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Chapter 13 · Period 1 of 8 · 40 min

Mystery Box: Who Is the Hidden Multiplier?

Lesson

Students frequently assume that a larger number must have more factors; the arrays for 37 (only 1 × 37) often surprise them, and this moment is worth pausing on. Watch for students who list 1 and the number itself as an array without drawing it — insist they build it with counters so the concrete reality sticks. The common error is writing a factor pair only once (e.g., 2 × 5 but not 5 × 2) when both orientations of the array are distinct and worth recording.

0–10The teacher draws a mystery box on the board with four numbers entering and four transformed numbers coming out. Students discuss what rule the box might be applying — Find the hidden numbers. (a)(b)(c), p.164 ("Guess the multiplier from four numbers coming out...."). Pairs share guesses and test whether a second multiplier could also work.
10–28Students use squared paper or counters to build every possible rectangular arrangement for each number in Let Us Do (a)–(i), p.165 ("Make different arrays for 10, 14, 13, 20, 25, 32, 37, 46, 54...."). They record each arrangement as a multiplication sentence and list the factors they find. Numbers like 13 and 37 prompt the class to notice that only one non-trivial arrangement is possible, raising the question of what makes these numbers special.
28–40Groups share their factor lists for 13 and 37. The teacher steers a whole-class discussion: what is it about these numbers that limits their arrays? Students articulate in their own words what it means for a number to have exactly two factors, consolidating the concept of prime numbers as an observation rather than a definition.
Assessment

Let Us Do (a)–(i), p.165 — Make different arrays for 10, 14, 13, 20, 25, 32, 37, 46, 54 and identify the factors in each case.

Priya tries to arrange 17 tiles into a rectangular array with more than one row and more than one column. How many such arrangements can she make?

  • A Exactly one arrangement.
  • B Four arrangements.
  • C None — 17 cannot be arranged that way.
  • D Two arrangements.

C. None — 17 cannot be arranged that way.

What each wrong option reveals

  • A The student may be thinking of the 1 × 17 arrangement and not applying the 'more than one row and column' condition.
  • B The student likely guessed without checking factor pairs.
  • D The student may be confusing 17 with a number like 15, which has arrangements 3 × 5 and 5 × 3.
Chapter 13 · 8 periods of 40 min

Also saved as 5 periods.

  • 2Rabbit and Frog Race: Who Lands Together?
    Lesson
  • 3Who Reaches the Food? Reasoning With Common Multiples
    Lesson
  • 4Mowgli's Trail in Reverse: Finding Jump Sizes That Fit
    Lesson
  • 5Common Factors of Pairs: Listing and Checking
    Lesson
  • 6True or False? Reasoning About Factors and Multiples
    Lesson
  • 7Venn Diagram Sort: Divisible by 2, 5, or 10?
    Lesson
  • 8Bringing It All Together: Factors, Multiples and the Jumps Between
    Lesson

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