INSIDE THE MATH CLASSROOM:
Owen Herbert, eighth grade math teacher and associate dean of instruction, Brooklyn Ascend Middle School, Brooklyn
Middle school students need to reason and construct arguments, not just memorize formulas, to truly understand mathematics and be prepared for high school. At Ascend, our eighth grade Common Corealigned curriculum gives students opportunities to experience strategies, refine their thinking, and develop and test conjectures.
In eighth grade geometry, for example, students solve real-world problems related to the volume of a cone. Because a cone and a cylinder have similar features, we teach students to explore the relationship between the two figures. Ultimately, we develop and foster their thinking so that they are able to derive one formula from the other. There is no need for memorizing — and potentially forgetting — a formula if students truly understand how the formula came to be.
Students had previously learned that the
formula for the area of a circle is Πr . With that prior knowledge, I poured water into a cylindrical cup and asked the students what happened to the water. They realized that the water first had to cover the area of the bottom of the cup, and they quickly understood that some part of the formula had to include the area of the circular bottom of the cup, namely Πr .
I asked the students a series of questions to develop their thinking even further. What happens after the water covers the bottom of the cup? The students said that the water started to rise. Then I asked, how much water will be in the cup after 1 inch? After some discussion, students concluded that the amount of water would be the area of the base times the height of the water ( Πr x 1 inch).
In groups, students then discussed how much water would be in the cup when the water was at 2 inches, 3 inches, 4 inches, and 6 inches. Each group developed the formula for the volume of any cylinder: Πr h.
Later, students used their discoveries about cylinders to develop the formula for the volume of a cone, and moved on to apply these conceptual understandings to prisms and pyramids. Demonstration and intense group discussion paved the way to conceptual understanding.
Sample questions:
Students will complete 51 multi tiple-choice questions on the 2017 exam, as well as 10 open-response questions that require them to demonstrate their use of mathematical methods.
1) Kevin moved from a city to a small town. The populaton of the city is 6 x 10 , which is about 15 times as great as the small town. Which expression could represent the approximate population of the small town? A. 4 x 10 B. 4 x 10 C. 9 x 10 D. 9 x 10
2) Which equation represents aa nonlinear function? A. y = -3x + 1 B.y = x + 1
C. y= x + 1
2 D.y = 2x + ½
3) Triangle ABC is translated to create triangle DFG, as shown below. In these triangles, side AB is congruent to side DF, and side BC is congruent to side FG. Determine the values of x and y.
ANSWERS 1. B; 2. B; 3. Two points possible.