDAS," and two word problems require students to record expressions (Ellen: 12 × (2 + 5); Devon: 40 ÷ 5 + 2). In the PEMDAS Dice Game and discussion prompts, students construct numerical expressions from given digits and place parentheses to change results (e.g., (4 × 7) + 2, 4 + 3 × (8 - 6)). The "Think About It!" task has students list all operations in an expression in the correct order, which requires students to interpret the structure of a numerical expression without performing full evaluation.
Lesson 13
Unit Test
Students translate verbal phrases into symbolic expressions on multiple pages (e.g., tasks asking for "6 times the sum of 5 and 3" and "8 less than the quotient of 45 and 9" and the matching exercise that pairs 5 × (9 + 6) with "5 times the sum of 9 and 6"). Students match given mathematical expressions to their corresponding verbal descriptions, demonstrating interpretation of numerical expressions (the matching activity pairs expressions like (5 × 9) − 6 with "6 less than the product of 5 and 9"). Students draw parentheses to make expressions true and write expressions from statements, practicing the use of grouping symbols to record calculations with numbers.
Final Project
Teaching Presentations
The Parent Plan section explicitly lists the skill "Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them." The Order of Operations/PEMDAS planning sheet asks students to write a "Mathematical expression to be solved:" and to fill a table with "Steps Using Numbers" and corresponding "Steps in Words." The Decimal Operations planning sheets require students to create real‑world problems and write numerical steps alongside verbal descriptions.
Unit 3: Measurement
Lesson 1
Reviewing Customary and Metric Units
Students complete problems that require working with numerical expressions, such as evaluating (15 + 3) + (6 x 4) - 10 and computing products like 266.3 × 3.9 and 37 times 10 to the 6th. Students set up and use simple equations (for example, 90 ÷ 2.5 = N) and match units and abbreviations, which involves reading and using numeric expressions. Several tasks present verbal-number phrases (e.g., "What is 37 times 10 to the power of 6?") that connect language to calculations.
Lesson 2
Converting Units of Measurement
Students write and evaluate numeric calculations such as 36 ÷ 12 = 3 and 4 × 2000 = 8000 when converting units (Activity 4 examples and Answer Key). Students work with powers of ten and write expressions for metric conversions (e.g., 35 meters × 10^3 = 35,000 millimeters) and complete arithmetic expressions in the Basic Skills Review (including 5 × (6 + 4) × (7 − 3)). The activities require students to set up multiplication or division expressions to perform unit conversions.
Lesson 3
Measurement Problem Solving
Students set up and compute numerical expressions to perform unit conversions and solve problems (for example, "4 pints = 4 x 2 = 8 cups", "3 meters 45 centimeters → 100 x 3 + 45 = 345 cm", and "48 in. x 15 = 720 in.; 720 ÷ 36 = 20 yards"). The Basic Skills Review and several answer keys present and require evaluating compound numerical expressions such as "196 ÷ 7 – (2.5 × 8) + 11" and "12 × (3 + 11) × 9." Multiple activity problems ask students to write multiplication and division calculations (e.g., 48 × 15, 140 × 15) to record and compute quantities.
Lesson 4
Working With Time
Students set up and carry out numerical calculations to convert units (for example, using 4 x 60 = 240 to convert hours to minutes and 3.5 weeks x 7 = 24.5 days). Word problems require students to choose operations and record addition or subtraction of time amounts (for example, 2 hours 37 minutes + 1 hour 52 minutes = 4 hours 29 minutes). The answer keys and parent notes show students writing simple numeric calculations for conversions and sums (e.g., 600 years / 10 = 60 decades; 5 hours x 60 = 300 minutes + 16 = 316 minutes).
Lesson 5
Working With Line Plots
Students create line plots from given numerical data and answer computation questions such as "What is the total weight of all the packages that weigh 4 pounds?" and "What is the total weight of all the packages that weigh 4 1/4 pounds?" The lesson's Skills section explicitly states that students should "use operations on fractions to solve problems involving information presented in line plots," and several activity questions require combining repeated values (summing or finding total weight) and counting multiplicities from the plot.
Final Project
Measurement Book
Students are asked to create and solve measurement word problems (Option 2) and to include solutions, which requires recording calculations using addition, subtraction, multiplication, or division. Page requirements for Pages 2–7 ask students to show equivalent measurements and provide examples of converting between units using words, images, or word problems (e.g., 60 seconds = 1 minute; 300 minutes = 5 hours). Page 8 requires students to create conversion questions, including two that convert between units in the same system, and to produce a separate answer sheet demonstrating their calculations.
Unit 4: Adding and Subtracting Fractions
Lesson 1
Reviewing Fractions
Students repeatedly multiply and divide numerators and denominators to create equivalent fractions (for example, 1/2 = 3/6 with ×3 shown on numerator and denominator). They complete equivalent-fraction chains and write equations showing equality (e.g., 4/6 = 2/3 = 6/9) and use fraction strips to demonstrate how multiplying parts produces equivalent fractions. Activities ask students to write equivalent fractions by performing the same multiplication or division on both numerator and denominator.
Lesson 3
Improper Fractions and Mixed Numbers
Students write and use arithmetic operations when converting between forms: the lesson shows 11/3 = 11 ÷ 3 and the mixed-number example 3 1/4 = 3 × 4 = 12, then 12 + 1 = 13, and the conversion steps list "Multiply the whole number by the denominator" and "Add the numerator." Students are prompted to "Think: How many parts do I have in 3 wholes?" which asks them to interpret the multiplication operation conceptually.
Lesson 5
Working With Unlike Denominators
Students write and compute multiplication expressions when finding common denominators (for example, completing 5 x 3 = 15). Students set up equivalent-fraction expressions by multiplying numerator and denominator (for example, 2/5 x 3/3 and 1/3 x 5/5 and the resulting 6/15 + 5/15). Students complete blanks that require forming these numeric multiplication expressions while applying them to add unlike fractions.
Lesson 9
Problem Solving With Fractions
Students are instructed to "write out each problem" for the walking contest (Dee, Jay, and Kay), which requires translating the verbal questions into numeric addition and subtraction expressions with mixed numbers. The cookie recipe questions (total sugar, flour needed, walnuts left, two batches of butter) require students to set up and compute expressions such as 1/2 + 2/3 and 2 1/4 - 1/2. Activity 3 asks students to use fraction boxes to "create one addition sentence and one subtraction sentence," and the Basic Skills Review includes compound numerical expressions with parentheses (e.g., 11 x (21 + 7) + (6 x 7 x 3)).
Unit 5: Multiplying Fractions
Lesson 1
Multiplying Fractions and Whole Numbers
Students write multiplication expressions for group problems (for example, problems like "6 groups of 1/4" are written as 6 × 1/4 = 6/4, and worksheets list expressions such as 7 × 1/2 and 5 × 1/4). Tables require students to record repeated addition, draw pictures, and write the multiplication expression and product for each problem. Tasks such as "Prove it!" ask students to use multiplication to show equivalence (7 × 3/8 is the same as 3 × 7/8), and "What's missing?" asks students to fill boxes in expressions like 3/8 = □ × 3.
Lesson 2
What Does Multiplying by a Fraction Mean?
Students are asked to write fractional parts of whole numbers as multiplication expressions (for example, to write 1/2 of 8 and 1/2 of 12 as multiplication problems such as 1/2 × 8). Several activities and the quiz require students to match multiplication expressions with descriptions or visual representations (e.g., 2/5 × 15, 1/6 × 30, 2/3 × 12). The scaling practice asks students to interpret expressions without full calculation by deciding whether n <, =, or > the whole-number factor.
Lesson 3
Using Area Models
Students translate a verbal context (the painting story) into the expression 3/4 of 1/2 and produce the numerical sentence 3/4 × 1/2 = 3/8. Students write and compute fraction multiplication expressions repeatedly (e.g., 2/3 × 1/4 = 2/12) during paper-folding and area-model activities. Students match written fraction multiplication expressions to area-model diagrams and create multiplication problems in the "Find the Product" activity.
Lesson 4
Area Models to Algorithm
Students create and write fraction multiplication expressions in multiple activities (e.g., drawing four cards to create two fractions and then writing __/___ × __/___ = ___/___). Activity pages and instructions ask students to "write a number sentence" and to use the algorithm (multiply numerators and denominators) to record products. The Parent Plan and Skills list explicitly state that students should "Interpret the product (a/b)(x) q as a parts of a partition of q into b equal parts," showing interpretation of multiplication in context.
Lesson 5
Multiplying Fractions Trick
Students translate word problems into fraction multiplication expressions (for example, modeling "what fraction play the trumpet" as 5/8 × 7/12) and complete multiple worksheets that require writing and simplifying multiplication expressions with fractions. Students practice setting up and simplifying numeric multiplication expressions (numerators and denominators) and deciding when cancellation can be applied before multiplying.
Lesson 7
More Fraction Multiplication
Students complete input/output tables labeled with rules such as "Multiply by 2/3," "Multiply by 2 1/3," and "Multiply by 1 1/2," applying a multiplicative rule to given numbers. Students determine an unknown rule in a table (fill in the blank rule) from given IN and OUT pairs, which requires interpreting the multiplicative relationship between numbers. Word problems use the phrase "of" to indicate multiplication (e.g., 2/3 of 24 → 2/3 × 24) and students follow multi-step verbal scenarios that translate into multiplication and subtraction steps. The Always/Sometimes/Never activity has students reason about how multiplying by fractions greater than, less than, or equal to 1 changes a quantity.
Final Project
Product Sorting Game
Students create and write 20 multiplication problems (5 whole-number × fraction, 5 fraction × fraction, 5 mixed-number problems, and 5 rectangular area problems) on the "Fraction Multiplication Problems" sheets and copy each problem onto individual problem cards. Students record each problem in a "Problem" column and write the product in a "Product" column, so they produce numerical multiplication expressions (including fractions and mixed numbers) that record calculations. Students also design categories based on product size (less than 1, equal to 1, greater than 1 and less than 2, 2 or greater) and place problem cards into those categories during play.
Unit 6: Geometry
Lesson 3
Graphing Review
Students answer questions from a bar graph that require them to compute totals and differences; the answer key shows arithmetic written as products and sums (e.g., 2 × 12 = 24; 7 × 12 = 84; 24 + 84 = 108). Students also answer questions from a line plot that require counting and summing X marks to find totals (e.g., How many patients grew 3 inches or more? = 52). Activity 2 has students collect numerical data and create a line plot, which requires organizing and recording numeric values.
Lesson 6
More Work With Ordered Pairs
Students create input/output tables labeled with a rule such as "Multiply by 2" or "Multiply by 4," convert those tables into ordered pairs, and plot the resulting lines (Patterns on a Coordinate Plane). Students compare two data sets (saving $2/week vs $4/week) and are led to state the relationship that "The second set is double the first set." The Basic Skills Review and other pages include numerical expressions with parentheses (for example, 9 × (64 + 8) ÷ (3 × 7)) that students compute.
Unit 7: Dividing Fractions
Lesson 1
Reviewing Division
Students are asked to "Create Division Sentences" using a given set of numbers, which requires them to write numerical division expressions (division sentences) that record calculations. The activity asking students to place > or < between pairs like 360 ÷ 5 and 360 ÷ 10 has students compare and interpret numerical division expressions without computing every value. Multiple exercises (mental division problems, long-division practice, and matching division problems to quotients) require students to read and work with division expressions.
Lesson 2
Getting Ready to Divide Fractions
Students write numerical expressions for division as fractions (e.g., recording 4 sandwiches ÷ 1 person as 4/1, 4 ÷ 3 as 4/3, 4 ÷ 6 as 4/6) and complete tables showing how a fixed number of items is divided among 1–10 people. Students label dividend, divisor, and quotient (cookie number as dividend, people as divisor) and convert improper fractions to mixed numbers. Students answer conceptual questions about when a quotient is less than, equal to, or greater than 1 without computing large numerical results.
Lesson 3
Dividing Unit Fractions by Whole Numbers
Students translate contextual scenarios into numerical division expressions such as 1/2 ÷ 3, 1/5 ÷ 4, and 1/3 ÷ 5 in the visual activities and word problems. They match visuals to problems and write and manipulate expressions using the Keep–Switch–Flip–Solve procedure to rewrite division expressions as multiplication by a reciprocal. The quiz asks students to write reciprocals and to solve given expressions using the prescribed steps.
Lesson 4
Dividing Whole Numbers by Unit Fractions
Students are asked to write math sentences for word problems (e.g., "Write a math sentence for each word problem and then find the answer" on the More Dividing sheet) and repeatedly translate contextual situations into division expressions such as 4 ÷ 1/3, 5 ÷ 1/2, and similar. Multiple Student Activity Pages and answer keys show problems written as numerical expressions (for example, 5 ÷ 1/2 = 10, 4 ÷ 1/3 = 12) and students use the Change–Switch–Flip–Solve steps to rewrite and compute these expressions. The lesson also asks conceptual questions about whether a quotient is larger or smaller when dividing by a fraction versus a whole number, prompting students to reason about the meaning of expressions.
Lesson 5
More Division Practice
Students sort cards matching division expressions, word problems, and visual representations, showing practice interpreting numerical division expressions in context (Activity 1). In Activity 2 students create a situation and a multiplication problem for given division problems, and the answer key explicitly maps division expressions to multiplication expressions (for example, 4 ÷ 1/3 -> 4 × 3). The Parent Plan instructs students to use and explain the term "reciprocal" and to write the related multiplication expression for each division expression.
Lesson 6
Unit Review and Test
Students are asked to "write the division math sentence before solving each word problem" (e.g., Darcy, Marcus, Danny problems), which requires translating verbal situations into numerical expressions. Several activities ask students to match division expressions to real-world scenarios ("Draw lines to match each division problem to its situation" and Unit Test matching items), requiring interpretation of expressions as situations without necessarily computing them. Students are also prompted to "write a situation/word problem for each division problem," which has them record calculations with numbers in context.
Final Project
Fraction Division Pamphlets
Students are prompted to "Show the step using numbers" and to "Show the corresponding division word sentence," which requires translating between verbal descriptions and numerical expressions. The parent/skills section explicitly says students will "use ... equations to represent the problem." Students must "Explain the step in detail in your own words," indicating they will interpret numeric steps without necessarily computing them.
Unit 8: Volume
Lesson 1
Perimeter and Area Review
Students are asked to write formulas and numeric examples for perimeter and area (e.g., Perimeter = (2 × length) + (2 × width) with example (2 × 8) + (2 × 3) = 22 ft, and Area = length × width with example 8 × 3 = 24 sq ft). Students write and use the shortcut for a square's perimeter as 4 × side and are shown that multiplication is repeated addition (adding 6 + 6 + 6 + 6 or using 6 × 4). Student sheets include blanks labeled "Formula" and "Example," requiring students to record calculations with numbers for given dimensions. The quiz asks students to draw shapes given a numeric perimeter or area, requiring them to represent and use numeric expressions for those measures.
Lesson 3
From Area to Volume
Students complete Basic Skills Review items that require writing and working with numerical expressions: the lemonade problem requires using 1/8 × 8 to find sugar for a gallon, the comparison problem presents the expressions 13 × 11 − 54 and 156 ÷ 12 × 6.5 for students to compare, and an order-of-operations item (72 − 10 + 2 × (3 × 4)) requires understanding parentheses and operation order. Students are asked to show work and label units, so they practice forming and manipulating expressions with numbers.
Lesson 4
Volume and Unit Cubes
Students are asked to record calculations as multiplications when finding volume: the "Finding Volume" page directs students to multiply the number of cubes in one layer by the number of layers and to show the multiplication. Students record dimensions and check that Length × Width × Height equals the number of cubes on the "Making Prisms" sheet and in the Parent Plan examples. Day 3 explicitly shows repeated addition converted to multiplication (154 + 154 + … = 154 × 6), so students translate repeated addition into a multiplication expression.
Lesson 5
Unit Cubes to Algorithm
Students build rectangular prisms with 24 unit cubes, record dimensions in a table, and write multiplication expressions such as 8 × 3 × 1 = 24 to show the relationship between dimensions and volume. Students write and use the formula l × w × h to compute volume and complete problems using that product. The lesson prompts students to test whether adding or multiplying the dimensions yields 24 and directs them to read the "In Any Order" section to reinforce that multiplication can be done in any order.
Lesson 6
Practicing the Algorithm
Students write multiplication sentences to record volume calculations (e.g., blanks and answer keys showing 10 × 3 × 5 = 150 and other V = l × w × h equations). Students place parentheses around factors and evaluate them in examples ((5 × 4) × 6 = 20 × 6 = 120), demonstrating use of associative/commutative properties. Students also compare expressions for the same box in different orders (Joel's 4 × 8 × 10 vs Kate's 8 × 10 × 4) and conclude the volumes are the same without recalculating each product.
Lesson 7
Extending Work With Volume
Students are asked to write mathematical expressions on the "Think About It!" page, where they must "write 3 different mathematical expressions to show how to find the volume of the prism, using both formulas at least once" (e.g., 16 × 5 = 80 and 4 × 4 × 5 = 80). The "Something's Missing" activities require students to write the formula V = l × w × h, plug in known numbers (e.g., 400 = 10 × 8 × h), and reason about what factor would make the equation true. The lesson also shows substituting (l × w) for B to form B × h = V, asking students to recognize and use equivalent numerical expressions for the same calculation.
Lesson 8
Problem Solving
Students write and use multiplication expressions for volume such as V = l × w × h and compute volumes in problems (e.g., 20 × 20 × 15 = 6,000 and 10 × 10 × 10 = 1,000). Students write math sentences for scaled dimensions (e.g., doubling one dimension: 12 × 4 × 3 = 144) and complete a table showing original, doubled, and tripled dimensions with corresponding volume calculations. Students reason about multiplicative relationships (e.g., Noah's prism had twice the volume and twice the length) and are asked to write math sentences to show those relationships.
Lesson 9
Unit Review and Test
Students are asked to "write two formulas for finding the volume of a rectangular prism," and to "write out the math sentences for each problem," which requires recording calculations as numerical expressions (for example, l × w × h = V and b × h = V). The answer key shows many explicit numerical expressions such as 12 × 5 × 4 = 240, 6 × 3 × 7 = 126, and composite expressions used to find total volume (30 + 36 = 66). Students are also prompted to "prove" solutions with a math sentence (e.g., showing doubling a dimension doubles volume), which requires forming expressions that represent operations on numbers.
Final Project
Volume Town
Students record each building's measurements in a 'Dimensions (L x W x H)' column, explicitly using the form l × w × h. Students compute volumes using the formula l × w × h = V and write the results in the 'Volume (cubic units)' column. Students also add the volumes of non-overlapping rectangular prisms to find total volume for composite buildings.
Unit 9: Skills Review
Lesson 1
Fraction Operations
Students complete input/output tables labeled with rules such as "Multiply by 3/4," "Divide by 4," "Multiply by 1 1/2," and "Divide by 1 1/3," filling in outputs for given inputs. Students solve word problems that require setting up and calculating expressions (e.g., area of a rectangle twice as long as its width, division of chocolate among friends, dividing ribbon into 1/3-inch pieces). Students review PEMDAS and play an order-of-operations game that asks them to select numbers and operators to evaluate expressions.
Lesson 2
Decimal Operations
Students solve concrete arithmetic problems that require writing and computing numerical expressions, for example adding snack prices ($1.89 + $0.99 = $2.88) and computing change, and solving multiplying/dividing decimal word problems such as 19.5 × 2, 19.5 × 5, 19.5 × 7, 581.7 ÷ 7, and 83.1 × 4. The multiplying/dividing problems and the parent answer keys explicitly show students forming and evaluating expressions for those scenarios. The representing-decimals and powers-of-10 activities ask students to write numbers in expanded and exponent form, showing practice in expressing numerical structure.
Lesson 3
Measurement
Students write numeric multiplication sentences to record calculations for volume (e.g., 8 × 4 × 5 = 160, 3 × 6 × 1 = 18) and use formulas such as V = b × h and V = l × w × h. Students set up and compute unit-conversion calculations in word problems (e.g., 8 gallons × 4 quarts/gallon = 32 quarts; 3 feet × 12 inches/foot = 36 inches). The activity directions explicitly ask students to "write a math sentence and find the volume" for each prism.