This teaching unit addresses four standards under the state of Florida’s strand/Big Idea 3 in 6th Grade Mathematics: Write, interpret and use mathematical expressions and equations. In informal terms, the teaching unit will serve as the very first introduction for sixth grade math students to vocabulary, concepts and basic operations with “unknowns” in algebra.
Standards Covered/Assessed Within the Unit
This unit will specifically address four standards under the Big Idea 3. Key learning objectives within the four standards are outlined below:
* Standard 1 – M.A.6.A.3.1: Write and evaluate mathematical expressions that correspond to given situations. * Learning Objective 1 – Students can define and use correctly/conversationally the terms, “variable, constant, algebraic expression and evaluate”. * Learning Objective 2 – Students can translate between words and math, creating mathematical expressions given situations expressed in language. * Learning Objective 3 – Students can evaluate single-variable expressions when given the value for the variable with 100% proficiency. * Standard 2 – MA.6.A.3.2: Write, solve and graph one and two-step linear equations and inequalities. * Learning Objective 1 – Students can determine if values for variables are or are not solutions to equations with 80% proficiency. * Learning Objective 2 – Students can solve one-step equations using inverse operations with 80% proficiency.
* Standard 3 – MA.6.A.3.4: Solve problems given a formula. * Learning Objective 1 – Students can substitute values for multiple variables into a multi-variable equation, such as a commonly used formula, to derive the solution with 80% proficiency. * Learning Objective 2 – Students can find the area of a rectangle, given the formula and the measurements of the rectangle’s sides. * Standard 4 – MA.6.A.3.5: Apply the commutative, associative and distributive properties to show that two expressions are equal. * Learning Objective 1 – Students can define the commutative, associative and distributive properties, and can verbalize which properties were used when viewing a set of equivalent expressions. * Learning Objective 2 – Students can use the distributive property to find the product of two numbers. * Learning Objective 3 – Students can use the commutative and associative properties to simplify numerical expressions using mental math.
The Unit will begin with a pretest of students’ background knowledge and familiarity with content in the standards being taught and assessed. The pretest and answer key follow.
Formative assessments are used so frequently in my classroom that they often “feel” like instruction. Following are just several of the “assessments in disguise” (formative assessments) that correspond to the learning objectives, standards and overall “Big Idea #3” of the Introduction to Algebra unit.
Formative Assessment 1
Standard #1 within Big Idea 3: MA.6.A.3.1: Write and evaluate mathematical expressions that correspond to given situations. Measureable objective being formatively assessed within standard #1: 1. Students can define and use correctly & conversationally the terms “variable, constant, algebraic expression and evaluate”. a. Formative Assessment – Assessing this objective will include a “who am I?” exercise, where students will be given stickers on their backs containing the new vocabulary words (one per student). I’ll give the kids about five minutes to walk around and talk to their classmates to try to find out “who they are” (which vocabulary word is on their backs). The students describing the term to the guessers are not allowed to use the term in their definition.
Assessment occurs as I circulate & hear the conversations. Also – a reward system is in place in my classroom, rewarding kids when they use math vocabulary in class. The students can earn tickets each time they are ‘caught’ using a word from a mathematician’s vocabulary – sometimes this alone can act as a formative assessment. If someone says, “I timesed it!”, I quickly glance at the roll of tickets and the student almost always corrects himself with “I multiplied it!” The ticket method of formative assessment will be ongoing during the algebra unit, rewarding students when they show their abilities in using algebraic terminology correctly and conversationally (this ties directly to the learning objective).
Formative Assessment 2
Standard #2 within Big Idea 3: MA.6.A.3.2: Write, solve and graph one and two-step linear equations and inequalities. Measureable objectives within standard #2:
1. Students can determine if values for variables are or are not solutions to equations. a. Formative Assessment – During a session of rotating, small-group-work, groups of four students work on an interactive learning and practice tutorial, provided by Holt online: http://my.hrw.com/math06_07/nsmedia/interactivities/mia122/mia122.html At the end of the interactive tutorial, there is a quiz. Since students log in with a unique code, results are delivered to me individually and immediately. The actual quiz questions and their answers follow. The “hint” sections are links for students to visit providing virtual tutoring (videos of instructors conducting brief, 5-minute lessons, re-teaching the material). This is a wonderful formative assessment, providing immediate feedback!
The following summative assessments have been designed to assess the “sum” of a student’s learning over the course of several weeks of work within the unit.
Summative Assessment 1
This assessment will assess students’ knowledge of content within each of the four defined standards: M.A.6.A.3.1, M.A.6.A.3.2, M.A.6.A.3.4 and M.A.6.A.3.5. Students will visit the computer labs and take a test online. Link to test is provided here:
The printed test questions along with an answer sheet follow.
Summative Assessment 2
This assessment will assess students’ knowledge of content within each of the four defined standards: M.A.6.A.3.1, M.A.6.A.3.2, M.A.6.A.3.4 and M.A.6.A.3.5. Students will complete an extended-response test of their knowledge within the four content standards. Student responses will be scored according to the attached rubric.
This self-assessment consists of a set of flash cards students can cut out and use to help them review many concepts for a semester exam, including concepts covered under the four standards addressed in this unit, namely MA.6.A.3.1 and MA.6.A.3.5. The flash cards are accompanied by a student study log. The students will quiz themselves, making piles of the cards they knew well and the cards they aren’t completely comfortable with. The log records the number of cards in the “knew well” pile. The more times students review with the flash cards, the higher their number of “known” cards will be. The study log documents and shows the student their own progress and challenges them to “beat” their last score. The included cards that address specific standards within this unit have been labeled in the following pages.
Self Assessment 2
Student portfolios contain a comprehensive record of notes, journal entries, practice, activities and informal assessments for each student’s interaction with all four standards covered in this unit (MA.6.A.3.1, MA.6.A.3.2, MA.6.A.3.4 and MA.6.A.3.5).
The standardized assessments included in this section address all four content standards being covered in this unit (MA.6.A.3.1, MA.6.A.3.2, MA.6.A.3.4 and MA.6.A.3.5). The first standardized assessment is one that is created and provided by our county. It is a semester exam, administered to every sixth grade math student county-wide near the end of the first semester of the sixth grade. Results of this exam are important to teachers, as we compare and discuss the results as they relate to variations in pacing and instructional strategies in schools across the county. The second standardized assessment is a “predictive benchmark assessment” administered through Discovery Education. Every sixth grade math student in our county takes a series of three of these assessments throughout the year to provide a snapshot of the mastery level of all content standards at three different times during the school year. The results, especially of the second and third tests, illuminate gaps in knowledge across the group of students as well as identifying individual patterns of strength or weakness among individual students.
Standardized Assessment 1
This is a sample Semester One Exam, administered by our county. The answer key is attached.
Standardized Assessment 2
This is a test generated by Discovery Education as described in this section’s opening statements. Each test question has to be printed individually, so I have provided a sample question below from one of the tests addressing standard MA.6.A.3.1, learning objective: Students can translate between words and math, creating mathematical expressions (and equations) given situations expressed in language.
44. Terry is planting seeds in small pots. He has 120 seeds to plant. The same number of seeds must be planted in each of his 20 pots. Which equation can be used to find n, the number of seeds that Terry should plant in each pot? A. 120 – n = 20 B. 120 ÷ n = 20 C. 120n = 20 D. 120 + n = 20 Also included is a breakdown of the test results by standard assessed. The standards covered within this unit have been circled. This standardized assessment gives us an “across the county” look at how students are grasping the content within the standards.
Explain how different assessments may be used for various purposes:
Assessments are ways to provide feedback – feedback to students, teachers and stakeholders. “Feedback is not advice, praise or evaluation. Feedback is about information about how we are doing in our efforts to reach a goal” (Wiggins and McTighe 2005, p. 10). Feedback in the form of meaningful assessments in the classroom provide a learning experience for students that is enhanced in concrete ways. “Unless teachers systematically monitor students’ progress via some type of assessment, there’s too much change teachers will improperly conclude progress is taking place when, in fact, such progress is not (Popham 2011, p. 10). Pretests can be given to assess students’ levels of knowledge prior to beginning a section or unit. “By measuring students’ current status, teachers can discern (1) where to put their instructional energies to ameliorate a student’s weaknesses and (2) what already mastered skills or knowledge can be instructionally avoided (Popham 2011, p. 9).
Formative assessments are used effectively in classrooms as well. A formative assessment is a planned process through which assessment-illuminated evidence of students’ learning is used by educators to adjust their educational strategies or by students adjusting their own learning tactics. A formative assessment is a “process, not a test” (Popham 2011, p. 270). Formative assessments focus on improving teachers’ ongoing instruction, and they help students adjust their current learning tactics. They’re about making adjustments during instruction – as opposed to waiting until the end of a unit or section to see if children are learning what teachers are teaching. They involve a great deal of careful planning, even more constant monitoring in the classroom, and are irrefutably effective (Popham 2011).
Self-assessments are another type of assessment used in the classroom. These types of assessments are designed to demonstrate understanding of specific content areas through various formats, such as performance assessments and the creation of portfolios. They also allow kids to provide a demonstration of their mastery of an objective or content benchmark through real-life projects, experiments and models. These are great assessments for diverse groups of students.
Summative assessments are examples of another form of measuring learning in the classroom. They are assessments used as cumulative measurements to evaluate a student’s progress over an established period of time. They’re used to make decisions about already completed instructional activities. They’re also used to help assign a grade to a student, based on how much he or she is shown to have learned during a semester or school year. The terms “summative” and “formative” are often confused – but the great distinction lies within how each is used. Summative assessments are more fully mature, final looks at what one has learned; whereas formative assessments appraise the “under construction” instructional work that’s going on (Taylor & Nolen 2008, Ch. 12).
Standardized assessments are also used as measurements of student learning. “Standardized tests are defined as assessment instruments administered, scored, and interpreted in a standard, predetermined manner” (Popham 2011, p. 330). These tests are typically created by commercial, for profit companies. Standardized tests are mandated by the government. The basic premise behind the legislation that makes these tests mandatory is the belief that “state and national tests will force schools to become better places, make teachers more accountable to children, and make schools more accountable to the public” (Taylor and Nolen 2008, p. 3). The “big picture” uses of the results of standardized tests are to provide a snapshot of today’s students’ skills and abilities, and to provide educators and the public with a common standard of measure. These types of tests are most commonly and effectively used to evaluate school programs, report on students’ progress, diagnose students’ strengths and weaknesses, select students for special programs, place students in special groups, and to certify student achievement. C. Shiermeyer (EDD/544, Week 2 post, October 30, 2012). Most recently, results of these types of tests have also been used to evaluate individual teachers’ performance.
Explain how assessment strategies are linked to specific instructional approaches to meet the needs of diverse learners:
Some examples of how various assessment strategies might influence instruction to better meet the needs of diverse groups are outlined below:
1. Identify different performances – I think the key word here is “different”, and the key strategy is “providing choice”. Identifying several options for student choice allows students the freedom to select the option they feel best suits their individual needs and minimizes bias presence. 2. Let students choose the performance – Performance assessments in nature provide students with opportunities to uniquely create responses that demonstrate mastery of skills. Since each student would be conducting his or her performance in an original manner, the performance would be bias-free (and as long as the evaluation of such performance was also without bias – this would be a perfect option). (Popham 2011). 3. Allow students to choose the topic or focus of the performance – This even further narrows the possibility for bias, in that the very focus of the assessment can be chosen by the student, and I make a broad assumption that each student would naturally choose something very familiar to him or her, and something that would allow the individual to show mastery of the skill in an original, unique and individual manner free from bias.
For example, in assessing kids’ knowledge of collecting and displaying data, a survey project could be assigned as a performance assessment. Rather than dictating the focus of the surveys, allowing the students to choose topics they’re interested in doesn’t reduce the validity or reliability of the assessment at all (no matter what the subject of the survey is, the kids are still being assessed on their abilities to collect and display data). The difference is that each child is free to choose the focus of his or her own project so that bias is minimized and an authentic assessment is assigned. 4. Allow students to have a reader, translator or scribe – This provides an accommodation for several groups of learners (ELL or other students with accommodations as described in an IEP). “Assessment accommodations, however, must never alter the fundamental nature of the skill or knowledge being assessed” (Taylor & Nolen 2008, Ch. 5). 5. Eliminate unnecessary language and use simple sentence structure in performance directions – Stating instructions as simply as possible minimizes the possibility of bias. The “KISS” theory comes to mind. Self-assessments could provide another way for students to demonstrate mastery of skills in the absence of bias.
Explain the purpose of utilizing assessment evaluation in a teaching unit:
“A well-constructed test, if used with the wrong group of students or if administered under unsuitable circumstances, can lead to a set of unsound and thoroughly invalid inferences. Test-based inferences may or may not be valid. It is the test-base inference with which teachers ought to be concerned” (Popham 2011, Ch. 4, pg. 1). This quote demonstrates the purpose of evaluating assessments in education.
How would summative assessments for the teaching unit be designed showing attention to the formative assessments?
Summative assessments are created first, using the “working backwards” strategy in instructional and assessment design. From the summative assessments, the content being tested is broken down into digestible chunks of information (building blocks of content). From these building blocks, formative assessments are designed to administer in the classroom to let teachers and students know if the required learning is actually occurring or if adjustments need to be made to the pacing or instructional strategies being used.
Bruff, D. (2012). Classroom Response System “Clickers”. Retrieved from the
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Marzano, R. J. (2007). The Art and Science of Teaching: A Comprehensive
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Popham, W. J. (2011). Classroom Assessment: What Teachers Need to Know
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C. Shiermeyer (EDD/544, Week 2 post, October 30, 2012)
Taylor, C. S. & Nolen, S. B. (2008). Classroom Assessment: Supporting
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Wiggins, G. and McTighe, J (2005). Understanding by Design (2nd ed.). Pearson; Expanded edition (July 24, 2005).