Breaking Apart Arrays: An Inquiry Based Math Lesson

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In this inquiry-based activity for grades 1-4, students will apply distributive property to decompose units.  

Breaking apart arrays is another effective strategy for students who are learning multiplication, and helps model distributive property.   For example: students usually learn their twos and their fives sooner than the larger times tables, so that when faced with a problem like 6 X 7, students can instead look at the 7 as a (5+2) instead.  Like this: 6 X (5+2) OR (6 X 2) + (6 X 5) = 12 + 30 = 42.  

It's also important that students know they can break apart a multiplication problem (an array) in order to make the problem simpler to solve.  

  • You can start by asking students a simple question and entertain answers and discuss openly as a whole class or in small groups.
  • Display a 6X6 array for all the students to see.  You can do this by drawing it on the whiteboard, handing out counters or legos and students form it, or giving students some graph paper and instruct them to sketch it out.  
  • I then would ask students the following question:  How can you display this problem differently, while still arriving at the same answer?    This can be such a valuable group discussion and collaboration time!  This would be a great time to show your students the following model shown below, making sure the students know that this is only one of the many possible solutions. If students are more independent, or if you want to use this as seat work, simply print off this tutorial PDF for each student.  (This is also found on the first page in the Array Break Apart Worksheet packet.)

  • You can also show them how this looks as an equation:  6 X (2 + 4) = 36 OR (6 X 2) + (6 X 4) = 36.  
  • Now, further the inquiry by asking students to list the other ways this 6 X 6 array can be displayed.  See how many each student/group can find!  
  • Ask for volunteers to come up and show other ways of breaking apart the array.

Here are just a couple of the many ways it can be broken up:


  • Ask: What other ways can the students break this array apart?  Can students correctly write out the distribution property?
  • When you have checked for understanding, give students the following worksheet pages 2 and 3 in order to give students practice breaking apart arrays on their own or in groups. You may also wish to print off the tutorial page for students to place in their math notebooks.  I have also included Array Break Apart Page 4 for more advanced students, as well as a colored-version of the entire set.  

Array Break Apart Tutorial Page 1                                 Array Break Apart Tutorial Page 1, Color

Array Break Apart Page 2                                              Array Break Apart Page 2, Color

Array Break Apart Page 3                                              Array Break Apart Page 3, Color

Array Break Apart Page 4 (for advanced students)       Array Break Apart Page 4, Color (for advanced students)

This free PDF worksheet packet is the perfect addition to math centers.   It's ideal to complete either alone or in pairs and can initiate great conversations as each array has several different ways in which to break apart.  

Make sure to provide students with ample counters and graph paper and other manipulatives to help students understand the distribution property!  

When finished, draw all the students in together as a group to discuss their findings.  Summarizing the mathematical experience is critical to an inquiry-based math lesson.  

Here are some questions you can ask:

  1. How did you break up each array?  
  2. Was there a method you followed to make your choices easier?
  3. How does the array broken apart demonstrate distributive property?
  4. How is this method of breaking apart arrays helpful to a student who is still learning their multiplication facts? 
  5. How many times can you break up an array? A specific array?


What other questions would YOU ask?