Standards and Backwards Mapping


This post is an example of how I use backwards design for the creation of a set of lessons for Algebra 1 which is taught in Eighth grade in my school. We are located in Beijing, China and we are teaching California Common Core Math Standards for the traditional High School track (the 2013 Electronic Edition). There are no standards from the Chinese government that address the teaching of Mathematics in Private schools, although there is talk of requiring the material be taught in Mandarin till Grade 9. That is a battle being fought at a much higher level than mine, so I am not dealing with that wrinkle here.

Next year, our students will be studying the Cambridge International Examinations IGCSE Mathematics Syllabus 0580, so we do adjust the standards in G8 to align with the requirements of IGCSE. Specifically, the students take written exams in two papers at the end of tenth grade for a total of four hours. In addition, some of the terminology is different and Cambridge forbids the use of graphing calculators. The fluency in English, and temperament required to sit for a ninety minute exam and a 2 ½ hour exam are above what is outlined in the standards, so we have to present problems of greater complexity to scaffold the skills needed to take high-stakes exams of this nature.

After the IGCSE, exams, the students start the two year International Baccalaureate Diploma Programme which is also a strongly exam-based assessment as well as requiring students to write a 6-10 page research paper about Mathematics.

Our students are 75% Chinese citizens, 15% Korean, and 10% from other countries (mainly children of teachers at the school). Most of the students are ESL learners, and some are coming into eighth grade straight from Chinese Public Schools and have almost no English knowledge. Getting these students to write research papers in Math at the end of Grade 12 requires a lot of foundational work be done in Grade 8 to ensure they have the basic Mathematical English they need to deal with the rigors of subsequent work. Consequently, I chose standards that help develop student ability to comprehend and write Mathematical English as well as master the Algebraic skills.

The standards for Math are organized in three levels: Standards for Mathematical Practice that sit at a top level, the general standards which connect to other grades and courses, and the specific standards for teaching. The following are taken from the 2013 document California Common Core State Standards Mathermatics Electronic Edition.



Standards for Mathematical Practice

1.   Make sense of problems and persevere in solving them.

2.   Reason abstractly and quantitatively.

3.   Construct viable arguments and critique the reasoning of others.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

8.   Look for and express regularity in repeated reasoning.

GENERAL STANDARDS:

Creating Equations

·         Create equations that describe numbers or relationships.

Reasoning with Equations and Inequalities

·         Understand solving equations as a process of reasoning and explain the reasoning.

·         Solve equations and inequalities in one variable.

DETAILED TEACHING STANDARDS:

1.   Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.

a.   Include equations arising from linear and quadratic functions, and simple rational and exponential functions. CA *

2.   Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. *

“CA” denotes a standard added by the state of California, while the asterisks denote the requirement that students be able to solve contextual real-world problems involving the standards.

The specific proficiencies and Meta-Level understandings I want my students to have are:

1)   The ability to write Algebraic models from descriptions/situations written in English as fast as the student can read.


2)   Solving linear, absolute value, quadratic, rational and exponential equations.


3)   Recognizing that creating an efficient solution to any of the equations in 2) above is an interaction between one’s individual skills and the complexity of the problem.  


4)   Creating detailed Algebraic solutions using formal Algebraic properties of equality, operations and manipulation for linear equations.


5)   Creating formal graphs/sketches of systems of equations and finding the point of intersection of the curves.


6)   Understanding and demonstrating that solutions to non-linear equations are contingent on rewriting the equations in forms that are amenable to manipulation.


7)   Extend the concepts to the creation of inequalities involving the same types of expressions.


8)   Using technology (MS Excel and websites) to graph solutions. Graphing calculator use is not allowed in the IGCSE syllabus the students follow for grades 9-10, so we do not use Graphing calculators in this course.

Activities I will use to develop the skill of writing Algebraic Sentences from English ones are:
·         Having students brainstorm words that relate to the four mathematical operations (+, ¸,´, and –) and how the words “by a factor of” converts addition and subtraction words to multiplication and division phrases. Helping them create a comprehensive list adding more technical terms like incremented and decremented.
  •  Having them work through worksheets of increasing linguistic and Algebraic complexity as shown below



·         Once they are able to write equations from written text reliably, I will then give them problems of situational complexity that they work on in pairs. Some sample is shown below



The assessment for translating between English and Algebra is done through traditional tests and exams and one complex task at the end of the unit. A sample from a relevant test is shown below



Activities I use to develop their graphing skills are worksheets and a tutorial I developed for graphing in Microsoft Excel. The tutorial will teach students how to graph polynomial functions. Screen shots of the first few pages are shown below.


  • Graphing is assessed through a quiz and on the graphs they create afterwards throughout the school year. A sample quiz is shown below:

Towards the end of the unit the students are given this project to do where they work individually except for set times when they are encouraged to use the messaging feature in the school platform Managebac to discuss the assignment. This assesses the proficiencies discussed above as well as others I have not shown the activities for. The rubric is shown later and used when students reflect on the task. The assessment is shown below:




 




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