Students practice multiplying and dividing by 10 with whole numbers and decimals (e.g., problems like 3 × 10, 300 ÷ 10, 6.5 × 10 = 65, 1.2 ÷ 10 = 0.12). Students identify that multiplying by 10 makes a number 10 times greater and that dividing by 10 (finding 1/10) makes a number smaller, and they compare 1/10 and 1/100 using a cake example. The materials ask students to compute and explain what happens to place value and the decimal point when multiplying or dividing by 10.

Students multiply whole numbers by unit fractions in expanded notation (e.g., (8 × 1/10) = 8/10 and (2 × 1/100) = 2/100) and write decimals as sums of these products (e.g., 0.82 = (8 × 0.1) + (2 × 0.01)). Students use base-ten grids to show how moving one decimal place to the right scales a value by one tenth (e.g., 0.6 compared to 0.06 and 0.006). Students convert decimals to fractional forms such as 563/1000, reinforcing that decimal place-value represents multiplication by unit fractions.

Students practice multiplying and dividing whole numbers and decimals by powers of 10 (e.g., problems such as 0.01 × 10 = 0.1, 0.1 × 10 = 1, 1,000 ÷ 10 = 100, 10 × 10,000,000 = 100,000,000). Students use a number line with exponential notations (10^0 to 10^6) and complete matching and cut‑and‑paste activities that show how multiplying by 10 adds zeros or shifts the decimal point. Discussion prompts ask students to notice patterns (number of 10s and number of zeros) and to state what happens to whole numbers and decimals when multiplied or divided by 10.

Students complete a fraction-equivalence problem (1/5 = ___/15) in the Basic Skills Review, producing 3/15, which requires multiplying numerator and denominator by the same number. The parent notes and student materials explicitly show that students can add zeros to decimals to show equivalence (for example, 3.4 = 3.40), demonstrating an understanding of creating equivalent numeric representations by multiplying by powers of 10.

Students draw and use base-10 blocks to model multiplication by whole numbers (e.g., 2 × 0.45, 4 × 0.36, 3 × 0.27) and they exchange hundredths for tenths to find products. Students use grids to model decimal × decimal (e.g., 0.3 × 0.5 = 0.15 and practice problems like 0.4 × 0.6), counting overlapping hundredths to show the product is smaller. Students are prompted to answer reflective questions about whether products are greater or smaller when multiplying by whole numbers versus multiplying two decimals, and they explicitly note that multiplying by a whole number > 1 produces a larger product while multiplying two decimals (parts) produces a smaller product.

Students perform many multiplication tasks that involve multiplying by numbers less than 1 (for example, Parker: 2.9 × $0.90; donut cost: $0.15 × 6 or × 10; coin totals: counts × $0.01, $0.05, $0.25, etc.). Students also multiply by whole numbers greater than 1 in several places (for example, Maggie's dog: 20.6 × 7 days; hourly pay × 8 hours or × 5 days; pay × multiple weeks). The Coinstar and Money Matters activities require students to compute totals by multiplying counts or rates by unit values and then add results.

Students are asked to decide whether to multiply or divide when converting units (e.g., "Going from a large unit to a small unit, MULTIPLY!" and practice problems like 4 tons = ? pounds where they compute 4 × 2000 = 8000). Students multiply by whole-number conversion factors (12 inches = 1 foot, 4 quarts = 1 gallon, powers of ten in metric conversions such as 35 meters × 10^3 = 35,000 mm) and use moving the decimal for multiplying/dividing by powers of 10. Students also practice interpreting multiplication and division in measurement contexts (e.g., converting inches to feet by dividing 36 ÷ 12 = 3).

Students perform unit conversions by multiplying and dividing: for example, they compute 4 pints = 4 × 2 = 8 cups and convert 3 meters to centimeters by multiplying 3 × 100 = 300 cm. Problems and answer keys show students multiplying quantities by whole-number conversion factors (e.g., 6 qt × 2 = 12 pints) and dividing larger counts to get larger units (e.g., 82 ÷ 16 = 5 R 2 to find pounds). The materials also note equivalences such as 1 pint = 1/2 quart in a practice problem.

Students are instructed to multiply when converting from a larger unit to a smaller unit (e.g., 4 hours × 60 = 240 minutes) and to divide when converting from a smaller unit to a larger unit (e.g., 540 seconds ÷ 60 = 9 minutes). Students complete conversion problems that use multiplication by whole-number factors (e.g., 3.5 weeks × 7 = 24.5 days; 80 pounds × 16 = 1,280 ounces) and problems that produce decimal results (e.g., 330 seconds → 5.5 minutes; 4.7 days × 24 = 112.8 hours).

Students perform many unit conversions that require multiplying and dividing by whole numbers and powers of ten (e.g., converting 2 L to 2,000 mL by ×1,000; converting 4 kg to 4,000 g by ×1,000; moving the decimal when changing metric units). Students solve word problems that use multiplication by whole numbers greater than 1 (doubling a muffin recipe, multiplying 12 balloons × 42 in. to get total inches) and problems that involve partitioning or dividing (sharing 2 L among 10 people to get 200 mL each), which implicitly involve multiplication by fractions like 1/10. The metric conversion chart and practice emphasize multiplying or dividing by factors of ten and converting by multiplying or dividing, reinforcing scaling with integer and decimal factors.

Students create and use fraction strips and complete activities that show examples such as 1/3 = 2/6 and 1/2 = 3/6 to demonstrate how multiplying numerator and denominator by the same number produces equivalent fractions. The materials explicitly prompt students to notice that "both are being multiplied by 3" (for 1/2 → 3/6) and that "the same thing that happens to the numerator must also happen to the denominator." Students also build equivalent-fraction chains and write equivalent fractions by multiplying or dividing numerator and denominator by the same factor.

Students practice creating equivalent fractions by multiplying numerator and denominator by the same number (e.g., the lesson shows 3/4 → 15/20 and 2/5 → 8/20) and use those equivalent fractions to compare sizes. The Changing Denominators method and its examples explicitly have students multiply both parts of a fraction to build a common denominator. The student activity pages and answer keys instruct students to multiply numerator and denominator (n×a)/(n×b) when making equivalent fractions for comparison.

Students shade models and convert improper fractions greater than 1 to mixed numbers, showing they recognize when a fraction represents more than one whole. Students follow procedures that include multiplying the whole number by the denominator when converting a mixed number to an improper fraction. Students are asked to create equivalent fractions to compare values and to use those equivalents when ordering fractions.

Students convert fractions to equivalent fractions by multiplying numerator and denominator (for example 2/5 × 3/3 and 1/3 × 5/5) on the "Easiest Common Denominator" sheet. Students are asked to use and show unit fractions with fraction strips to make sums, practicing decomposing and recomposing amounts. The Parent Plan directs students to see that fractions with the same numerator and denominator equal 1 and to recognize that multiplying by such a fraction does not change the original value.

Students create equivalent fractions by multiplying numerator and denominator by the same number (for example, 5/8 becomes 25/40 and 2/5 becomes 16/40) when finding a least common denominator. The lesson explicitly shows that 2/3 is the same as 4/6 using a fraction chart and has students convert fractions to equivalent forms (e.g., changing denominators to 24 or 40) before adding. Students list multiples to find least common multiples and then use those multiples to produce equivalent fractions for addition.

Students convert fractions to equivalent fractions by multiplying the numerator and denominator by the same whole number (for example, rewriting 3/4 as 6/8). Students use a laminated fraction chart to see and reason that 3/4 equals 6/8 and follow steps that explicitly tell them to "multiply the numerator (top number) and denominator (bottom number) by the same number" to make common denominators. The lesson also instructs students to use either the easiest common denominator or least common denominator methods and to "make equivalent fractions using the common denominator."

Students repeatedly create equivalent fractions and find common denominators (e.g., changing 2/3 and 1/5 to 10/15 and 3/15) when adding mixed numbers. The lesson explicitly instructs students to multiply numerator and denominator to create equivalent fractions and to "Find the common denominator and create equivalent fractions." Day 2 shows students converting mixed numbers to improper fractions and finding common denominators by creating equivalent fractions (e.g., 7/5 -> 14/10).

Students use repeated addition and pictures to compute whole-number × fraction products (e.g., 4 × 1/2 = 2 and 6 × 1/4 = 6/4), practice the algorithm of multiplying the numerator by the whole number while keeping the denominator the same, and use fraction strips to trade parts for wholes (trading four 1/4 strips for one whole). Activities ask students to simplify or convert improper fractions to mixed numbers and to explain their work to a parent, and matching/prove-it tasks have students manipulate products such as 7 × 1/2 = 7/2 and 9 × 3/4 = 27/4. These tasks give students multiple representations (repeated addition, pictures, algorithm) for multiplying by fractions and whole numbers.

Students practice comparing products to the whole-number factor on multiple activities (e.g., the "Greater Than, Less Than, or Equal To" sheet and the "Scaling Practice" sheet) where they categorize whether products are greater than, less than, or equal to the whole number. Students solve and interpret examples showing multiplication by fractions less than 1 (e.g., 1/2 of 8 or 1/3 of 12) and fractions greater than 1 (e.g., 3/2 × 112, 5/3 × 25) and are prompted to decide and justify whether the product is larger or smaller. Students are asked reflective questions and "Think About It!" prompts that require them to explain in words when multiplying by a fraction makes a number grow, shrink, or stay the same and to relate this to familiar whole-number multiplication cases.

Students use fraction strips and area models (folding paper and grid overlaps) to compute products such as 3/4 of 1/2 = 3/8 and 1/3 × 1/2 = 1/6, showing how to find 'parts of a part' by counting overlapping shaded regions. Students fold a whole into denominators and shade numerators, then unfold to count total parts and shaded parts, demonstrating that multiplying by a fraction less than 1 produces a smaller portion of the whole. Students practice multiple problems with unit fractions and nonunit fractions using area models and an interactive tool to visualize the overlap and compute products.

Students use number line and area models to find products of fractions, for example using a number line to show 3/4 × 1/2 = 3/8 and comparing that to the area model result. The text explicitly states that "fraction multiplication with proper fractions involves finding part of a part, so the answer is less than either of the parts," and students complete practice problems multiplying proper fractions and whole numbers by fractions. Students also use equivalent fractions in examples (e.g., 2/3 = 8/12) when adding, subtracting, and interpreting parts of models.

Students practice multiplying mixed numbers by converting mixed numbers to improper fractions, canceling, and multiplying numerators and denominators in Activities 2 and 3. Students solve practice problems and word problems that involve multiplying by fractions less than 1 (e.g., the team ate 3/4 of 5 1/2 bags) and by whole numbers greater than 1 (several problems multiply mixed numbers by 3, 4, 6, 9). The lesson also shows equivalence between mixed numbers and improper fractions (e.g., 1 2/5 = 7/5) when comparing amounts.

Students practice multiplying by fractions less than 1 and greater than 1 in multiple input/output table activities (rules: multiply by 2/3, 1/4, 1 1/2, 2 1/3) and complete problems that produce outputs smaller or larger than the inputs. The Always/Sometimes/Never activity asks students to decide whether products are greater than, less than, or equal to 1 and to place statement cards after testing sample problems, prompting students to draw general conclusions about multiplying by fractions >1, <1, and =1. Parent prompts and wrap-up questions ask students to explain when products are less than 1 and note that multiplying by improper fractions or mixed numbers can yield products greater than the original number.

Students set up and compute areas by multiplying mixed numbers and proper fractions in multiple activities (e.g., tiling a 5 1/4 × 4 rectangle to get 21 sq in, finding areas for 3 1/2 × 2 1/2 = 8 3/4, and computing 2 1/3 × 1/2 = 1 1/6). Students use visual tiling models to build and count unit and partial tiles, and they translate those counts into fraction products on student activity pages and answer keys. The lesson includes problems where one factor is greater than 1 (mixed numbers) and others where a factor is a proper fraction, so students produce examples of products that are larger or smaller than the original factors.

Students complete IN/OUT tables that require multiplying given numbers by 1 1/2 and by 3/4, producing results that show outputs larger when multiplied by 1 1/2 and smaller when multiplied by 3/4. Students evaluate always/sometimes/never statements that ask about whether multiplying by a number greater than 1 yields an answer greater than 1 and whether multiplying by a fraction greater than 1 yields a product greater than the original fraction. Students use and match visual models (number lines, area/grid models, bar models) to compute and interpret products such as 5 × 1/4, 2/3 × 6, and 1/3 × 1/2, giving concrete examples of scaling.

Students are required to create 20 multiplication problems and sort them into four product-size categories: less than 1, equal to 1, greater than 1 and less than 2, and 2 or greater, which requires them to compute and recognize when products are smaller than, equal to, or greater than 1. The project requires students to make problems that include whole-number-by-fraction, fraction-by-fraction, mixed-number, and fractional-area (rectangle) multiplications, so students practice multiplying various types of fractions and mixed numbers. The materials remind students that some numbers can be represented in different ways (for example, 4/4 = 1 and mixed numbers can be written as improper fractions), providing an implicit connection to equivalence and alternative representations.

Students create input/output tables and plot patterns where the rule is "multiply by 2" and "multiply by 4" (Activity 2), and they compare two lines showing $2/week versus $4/week savings. The materials mention a daisies example that grows 1/2 inch per day, giving an implicit example of multiplication by a fraction less than 1. The answer keys and questions ask students to identify the relationship between two data sets (e.g., "The second set is double the first set"), and students compute products of mixed numbers by whole numbers in the Basic Skills Review.

Students physically cut 4 sandwiches into thirds, sixths, fifths, and tenths and record results such as 4/3, 4/6 (2/3), 4/5, and 4/10, showing how changing the denominator changes the fraction. Students fill a table that identifies the numerator as the number of sandwiches and the denominator as the number of people and answer prompts about when a quotient is greater than, equal to, or less than 1. The activities explicitly show creating equivalent fractions by scaling numerator and denominator (for example, cutting fifths in half to make tenths and showing 4/5 = 8/10 or 2/5 = 4/10).

Students solve and visualize problems like 4 ÷ 1/2 = 8 and 4 ÷ 1/3 = 12 using diagrams that show how many fractional parts fit into wholes. Students answer explicit questions (e.g., "When dividing a whole number dividend by a fraction divisor, is the quotient larger or smaller?") and justify that dividing by a fraction gives a larger quotient. Students learn and practice the procedure Change–Switch–Flip–Solve, converting expressions like 3 ÷ 1/4 into 3/1 × 4/1, and use reciprocals to turn division by a unit fraction into multiplication by a whole number.

Students solve scaling problems that involve multiplying fractions by whole numbers (e.g., the lemonade recipe: 1/8 cup per pint scaled to 8 pints to get 1 cup) and compute products of mixed numbers and whole numbers (4 2/3 × 5). Students also work with fraction arithmetic on the Basic Skills Review (e.g., 1/8 ÷ 7 and problems involving division by unit fractions and multiplication with whole numbers). These tasks give students practice carrying out multiplication where a quantity is scaled by a whole number.

Students manipulate whole-number scaling in Activity 2 by doubling and tripling the length, width, and height and computing the resulting volumes (e.g., 6×4×3 = 72, doubled dims 12×8×6 = 576, tripled dims 18×12×9 = 1,944). Students solve a problem where one prism has twice the volume and is twice as long to determine the missing height, applying multiplication by 2 to relate dimensions and volume. The parent plan and wrap-up questions explicitly prompt students to double or triple a single dimension to double or triple the volume, supporting reasoning about scaling by whole numbers.

Students repeatedly compute volumes by multiplying dimensions (l × w × h and b × h) and count unit cubes to verify volume, as shown in multiple practice and test problems. Students are asked to double the volume while keeping height the same (Joey/Travis tasks) and to prove by picture and math sentence that doubling a length or width doubles the volume. Students use multiplication with whole-number scaling in dice-generated prism activities and in using algorithms to find volumes and missing dimensions.

Students complete input/output tables that require multiplying whole and mixed numbers by 3/4 (a fraction less than 1) and by 1 1/2 (a fraction greater than 1). The answer key shows specific results (e.g., 2 × 3/4 = 1 1/2, 2 × 1 1/2 = 3), so students practice computing products that are smaller when multiplied by 3/4 and larger when multiplied by 1 1/2. Students also solve word problems that involve scaling (e.g., finding area by doubling a length and multiplying mixed numbers).

Students convert measurements by multiplying with conversion factors (for example, Brigid's bookcase: 3 ft × 12 inches/ft = 36 inches; Ace: 8 gallons × 4 quarts/gallon = 32 quarts). The matching activity and answer key show conversions such as 2 kg to 2,000 g and 3.5 liters to 3,500 ml, which require multiplying by whole-number or decimal factors. Students also compute volumes using multiplication (V = l × w × h and V = base × height), reinforcing multiplicative scaling of dimensions.