Find factor pairs and explain rules in number and shape patterns

Learning goal: Find all factor pairs of numbers from one through one hundred, distinguish primes and composites, and generate and explain number or shape patterns. A factor divides a whole number into equal whole-number groups without a remainder. A multiple is a product formed using that factor and another whole number.

For twenty-four tiles, possible rectangular arrays include one by twenty-four, two by twelve, three by eight and four by six. These give the factor pairs 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Reversing each array gives the same pair in the opposite order. List each pair once while recognizing that both numbers are factors.

A systematic search prevents missing a pair. Test possible smaller factors in order. Five does not divide twenty-four evenly. At six, the partner is four, a pair already found in reverse. Once the factor pairs meet and begin reversing, no new pairs remain. For a square such as thirty-six, include the equal pair six and six once.

Twenty-four is a multiple of each of its factors because a whole number can multiply each factor to make twenty-four. It is a multiple of eight because 3 × 8 = 24. It is not a multiple of five because no whole number multiplied by five gives twenty-four. Use known products, grouping or division to justify a multiple claim.

A prime number is a whole number greater than one with exactly two positive factors: one and itself. Thirteen has only the pair one and thirteen, so it is prime. A composite number has more than two positive factors; twenty-four is composite. One is neither prime nor composite because it has only one positive factor. Two is prime even though it is even; the claim that every even number is composite has this exception.

A pattern rule tells how to generate terms. Begin at one and repeatedly add three: 1, 4, 7, 10, 13, 16. An additional feature is that odd and even values alternate. Adding an odd number changes parity: an odd amount added to an odd amount gives an even total, while an odd amount added to an even amount gives an odd total. The alternation follows from the rule, not just the first two examples.

Shape patterns can use a repeating rule. Start with a triangle and repeat triangle, square, square. The first nine shapes are triangle, square, square, triangle, square, square, triangle, square, square. Triangles appear at positions one, four and seven, advancing by three places. This position pattern is an additional feature not stated merely by naming the next shape.

Different rules can fit a short list, so state the generating rule rather than assuming a few terms determine it uniquely. When a rule is given, follow it consistently and explain any predicted feature. Check both number and shape patterns; being able to continue one familiar sequence is not evidence about every rule.

Worked factor search: For thirty-six, the pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Five does not work. The positive factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Since there are more than two, thirty-six is composite.

Worked pattern prediction: In the repeating triangle-square-square pattern, positions 10 and 13 will contain triangles because each new triangle occurs three positions after the previous one. This conclusion uses the repeated three-shape unit and can be checked by extending the sequence.

Guided practice: Build arrays for several numbers, including a prime, a composite and a square number. Record factor pairs systematically and connect them to multiple statements. Generate a number pattern and a shape pattern from explicit rules, then identify and explain an additional feature.

Independent evidence: Find all factor pairs for an unfamiliar number within one hundred, justify prime or composite status and decide whether it is a multiple of a one-digit number. Create a pattern from a rule and support a prediction with reasoning. Include one as a special case and avoid treating a repeated factor pair as new evidence.