Students are repeatedly asked to identify the digit in the ones, tens, hundreds, and thousands places and to state the value of that digit (for example, asking "What is the value of 6 in this number? (6,000)"). Students write numbers in expanded notation that explicitly show each digit multiplied by its place value (for example, (8×1,000)+(9×100)+(5×10)+(0×1)). Activities prompt students to find "What is the value of the digit in the tens place?" and "How many 100s make up this number?", and worksheets include practice writing numbers in expanded form and expanded notation.

Students use base-10 blocks and a vertical sequence (1, 10, 100, 1,000, etc.) to observe that each number is multiplied by 10 to get the next. The activity pages require students to convert between places (e.g., 2 hundreds = 20 tens; 1 thousand = 100 tens) and answer questions such as "How many tens equal the number of hundreds in this number?" The lesson explicitly prompts students to recognize and state that each place is ten times bigger than the place to its right and has them write values and expanded forms for five- and six-digit numbers.

The Skills section explicitly states students should "Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right." The dot-visual activity has students compute 10, 100, 1,000, 10,000 by repeatedly multiplying by 10 and shows that multiplying by 10 adds a zero (examples: 6×10=60, 60×10=600, 600×10=6,000). The flipbook and student pages list equivalent counts for a million (e.g., 1,000,000 ones; 100,000 tens; 10,000 hundreds; 1,000 thousands), which requires students to connect each place to the next by factors of ten.

Students place M&Ms into columns labeled Millions, Hundred Thousands, Ten Thousands, Thousands, Hundreds, Tens, and Ones and record counts to form multi-digit numbers, showing explicit work with place-value positions. Students write large numbers on a laminated place-value mat (for example, 1,366,432 and 2,187,122) and answer questions that compare numbers by looking at digits in specific places (millions vs. hundred-thousands). Students also write numbers in expanded form (the answer key shows Phoenix's population in expanded form), which decomposes numbers by place value.

Students practice place-value reasoning when they round multi-digit numbers to various places (examples: rounding 5,632 to tens/hundreds/thousands; rounding 3,489, 45,823, and 595,456 to thousands/ten-thousands/hundreds-of-thousands). Students write and identify digits in specific places (tasks asking for a seven-digit number with 3 in the hundred-thousands place and another with 7 in the ten-thousands place). Students use the method of underlining a digit and looking at the digit to its right to decide rounding, and they reason about how digits shift when rounding (e.g., 595,456 rounding up to 600,000 and to 1,000,000).

Students practice lining up digits, carrying (composing tens) and borrowing across place-value positions when adding and subtracting multi-digit numbers. Students use place value understanding to round multi-digit numbers to the nearest hundred or thousand to estimate sums (e.g., rounding 637 and 748 to 600 and 700). The image and text show carrying a 1 from the hundreds place into the thousands place (6+7=13, bring down 1 to thousands).

Students are asked to explore "How does place value work?" and to teach others about place value, and the skills list requires reading and writing multi-digit whole numbers and writing numbers in expanded form. The puzzle requirements force students to create at least three clues that require writing numbers in expanded form and in word form, which has students break numbers into place-value components. The Making the Puzzle instructions and Clues and Answers sheets require students to produce and check numerical answers (including big-number addition and subtraction) that depend on understanding place value.

Students identify place-value values in multi-digit numbers (for example, the review asks "What number does the 7 in 378,241 represent? (70,000)") and practice rounding to a given place (round 345,299 to the greatest place). Students model division with concrete counters (e.g., using 30 counters to show 30 ÷ 3 = 10) and use multiplication to check division (many problems require multiplying the quotient by the divisor and adding any remainder to recover the dividend).

Students are asked to identify place-value magnitudes in the Basic Skills Review (e.g., "What number does the 9 in 983,110 represent? (900,000)") and to round large numbers to a given place (e.g., "Round 785,176 to the greatest place. (800,000)"). The Basic Skills Review also includes a multiplication item (7 x 80 = 560) that involves a factor that is a multiple of ten.

The answer key explicitly shows 14 × 8 = (10 × 8) + (4 × 8) as an example of the distributive property, which breaks a two-digit number into tens and ones. Students complete multiple division and long-division problems (e.g., 28 ÷ 7, 35 ÷ 5, 56 ÷ 7 and other quotient exercises), providing practice with division of multi-digit numbers. Function-table exercises require students to multiply inputs by a constant (rules: multiply by 3 or multiply by 4), reinforcing multiplicative scaling of whole numbers.

Students repeatedly practice multiplying single digits and multi-digit numbers by 10, 100, and 1000 (e.g., card activity, worksheets, and answer key problems like 8×100=800, 3×1000=3000). The table with rows "3, 30, 300, 3000" and problems that rewrite numbers as factors (e.g., 700 = 7×100; Sal's steps rewriting 8×60 as 8×6×10) show students representing numbers by place-value groups and shifting digits when multiplying by powers of ten. Worked examples such as "3 ones × 4 tens = 12 tens = 120" make explicit the idea of grouping by place value.

Students decompose problems like 20 × 40 into (2 × 10)(4 × 10) and rewrite them as (2 × 4)(10 × 10) to get 8 × 100 = 800. Activity sheets present sequences such as 3, 30, 300, 3000 × 6 and examples like 7 × 4, 7 × 40, 7 × 400 that require students to shift zeros and observe products scaling by 10. The matching activity pairs equivalent products (for example, 30 × 40 and 6 × 200) that require students to reassign place-value factors to produce the same product.

Students represent numbers with base-10 blocks (e.g., 3 squares, 5 rods, 6 dots to show 356) and are asked to break numbers into tens and ones (for example, 17 into 10 and 7). The materials show area-model decompositions such as (20+2)(10+6) and compute partial products (e.g., 200 + 140 + 12 = 352). The answer key explicitly asks about 10x10 = 100 and 10x1 = 10, and students place 10-rods and 1-units in arrays to calculate products.

Students break multi-digit numbers into expanded form (for example, 64 into 60 + 4 and 72 into 70 + 2) and compute partial products such as 60×70 = 4200, 60×2 = 120, and 4×70 = 280. Students are instructed to write factors in expanded form across the top and left of an area model and then find and add partial products (steps explicitly include "Break down each factor using expanded form" and "Find the area of each space"). Basic skills problems include multiplying by powers of ten (for example, 59×100 = 5900) and problems that use multiplication by 10s and 100s in the answer key and activities.

Students are asked to explain why a 0 placeholder is needed when multiplying by a tens digit and are told that this is because the digit in the tens place represents 30 (or 50) — i.e., multiplying by the tens place makes the answer ten times bigger. The student activity page explicitly states: "The zero shows that we are technically multiplying 37 by 50 in this step." The lesson also prompts students to justify whether a product is closer to 16 or 160, and asks students to explain how they can use what they know about place value when finding products.

Students decompose multiplication problems into tens and ones in the "What's Missing?" activity (e.g., 82 x 16 as 82 x 10 = 820 and 82 x 6 = 492). The Basic Skills Review asks students to multiply by powers of ten (73 x 1000 = 73,000; 900 x 70 = 63,000) and to solve for a missing factor in 6000 x N = 240,000 (N = 40), which requires reasoning about place-value shifts. Expanded-form examples (43 x 27 = 43 x 20 + 43 x 7) and grid work for two-digit multiplication repeatedly use place-value-based strategies.

The Skills section explicitly asks students to multiply using strategies based on place value and to use expanded form, arrays, and area models. The student tasks include an expanded-form problem (81 x 35), area-model and array prompts (46 x 73, 36 x 7), and several problems that multiply by tens/hundreds (10 x 70, 70 x 40, 8000 x 3, 9 x 400). These items require students to decompose numbers into tens and ones and to reason about products that involve tens and hundreds.

Students complete tables and problems that include division by 10 (for example the row showing 100 ÷ 10 with quotient 10). Students practice writing and interpreting division in multiple forms (e.g., 18 ÷ 6, 6)18, 18/6) and relate division to multiplication (e.g., writing multiplication facts and corresponding division statements). Students solve quotients for a variety of whole-number division problems, some of which involve dividing by powers of ten.

The lesson's Skills statement explicitly says students will find quotients using strategies based on place value and the relationship between multiplication and division. In Activity 2 students are instructed to "divide the tens place" first and then "bring down the ones place," with worked examples (e.g., 72 ÷ 4 broken into tens and ones) that require reasoning about tens and ones. Division tables and practice problems (for example 80 ÷ 4 = 20 and 32 ÷ 8 = 4) give students opportunities to compute quotients where place-value grouping (tens vs ones) affects the result.

Students are shown and practice the "remove the zeros" method (example: 1600 ÷ 8 → 16 ÷ 8 = 2, then add zeros to get 200) and complete many problems that divide large numbers with zeros (e.g., 16,000 ÷ 4, 320,000 ÷ 8, 50,000 ÷ 2). The lesson's skills statement directs students to use strategies based on place value and the activities have students simplify by powers of ten before dividing. Students also practice standard long division with one-digit divisors on multi-digit dividends.

The lesson explicitly asks students to use "strategies based on place value" to find whole-number quotients and remainders and models long-division problems (e.g., 72 ÷ 4 = 18; 936 ÷ 4 = 234; 340 ÷ 4 = 85) that require working with multi-digit dividends and one-digit divisors. Activities prompt students to set up long division, use multiplication to check answers, and create multi-digit division sentences (card activity) that engage them in dividing numbers by one-digit divisors using place-value procedures.

The skills list explicitly tells students to find quotients using strategies based on place value and the relationship between multiplication and division. Students solve and check division problems with large multiples of ten (for example answer key and problems: 63,000 ÷ 7 = 9,000; 360,000 ÷ 6 = 60,000; 50 ÷ 10 = 5; 3,500 ÷ 5 = 700). The unit test table and practice items require students to compute quotients for numbers in the thousands and hundreds, encouraging use of place-value reasoning when dividing by one-digit divisors.

Students are asked to find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using strategies based on place value (Skills section). The project requires students to include "Steps to Divide" for dividing 2 digits by 1 digit and 4 digits by 1 digit and to solve real-world long-division problems. Students are prompted to go through the steps of long division using the problems shown on their poster, which will require applying division procedures to multi-digit numbers.

Students write and analyze whole-number place values in the Introduction (e.g., identifying the thousands, hundreds, tens, and asking for the value of digits in 98,462). The expanded form activities explicitly break numbers into (digit × place value) terms (e.g., 562 = (5×100)+(6×10)+(2×1)), showing the multiplicative factors of 10 between adjacent places. Activity 2 includes an interactive where students click "(x)10" and "÷10" to watch a number grow or shrink by factors of ten, reinforcing the ten-times relationship between adjacent places.

Students read and say 2/10 and 2/100 and convert them to 0.2 and 0.02, practicing equivalence between tenths and hundredths. Students complete a Base-10 Fractions and Decimals table (e.g., 1/10 = 0.1, 5/100 = 0.05) and create equivalent fractions with denominators 10 or 100. Students use steps that convert fractions by scaling denominators and numerators by 10 (for example, 2/5 → 4/10 → 0.4 and 11/20 → 55/100 → 0.55).

Students label and locate tenths and hundredths on number lines and grids (e.g., labeling 0.1, 0.01 and placing 0.15, 0.51, 3.2, 3.62). The lesson has students use a 10×10 grid to show that 10/100 = 0.10 and that 0.1 = 0.10 (ten hundredths equals one tenth) and practices writing mixed numbers as decimals (e.g., 1.23). The materials ask students to note that 0.2 = 0.20 and to count hundredths up to a tenth, reinforcing the multiplicative relation between adjacent decimal places.

Students are asked to name the place to the right of the hundredths as the thousandths and to decide which is greater, one hundredth or one thousandth, showing attention to relative place-value sizes. The lesson tells students that "the value of the places gets smaller as you move to the right" and has them compare numbers by checking ones, tenths, and hundredths in a flowchart. Students practice adding a zero (e.g., writing 0.50) to show equivalent values and use grids and comparison activities to reason about which place values make a number larger.

Students write numbers in expanded form (e.g., 23.87 as 20 + 3 + 0.8 + 0.07; 14.08 as 10 + 4 + 0.08) and convert place-value expressions to decimal form (e.g., 7 + 6/10 + 2/100 = 7.62). The materials include equivalence tasks that map places across scales (e.g., 8/10 = 80/100, matching fractions to decimals) and ordering/comparing decimals which requires attention to tenths, hundredths, ones, tens, etc.

The lesson repeatedly states that the metric system is based on 10 and instructs students to multiply or divide by 10 when converting between metric units (e.g., "to go from gram to decigram, he will multiply 1 gram by 10 to get 10 decigrams"). It explicitly has students move the decimal point and perform sequences of multiplying/dividing by 10 (for example, converting 1 gram to decigrams, centigrams, milligrams and converting 100 meters through kilometers, hectometers, dekameters, etc.). The materials ask students to start at a base unit and use arrows to multiply or divide by 10 on conversion charts, reinforcing the ten-times relationships between adjacent metric units.

Students practice multiplying and dividing by 10 as a rule for metric conversions (King Henry sheet: "MULTIPLY numbers by 10..." and "DIVIDE numbers by 10..."). Students complete conversion problems that require shifting digits by powers of ten (e.g., converting 3 m to 300 cm, 40 m to 4000 cm, 750 cm to 7.5 m) and fill tables that use multiplication/division by 10, 100, 1000. Students order index cards (1 mm, 10 mm, 10 cm, 1 m, etc.) and write equivalent measures, reinforcing how values change by factors of ten.

The lesson asks students to note that the metric system is based on 10 and tells them to multiply when moving from larger units to smaller units and divide when moving from smaller units to larger units. Several comparison problems require converting between metric units that are powers of ten (for example, 5 liters and 5000 milliliters; 50 meters and 5000 centimeters; 10 meters and 10,000 centimeters). Student recipe and measurement problems require students to convert between units like kilograms and grams (5 kilograms = 5000 grams) and milliliters and liters (5000 ml = 5 liters).

Students are asked to read and write multi-digit numbers and identify specific place values (e.g., which number has a 1 in the ten thousands place and which has a 7 in the hundred thousands place). The listed skills include reading/writing numbers using base-ten numerals, number names, and expanded form and comparing multi-digit numbers based on meanings of the digits in each place. The wrap-up activity has students compare pairs of large numbers and identify which is greater, requiring attention to digit place values.

The lesson lists the skill "Use place value understanding to round multi-digit whole numbers to any place" and directs students to write and round large numbers (e.g., 395,578; 2,333,598) to the nearest hundred thousand. Students are instructed to underline the digit in the hundred-thousands place and look to the digit to its right to determine rounding, which requires identifying place-value positions. The rounding tasks require students to identify and reason about digits in different places when producing rounded numbers.