# Unifying Themes

This section describes two of the unifying mathematical themes in Connected Mathematics that cut across the strands and grades.

## Proportional Reasoning

Proportional reasoning is a unifying theme throughout all three grades in *Connected Mathematics*. In Grade 6, ratios make their first appearance in the number and operations unit, *Comparing Bits and Pieces*. A ratio is a comparison of two quantities that depend on multiplication or division. This extends the work of thinking comparatively using differences where a comparison of two quantities depends solely on using addition or subtraction. With ratios, contexts involve “for every” statements. For example, “for every $60 the sixth-graders raise in donations, the seventh-graders raise $90.” The ratio of donations raised by the sixth-graders to the seventh-graders is 60 to 90 or for every $1 the sixth-graders raise in donations, the seventh-graders raise $1.5. This later ratio is called a unit rate because the comparison involves $1. While it is also true that the seventh-grade raises $30 more than the sixth-grade, this is not a ratio. It is a difference comparison. In this unit, students use ratios to compare quantities, learn about unit rates and rate tables, and develop strategies for working with percentages. They also explore the differences between fractions and ratios.

In Grade 7, proportional reasoning plays a major role in developing understanding of important concepts in the geometry, number, algebra, probability, and data units. In the geometry unit, *Stretching and Shrinking,* proportional reasoning is enhanced through the development of similarity, scale drawings, and scale factors. A proportion is a statement of equality between two ratios. Equivalent ratios are used to compare lengths within a figure which describes the general shape of the figure. Equivalent ratios can also be used to look at ratios of corresponding sides across two figures. These ratios are called the scale factor between two similar figures. Using similar geometric shapes as a context to explore proportional reasoning provides an alternative visual entry point to proportionality to those presented in numerical proportional contexts. The visual approach provides more access for some students that may not be available in a number context. It also enriches understanding of proportionality for students who are inclined to use numeric representations. Similarity is followed by the unit, *Comparing and Scaling*, which brings proportional reasoning back to reasoning about numeric situations. The emphasis is formalizing proportional reasoning by using equivalent ratios written as fractions (a/b = c/d) or rates (a is to b as c is to d) to solve a proportion. Students explore equivalent forms for representing ratios or rates when they use part- to-part or part-to-whole reasoning. Scaling rates or ratios can be used to solve proportions. The constant of proportionality is brought to the foreground. For example, if 10 pizzas cost $120, then 1 pizza costs $12 or C = 12*n* (C is the cost and *n* is the number of pizzas. The constant of proportionality is 12 and the relationship between cost and the number of pizzas is represented by a straight line graph. In the algebra unit, *Moving Straight Ahead*, a unit about linear functions, students relate a constant rate of change in a table to a constant slope on a graph and to the constant of proportionality in equations of the form *y* = *mx*. In addition, proportional reasoning is enriched in the data analysis and probability units. In the probability unit, *What Do You Expect, *students use proportional reasoning to understand and answer questions in experimental or theoretical probabilistic situations. In the statistics unit, *Samples and Populations*, students make predictions about population characteristics by scaling up information from random samples.

In Grade 8, students employ proportional reasoning to solve problems in *Thinking with Mathematical Models *(linear and inverse variation). In this unit students analyze two-way tables that are used to compare preferences between age and preference of roller coaster type. In *Butterflies, Pinwheels, and Wallpaper* (symmetry and transformations), similarity and hence proportional reasoning, is revisited through symmetry and transformations, hence deepening the understanding of similarity and proportional reasoning.

## Mathematical Modeling

The CCSSM say “Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards.” In some sense, mathematical modeling is what every Unit in CMP is about. When students write number sentences in the Grade 6 Number and Operations Units, or write symbolic expressions for contextual situations in Algebra Units, they are creating a model of a situation. Not only does this kind of model capture something important about quantities in the situation, but it can be manipulated into equivalent forms, to expose new understandings or solve problems. Graphs and tables can also be descriptive or analytic models. In Data and Probability Units, students choose variables, represent data with an appropriate graphical model, and analyze the distribution using statistical models. Modeling gets explicit focus in *Thinking With Mathematical Models*. Students (1) identify the variables in the situation, (2) formulate graphical and algebraic models for the relationships between variables, (3) analyze the models and make predictions, (4) improve on the models by using a line of best fit, and (5) measure how well their statistical model fits the data.

One particular aspect of modeling is quantitative reasoning. As with modeling, and consistent with CCSSM guidelines, CMP integrates quantitative reasoning throughout the curriculum. When students create algebraic models they have to define their variables, choose the units with which to measure the variables, and interpret the formulas or equations consistently with the units chosen. When students create graphical models they must select an appropriate scale for the situation. When students collect data and model a situation with a descriptive graph, they must define the quantities carefully. When students report a statistic or a conclusion they must consider what level of accuracy is appropriate for the context.