In this interactive from Annenberg Learner’s Learning Math, students learn to see the mathematical patterns that are all around us, even in staircases ...
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Students will be able to
- use logic and modeling to predict future iterations of a geometric pattern
- generate a rule governing a geometric pattern
- use variables to represent an unknown quantity
- represent a mathematical relationship in multiple ways
Common Core State Standards: 5.OA.A.3, 6.EE.C.9, 7.EE.B.4, 8.F.B.4
Vocabulary: Pattern, independent value, dependent value, recursive rule
Materials: Finding Patterns to Make Predictions worksheet
Preparation: On your whiteboard, prepare the following pattern:
1. Introduction (5 minutes, whole group)
Ask students to make observations about the pattern shown on the board. In each successive step, what is happening? Can they predict what the 5th step will look like? How about the 10th step? As much as possible, encourage students to explain the reasoning behind their logic.
During this discussion, look for an opportunity to show students that this pattern can also be represented in a table, featuring the independent value (also called the term, or n) and the dependent value (the number of dots in each term).
Using all of this data—the visual representation, the table of values, and the students’ descriptions of the pattern—help students develop a general rule for the pattern. A recursive rule (a rule that can be repeatedly applied to find the next term of a sequence), such as “Each term adds two dots to the previous total,” is fine to start with. But help students see the relationship between each term, n, and the total number of dots. In this pattern, each term is one dot short of a rectangle—so one general rule is “two times the term, minus one dot,” or simply 2n – 1. Students may have other ways of expressing the pattern.
Throughout this introduction, emphasize the importance of multiple representations in understanding a pattern like this.
2. Activity (15 minutes, pairs)
Explain to students that they will be looking at another pattern, and that their task will be to predict how many blocks will be contained in the 7th, 10th, and nth terms. To answer these questions, they will use the interactive to build the step pattern themselves. They will also be working in pairs and examining the pattern through multiple representations.
Hand out the worksheet and have students begin working with the interactive. Ask students to record all of their information on Page 1 of the worksheet. When they have identified an algebraic rule for the pattern, they can turn to Page 2 and create their own geometric pattern for a classmate to solve.
3. Conclusion (10 minutes, whole group)
Bring everyone back together and review the steps problem. Start at a general, observational level as in the introduction: Ask students what they noticed about the pattern, both visually and when they constructed the table of values.
Then ask the following questions:
- If someone tells you how many blocks there are in staircase n, describe how you could use that to find the number of blocks in staircase n + 1.
- Suppose there are 37,401 blocks in the 273rd staircase. How many blocks are there in the 275th?
Then, ask students whether they found a general pattern that governed the number of blocks in any term. Test out these patterns to see whether they work. If time allows, share this solution:
The geometric solution to this problem is the most elegant: If you take any staircase with bottom row length n, form a second identical one, rotate it, and put it together with the first, you form a rectangle of dimensions n by n + 1. Therefore, the area of the original staircase is 1/2(n)(n+1).
Common Core State Standards
CCSS.Math.Cont.5.OA (Grade 5 ): Operations and Algebraic Thinking
CCSS.Math.Cont.5.OA.B (Grade 5 ): Analyze patterns and relationships.
- CCSS.Math.Cont.5.OA.B (Grade 5 ): Analyze patterns and relationships.
CCSS.Math.Cont.6.EE.C.9 ( Grade 6 ): Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
CCSS.Math.Cont.6.EE (Grade 6 ): Expressions and Equations
CCSS.Math.Cont.6.EE.C (Grade 6 ): Represent and analyze quantitative relationships between dependent and independent variables.
- CCSS.Math.Cont.6.EE.C (Grade 6 ): Represent and analyze quantitative relationships between dependent and independent variables.
CCSS.Math.Cont.8.F.B.4 ( Grade 8 ): Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
CCSS.Math.Cont.8.F (Grade 8 ): Functions
CCSS.Math.Cont.8.F.B (Grade 8 ): Use functions to model relationships between quantities.
- CCSS.Math.Cont.8.F.B (Grade 8 ): Use functions to model relationships between quantities.
Learning Math: Patterns, Functions and Algebra, "Counting Stairs"
See this activity on Annenberg Learner's professional development course for teachers.