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Utilizing Components

Application, Connection, and Extension (ACE)

The Application, Connection, and Extension (ACE) exercises provide additional learning opportunities for your students. Initially most teachers used them as homework assignments. Over the course of implementing this curriculum, some teachers have expanded their flexibility in utilizing ACE exercises in other ways, such as a “bell ringer” or “opener.” A rule of thumb is to answer the exercises yourself with the level of detail that you expect from your students while timing yourself. Take that time and multiply it by three or four to estimate how long it will take your students to complete. There are suggestions in the teacher support materials for which ACE exercises align with each of the Investigation’s Problems.

Application exercises tend to be very similar to the in-class Problem using a similar context or a new context but the same mathematical concepts or skills.

Connection exercises review prior concepts or skills that are mathematically connected to the daily lesson Problems. The connections can also be to “real-world problems.”

Extension exercises take the concepts and skills from the Investigation to the next level or expand on the Problem. Often these questions have a greater depth and openness to them. Many students can and will benefit from doing some of these Problems. Students who are ready will typically find these questions interesting and worthy of their thinking time.

For more information on the ACE exercises, please visit Organization in Student Materials.

Suggestions for Implementing ACE Exercises

You can use the ACE exercises for homework. Some questions to consider.

  • Why homework?
  • What do I intend for my students to gain from the homework assignment?
  • How will it further our collective learning?
  • How will students get feedback about their work on the ACE questions?

Consider how much time you will devote to homework correction both during class and outside of class and the time you will need to spend planning and implementing the daily lessons. You may also consider offering different ACE assignments to different students that align with where they are in their learning.

Often you can select an ACE exercise to use as an entry activity or exit slip. One procedure is to have students respond to the question on mini-whiteboards they hold up for a quick assessment. Another procedure is to have students respond on a sticky note and stick it on the door as they exit class. You can readily sort their responses to assess who is ready to build on the concept or skill and who may need some additional practice or support.

You can select various ACE exercises as additional learning opportunities for students. These exercises may be chosen by you or selected by individual students. To help students choose, you may point out that the ACE exercises are designed to help students become more confident in their understandings. Students who believe that they would benefit from additional practice choose from the exercises that align with the day’s lesson. You may also use the ACE exercises to differentiate instruction for various learners. You can assign different ACE to individual students who need additional practice to solidify their understandings or need an additional challenge to deepen their understanding beyond the class discussion.

Some CMP teachers allow students to ask questions about the previous night’s homework. The students are then given an opportunity to revise their work before turning it in. This allows students, and families, to see the homework as a learning opportunity. When assigning homework, you might suggest that if a student struggles to answer an ACE exercise, he or she write a question about it. Question should be detailed, such as, “What are ‘increments of 5 campers?’” or “Which variable should go on the x-axis?” Questions such as these focus the student on the area of difficulty, let you know the student’s thoughts about the problem, and give you insight into the difficulty the student may be having.

Mathematical Reflections

At the close of each Investigation, you can lead a whole class conversation around the Mathematical Reflection questions. Students should have the opportunity to discuss their understandings and then record their individual responses to the questions in their notebooks or journals.

You can also post each question on a sheet of chart paper or on a section of your whiteboard. Each student or small group of students can record their responses and respond to each other in “chalk talk” format. For a chalk talk, your writing does the talking instead of talking aloud. Students write an initial response and then their peers add their thinking in the form of new ideas and connections.

Some teachers use the Mathematical Reflections as an organizer for note taking during the Investigation. As part of the Summarize of Problems, students record key ideas from the daily lessons that are related to the reflection questions. Then at the close of the Investigation, students synthesize their notes into responses to the questions.

Mathematical Practices

You can plan for opportunities in which students can reflect on their experiences and name the Mathematical Practices that they used. At the close of each Investigation, various Mathematical Practices are highlighted for students to recognize how those practices were used during the Investigation. Take a few minutes during the Summarize of the Problem to ask students to reflect on which of the Mathematical Practices were used during the class period. You will enhance student understanding of their learning and processing mathematically.


Essential vocabulary terms are developed during each Unit. You can plan time for students to record their definitions with specific examples for the terms as they occur in the Unit. Students create their own mathematical dictionaries ?to reference from year to year in middle school and beyond. Encourage students to view their lists as working glossaries that they can add to and refine as they gain new insight and encounter new examples. Revising and updating descriptions and examples can help students improve their working knowledge of the vocabulary. You might find it helpful to have students occasionally work in a group or as a whole class to discuss the descriptions they have written.

Some teachers ask students to create a blank dictionary on lined paper with one to three pages for each letter of the alphabet. Students then add the essential vocabulary terms as the year progresses.

Other teachers prepare terms in alphabetical order by Unit. Students add their definitions and a specific example for each vocabulary word.

Displaying each essential vocabulary term with student displays of learning on the memory wall (described in Preparing Your Classroom Environment) will enable students to have a readily accessible reference when engaged in discourse around Problems.