**Upper Elementary:**

Forwards: Our forwards are underway into an exploration of all things fractional. We have studied addition and subtraction of fractions, both through the use of models and arithmetic. Students are able to add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. Students are busy solving word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Students have discovered the use of benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognizing that an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Our next studies will involve multiplication and division of fractions.

Midfielders and Backs: Students have been working to understand how to solve equations and inequalities as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Furthermore, students are using variables to represent numbers and write expressions when solving a real-world or mathematical problems; understanding that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Most recently, our midfielders and backs have been solving real-world and mathematical problems by writing and solving equations of the form

*x*+

*p*=

*q*and

*px*=

*q*for cases in which

*p*,

*q*and

*x*can be any rational number.

**Middle School:**

Middle School students have been solving problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Students are able to draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Our focus has been on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Additionally, we have explored supplementary, complementary, vertical, and adjacent angles in multi-step problems. Students are able to to write and solve simple equations for an unknown angle in a figure. Also, students have been solving real-life and mathematical problems involving angle measure and area. Most recently, we have been exploring the formulas for the area and circumference of a circle. Collectively, we and have been putting all of our formulas to good use in determining the area of real-world composite figures.