Divisibility Rule for 4 and 5

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SWBAT discover the divisibility rule for 4 and 5.

Big Idea

Using calculators allows students to experiment and to discover patterns, in this case resulting in a divisibility rule for the number 4.

Do Now

10 minutes

For today's Do Now, I ask my students to complete the table based on the divisibility rules that they have learned so far. I remind my students that they don't need to use long division, but instead use the "shortcuts" or divisibility rules since they do not need to know the quotient. 

















If they have forgotten the rules or look puzzled, I will refer them to their notes. These rules were covered in the previous lesson

Group Work

25 minutes

For today's group work, I plan to arrange my students in heterogeneous groups of four students. Each student will receive a calculator, but there will only be one 4 Divisibility Rule worksheet per group. Although each student has a calculator to find the quotient, students will be directed to work together to complete the worksheet. They must agree on the quotient before completing each question. I think that the calculator will help all of the students in each group work at about the same pace. 

For Problem 13 on the worksheet, my students may need some clarification.  I will explain to them that they need to separate the numbers into 2 categories: (a) the numbers that are divisible by 4 (the quotient was a whole number) and (b) the numbers that aren't divisible by 4 (the quotient has a decimal or remainder).

As each group finishes Problem 13, I will ask them to begin developing a divisibility rule for 4 based on any patterns they observe on the worksheet.

Mini Lesson

10 minutes

During today's Mini-Lesson we will review the answers for the worksheet. I will encourage my students to use their calculators to double check their answers as we share. After every few answers, I will ask students to share ideas for a Divisibility Rule for 4.  Based on Column A of Problem 13, many students may notice that all of the numbers are even. This is often our starting point. When this comes up, I will urge them to explore further. This usually sparks more exploration with the calculator. 

After a little prompting, there are usually a few students who notice that all of the numbers in Column A end in a multiple of 4.  Students may not word it succinctly, but this will ultimately lead to the divisibility rule for 4: 

If the the last two digits of a number are themselves divisible by 4, then the number is divisible by 4.

We've now found Divisibility Rules for 2, 3 and 4, so I'll ask, "Before we complete another investigation, does anyone know a Divisibility Rule for 5?" Most of my students are already aware of the divisibility rule for 5, so I will ask students to explain the rule and give examples that show why it works. We'll end up with a rule like: 

A number is divisible by 5, if the ones digit is 0 or 5.

Independent Practice

10 minutes

For a brief period of Independent Practice, I plan to give my students the table below for the Divisibility Rules for 4 and 5. My students have seen similar tables, and, they are aware of how to complete it.  I remind students that they should be using the rules as shortcuts, rather than using long division.


















As students complete the table, I will encourage them to come up with some examples of numbers that are divisible by 4 or by 5.

Lesson Review

5 minutes

To review what we have learned in this lesson, I will select a few students to share their examples of numbers that are divisible by 4 or by 5. I will ask students to explain why their example works. If time allows, I may make this a competition.

Can you come up with a number that is larger than the previous student's number?

This encourages students to use the rules to quickly develop a number that will work, and, it makes it more reasonable to keep asking students for explanations as to why their number works.