## Learning Math: Patterns, Functions, and Algebra

# Classroom Case Studies, Grades 3-5 Part E: Critiquing Student Lessons (20 minutes)

## Session 10: 3-5, Part E

In the grades 3-5 curriculum, students are frequently asked to think about patterns, but often their “pattern sniffing” skills end with simply finding the next object. **Note 8**

### Problems

**Problem E1**

a. |
What algebraic ideas are in this lesson? |

b. |
How are patterns used in this lesson? |

c. |
What mathematics do you think students would learn from this lesson? |

d. |
Are there any misconceptions that students might develop from this lesson? |

e. |
How would you modify the problem, or what additional questions might you ask, to incorporate the framework for analyzing patterns? |

**Problem E2**

a. |
What algebraic ideas are in this lesson? |

b. |
How are patterns used in this lesson? |

c. |
What mathematics do you think students would learn from this lesson? |

d. |
Are there any misconceptions that students might develop from this lesson? |

e. |
How would you modify the problem, or what additional questions might you ask, to incorporate the framework for analyzing patterns? |

### Notes

**Note 8**

In the lesson on dividing fractions, students might see a pattern of multiplying by the reciprocal. Students should, however, be asked if they understand why this method works. For this particular set of problems, this is a crucial question in their understanding of operations with fractions. Observing the pattern is not enough.

The same questions need to be asked regarding the problem from *Investigations in Number, Data and Space.* One of the key components in analyzing patterns is determining why they continue as they do, even “down the line.”

### Solutions

**Problem E1**

a. |
There is no algebra specifically shown in this problem. It does, however, examine some patterns. |

b. |
The patterns used in this problem compare dividing by a fraction and multiplying by its reciprocal. |

c. |
The lesson assumes that students will learn that dividing by a fraction and multiplying by its reciprocal produce the same result. |

d. |
No example shows that dividing by a whole number is the same as multiplying by its reciprocal. This could lead to confusion. There is no attempt to show why the process works. |

e. |
The lesson should include dividing by a whole number, and it should include an explanation of why the process works. |

**Problem E2**

a. |
There is no algebra specifically shown in this problem. It does, however, examine some patterns. |

b. |
The patterns used in this problem show two things: a) The units digit of a sum remains the same if the units digits of the addends remain the same; and b) The units digit of a sum increases by one as one addend remains the same and the other addend increases by one. |

c. |
The lesson assumes that students will learn the patterns described above. |

d. |
Students may not notice the patterns, or overgeneralize to think that all sets of problems will show similar patterns. |

e. |
The lesson should ask children to generalize and describe the patterns they see. |