Teaching Through Problem Solving

These are problems that have been investigated in the classroom, and teacher annotated. We decided to organize the problems by "Big Ideas," taken from the research of Marian Small (More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction), and they are those enduring understandings that kids need to have if they really understand a concept. Students need many opportunities to explore and investigate concepts if we want them to have more than a rote or memorized understanding of them. So we have tried to label the important math, make explicit those big ideas we think kids really need to "get", and show you a range of strategies that you may see students using, so that you know where to push them next. Even though the students are investigating and building their understanding through working collaboratively with peers, the teacher is key in monitoring their strategies, extending and challenging their thinking with probing and prompting questions, and then intentionally orchestrating the debrief/consolidation of the problem to bring out those big ideas and that learning goal. The debrief is the most important part of the process, that is where the teaching happens.
Marian Small speaks about the importance of consolidation: http://www.curriculum.org/k-12/en/videos/instructional-ideas
Math Study Groups: Analyzing and Interpreting Students' Thinking, LNS webcast:
http://www.curriculum.org/k-12/en/videos/analyzing-and-interpreting-students-thinking

Grades 7-9

Number Sense and Numeration

Numbers tell how many or how much
Classifying numbers provides information about the characteristics of those numbers
There are many equivalent representations for a number or numerical relationship. Each representation may emphasize something different about that number or relationship
Numbers are compared in many ways. Sometimes they are compared to each other. Other times, they are compared to benchmark numbers
The operations of addition, subtraction, multiplication, and division hold the same fundamental meanings no matter the domain to which they are applied. Each operation has an inverse operation
There are many algorithms for performing a given operation

Measurement

A measurement is an explicit or implicit comparison
The unit or tool chosen for a measurement can affect its numerical value as well as the precision of the measure
The same object can be described using different measurements; sometimes the measurements are related, and other times they are independent
Knowing the measurements of one shape can sometimes provide information about the measurements of another shape
Measurement formulas allow us to rely on measurement that are simpler to access to calculate measurements that are more complex to access

Geometry and Spatial Sense

Both quantitative and qualitative attributes of a geometric object can impact measurements associated with that object
Many properties and attributes that apply to 2-D shapes also apply to 3-D shapes
How a shape can be composed and decomposed, or its relationship to other shapes, provides insight into the properties of the shape
There are many representations of a geometric object or a relationship between geometric objects
Locations of objects can be described in a variety of ways

Algebra and Patterning

Algebraic reasoning is a process of describing and analyzing generalized mathematical relationships and change using words and symbols
Comparing mathematical patterns or relationships either algebraically or graphically helps us see that there are classes of relationships with common characteristics and helps us describe each member of the class
The same algebraic expression or pattern can be related to different situations, and different algebraic expressions can describe the same real-world situation, sometimes with limitations based on context
Many equivalent representations can describe the same pattern or generalization. Each representation may give more insight into certain characteristics of the situation or generalization
Limited information about a mathematical pattern or relationship can sometimes, but not always, allow us to predict other information about that pattern or relationship
The principles and processes that underlie operations with numbers and solving equations involving numbers apply equally to algebraic situations
The transformations that are fundamental to determining the geometric relationships between shapes apply equally to algebraic situations

Data Management and Probability

To collect good data, it is necessary to decide what collection method is most suitable and how to best pose any questions required to collect the data
Visual displays quickly reveal information about data
Not only can information be read from visual displays, but conclusions can be drawn and inferences made
How data are displayed can affect what conclusions are drawn from the data
A summary statistic can meaningfully describe a set of data
Sometimes it is reasonable to generalize from a sample of collected data
There are a variety of ways to calculate a probability, both theoretical and experimental
There are a variety of representations of a probability distribution