In this very first module, we will be focusing on 6th and 7th grade mathematics. We will start with a lesson focusing on Unit Rate and Proportional Reasoning, which can be a challenging concept for middle school students to grasp.

This module focuses on the following CAHSEE released test question:

The diameter of a tree trunk varies directly with the age of the tree. A 45-year-old tree has a trunk diameter of 18 inches. What is the age of a tree that has a trunk diameter of 20 inches?

A. 47 years
B. 50 years
C. 63 years
D. 90 years

This type of question is answered correctly by only 35% of students who take the CAHSEE. As you progress through this module, you will deepen your content knowledge on order of operations with fractions and develop instructional strategies that assist students in mastering these kinds of problems.

The major California Mathematics Content standards addressed in this question include:

Number Sense (6th) 1.3 Use Proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems; understand it as the multiplication of both side of an equation by a multiplicative inverse.

Algebra and Functions (6th) 2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity.

Algebra and Functions (7th) 4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation.


In Module 1.1: Unit Rate and Proportional Reasoning, you will:

  • Deepen your understanding of mathematical concepts in Algebra, including:
    • Using unit rate, ratios, and proportions; and
    • Solving direct variation problems through unit rates and proportions.
  • Develop skills in new instructional strategies, including
    • The importance of supporting English learners with hands-on activities and realia;
    • Use of poetry and literature to support teaching and learning of mathematics; and
    • Teaching vocabulary.