Appendix D: Assessment and Evaluation - Samples
Sample 7: Principles of Mathematics 10

Topic: Radicals

Prescribed Learning Outcomes:

Problem Solving

It is expected that the student will:

• analyse problems and identify the significant elements
• demonstrate the ability to work individually and co-operatively to solve problems
• determine that their solutions are correct and reasonable
• clearly explain the solution to a problem and justify the processes used to solve it

Number (Number Concepts)

It is expected that students will:

• classify numbers as natural, whole, integer, rational or irrational, and show that these number sets are nested within the real number system.

Number (Number Operations)

It is expected that students will:

• communicate a set of instructions used to solve an arithmetic problem.
• perform arithmetic operations on irrational numbers, using appropriate decimal approximations.
• Perform operations on irrational numbers of monomial and binomial form, using exact values.

Unit Focus

The teacher provided students with concrete examples to help them realize that the study of radicals can help them extend their understanding of irrational numbers. The teacher provided small-group instruction and guided practice to help students gain the skills necessary to do the work in the unit. In order to demonstrate their abilities to solve problems involving radicals, and use technology in solving problems, students participated in team competitions and special projects.

Planning the Unit

To develop the unit, the teacher:

• identified the prescribed learning outcomes for this unit
• examined the specific prerequisite knowledge and skills needed to achieve these outcomes
• determined which prerequisites were already in place and which should be reviewed
• planned a variety of activities to help the students achieve the outcomes
• looked for ways to connect students' learning to other desirable learning outcomes, including those associated with attitudes, work groups, and communication skills
• identified criteria to use to evaluate students' learning
• designed assessment that was an integral part of the instructional process

The Unit

Review of Squares and Square Roots

• The teacher used numerical examples to ensure that all students had a firm understanding of squares and square roots (e.g., ).
• Students responded to questions like the following: - What is the square root of ________? - What is the square of _______? - What is_______ squared?

• The students and teacher worked together to develop a table of perfect squares to 1000 that students could keep and use through the remainder of the unit.
• Do the same with perfect cubes, fourth powers, fifth powers (see table).

Table of Perfect Squares, Cubes, etc to...

n	1	2	3	4	5	6	7	8	9	10
	1	4	9	16	25	36	49	64	81	100

Rules for Working with Squares and Square Roots

• The teacher asked "what if?" questions like the following to help students discover that it is possible to take the rot of only a positive real number.
  • What if I wanted to find the square root of -25?
  • Try finding using your calculator. What happens? Why?
• The teacher used numerical examples and "what if"? questions to help students develop the rule for multiplying square roots:
  For example, the teacher asked "What if you want to multiply ?
  is the same as
• A similar process was used to help students develop the rule .
• Students worked as a class to practise a number of these types of problems until they demonstrated clear understanding and could easily apply the rules.

Simplifying Radicals

• The teacher guided students to discover how to use the product rule in reverse to simplify radials.
  For example:
• The teacher made it clear that when simplifying radicals, students should start by determining if any of the squares from their table of perfect squares was a factor of the radicand. If they found a factor with a nice square root (perfect square factor), they were to take it out from under the square root sign.
• The teacher had students do the same with perfect cubes.
• Students did the same with simplifying radicals with cube roots, fourth roots, etc. For example, given , the teacher asked students to find a factor of 54 in their list of perfect squares, then rewrite , and factor out the cube root of 27 to get . The teacher ensured that students understood that the small index 3 in the symbol is attached to the root sign.
• Students who found it difficult to recognize perfect squares were instructed to factor the radicand completely and use this prime factorization to rewrite the expression in its simplest radical form.
  For example:
• Students used their calculators to convert their final answers to decimal form.
• Students practised simplifying radicals, compared their answers with those of their peers, and resolved any differences.

Multiplying, Dividing, Adding, and Subtracting Radicals

• At this point, students were ready to move on to multiplying radicals. To help students understand that they should work first with coefficients and then with radicals, the teacher guided them as they worked through increasingly complex examples. For example:
• The class discovered how simplifying radicals before multiplying made their work easier.
  is difficult to simplify.
  Is much easier.
• Individually, students solved problems to practise multiplying radicals. They were encouraged to compare their answers with those of their peers and resolve any differences.
• From multiplication, the class moved to division and rationalizing denominators. The teacher used numerical examples of increasing complexity. The more complex examples illustrated the advantage of simplifying radicals before dividing.
• Students were told that rationalizing denominators involves both removing radicals from denominators and removing denominators from within radicals.
• Together, the class worked through an example of a problem that required removing a radical from the denominator. For example:   or  
• Students worked in small groups to solve complex problems requiring the multiplication or division of radicals, or a combination of both operations. They displayed their answers in both simplest radical and decimal form.
• In a discussion about adding and subtracting radicals, the teacher asked questions that prompted students to recall the algebraic rules for adding and subtracting like and unlike terms and to apply these rules to problems involving radicals. The class worked with the teacher through several examples of increasing difficulty and discussed the advantages of simplifying the radical before performing operations.
• Students practised adding and subtracting radicals by working independently on problems prepared by the teachers. They displayed their answers both in simplest radical and decimal forms.
• Students worked in small groups to solve application problems involving all the skills they learned in this unit.
• Throughout the unit the teacher used numerical examples such as to encourage students to recognize immediately that without having to work through the problem each time.
• As an alternative approach, the teacher drew on students' proficiency with basic algebra skills to develop the rules for working with radicals. The teacher provided opportunities that helped students realize that the rules for dealing with radicals are parallel to those they used in algebra.

Working with Radicals

Operation	Algebra Skill (Collect like terms)	Radical (You may have to simplify first!)
Addition and Subtraction
Multiplication
Division

Defining the Criteria

Mathematical Thinking

To what extend do students:

• Apply the rules for simplifying radicals
• solve problems requiring the multiplication of radical
• solve problems requiring the addition of radicals
• solve problems requiring the subtraction of radicals
• solve application problems involving radicals
• demonstrate an understanding of the processes involved in performing operations with radicals by developing complex problems for other students to solve
• explain the steps involved in completing the operations and procedures addressed in the unit
• report answers both in simplest radical and decimal forms

Attitudes

To what extend do students:

• show confidence in their abilities to solve problems involving radicals
• show flexibility in dealing with challenges and in using various resources to solve problems
• express their enjoyment of learning
• show interest, participate in activities, and volunteer responses

Communication Skills

To what extent do students:

• communicate ideas clearly and understandably
• explain their reasoning to others
• help other students
• accurately use the language and symbols of mathematics
• present ideas clearly and logically
• use examples to clarify explanations or arguments

Assessing and Evaluating Students

Observation and Questioning

Students' understanding, attitudes, and communication skills were assessed informally throughout the unit. The teacher:

• observed students during class discussions and while working individually and in small groups to monitor the extent to which they were meeting the criteria
• took note of which students seemed to be having difficulty
• asked questions to further assess students' understanding of concepts and to help ensure that the pace of instruction was appropriate
• observed students at work for evidence of understanding of concepts
• evaluated the quality of questions they asked and listened to their discussion in small groups
• moved through the classroom as the students worked independently and in groups to review their work and provide individual assistance as needed.

Team Competition

Students worked in teams to develop and solve a specified number of problems. Teams took turns writing one of their problems on the board. Students in the other groups were given three minutes to solve it. (Time limits may need adjusting.) Each group that correctly solved the problem in the specified time received one point. If none of the teams solved the problem, the team that developed the problem received one point. If none of the teams solved the problem, the team that developed the problem received one point. Any team that created a problem that was not solvable using the information learned in this unit or that incorrectly solved the problem, lost one point. Play rotated between the groups until all of the problems had been presented. Each group received a mark for the activity; the mark was then assigned to each student in the group. Marks were based on the number of points the group accrued plus the teacher's 1-5 rating of the complexity of the problems developed by the group.

Project

To assess individual student's understanding of the concepts covered in the unit, the teacher directed them to prepare detailed, step-by-step directions on how to complete the operations they had discussed. Students chose two topics from each of the following categories:

• Category 1:
  - simplifying radicals
  - removing radicals from the denominator
  - converting answers from simplest radical to decimal form
• Category 2:
  - adding radicals
  - subtracting radicals
  - multiplying radicals
  - dividing radicals and solving combination problems

Students prepared detailed instructions for completing the operations they chose. Students were asked to think of the teacher as a new students who did not know how to do the operations or procedures they were describing. Students were told to use whatever resources were available in preparing their instructions and to build in examples whenever possible. The teacher rated students' instructions on the following scale.

• Students received copies of the rating scale when they started the assignment. At the end of the project they used the scale for self-assessments. Students compared their ratings with those of the teacher during a brief conference. Finally the teacher compiled instructions for each topic and gave them to students for future reference.

Project Scale

Attitude Scale

Students completed the following rating scale. Students then summarized the class results and discussed their meaning.

Please complete the following rating scale to summarize your feelings about the activities you participated in and the things you learned in this unit. Circle the number that best describes your feelings.

Attitude Scale

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Revised: September 2001

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