Big Idea 1
Seeing Patterns inside Numbers

Numbers make up our world, and they are used throughout our lives, whatever our age, job, or level of interest. But many people develop a narrow relationship with numbers, seeing them as something to use in calculations, rather than as a fascinating set of ideas that can enrich their world. Our first big idea invites students to become captivated by numbers and to get to know numbers deeply. What is enchanting about numbers is that they are all made up of different arrangements, have different factors, can be seen differently, and have their own intricate system to be explored.

When we first came across Brent Vorgey's number visual (Figure 1.1), we were enthralled, as we immediately saw the creativity, beauty, and insights that the visual representations revealed.

Figure 1.1

In our Visualize activity, we invite students to explore this depiction of numbers and to see what patterns are uncovered by the visual representations. We invite them to see what primes look like and to see the different factors inside numbers. We invite them to investigate patterns among the numbers, seeing what their positioning on the diagram reveals. Also, we invite students to see numbers visually and to develop a realization that numbers contain all sorts of information that make them different from each other, special, and interesting.

In the Play activity, we extend students' time with the number visuals in a more playful setting. Students play a game with the number visual page as a game board, and move between visual and numerical representations. This, as with the other two tasks in this big idea, encourages important connections between different areas of the brain.

In the Investigate activity, we invite students to think carefully about number flexibility. One of the ways that numbers are different from one another is the number of factors they have and the degree of flexibility they give us when using them. For example, 24 is a very flexible number, as it can be broken up in all sorts of different ways. This makes it a useful number for packaging, for designing, and for measuring time. In this activity, we invite students to give value to different numbers according to their flexibility, helping them develop an appreciation for these numbers. The activity also invites students to make equal groups and gives teachers an opportunity to discuss whether students are thinking additively or multiplicatively and what those differences mean.

All three activities give students an opportunity to develop new insights into the numbers that they will use for the rest of their lives.

Brain science tells us that when students are engaging with numbers as symbols, such as the numeral 4, and with numbers as visuals, as shown in Figure 1.2, they are connecting between different areas of the brain, and such connections are critical for mathematics learning and achievement. The activities in this big idea will invite a lot of brain connecting, with students developing pathways that will help them as they go forward in their mathematical careers.

Jo Boaler

Figure 1.2

Visualizing Numbers

Snapshot

In this activity, students work with the number visual page to explore the patterns that they can see inside of numbers. In this activity, we open the door to understanding factors, multiples, and primes, as well as other number patterns.

Connection to CCSS

4.OA.4

Agenda

• Number visual page, one per group
• Colors
• Scissors
• Optional: posters or large paper

To the Teacher

The length of this lesson depends largely on how long students would like to explore patterns. We've found that some students want to explore patterns in depth, and they should be given time to do that. Follow your students' lead and interest. This activity can easily be spread across multiple days.

Activity

Launch

Numbers can be represented in lots of different ways. For example, 6 can be written as a numeral, but 6 can also be shown in other ways, as in Figure 1.3.

Figure 1.3

When you launch this activity, you may want to share some of these ways with students or have them generate ways that numbers are represented in their world. Give each student a copy of the number visual page and ask them to notice what numbers are shown. Have them record the actual number value by each visual. There are patterns all over this page. Ask students, what patterns do you notice?

Explore

Ask students to investigate the patterns in the number visual page.

• What patterns do you see?

Provide students with colors (colored pencils, pens, or markers).

• How can you use color to show the patterns within these numbers?

Students might notice equal-size groups within some numbers. For instance, 4, 8, 12, 16, 20, 24, 28, and 32 all have square clusters of 4 inside them. Students might notice that some numbers have no groups inside them; numbers like 11, 13, 17, and 19 are circles. Students might notice how some numbers grow outward from a central pattern. For instance, 6 has a group of 3 in the center, and each corner has been added onto with one dot. Students might also notice multiple numbers inside one number. For instance, 18 has three groups of 6, but also has 9 pairs. Some of these patterns are shown in Figure 1.4 as an example of how students might use color to highlight different patterns they see.

Figure 1.4

Discuss

Ask students to share the different ways they have color-coded their numbers to reveal patterns. What do different ways of coloring show? You may want to focus discussion on a single number to compare the different patterns inside. For instance, you could look at the different patterns inside of 12 that different ways of coloring make clear.

What do different numbers have in common? If you focused on a particular number, you might ask, What other numbers are like this number? How are they alike? If students notice the clusters within each number, give them the term factor to describe these clusters. For instance, if students see the three clusters of 4 inside 12, you can say that 4 is a factor of 12, or that 12 has 4 as a factor.

Explore

Now ask students to return to their color-coded number visuals and look for patterns that different numbers share. Provide students with a new number visual page and scissors to cut this new page apart so that they can group, sort, or web numbers by common features and color-code those features. Students should work with a partner or in a small group to find patterns.

• What patterns do different numbers share?
• How can you group or arrange the numbers to show what numbers have in common?

You might have students glue or tape their arrangements onto a poster to make sharing easier. This way, they could label the groups or the relationships between the numbers.

Discuss

Ask students to share the patterns they notice between numbers. You may want to have students hang their posters and do a gallery walk, or ask each group to share what they have found. In either case, discuss as a class the following questions:

• What patterns do different numbers share?
• What are you wondering now about these numbers?
• What are you wondering about the numbers we haven't looked at yet?

If students notice the clusters that different number share, be sure to tell them that we say that they share a factor. If students notice the circles and the lack of clusters within, be sure to probe what this means. You can name these numbers, where no equal groups are possible, as prime.

Look-Fors

• Do students notice that numbers are inside of other numbers? For instance, does anyone see three clusters of 4 inside of 12? One goal of this activity is for students to see the building blocks of numbers.
• Do students notice that some numbers are made only of individual dots? This is the beginning of noticing primes.
• Are students thinking multiplicatively or additively? Although numbers can be broken apart through addition,...