Third Grade - MATH
5: Math
Unit 1: Multiplication and Division I
Lesson 5
Multiplication and the Abacus
The lesson's Skills section explicitly states students will "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." The opening Questions to Explore asks, "What are some strategies we can use to multiply and divide?" indicating division is a topic to consider. Multiple activities have students model multiplication sentences and count groups on the abacus (e.g., showing arrays, writing multiplication sentences, and finding products).
Lesson 7
Multiples of 2 and 3
The Skills section explicitly lists "Determine the unknown whole number in a multiplication or division equation relating three whole numbers," and the Questions to Explore include "What are some strategies we can use to multiply and divide?" These statements indicate students are expected to think about division in relation to multiplication and to work with unknowns in multiplication/division equations.
Lesson 8
More Practice With 2 and 3
The Skills section includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." The Student Activity Pages give explicit missing-factor problems such as "__ x 2 = 14" and "__ x 3 = 21" and instruct students to use the number line to determine how many jumps are needed to reach the given product. Activities require students to find missing factors repeatedly for the 2 and 3 fact families and to write products and missing factors on number lines.
Lesson 9
Multiples of 4 and 5
The lesson's Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers," which names the idea of solving for an unknown factor. Students practice finding and recording products (e.g., completing multiplication sentences like 1×4, 4×3, 9×4; filling missing products on multiple sheets) and use tools (multiplication table, number line, abacus) to check products. Students also practice thinking about multiplication structure (doubles, repeated addition) that underlies the inverse relationship between multiplication and division.
Lesson 10
More Practice With 2, 3, 4, and 5
The Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers," which names the concept of finding an unknown factor. In Activity 1, students cut out numbers and sort them by placing each drawn number with the appropriate factor card (2, 3, 4, or 5), and they are asked to consider numbers like 8 or 10 that match more than one factor. Students also label and glue lists titled "Multiples of 2/3/4/5," reinforcing connections between numbers and their factors.
Lesson 11
Multiples of 10
Students practice reversing multiplication for the factor 10 when, during Activity 2, they are given a multiple of 10 (for example, hearing "sixty") and asked to say the number being multiplied by 10 (answering "six"). The Skills list explicitly states that students should "determine the unknown whole number in a multiplication or division equation relating three whole numbers," and several activities require students to identify missing products or the factor when one factor is 10 (coloring multiples, filling in 1x10, 2x10, etc.). Students also use manipulatives and number sentences to represent groups of 10 and write multiplication sentences that connect products and factors.
Lesson 12
Problem Solving With Multiplication
The Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." Activity 1 (Multiplication Targets) has students cover the second factor and answer questions like "What do you multiply by 2 to get 10?" which requires finding the missing factor given a product and one factor. Activity 4 (Input/Output Machine) asks students to infer the machine's rule (e.g., given input 5 and output 25, determine ×5), which requires finding the multiplier that relates input and output.
Lesson 13
What Is Division?
The lesson's Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers" and "Understand division as an unknown-factor problem." In Activity 3 students complete the "Connecting Multiplication and Division" sheet where they rewrite multiplication facts (e.g., 4 × 6 = 24) as division sentences (24 ÷ 6 = 4) and fill in missing numbers. The "Showing Division" and other worksheets require students to compute and fill missing quotients (e.g., 24 ÷ 4 = __) and to draw groups to model and find the missing number.
Lesson 14
Multiplication and Division Fact Families
The Skills list explicitly states that students will "understand division as an unknown-factor problem." In the Introduction, students model 20 ÷ 4 = 5 with counters, then rearrange the counters to show 20 ÷ 5 = 4 and write the corresponding division sentences. In Activity 1 and the student pages, students convert multiplication facts (e.g., 3 × 4 = 12) into division sentences (12 ÷ 3 = 4 and 12 ÷ 4 = 3) and complete a rolling-for-fact-families sheet practicing this translation. In Activity 2, students make triangular fact-family cards and practice covering one number to determine the missing factor, directly practicing finding an unknown factor.
Lesson 15
Unit Test
Students practice finding a missing number on multiplication/division fact family cards by covering one number and being asked, "What number is missing?" and to state both a multiplication sentence and a division sentence. The Unit Test requires students to write the fact family for given numbers (e.g., 3, 7, 21) including division sentences (21 ÷ 7 = 3) and to fill in fact-family blanks for 4, 9, 36. Several test items ask students to fill in missing factors (e.g., 2 × __ = 8) and to write division sentences that correspond to multiplication sentences.
Final Project
Multiples Mini-Posters
The lesson's Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers," and it lists "Fluently multiply and divide within 100." The Steps to Success require students to show "5 different ways to show the multiple" and to include a multiplication word problem where the multiple is the answer. The evaluation and sharing steps ask students to define "multiple" and explain how they showed each multiple, which reinforces factor relationships.
Unit 2: Place Value
Lesson 1
Three-Digit Place Value Review
The Skills section lists "Know from memory all products of two one-digit numbers," indicating practice with basic multiplication facts. The Wrapping Up directs students to work with "multiplication flashcards and/or the multiplication and division fact family cards for factors 2, 3, 4, 5, and 10," which links multiplication and division relationships. These elements provide opportunities for students to see division alongside multiplication within fact families.
Lesson 2
Numbers to 10,000
Students practice and are expected to know from memory all products of two one-digit numbers (Skills). Students play times-table games (Hit the Button) to rehearse multiplication facts for factors up to 10 (Activity 6). Students complete the "Which Numbers?" worksheet that asks for pairs whose product is a given number and includes at least one question framed in terms of a quotient ("The quotient of which two numbers is 4?").
Unit 3: Measurement
Lesson 4
Working With Weight
The lesson lists practice with multiplication flashcards and "multiplication and division fact family cards for factors 2, 3, 4, 5, and 10," indicating students will work with paired multiplication and division facts. No other part of the lesson provides worked examples or tasks explicitly about division problems.
Lesson 5
More Work With Weight
Students are asked to divide their weight by the weight of an item to find how many of that item equal their weight (example: 51 ÷ 2 to find cartons of milk). The "Weighty Things" activity requires students to fill in blanks such as "I weigh the same as ___ tires" where they must compute their weight divided by 15. The ping-pong-ball prompt and similar items require students to set up multiplication and division relations (e.g., 168 × 51 for ping-pong balls or dividing by an item weight to find a quantity).
Lesson 7
More Work With Volume
Students practice repeated grouping and multiplication when they add 10 milliliters ten times to make 100 milliliters and when they pour ten 100-ml measures to make 1 liter (1000 ml). The Basic Skills Review includes a problem that asks for half of 80 gallons, which requires dividing by 2. The Wrap-Up instructs providing time for students to work with multiplication and division fact family cards for factors 2, 3, 4, 5, and 10.
Lesson 8
Some Weighty Problem Solving
Students are asked to deduce individual shape weights from total weights in the "What Do They Weigh?" activity (for example, three identical shapes total 9 pounds), which requires finding a single-shape weight from a given total. Several problems ask students to reason from pictured totals to find unknown individual weights using trial-and-error and combinations of add/subtract facts. The Skills list also states that students will "add, subtract, multiply, or divide to solve one-step word problems involving weights," implying use of inverse operations to find missing values.
Unit 4: Multiplication and Division II
Lesson 1
Mastering More Multiplication Facts
The lesson's Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." Students are asked to use the terms "factor" and "product" when locating products on a multiplication table. Students also create multiplication and division fact family cards (triangles showing product at the top and factors), which reinforces the relationship between multiplication and division.
Lesson 2
Multiples of 6 and 7
The lesson's skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." Students complete worksheets with missing factors such as 7 x __ = 42, __ x 7 = 56, and 5 x __ = 35 on the Multiples of 7 activity. Directions tell students to note when they are filling in factors versus products, and the abacus and weeks/days problems ask students to form groups (e.g., use the abacus to show groups of 7 to find how many weeks are in 35 days).
Lesson 3
Multiples of 8 and 9
The Skills list explicitly includes "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." The student pages contain fill-in-the-blank division and multiplication problems such as "__ ÷ 8 = 7" and "8 x __ = 32" that require finding the missing factor or dividend. A word problem asks how many packages of 8 are needed for 32 hot dogs, which requires finding the number that, when multiplied by 8, gives 32.
Lesson 4
Practicing All of the Multiples
The Skills statement explicitly says students will "Determine the unknown whole number in a multiplication or division equation relating three whole numbers." Activity 1 has students create multiplication/division fact family cards, place × and ÷ on each card, hide one corner, and use the two visible numbers to figure the missing number (example given with 8, 7, 56). Activity 2 has students select two or three factors that multiply to equal a given product, which has students find factors that make a target product.
Lesson 5
Associative Property of Multiplication
Students use letters to represent unknown numbers and solve multiplication equations such as 3×n = 9 and 6×3 = n. Students set up and solve multiplicative equations with an unknown factor in contexts like (6×1)×2 = 6×(1×n) (n = 2) and (3×2)×3 = n×(2×3) (n = 3). Students complete practice problems that require filling in missing multiplicative factors (e.g., 8×n = 0, n = 0).
Lesson 7
Reviewing Properties of Multiplication
The lesson lists fluency goals that include multiplying and dividing within 100 and asks students to work with fact family cards where they "hide one number... and figure out the missing factor or product." The wrapping-up activity has students find a missing factor or product from fact family cards, which requires using multiplication facts to determine an unknown related number. The skill list and the fact-family practice both connect multiplication and division concepts in students' practice.
Lesson 8
Revisiting Division
Students use fact-family cards and complete the "What's Missing?" activity where a number is hidden and they are asked to find the missing factor (e.g., hide 2 in 16, 8, 2 and prompt 16 ÷ 8 = ? and 8 × ? = 16). Student pages explicitly show problems like 18 ÷ q = 2 with the companion equation q × 2 = 18 and instruct students to write the missing factor and then the corresponding division sentence. Activities require students to write four number sentences for fact families (two multiplication and two division sentences), and prompts tell students to think "What should I multiply the small number on the card by to equal the large number?"
Lesson 9
Multiplication and Division Facts Practice
The Skills section explicitly lists "Determine the unknown whole number in a multiplication or division equation relating three whole numbers" and "Understand division as an unknown-factor problem." In the domino activity, students create multiplication/division fact families and use a drawn number either as a factor or as a product (e.g., using 6 as product in 2×3=6). In the Input/Output Machine activity students reverse multiplication to solve ÷2 and ÷3 problems (the directions ask, "If you know the divisor and the quotient, how might you figure out the dividend?") and the Factor Family Reunion game has students match multiplication sentences to missing factors.
Lesson 10
Problem Solving Using Multiple Operations
The lesson includes explicit practice writing and solving division sentences (More Problems to Solve shows 16 ÷ 4 = 4 and the Dividend Challenge directs students to write many division sentences for given dividends). The skills statement and activities require students to use multiplication and division within 100 to solve word problems and two-step problems. The Chinese Checkerboard and some problem pages ask students to make equal groups and to multiply group sizes to find totals, linking multiplication and division procedures.
Lesson 11
Multiplying by Multiples of 10
Students solve several real-world division problems that require finding equal shares (e.g., Callie's 20 bags of 20 pieces: 400 ÷ 10 = 40; Jonathan's 42 books shared with 7 people; Amy's 120 pencils for 40 students; Lizzy's 40 apples divided into 8 bags). One problem gives multiple-choice options (10, 20, or 40) for the quotient, which prompts students to reason about which number times the divisor yields the total. The activity pages explicitly instruct students to write the number sentence and answer for each division scenario.
Lesson 12
Unit Test
Students are asked to write and solve number sentences with n in both multiplication and division positions (e.g., n ÷ 3 = 6; 9 ÷ 9 = n; 10 ÷ n = 10). Multiple word problems require students to set up division as an unknown (Marley: 16 ÷ 4 = n; Ellen: 36 ÷ 4 = n) and then solve for n. Fact-family items and answer keys show conversion between multiplication and division (e.g., 7×9=63, 63÷7=9, 63÷9=7), and students are prompted to decide whether situations call for multiplication or division.
Final Project
Planning a Picnic
Students compute quantities that require dividing totals by package or group sizes (e.g., 60 people ÷ 15 plates per package → 4 packages; 24 children ÷ 2 per pair → 12 ropes; 24 children ÷ 4 per team → 6 teams). The seating-arrangement tasks ask students to compute capacities (blankets × 4, small tables × 6, large tables × 10) to reach exactly 60 people. The materials direct students to use multiplication and division within 100 and to refer to an Interactive Notebook for information about factors.
Unit 5: Area and Perimeter
Lesson 3
Calculating Perimeter
Students solve at least one explicit division word problem in the Basic Skills Review (#2: 30 pieces of gum shared with 5 friends). Students practice and use multiplication repeatedly to find perimeters (examples and instructions tell students to use multiplication for regular polygons and to use repeated addition for squares). Several activity problems require writing number sentences (e.g., 4 × 6 = 24) that show connections between side length, number of sides, and total perimeter.
Lesson 4
More Work With Perimeter
Students write and use an equation 6(x) = 24 when working with a regular hexagon and determine the side length from that equation. Students find missing side lengths of rectangles by recognizing equal opposite sides (e.g., subtract known sides from the perimeter to get 10, then determine each of the two equal sides is 5). The activities repeatedly ask students to determine an unknown factor (side length) that, when multiplied by the number of sides or paired side counts, produces the given perimeter.
Lesson 5
Problem Solving With Perimeter
The Skills section explicitly says students will practice "finding an unknown side length," which requires dividing a total perimeter by the number of sides. The Practicing Perimeter activity asks students to draw tables to match given perimeters (e.g., a square with perimeter 20 units where each side is 5 units), so students must compute side length by dividing 20 by 4. The Basic Skills Review includes an explicit division problem (50 ÷ 8) requiring students to partition a total quantity into equal shares.
Lesson 6
What Is Area?
Students complete a Basic Skills Review problem that requires dividing 45 by 6 (answer key shows 45 ÷ 6 = 7 R 3), providing direct practice with division. Students create arrays when they roll two dice (e.g., 4 across and 6 down) and state the area as 24 square units, which has them reason with multiplication as repeated groups. Students also practice multiplication facts (e.g., 80 × 9) and use tiles to compose given totals, reinforcing the relationship between groups and totals.
Lesson 7
Calculating Area
Students repeatedly multiply side lengths to find area (e.g., writing 3, 4, 12 and inserting × and = to show multiplication sentences). In Activity 4 and the "Make These Areas!" tasks, students create polygons to match given areas and write multiplication equations such as 2×5=10 and 3×4=12. The "Think About It!" prompt asks students to find the side length of a square with area 25 (answer 5), which requires reasoning about the factor(s) that produce a given product.
Lesson 8
More Work With Area
Students solve multiplication equations with unknown factors in the Basic Skills Review (e.g., 60 × b = 540 and a × 7 = 420), which requires finding a missing factor. The review also includes a division word problem about sharing 39 pieces of candy equally among 7 friends (39 ÷ 7). Several activities ask students to use multiplication to compute areas of rectangles and composite shapes, reinforcing the multiplication–division relationship.
Lesson 11
Unit Test
Students are asked to find a missing side when area is given (e.g., a rectangle with area 48 sq. in. and one side 6 in. leading to the other side being 8 in.; Problem 18: area 32 sq. ft with one side 4 ft). The materials instruct students that finding area requires multiplying two sides and prompt them to use known information to find unknown side lengths (guess-and-check and inverse reasoning). Multiple activity items and the unit review require students to determine side lengths from given area or perimeter, directly requiring them to find the factor that makes the product.
Final Project
Shapesville
Students are asked to determine unknown side lengths from given areas (e.g., Building #1 and Building #3 require finding a missing width or height from area and a known other side). Students compute equal side lengths from given perimeters (e.g., Building #2 requires finding the side length when perimeter and number of equal sides are given). Day 2 asks students to explain how they found perimeters and areas for their own designs, reinforcing solving for an unknown factor in these contexts.
Unit 6: Fractions
Lesson 2
Naming Fractions
Students encounter explicit unknown-factor multiplication items in the Basic Skills Review (e.g., "4 × b = 320. What is b? (80)" and "a × 30 = 270. What is a? (9)"). Students also solve a division word problem when sharing donuts (55 ÷ 9 = 6 remainder 1), which requires finding how many each person gets from a total amount.
Lesson 3
Fraction Parts
Students encounter explicit unknown-factor multiplication equations in the Basic Skills Review (#16), e.g., "4×b=120. What is b? (30)" and "a×30=180. What is a? (6)." Students also solve a division word problem that requires finding how many each person gets: "43 ÷ 8 = 5 ... 3 left over." The lesson includes multiplication fact practice (flashcards) that supports finding missing factors.
Lesson 4
More Fraction Exploration
Students solve whole-number division in a sharing context (44 ÷ 7) in the Basic Skills Review. Students are given multiplication equations with unknowns to solve (60 × b = 360 and a × 9 = 270). The lesson also tells students that letters can represent missing or unknown numbers, framing work with unknowns.
Lesson 6
Fractions on Number Lines
Students are given explicit multiplication-equation problems that require finding a missing factor: 40 × b = 360 (b = 9) and a × 3 = 270 (a = 90) in the Basic Skills Review. Students solve a sharing problem that involves dividing 31 pieces among 7 friends and interpreting a remainder (31 ÷ 7 = 4 R3). The student activity pages include these problems for students to compute answers on their own.
Lesson 7
Equivalent Fractions
The Basic Skills Review includes equations of the form 50 × b = 450 and a × 7 = 420 that require students to find the unknown multiplicative factor. The review answer key shows students computing 60 ÷ 7 = 8 remainder 4 in a sharing word problem, so students perform division computations. Students also complete arithmetic problems that involve solving for unknowns in multiplication expressions.
Lesson 8
Fractions in Our World
Students count how many equal parts make a whole in the time activity (e.g., answer "How many five-minute parts are there in one hour? (12)") and convert minutes to fractions of an hour (e.g., 45 minutes = 9/12 or 3/4). In the money activity students complete a chart that asks "How many in a dollar" for each coin and write the number of coins that make a dollar (e.g., quarter → 4, dime → 10). These tasks require students to determine the number of equal-sized parts that compose a whole amount.
Unit 7: Geometry
Lesson 3
What Is a Quadrilateral?
The Basic Skills Review #20 includes the equation "a × 5 = 450" which requires finding the unknown factor a, and it also includes a division problem 68 ÷ 7 (Deena sharing jellybeans) with quotient and remainder. Students are asked to complete the Basic Skills Review sheet, which requires solving these operations.
Lesson 4
Getting To Know Specific Quadrilaterals
Students solve division and multiplication problems in the Basic Skills Review #21: they compute 33 ÷ 5 = 6 R3 for sharing cookies and solve the equation a × 8 = 400 (a = 50). The worksheet explicitly presents a multiplication equation with an unknown factor (a × 8 = 400) that requires finding the missing factor. The cookie-sharing problem requires performing division to determine how many items each friend receives and the remainder.
Lesson 6
Dividing Shapes
Students complete the Basic Skills Review #22 which includes the equation a × 4 = 320, asking students to find the unknown factor a. Students also solve division computations (52 ÷ 6 = 8 R4) and are asked to find quotients in context (e.g., "What's 12 divided by 3?") when partitioning areas into equal parts.
Lesson 7
Some Geometric Problem Solving
Students are asked to determine how many smaller shapes make one hexagon (e.g., "How many trapezoids make 1 hexagon?", "How many rhombuses make 1 hexagon?", "How many triangles make 1 hexagon?") and to use the interactive tool to place shapes to show equivalence (the answer key shows three rhombuses = one hexagon). Students name unit fractions for parts of a whole (e.g., 1 trapezoid = 1/2 of a hexagon, 1 rhombus = 1/3 of a hexagon), which involves reasoning about how many equal pieces compose a whole.
Unit 8: Graphing Data
Lesson 2
Pictographs
Students read scaled pictographs and compute how many picture units represent a total (e.g., 40 books with each book-icon = 10 leads students to answer 4). Students convert totals to mixed/decimal picture units (e.g., 65 apples → 6 and 1/2 icons; 85 games → 4 and 1/4 tennis balls). Students are asked to explain the scale using division language ("a quarter of the tennis ball represent 5 because 20 is divided by 4 = 5"), linking partitioning to the unit value. Students create their own pictographs and must decide and justify a scale by answering "How many will each picture represent?", which requires dividing totals by the chosen unit.
Lesson 3
Bar Graphs
The Basic Skills Review includes a division word problem where students compute 36 ÷ 6 to share cookies among friends, with the answer shown as 36 ÷ 6 = 6. The same review also presents an equation in unknown-factor form, 6 × b = 420, asking students to solve for b (answer b = 70). These items require students to perform division and to find a missing factor in a multiplication equation.
Lesson 4
Line Plots
Students solve problems on the "Basic Skills Review #24" sheet that require working with multiplication and division facts. The sheet includes the equation 60 × b = 420 (solve for b) and the division problem 41 ÷ 5 = 8 R1 (sharing equally among 5 friends), which require finding unknown factors and quotients.
Lesson 5
More Work With Graphing
The Basic Skills Review #25 asks students to divide 73 by 8 to determine how many cookies each friend receives and the remainder (73 ÷ 8 = 9 R1), requiring students to perform division in an applied sharing context. The same review also includes an equation 7 × b = 420, asking students to find b, which requires finding the unknown factor that makes the multiplication true (b = 60). Both explicit items require students to connect multiplication and division operations when solving for unknowns.
Lesson 6
Unit Test
The Unit 8 Test includes the item, "If Mark read 35 books in May, how many books does a picture of one book equal? (10)," which requires finding a unit value from a total represented in a pictograph. The materials also ask students to identify the scale for graphs (e.g., "What is the scale for this graph? (2)") and to use that scale to compute totals (e.g., determining how many books Mark read in August). Several questions require computing counts from symbols or scaled axes, which involves dividing or relating totals to unit values.
Unit 9: Skills Review
Lesson 1
Reviewing Multiplication and Division
In Activity 2 students draw a card to place a number in the output column for a ÷3 or ÷5 rule, then use multiplication facts to figure out the input number (e.g., they multiply the output by the divisor to find the dividend). The instructions explicitly ask students, "If you know the divisor and the quotient, how might you figure out the dividend?" and direct them to picture a fact-family triangle. Students also practice the inverse relation between multiplication and division by reversing the x10 activity and by replacing the rule with ÷3 and ÷5.
Lesson 3
Reviewing Perimeter and Area
Students are asked to write number sentences using multiplication for regular polygons (e.g., panels show 3 × 8 = 24, 4 × 7 = 28) and to compute perimeters of shapes by multiplying a side length by the number of equal sides. Students draw rectangles with given areas and identify possible side-length pairs for those areas (e.g., for area 12: 1 × 12, 2 × 6, 3 × 4). Students use multiplication to find area (length × width) and to list factor pairs that produce a target area.
Lesson 4
Reviewing Geometry
In Activity 2 students draw a rectangle with area 15 square centimeters and are asked to divide it into thirds so that each part has five squares, and then to show two-thirds. The activity repeats with other shapes (one-fourth of a 16-square square and two-thirds of a 9-square square), which requires students to partition a total area into equal-size parts and identify the size of each part.