Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Collaborating to Extend Mathematical Understanding


Collaborating to Extend Mathematical Understanding

Derek Boyd explains how he differentiates activities to teach specific mathematical concepts. 

Teacher: Derek Boyd

School: MetWest High School, Oakland, CA

Grade: 9

Discipline: Mathematics (Geometry)

Lesson Topic: Angle-side-angle (ASA) and angle-angle-side (AAS) triangle congruence theorems

Lesson Month: January

Number of Students: 16

Featured Lesson’s Student Goals:

  • Content objectives – Learn how to identify congruent triangles; recognize triangle congruence theorems and identify the information needed to make them true
  • Literacy/language objectives – Effectively explain in writing, using at least three academic vocabulary words, the identification of specific congruence theorems
  • Engagement/interaction objectives – Collaborate with other students to achieve objectives

Standards Addressed:

Common Core State Standards for Mathematics

    Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
    Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Common Core State Standards for English Language Arts

    Write informative/explanatory texts to examine and convey complex ideas, concepts, and information clearly and accurately through the effective selection, organization, and analysis of content.

Instruction Details:

The Unit
This six-day unit on triangle congruence theorems was taught about three-quarters of the way into the year’s curriculum. The lesson on theorems fell in the middle of the unit.

Before the Video
Prior to this lesson, students had learned how to identify side-side-side (SSS) and side-angle-side (SAS) triangle congruence theorems.

During the Video
Students began class with a “Do Now”—they had five minutes to individually answer four questions, two reviewing content from past units to keep concepts fresh and two on current material. Afterward, four students volunteered to answer the same questions on the whiteboard. Then, Mr. Boyd introduced the two new triangle congruence theorems, angle-side-angle (ASA) and angle-angle-side (AAS), through guided notes. During this time, Mr. Boyd gave direct instruction while students took notes. Then, students had independent practice time to complete a worksheet. When they finished, students moved on to solving problems Mr. Boyd had posted around the room. Students were given incomplete proofs and theorems and had to identify the required missing information and then write explanations articulating what it was, why it was missing, where it needed to be, and why it was required. Mr. Boyd then gave students an exit ticket to wrap up class, in which they wrote about triangle congruency. He assigned a worksheet as homework that required students to identify the four different types of triangle congruence theorems they had studied (SAS, SSS, ASA, AAS), solve equations that contained congruent triangles, and identify missing information in diagrams.

After the Video
Students continued to work on this unit, solving equations and solving for variables when present in congruent triangles.

Teacher Prep
Mr. Boyd prepared the vocabulary posted on the board, identified appropriate problems for students to solve around the room, and prepared the classwork and homework.

Prior Knowledge
To participate in this lesson, students needed to understand how to identify sides and angles of a triangle as well as certain triangle congruence theorems. They needed to know what congruence meant, how it is identified, and how to recognize when something is congruent and not similar. 

Differentiated Instruction
Mr. Boyd provided guided notes to students to help with retention of information and to serve as a reference during independent work. He walked around the room as students worked to provide one-on-one scaffolding. Mr. Boyd incorporated movement into the lesson by having students answer questions posted around the room. He offered graphic organizers (with ELL students in mind) so that they could unify concepts and vocabulary with diagrams and equations. For essay writing, Mr. Boyd modeled and provided sentence starters. He provided multiple opportunities for students to interact with mathematical words and their applications by having students solve problems, put them into graphs and tables, and then write about them.

Group Interaction
Students interacted with each other as they moved around the room to solve the posted problems. Mr. Boyd places great emphasis on collaboration and peers helping peers.

Resources and Tools


Formative Assessment
Mr. Boyd walked around the room to observe students’ attentiveness, engagement, and willingness to talk with him or with partners; to listen in on questions and conversations; and to watch how students interacted with the problems and what they were writing. 

Summative Assessment
Mr. Boyd assessed the lesson by the quality of the explanations that he received from the students on their exit ticket assignment and on their homework assignment. He also gave students a quiz several days after the lesson to assess understanding.

Impact of Assessment
The exit ticket writing showed Mr. Boyd how well students understood the concepts and if more review were needed or if he could move on to the concept. Although his lesson topics were generally set, Mr. Boyd allowed for 15- to 20-minute modifications based on what he gleaned from this assessment. The assessment showed that students understood the lesson on triangle congruence theorems well.