Do You Notice Sum-Thing?

• Lesson
6-8,9-12
1

This lesson invites students to investigate the patterns when a "plus sign" (a cross-shaped arrangement of five squares) is placed on the board in various locations. Students will conjecture about the pattern of the five displayed numbers, the sum of the five numbers, and any other patterns that they notice. Students may also explore similar patterns when other shapes, such as a 2 × 1 rectangle, are placed on the hundreds board.

This lesson is based on an activity created by Andrew Derer of the Math Science Innovation Center and Art Stoner of A+ Compass. Unlike the paper version described in this lesson, their original activity uses eleven different Hundreds Board Tiles, as shown below.

To begin the lesson, display one month of dates from a calendar. As a class, discuss the numerical patterns that appear among the dates.

To facilitate this discussion, choose a date in the middle of the month, and ask the following questions:

• What number is in the square above this one? [The number will be 7 less than the chosen number.]
• What number is in the square below this one? [The number will be 7 more than the chosen number.]
• What number is in the square to the left of this one? [The number will be 1 less than the chosen number.]
• What number is in the square to the right of this one? [The number will be 1 more than the chosen number.]
• What other patterns do you notice? [Answers will vary, but students might conjecture that a number up one and over one (i.e., "northeast") will be 6 less than the chosen number, as shown above, or that a number two squares below will be 14 more than the chosen number. Many other patterns can be identified.]

Explain to students that they will now investigate patterns on a hundreds board. They can look for patterns similar to those they notice on the calendar, or they can search for entirely different patterns. Distribute the Board + Plus Sign Activity Sheet and a pair of scissors to each student. (To minimize materials, you could allow students to work in pairs. Doing so will also allow students to discuss the patterns they notice during the exploration.)

This activity sheet contains a hundreds board at the top and a template to create a "plus sign" at the bottom. When prepared, the portion with the plus sign removed can be placed on the hundreds board to display five numbers in a cross‑shaped pattern, as shown below.

In the image above, the plus sign reveals five numbers, 23, 32, 33, 34, and 43. When the plus sign is moved, a different configuration of five numbers will be revealed. Students are to investigate the patterns that occur as the plus sign is moved around the board.

Allow 10‑15 minutes for students to explore the hundreds board with the plus sign. Ask students to record any observations and generate as many conjectures as possible. As they investigate, circulate and offer assistance as necessary. However, take care to allow students to explore on their own, and don't take away their fun by finding patterns for them. Instead, use some of the Questions for Students to prompt their thinking. As you circulate, also note some of the observations made by students. Note incorrect observations as well as correct ones. After students have had ample time to explore, ask them to share. If students are unwilling, present some of the observations that you noted while circulating. (You may wish to present these observations anonymously.)

Students will make many observations. Some of them may include:

• Top number is 20 less than the bottom number.
• Center number is the mean of the five numbers.
• Center number is also the median of the five numbers.
• The right number is 2 more than the left number.
• The sum of the five numbers is a multiple of 5.
• The product of the top and bottom numbers is 100 less than the square of the center number.
• The product of the left and right numbers is 1 less than the square of the center number.

In addition to the observations listed above, students will make other observations, some correct and some incorrect. Present all observations to the class for discussion, and allow them to collectively decide if each one is true or false. You can lead this discussion by asking students to consider if the observation is true in all cases; and if not, can they provide a counterexample? In addition, ask students to consider how they might prove that a particular observation is true.

Depending on the level of your class, you can present an algebraic proof for one of the observations, or you can allow them to generate an algebraic argument on their own. For example, students may conjecture that the sum of the numbers is five times the center number. Algebraically, if the center number is n, then the other four numbers are n – 10, n – 1, n + 1, and n + 10. Therefore, the sum of the five numbers is:

n + (n – 10) + (n – 1) + (n + 1) + (n + 10) = 5n,

which algebraically justifies the conjecture.

Offering Assistance

This activity works best if students are given the freedom to explore on their own. However, students who have not had similar opportunities for free exploration in the past may be less comfortable with an open investigation. Such students might benefit from the structure provided by the Patterns Activity Sheet.

It is worth noting, however, that many students are not able to explore independently, form conjectures, or justify results because they have not been given ample opportunities to do so. Teacher‑led coaching may be more beneficial than a well‑structured activity sheet in developing students who are comfortable with open‑ended exploration.

Question 1 on the activity sheet asks students to record the displayed numbers and to make three observations when the plus sign is placed on the board. It may be helpful to do a first example with students. Display a hundreds board, and show them which numbers are displayed when the plus sign is placed with 12 at the center. Ask students to suggest any observations about the numbers that are displayed. Do not limit or judge any of the responses. At this point, allow students to verbalize anything they notice.

As a management technique, it may be beneficial to allow students who respond less often to offer their observations first. If very active students respond first with advanced observations, other students may be reticent to participate.

During the conversation, students may make the following observations:

• The five numbers displayed are 2, 11, 12, 13, and 22.
• The sum of the five numbers is 60.
• The sum of the five numbers is even.
• The top number is 10 less than the center number, and the bottom number is 10 more than the center number.
• The three numbers in the middle row of the plus sign are consecutive integers.
• The three numbers in the center column are all even.
• The difference between the top and bottom numbers is 20.

Students may make other observations, too. After the class has compiled a list of observations about this first example, allow students to work alone or in pairs on the remainder of the activity sheet.

Circulate as students work, offering assistance as necessary. While walking around the class, take notes about the various strategies students are using and the conclusions they are reaching. With your goals for the lesson in mind, decide the order in which you will ask students to share their observations during an end-of-class discussion.

To conclude the lesson, review the activity sheet, but don't simply go over the answers. Be judicious in selecting students to present their conjectures. It is often through the course of these discussions that students identify and correct misconceptions. In particular, focus on Questions 2 and 3 during this review, which allow students to make conjectures about patterns. These open‑ended conjectures provide an excellent opportunity for students to communicate mathematically. In the course of justifying their conjectures, students often engage in reasoning and sense‑making at the highest levels.

References

This lesson adapted from an activity by Andrew Derer of the Math Science Innovation Center and Art Stoner of A+ Compass.

Assessment Options

1. Allow students to explore patterns using tiles with other shapes, such as a 3 × 1 rectangle or a U‑shaped arrangement of five squares. Ask students to identify at least three patterns using the provided shape, and require them to explain why they think that their conjecture is true.
2. Give a different tile to each student, and require every student to identify at least three patterns. Then, collect each student's list of observations as well as the tile they were using, and redistribute them to other members of the class. Have students prove (or disprove) the conjectures made by their classmates. Time permitting, allow each student to discuss the results with the student whose observations they reviewed.
Extensions
1. Ask students to consider the patterns that would arise if the shapes were used on boards with a different arrangement of numbers. For instance, what observations could be made if the plus sign were used on a calendar? What if it were used on a 30 × 40 grid of the numbers 1‑1200? Students can create their own number grids using the Dynamic Paper Tool and choosing Number Grids option in the drop down menu.
Dynamic Paper
1. Ask students to write expressions for the sum when x is used to represent different squares within the tile. For instance, five different expressions are possible for the plus sign, though each will be of the form 5x + k, where k is an integer. Similarly, two different expressions are possible for the 2 × 1 rectangle in either orientation, and each will be of the form 2x + k, where k is an integer.
2. Use several tiles, the hundreds board, and a series of statements to create a logic problem for students. For instance, tell students that you placed a 1 × n rectangle somewhere on the board, and then see if they can figure out exactly which rectangle was used and where it was placed based on the following clues:
• The difference between the least and greatest numbers is 20.
• The sum of the displayed numbers is a multiple of 6.
• The units digit of each number displayed by my shape is 8.
• The sum of the squares that touch my shape, either along a side or at a point, is 456.
Based on these clues, students could figure out that a 1 × 3 rectangle was placed vertically to display 28, 38, and 48. Use the same idea to generate more difficult problems for students to solve; or, better yet, have students create problems for one another.

Questions for Students

1. Is there a relationship between the number at the top and the number at the bottom? Is there a relationship between the number on the left and the number on the right? When the plus sign is moved to a new location, are the five numbers related in the same way?

[No matter where the plus sign is placed on the hundreds board, the top number is always 20 less than the bottom number. Similarly, the left number is always 2 less than the right number.]

2. If I tell you what number shows in the top of the plus sign, could you predict what the other four numbers would be?

[Algebraically, if the top number is n, then the other four numbers are n +9, n + 10, n + 11, and n + 20. For example, if the top number is 37, then the other four numbers would be 46, 47, 48, and 57.]

3. Is the sum of the five displayed numbers even or odd?

[The sum will be odd if the center number is odd, and the sum will be even if the center number is even. The sum of the five numbers is equal to five times the center number. This pattern may not hold for other tiles, such as the 2 × 1 rectangle in Question 7 on the activity sheet.]

4. When the sums of the five displayed numbers are plotted on a graph, as in Question 4 on the activity sheet, what observations can be made?

[The points form a linear pattern, but there are several gaps. No points occur when the value on the horizontal axis is 1–11, 20, 21, 30, 31, 40, 41, … 90, 91, because the plus sign is not able to fit on the graph when those numbers occur at the center.]

Teacher Reflection

1. How did you challenge the high achievers?
2. What modifications did you make to ensure that all students were successful?
3. Was your lesson at the proper developmental level for your students? If not, could the lesson be modified to make it more appropriate?
4. How did you ensure that students knew what was expected of them? If students were not aware of what was expected, how can you make the expectations clearer?

Learning Objectives

Students will:

• Make and test conjectures about patterns they observe when placing special tiles on a hundreds board.
• Justify conjectures with algebraic proofs, or refute conjectures by providing counterexamples.

NCTM Standards and Expectations

• Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
• Use graphs to analyze the nature of changes in quantities in linear relationships.
• Generalize patterns using explicitly defined and recursively defined functions.
• Draw reasonable conclusions about a situation being modeled.
• Approximate and interpret rates of change from graphical and numerical data.

Common Core State Standards – Mathematics

Grade 6, Expression/Equation

• CCSS.Math.Content.6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Grade 6, Stats & Probability

• CCSS.Math.Content.6.SP.A.3
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Grade 7, Expression/Equation

• CCSS.Math.Content.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.
• CCSS.Math.Practice.MP8
Look for and express regularity in repeated reasoning.