Students practice extending a numeric sequence in the Basic Skills Review where they fill in blanks for a sequence (150, 250, 350, 450, 550, 650), which requires recognizing and continuing a rule-based pattern (adding 100). Students also identify multiples of 3 by circling them, which involves recognizing a repeated numeric feature across a list of numbers. These activities provide some experience with number patterns and with detecting regularities in sequences.

Students write and extend the sequence 1, 10, 100, 1,000 and are asked to write the next two numbers (10,000 and 100,000). They are asked what operation produces each next term and led to identify the rule "multiply by 10." Students are prompted to note that a zero is added each time and to prove this by multiplying a one-digit number by 10 repeatedly on a calculator. The lesson also directs students to watch a video that discusses rules and patterns in place value.

Students use the million-dots activity to count by groups (10, 100, 1,000, 10,000, etc.) and are prompted to find that one row = 10, a small square = 100, a column = 1,000, and the larger square = 10,000. The lesson explicitly models and gives examples of multiplying by 10 (e.g., 6×10=60, 60×10=600, 600×10=6,000) and includes the skill statement that a digit in one place represents ten times what it represents in the place to its right. Students also generate multi-digit numbers by drawing digit cards and placing them in ones, tens, hundreds, etc., which produces different numbers for comparison and reading.

Students are asked to "Fill in the blanks: 1,320, ___, ___, 1,340" on the Basic Skills Review and the answer key shows the extended sequence 1,320, 1,330, 1,340, 1,350, 1,360, 1,370. Students are also asked to "Circle the multiples of 9" from a list of numbers, which asks them to recognize a numerical pattern (multiples). These items require students to continue or recognize numeric sequences.

Students are instructed in the Spinner Addition activity to "Spin the attached spinner," "Add the numbers after each spin," and "Write your answers in the grid," which causes them to generate a sequence of sums governed by the spinner rule. The Making One Million activity asks students to draw lines to connect pairs of numbers that sum to 1,000,000, requiring them to produce number pairs that follow a specific rule. The Nine-Digit Addition Challenge gives explicit tips and a suggested pattern (e.g., use smaller digits in the first number, medium in the second, bigger in the sum) that prompts students to construct multiple addition solutions following systematic choices of digits.

Students complete Subtraction Ladders where they solve the top subtraction, write the answer in a shaded box, and then use that answer as the first number for the next subtraction, thereby producing a sequence of numbers by repeatedly applying the given subtraction steps. Students work through multiple ladders with sequential subtractions, creating terms in resulting numeric sequences. Activity instructions explicitly tell students to use each result as the starting point for the next calculation, so students practice generating terms that follow a specified arithmetic rule (subtracting specified amounts).

Students are asked to list multiples of given numbers (e.g., multiples of 2 from 0 to 24, multiples of 5 from 0 to 60) and to write out the multiples for 7 using multiplication expressions (0×7, 1×7, 2×7, ...). The student pages include exercises that generate sequences of multiples (first five multiples of 7, first six multiples of 5, multiples of 8 and 6 in given ranges) and a table where students mark whether given numbers are multiples of 8 and 3. Riddle problems require students to use properties of multiples to identify specific numbers (e.g., find a number that is a multiple of 5 and 10 more than a multiple of 4).

Students follow the Sieve of Eratosthenes steps to cross out all multiples of 2, 3, 5, 7, and 11 on a 100 chart, which requires them to generate number sequences (counting by 2s, 3s, etc.) according to those rules. Activity 4 has students circle the remaining numbers as primes after applying those repeated-addition rules and suggests using different colors to mark the multiples, reinforcing the generated pattern of multiples. Activity 1 (Factorize) has students create rectangles with side lengths that multiply to a target number, producing shape configurations that reflect the rule for forming factor pairs.

Students create Number Chains of six numbers where each term must be a factor or multiple of the previous term, using number cards 1–48, and they are asked to explain the example chain by identifying which numbers are factors and which are multiples. In Activity 4 (Four Rules) students generate numbers that satisfy specific verbal rules (e.g., multiples of 6, primes, odd but not multiples of 5) and answer questions about overlaps between groups. One Four Rules prompt explicitly asks for reasoning about overlap (Can a number belong in both Mike's group and Tashi's group? Why or why not?), which requires students to identify and explain an implied feature (multiples of 6 are even and thus cannot be prime).

Students are asked to identify and apply rules for numeric patterns (e.g., recognizing 2, 4, 8, 16 as "multiply by 2" and writing their own multiplication-based pattern). Students generate terms from one- and two-step rules (examples: 2, 5, 11, 23 as multiply by 2 then add 1; start 3 with rule multiply by 3 then subtract 2 and find subsequent terms). Activity 1 and Activity 2 require students to follow given input→output rules to complete tables and to deduce rules from input/output pairs, and the "Think About It!" prompts ask students to compute later terms in sequences using given rules.

Students complete multiple In/Out tables and are asked to write the rule (e.g., tables showing In 1–5 with Outs 4, 8, 12, 16, 20 and rule: "multiply by 4"; another table showing In 1–5 with Outs 3, 6, 9, 12, 15 and rule: "multiply by 3"). Several items explicitly prompt students to "Complete the table. Write the rule" or "Find the rule and fill in the missing numbers," requiring students to generate terms that follow a given arithmetic rule. Problems require students to produce output values from a specified multiplication rule for a sequence of inputs.

The Basic Skills Review #8 asks students to continue a numeric sequence: "What comes next? 2, 3, 5, 9, 17," which has students generate the next term. The "My Own Flags" activity has students design shapes that satisfy given geometric rules (e.g., "2 acute angles, 2 obtuse angles, 2 colors" or "4 right angles"), requiring them to produce shapes that follow specified constraints. The Wrapping Up task asks students to draw angles with given measures (25°, 45°, 100°, 155°), so students produce examples that follow a stated rule (specific angle measures).

Students generate number patterns by writing the first 10 multiples for given numbers (Activity 1: multiples of 2, 4, 6, 8; Activity 2 and More Multiple Patterns: multiples of 3, 5, 7). Students are asked to record and describe features of those lists (e.g., digits in the ones place repeat, all listed multiples are even) and to explain relationships they observe (e.g., doubling multiples of 2 gives multiples of 4; adding corresponding multiples of 2 and 8 yields multiples of 10). Students also practice predicting whether the product of two numbers is even or odd and are asked to justify that rule informally using drawn number cards.

Students draw number cards and repeatedly state the product when the card value is multiplied by 10, 100, and 1000, generating sequences such as 3, 30, 300, 3000. Students complete tables and worksheet problems that produce ordered number patterns by repeatedly multiplying by 10, 100, and 1000. Students are asked to explain "What's the trick?" (noting removing zeros, multiplying, and adding zeros back) and to follow Sal's Steps which use associative and commutative properties to justify the procedure.

Students complete sequences such as 3 x 6, 30 x 6, 300 x 6, 3000 x 6 and fill in blanks on pages labeled "Fill in the blanks to complete the patterns." The materials include explicit pattern examples (e.g., 7 x 4 = 28, 7 x 40 = 280, 7 x 400 = 2800, 7 x 4000 = 28000) and ask students to extend those multiplication-by-10 patterns. Activities ask students to generate products for repeated multiplication by 10, 100, and 1000 (for example 3→30→300→3000 or 5 x 10, 5 x 100, etc.).

The Basic Skills Review includes the numeric sequence 3, 6, 15, 42, 123 and asks students "What will come next?" and "What's the rule?", with the provided rule ((×3, −3)). The worksheet presents a similar sequence prompt (3, 6, 15, 42, __) asking students to determine the next term and the rule. These items require students to analyze a sequence and state the rule that generates its terms.

Students are asked in the Basic Skills Review to determine what comes next in the sequence 2, 5, 11, 23, 47 and to state the rule; the answer key indicates the rule is "multiply by 2, then add 1." This requires students to analyze given terms, infer the generating rule, and produce the next term in the sequence. The activity asks students to articulate the rule that generated the shown number pattern.

Students are asked to use knowledge of numbers, multiplication, and patterns in the "Comparison and Patterns" section where they compare products without computing them, and in the "Even or Odd Products" section they determine whether given products will be ODD or EVEN. The unit test includes true/false items about parity (e.g., "When both factors are odd, the product is odd") and practice problems that require recognizing parity and multiplication properties. Several items ask students to reason about factors to decide product properties rather than compute every term.

Students are explicitly taught the rule that equivalent fractions are created by multiplying or dividing both numerator and denominator by the same number, and they practice this when asked to convert 3/4 to 9/12 (×3) and 10/12 to 5/6 (÷2). Multiple activities require students to generate equivalent fractions (Creating Equivalent Fractions cut-and-paste, dominoes matching, activity sheets with missing numerators/denominators). Visual tasks (geoboard, pie charts, interactive fraction tool) and repeated practice ask students to produce series of fractions that follow the given multiplication/division rule.

Students list multiples of denominators (for example, 3, 6, 9, 12, 15, 18) when using Jess's steps to find a common denominator and convert 2/3 to 10/15 and 3/5 to 9/15. Students are instructed to "find a common denominator" by listing multiples of the smaller denominator and to create equivalent fractions. Students also generate and order fractions (for example, putting 3/10, 11/30, 1/2, 12/15, 5/6 in order) which requires producing and using number sequences.

Students generate repeated-addition sequences for problems like 4 × 1/5 (writing 1/5 + 1/5 + 1/5 + 1/5 = 4/5) and complete activity pages that ask them to draw circles and write addition sentences for multiple repeated-fraction problems (e.g., 6 × 1/9, 3 × 1/5). Students are asked to compare results across problems and explicitly note what changes ("What do you notice happening to the numerator and the denominator in each one?"), and they sort products as less than, equal to, or greater than 1, observing emergent features such as when products become improper fractions.

Several items in the Student Activity Page require students to continue number patterns by filling in missing fractional terms (e.g., "__, 1/4, 3/8, 1/2" and "5/10, 6/10, __, 8/10"). Additional items ask students to provide missing numbers in sequences (e.g., "2/3, __, __, 1/3" and "4/9, 5/9, __, __"), which requires generating terms that follow a consistent rule. These tasks ask students to produce subsequent terms based on an implied additive step between fractions.

Students repeatedly convert fractions with denominator 10 to equivalent fractions with denominator 100 (e.g., 2/10 = 20/100, 7/10 = 70/100, ?/10 = 50/100) on the worksheet and fill-in activities. The comparing activity asks students to rewrite 3/10 as 30/100 to determine which is larger, and the adding activity explicitly directs students to change a denominator of 10 to 100 (3/10 -> 30/100) before adding. Students also number the steps for the procedure (change denominator, leave 100s, add numerators), showing repeated generation of equivalent examples using the same multiplicative rule.

Students are instructed on the "How the Metric System Works" chart to start at a base unit and repeatedly multiply or divide by 10 to convert units, with explicit examples (e.g., gram → decigram → centigram → milligram and the sequence from 100 meters to .1 kilometer, 1 hectometer, 10 dekameters, 1000 decimeters, 10,000 centimeters, 100,000 millimeters). The lesson also directs students to use the method of moving the decimal point right or left to perform these repeated 10× or 1/10× steps.

Students fill conversion charts and complete tables that produce sequences such as 12, 24, 36, 48 (inches to feet) and 3, 6, 9, 12 (feet to yards). The lesson explicitly asks students to explain how the pattern works (e.g., multiply by 12 to go from feet to inches, divide by 12 to go from inches to feet) and gives the general rules "Going from a LARGE unit to a small unit, MULTIPLY" and "Going from a small unit to a LARGE unit, DIVIDE." Students also order index cards from shortest to longest (1 mm, 10 mm, 10 cm, 1 m, …), which requires generating a sequence that follows the given unit-size rule.

Students complete conversion tables that list sequences such as 16, 32, 48, 64, 80 ounces = 1, 2, 3, 4, 5 pounds and 2000, 4000, 6000, 8000, 20000 pounds = 1, 2, 3, 4, 10 tons. Students fill in equivalent metric sequences (e.g., 1000 milligrams = 1 gram; 1000 grams = 1 kilogram) and convert problems like 25 kilograms = 25,000 grams and 300 grams = 0.3 kilograms. Students build a metric flip chart and complete matching and ordering exercises that require producing and using regularly spaced numeric conversions.

Students are asked to "Show 4 different ways to create a rectangle with an area of 24 square cm," and other tasks require finding rectangles that meet specified area or perimeter conditions (e.g., area of 48 sq cm with length 8 cm; perimeter of 48 cm with length 15 cm). Activity 3 directs students to use a one-square-centimeter grid and dry-erase markers to create rectangles and check their work. Several problems ask students to determine missing side lengths from given area or perimeter, which requires producing shapes that satisfy a numerical rule.

Students are asked to draw multiple rectangles that satisfy a given rule (e.g., area = 16 cm^2 and later area = 36 cm^2) and then compute the perimeter of each, thereby generating shape sets that follow the rule. Students are also asked to create several rectangles that share a given perimeter (e.g., 16 cm, 12 cm) and compute each shape's area, generating shapes that follow the 'same perimeter' rule. Students are prompted to observe that the square has the smallest perimeter for a given area and the largest area for a given perimeter and to test or "prove" this by creating additional examples.