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Connected Mathematics: Grade 8 Help
Click your Connected Mathematics: Grade 8 textbook below for homework help. Our answers explain actual Connected Mathematics: Grade 8 textbook homework problems. Each answer shows how to solve a textbook problem, one step at a time.
|Author:||Smith, Johnny one|
|Publisher:||Smity and smatty|
My child’s math class seems different than what I’m used to. What does a typical day look like in a CMP classroom?
CMP is a problem-centered mathematics curriculum, which means that students spend a significant amount of class time working on problem solving tasks. CMP lessons are designed around a Launch-Explore-Summarize pattern. In a typical lesson, the teacher starts by launching the task. Teachers help students connect to prior learning and work through important terms or aspects of the problem(s). Then, students explore the mathematical ideas by working on problems individually or with other students. Depending on the purpose of the task, the explore phase might involve just getting student ideas on the table for discussion, or it might involve students analyzing big ideas and putting multiple ideas together to construct core understandings. Teachers monitor students, attend to individual needs, and ask probing questions to enhance students learning from the problems. Finally, the teacher leads a discussion using students’ thinking and strategies to summarize the important ideas and skills for the lesson. For more information, visit the pages Learning in a Problem-Centered Curriculum and Instructional Model.
My child says the teacher rarely demonstrates methods. When he is stuck the teacher often asks more questions. Why is this?
CMP was designed to be compatible with cognitive science research about how people learn. Specifically, it supports communication about mathematics between students and between students and the teacher. The single mathematical standard that has been a guide for all the CMP curriculum development is that all students should be able to reason and communicate proficiently in mathematics. If students are to reason and communicate proficiently then these are the skills they must spend time practicing in the classroom. Sometimes students work individually and write explanations of their thinking, but often students discuss problems, argue with and convince each other, make conjectures and draw conclusions, and make summaries—things that mathematicians do! The teacher's role is to guide students as they work with problems to develop understanding of the embedded ideas, and explore powerful mathematical principles, strategies and algorithms. Of course, there are occasions when the teacher demonstrates mathematical ideas, but generally the teacher will try to help students draw as much sense as they can from a problem before intervening. For more on these ideas, see Communication as a Learning Tool.
If my child wants help with their homework, is it okay to just tell them my way of doing the problems?
When your child seems to need help with mathematics homework, as a caring parent/guardian, your first wish and impulse is to try to tell your child how to do the assignment or show them a method you recall. Teachers are caregivers too, and they are similarly tempted to show and tell, but a good way to support children to be independent learners is to have mathematical conversations in which the child is an active participant.
CMP is a problem-centered math curriculum. This means that important mathematical concepts are embedded in engaging problems. Students develop understanding, skill, and confidence in themselves as they explore problems, whether individually, in small groups, or with the whole class. You can support this approach in the way you interact with your children over their math homework. See Homework Support for specific ideas about helping your children with their homework. See the Learning in a Problem-Centered Curriculum section on this website for more about this curriculum approach and for the research base that supports this approach.
Is there a general way of helping my child with math no matter which unit we're in?
One of the best ways to help your child is to make it a habit to find out first what she does and does not understand, and what resources she has to draw on. We developed a set of questions (see Homework Support) to support you in starting up helpful conversations when working on math. You will want to personalize these to fit the relationship you have with your child and the particular circumstances of the classroom she is in. These can be useful questions regardless of how confident (or not) you feel in your math abilities!
I feel like I’ve forgotten a lot of the math I learned in school. How can I help my child?
You can still have meaningful mathematical conversation with your child! If you go to his textbook, you may be disappointed to find there are almost no examples shown, nor methods told. Instead there are questions for students to discuss, followed by more questions and problems for them to solve. Your child may have a mathematics notebook with a record of class ideas, strategies, examples, solutions, and vocabulary. Without being part of the classroom discussion it can be hard to reconstruct the mathematics embedded in the problems. To help you bridge this gap, an on-line summary of goals, a glossary of vocabulary words and concepts (with examples!), and sample solutions for homework problems are available for each unit. See Math Homework Help section and Glossary pages. For more on why there are so few worked problems, try Learning in a Problem-Centered Curriculum.
Why are there so few worked examples in CMP texts?
The overarching goal in CMP is for students to make sense of and take ownership of mathematical concepts. Students learn effectively when they take on the role of formulating, representing, clarifying, communicating, and reflecting on ideas. If the format of the texts included many worked examples, the student role becomes more about reproducing the examples with small modifications. This may help students become fluent at a particular skill, or with a particular strategy, but would hinder their development of independent reasoning skills that can be applied flexibly in new situations. This is not to say that worked examples aren’t valuable in CMP—but the work in these examples is done by students independently, in groups, or with the help of the teacher. Their work is captured in their notebooks, so students can refer to it whenever needed.
If there are few examples in the student textbook, then what should my child refer to when she is doing homework or studying for tests?
Your child should have notes about how they solved in-class problems, noting new vocabulary, summarizing each problem with the teacher's help, and completing a Mathematical Reflection on the entire Investigation). Different teachers will have students organize their notes in different ways, but a comprehensive notebook provides a record of what each student understands and can do. These notebooks are crucial for each student's success and can be a great resource when they are working on their math at home.
If there are few examples in the student textbook, then what should my child refer to when she is doing homework or studying for tests?
There is a consistent and substantial body of research indicating that when students work cooperatively on challenging problem solving tasks, their mathematical learning will be enhanced. Therefore, CMP materials have been designed to be suitable for use with both cooperative learning and individual learning formats. The teacher materials assist and advise teachers in planning for and implementing both kinds of lesson designs depending on the task. In fact, in any given lesson there is likely to be a blend of individual, group, and whole class activity.
When students are working in groups the classroom and the teacher’s role look different than when they are working individually. However, group work shouldn’t take away from students’ individual responsibility and accountability. Teachers can and do observe closely what individuals are doing and learning when they monitor group progress. Each student will be responsible at minimum for taking notes, asking questions, and summarizing the big ideas.
Depending on the particular problem, groups or individual students may be responsible for sharing or presenting ideas and leading class summaries.
How much algebra is there in CMP? What high school class will my child be ready for after CMP in 8th grade?
CMP3 students completing all 23 units in Grades 6, 7, and 8 will be prepared to take college-level courses, including calculus, by their senior year of high school. The Connected Mathematics Project was originally funded by the National Science Foundation and designed by its authors with one of its goals being to provide more algebra before high school. The NCTM Principles and Standards and Common Core State Standards guided the development of the algebra strand in the Connected Mathematics Project.
There are two paths through Grade 8: Grade 8 and Algebra I. Depending on the path and the design of your local district offerings, your child should be ready to enter into the first or second year of high school mathematics. Students who have been successful in CMP Algebra I can skip the first year of the high school program. Students who have been successful in the Grade 8 path of CMP3 are well prepared to take a traditional Algebra I course in high school. Obviously this decision can only be made based on your specific middle and high school programs; the best guides are the teachers involved.
Students who have been successful in all of the CMP algebra units will have met and mastered the ideas and skills that are part of a traditional Algebra 1. But, they also will have done very much more than this in their study of algebra in CMP. Their experience will have been a coherent functions approach to important mathematical relationships, especially linear, exponential, inverse proportion and quadratic, including solving linear, exponential and quadratic equations, and inverse and direct proportions. Therefore, CMP algebra units are an excellent preparation for a functions-based approach in Algebra 2. Because of this extensive and thorough study of algebraic ideas in CMP, many students entering a high school with a traditional curriculum in place may successfully proceed to Algebra 2.
You can find more information on algebra in the CMP at Units by Strand and Development of the Math Strands.
How much geometry is there in CMP? How well does CMP prepare my child for a high school Geometry course?
Six out of a total of 23 units in three grade levels are primarily on geometry and measurement. Geometric ideas are connected and reviewed in all other units. See Units by Strand and Development of the Math Strands for a list of topics and goals. The CMP authors were influenced by mathematics education research indicating a shift from a focus on shape and form to the related ideas of congruence, similarity, and symmetry transformations.
High school Geometry courses typically spend a semester on shape and measurement, two topics thoroughly explored in CMP, and a semester on proving. All CMP units require students to reason and communicate their reasoning, which means they have a strong foundation for writing proofs. In CMP geometry units, students are asked to reason about ideas that are part of the high school curriculum, such as similarity and congruence. At first the reasoning will be informal, but their ability to reason and their ability to communicate their reasoning continues to develop over each unit and year.
It doesn’t seem like my child spends much time practicing skills and procedures, like adding and subtracting fractions, decimals, and negative numbers. Are students in CMP classrooms developing the skills they need?
Two of the guiding principles of CMP are intertwining conceptual and procedural knowledge, and developing this knowledge as needed to solve problems. Theory and research on math teaching and learning indicates that mathematical understanding is fundamentally a web of connections among many different ideas. These ideas can be procedural (like algorithms or solving different kinds of problems step-by-step) or conceptual (for example, seeing how multiple ideas fit together to solve abstract problems). This research and theory has been influential on the authors of CMP and how they developed the curriculum. When considering the interplay between conceptual and procedural knowledge, they believe that conceptual understanding is an important foundation for procedural skill, not a secondary result of repeated procedural practice.
Therefore, CMP problems concentrate on conceptual development, but often draw on previous conceptual and procedural knowledge so that there is repeated practice embedded in new problems. For example, the unit Covering and Surrounding focuses on area as one of the major learning goals. This provides rich opportunities to review and practice prior work with fractions and decimals when students work with dimensions that are not whole numbers. Homework questions, particularly Connections questions, repeatedly connect new ideas to prior procedural knowledge. There are many practice problems distributed throughout all three grades for procedures or algorithms that students need to be fluent in (for example, fraction operations). This kind of practice within connections is an important and intentional aspect of CMP.
An example: You might be wondering what the difference is between a skill and a concept, between procedural knowledge and conceptual knowledge, and how these are assessed differently! We will use decimal multiplication to illustrate these types of understanding. Imagine a student is assigned the problem 4.56 x 2.35 to be done by hand. If she produces the answer 10.716, which is correct, we can be pleased because it appears she has mastered that skill. But, we cannot be sure what she actually understands about the important concept of place value and how that concept can be used in calculating the correct answer. If we want to find out what the student understands about place value we need to ask a different question.
Instead of assigning 4.56 x 2.35, we could start by giving the student the fact that 4.56 x 2.35 = 10.716 and then ask for the result of 4.56 x 0.235 without access to paper and pencil or a calculator. In this problem, she can reason that since one of the factors is a tenth of its original value the answer should also be a tenth of its original value, or 1.0716. Now she is using the place values for the digits in 2.35 and 0.235 to think about the calculation. Similarly, giving a student the fact that 4.56 x 2.35 = 10.716, and asking for the result of 456 x 235 without access to paper and pencil or calculator gives him the opportunity to consider place value.
One procedure that students often learn as a shortcut is to "count the decimal places in the two factors, and make the answer have the same number of decimal places." Another way to assess students’ conceptual understanding of decimal multiplication is to start by telling them that a broken calculator has produced the answer 4.56 x 2.35 = 10716, then asking students to place the decimal point. If a student applies the shortcut without thinking about place value, then she might state that the answer is 1.0716, which has four digits after the decimal point. But if she understands multiplication and estimation, then she is more likely to reason that the answer must be more than 4 x 2, and, therefore, the correct answer is 10.716.
In short, CMP values both skills and concepts, and the authors have created ways for students to develop concepts and practice skills, and for teachers to assess both.
In Evaluation Studies, you can find studies comparing CMP students and their peers in more traditional curricula. Many of these studies indicate that their performance is similar on tests of basic skills, and CMP students perform significantly better on assessments of conceptual understanding and problem solving.
Why does my child do so much writing in CMP?
The ability to communicate about mathematics and mathematical reasoning is an important aspect of the subject. Communication is an important theme in the NCTM Principles and Standards and the Common Core Standards for Mathematical Practice. You can visit Communication as Learning Tool for much more on the role of communicating about mathematics in CMP.
What is the research basis for CMP's design?
The design of CMP was influenced by two main bodies of research: cognitive science (the study of thought and learning, which draws on fields like psychology and philosophy) and mathematics education research that focuses on teaching and learning math. On this website, visit Learning a Problem-Centered Curriculum and Communication as Learning Tool for a summary on influential research ideas from cognitive sciences. See Development of the Math Strands for a summary of influential research in mathematics education.
How does CMP accommodate gifted students?
One of the important guiding principles of CMP is that the curriculum maintains high expectations of all students. All students should have access to appropriate levels of challenge and support. The Guide to Connected Mathematics 3: Understanding, Implementing, and Teaching (the Implementation and Teaching Guide in previous editions) pays specific attention to the needs of gifted students in the mathematics classroom. A National Council of Teachers of Mathematics publication, Providing Opportunities for the Mathematically Gifted K-12, proposed several essential components, many of which relate directly to the mathematics curriculum. CMP possesses these components described by NCTM's publication, notably:
- nurturing higher-order thinking processes in open-ended investigations,
- prompting students to communicate effectively,
- having problem-solving as a major focus,
- including mathematics problems that model real-life situations,
- encouraging experimentation and connections of mathematics to other subject areas.
Researchers in cognitive science have suggested that organizing the curriculum around key concepts and complex problems, promoting use rather than acquisition of knowledge, and pushing for abstraction are all appropriate modifications for mathematically gifted students. These features are already part of the organization of CMP. In particular, the Extension questions in the ACE homework ask for levels of abstraction that go beyond what the class may be asked to do in a lesson. But even within in-class problems, there are thought-provoking questions that can be answered at various levels of abstraction. For more information, visit Learning a Problem-Centered Curriculum.
Interestingly researchers stress opportunities for different ways of thinking and solving problems, and group interaction allows gifted students the opportunity to explain their ways of thinking to others, and to assume leadership roles. The focus should be on explaining why and how strategies work, and on classroom patterns of interaction that make learning, not the teacher's approval, the focus.
How does CMP accommodate students who are English Language Learners?
One of the important guiding principles of CMP is that the curriculum maintains high expectations of all students. All students should have access to appropriate levels of challenge and support. The Guide to Connected Mathematics 3 orDifferentiating on this web site pay detailed attention to the needs of English Language Learners (ELL) in the mathematics classroom. Several strategies are outlined and illustrated to support teachers as they meet the unique needs of ELL students. For example, teachers are encouraged to provide specific and mathematically meaningful feedback to ELL students, be explicit about school and classroom norms and model those norms, and use various forms of communication including pictures, graphs, and gestures. Each of the strategies described in the Guide addresses instructional delivery in some way. It’s important to note that none of these strategies involves modifying the cognitive demand of the mathematics in CMP. That is, every effort has been made to support all students in gaining a rich understanding of important mathematics.
In addition, student texts and assessments are available in Spanish, and the teacher materials provide family letters in both English and Spanish. You can talk to your child’s teacher about accessing these materials.
How does CMP accommodate students with special needs?
One of the important guiding principles of CMP is that the curriculum maintains high expectations of all students. All students should have access to appropriate levels of challenge and support. The Guide to Connected Mathematics 3 or Differentiating on this web site pay detailed attention to students with special needs in the mathematics classroom. Several strategies are outlined and illustrated to support teachers as they address the individual needs of students. For example, teachers are encouraged to emphasize the use of manipulatives and real-world problems and to ensure that all students are participating effectively in small group work. Each of the strategies in the Guide addresses instructional delivery in some way. The important thing to know, though, is that none of these strategies involves modifying the cognitive demand of the mathematics in CMP. That is, every effort has been made to support all students in gaining a rich understanding of important mathematics.
In addition to the supports in the Guide, special labsheets are provided for some homework problems to provide more structure and visual support for students who might need it.
Supports for students with special needs are often strategies that help all children learn. Blending group and individual work, student conversation and silent reflection, and conceptual development and procedural skills practice--characteristics that are integral to CMP's format—offer more ways for students with a diverse range of skills to be successful.