Standards-based assessment and Instruction

## A Stone Wall

Sarah wants to build a stone wall along one side of her garage. Sarah collects stones from the field behind her house. The first day, Sarah collects four small stones and five large stones. The second day, Sarah collects eight small stones and eight large stones. The third day, Sarah collects twelve small stones and eleven large stones. If this pattern continues, how many small and large stones does Sarah collect on the tenth day? Sarah realizes that she now has enough small and large stones for her stone wall. How many small and large stones does Sarah collect for the stone wall? Show all your mathematical thinking.

### Common Core Content Standards and Evidence

#### 5.OA Operations and Algebraic Thinking

Analyze patterns and relationships.

3.   Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

This task requires students to identify apparent relationships between two numerical patterns to generate the rules. Students also need to extend the patterns and determine the total for a given term in the numerical sequence.

### GO Math! Alignments

Chapter 9, Lessons: 9.5, 9.6, 9.7

### Common Core Standards of Mathematical Practice

MP.1 Make sense of problems and persevere in solving them.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.

### Underlying Mathematical Concepts

• Generate two numerical patterns using two given rules
• Skip counting
• Ordinal numbers
• Number sense to 220

### Possible Problem-Solving Strategies

• Model (manipulatives)
• Diagram/Key
• Table
• Number line
• Graph (Students may independently select graph paper.)

### Possible Mathematical Vocabulary/Symbolic Representation

• Model
• Diagram/Key
• Table
• Graph
• Number line
• Axis
• Pattern
• Multiples
• Input/Output
• Total
• Amount
• Odd/Even
• Greater than (>)/Less than (<)
• Day, week, month
• Monday, Tuesday ...
• Per
• Rules: 4 · d = s, (3 · d) + 2 = L
• Variables
• Linear function
• Ordinal numbers: 1st, 2nd, 3rd ...
• Dozen

### Possible Solutions

On the 10th day Sarah collects 40 small stones and 32 large stones. Sarah collects a total of 220 small stones and 185 large stones for the stone wall.

### Possible Connections

Below are some examples of mathematical connections. Your students may discover some that are not on this list.

• Generalize rules and verify: 4 · = s, (3 · d) + 2 = L (d is day, s is small stones, l is large stones).
• Sarah collects stones for 1 week and 3 days.
• Graph the table.
• There are 36 more small stones on the 10th day than the 1st day.
• Solve more than one way to verify the answer.
• Sarah found the same number of small and large stones on the 2nd day.
• Sarah found a dozen small stones on the 3rd day.
• On the 10th day Sarah found 8 less large stones than small ones.

### Scoring Rationales and Corresponding Anchor Papers

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