Students will create rules for ordering patterns of circles and squares. Students generate all possible messages with three place values, then create rules that explain how they ordered each message. Emphasis is placed on creating clear rules so that, if another group were to follow the rules, they would generate the same list in the same order. Using these rules, students then try to list all possible messages with four place values. As the lesson concludes, students share their rules with classmates.


Students will be able to:


Eventually, students will need to understand the binary number system which uses 1's and 0's rather than circles and squares. This lesson acts as a bridge to the next lesson where binary is formally introduced and practiced. In wrestling with the challenge of describing the rules of ordering patterns of circles and squares clearly, students will be primed to see how the binary number system solves many of these problems. Students may discover a system that is equivalent to the binary number system, which is a feat worth celebrating, but it is not expected that every student uncovers the rules for binary in this lesson.


Getting Started

Prompt: How many ways can you represent 7?

Students should brainstorm individually before sharing in small groups. It is important to allow ample brainstorming time - students may generate familiar responses at first, but may stretch their thinking and get more creative with additional time. Some examples may include:

After a short time, ask students to share some responses with the whole class. Use these responses to quickly generate a wide variety of representations.

Discussion Goal: Introduce the idea that the numeral '7' is just one commonly used symbol to represent the number seven. There are many ways to communicate this same number that may use different symbols or representations, all of which are valid. Sharing the variety of responses helps motivate the following activity that asks students to discover a system for using the circles and squares to represent numbers, including the number 7.




Prompt: With a partner, work out how many different pieces of information (made of up of circles and squares) you can represent with three place values. For example: circle-circle-circle and circle-square-circle can represent two different pieces of information.

Discuss: Give students time to work individually, then have them share with their neighbor and fill in any patterns they may have missed.

Do This: Confirm with the class that there are 8 possible patterns, but don't list all of them out. Ask students to share the 7th pattern in their list. Students will likely have different answers for this.


Circle Square Activity

Place students in groups of 2, making one group of 3 if necessary. Distribute Circle Square Patterns - Activity Guide - one for each group. Each group also gets the Shape Cutouts - Resource to use during the activity.

Challenge #1: Students again list all possible three place value patterns, but with an added focus on the order of their patterns.

Challenge #2: Students describe the rules or strategy they used to create their list. They are aiming for clear directions that can be followed by another group to reproduce their same list in the same order.

Challenge #3: Students extend their rules to generate all possible four place value patterns. This challenge has 2 parts: discovering all possible four place value combinations (there are 16) and extending the rules from the previous challenge so they work here as well. Most groups will need to change or add to their rules in order to accomplish Challenge #3.

Discuss: Select a few groups to share out their rules, highlighting groups with different strategies and rules they used for their lists. Emphasize the 7th item in each list, connecting it back to the warm-up activity as another way to represent the number 7.

Teaching Tip

Manipulatives: Students are given manipulatives to help visualize any rules they are using to move from one element of the list to the next. You might see students use all their manipulatives at once to create the different patterns, then discuss how to arrange them into an ordered list. You might see students representing one pattern at a time, then discussing the rules for "replacing" shapes to generate each of the next patterns. Students might not use the manipulatives at all, using pen & paper or whiteboards instead.

Many Possible Answers: It is okay for different groups to come up with different orders for their lists of patterns - this will help with the share-out discussion as you highlight different strategies.

Facilitating With Groups: You should act as a facilitator during this part of the activity, guiding students in describing the rules & strategies that they used to create their list. These strategies may be implicit and unconscious to the student, but you can ask questions to help students realize their own thinking that went into generating their list. Aim to help students clarify their thinking to make it easier for other groups to follow.

Group Dynamics: Be mindful of groups that appear to be dominated by a single student. Asking each student individually about their strategy can help bring students back together and reinforce the collaborative aspect of this activity.

Wrap Up


Prompt: How is counting in this circle/square system similar to how we count in our regular lives? How is it different?

Teaching Tip

Goal There are many ways to structure this discussion, especially if you have your own established share-out routines. Here are a few that could work for this particular discussion:

Assessment: Check for Understanding

Question: How would you explain a number system to someone who had never seen numbers before?

Standards Alignment