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Grades: 3‒6

Students develop their flexibility in thinking about multiples and common multiples through solving puzzle-like problems. Students are given information about the multiples and common multiples of two unknown numbers to use as clues in determining the identity of the numbers. An interactive grid serves as the unifying model for the problems, with students using it to check their work as well as to create puzzle challenges for each other.

  • Students use a picture of part of a number grid to determine the number of columns in the grid.
  • Students use a partial grid on which all multiples of an unknown number are shaded orange to form a conjecture about the value of the unknown number.
  • Students use a partial grid with multiples of one number in orange and multiples of another in blue to form a conjecture about the values of the two numbers.
  • Students verify their conjectures and determine whether they are unique.
  • Students examine shaded cells in a grid without numbers to determine the skip-count interval between them, and fill in the cells with a possible sequence of numbers.
Mathematical Practices

(1) Make sense of problems and persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable arguments and critique the reasoning of others; (5) Use appropriate tools strategically; (7) Look for and make use of structure; (8) Look for and express regularity in repeated reasoning.

Content Standards

3.OA5, 6, 7, 9; 4.OA4, 5; 6.NS4

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This activity is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License: If you adapt and/or share this activity, you must attribute it to "KCP Technologies, a McGraw-Hill Education Company." You may distribute it only non-commercially under the same or similar license.

This material is based upon work supported by the National Science Foundation under KCP Technologies Award ID 0918733, with grant period September 1, 2009 through August 31, 2013. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.