FTCE K-6 Subtest 4: Mathematics. General Exam Knowledge Competency# of Questions 1 Knowledge of Student Thinking and Instructional Practices About 13

FTCE K-6Subtest 4: Mathematics

General Exam Knowledge

Competency # of Questions

1 Knowledge of Student Thinking and

Instructional PracticesAbout 13

2 Knowledge of Operations, Algebraic Thinking, Counting, and Numbers in

Base Ten

About 14

3 Knowledge of Fractions, Ratios, and

IntegersAbout 9

4 Knowledge of Measurement, Data

Analysis, and StatisticsAbout 8

5 Knowledge of Geometric Concepts About 6

General Exam KnowledgeQuestion Types

Complete the statementExample – An acute angle is ___________.

a. Greater than 90 degrees and less than 180 degreesb. Greater than 0 degrees and less than 90 degreesc. Exactly 90 degreesd. Exactly 180 degrees

Which of the followingExample – Which of the following is a true statement?

a. The product of two negative numbers is negative.b. The product of one positive number and one negative number is positive.c. When dividing a positive number by a negative number, the results are negative.d. When dividing a negative number by positive number, the results are positive.

Negative questionsExample – All of the following are examples of obtuse angles except…

a. 110 degreesb. 90 degreesc. 135 degreesd. 91 degrees

Questions that include graphs, tables, or reading passages Example – Identify the coordinates of point T in the figure shown.

– (5, -4)– (-4, 5)– (-4,-5)– (-5, -4)

There are three levels of question difficulty.

EasyAverage Rigorous

Helpful Strategies

Questions You Should Ask1.What am I being asked to do?

(number facts, relationships, properties, comparisons, etc.)

2.What do I need to know to solve the problem?3.What information do I already have?4.What information is not necessary?

Let’s try with some sample questions…3 km is equivalent to ___ . (Easy - Measurement)a.300 cmb.300 mc.3,000 cmd.3,000 m

What is the median of the following set of numbers? 4, 5, 7, 9, 10, & 12 (Easy - Statistics)a.6b.7.5c.7.8d.8

Helpful Strategies






Helpful Strategies






What to Know about Effective

Math InstructionInstructional Strategies To Know

– Concrete to Pictorial to Abstract• Which of the following is the most appropriate learning progression for students’ mastery of

concepts of area measurement?A. First: Draw rows and columns to determine the area of a rectangle, given an incomplete array.

Second: Form rectangles by tiling with unit squares to make arrays.Third: Interpret area models to form rectangular arrays.Fourth: Find the area of a rectangle through multiplication of the side lengths.

B. First: Form rectangles by tiling with unit squares to make arrays.Second: Interpret area models to form rectangular arrays.Third: Draw rows and columns to determine the area of a rectangle, given an

incomplete array.Fourth: Find the area of a rectangle through multiplication of the side lengths.

C. First: Form rectangles by tiling with unit squares to make arrays.Second: Draw rows and columns to determine the area of a rectangle, given

an incomplete array.Third: Find the area of a rectangle through multiplication of the side lengths.Fourth: Interpret area models to form rectangular arrays.

D. First: Form rectangles by tiling with unit squares to make arrays.Second: Draw rows and columns to determine the area of a rectangle, given

an incomplete array.Third: Interpret area models to form rectangular arrays.Fourth: Find the area of a rectangle through multiplication of the side lengths.



– Concrete to Pictorial to Abstract• Which of the following is the most appropriate learning progression for students’ mastery of

concepts of area measurement?A. First: Draw rows and columns to determine the area of a rectangle, given an incomplete array.

Second: Form rectangles by tiling with unit squares to make arrays.Third: Interpret area models to form rectangular arrays.Fourth: Find the area of a rectangle through multiplication of the side lengths.

B. First: Form rectangles by tiling with unit squares to make arrays.Second: Interpret area models to form rectangular arrays.Third: Draw rows and columns to determine the area of a rectangle, given an

incomplete array.Fourth: Find the area of a rectangle through multiplication of the side lengths.

C. First: Form rectangles by tiling with unit squares to make arrays.Second: Draw rows and columns to determine the area of a rectangle, given

an incomplete array.Third: Find the area of a rectangle through multiplication of the side lengths.Fourth: Interpret area models to form rectangular arrays.

D. First: Form rectangles by tiling with unit squares to make arrays.Second: Draw rows and columns to determine the area of a rectangle, given

an incomplete array.Third: Interpret area models to form rectangular arrays.Fourth: Find the area of a rectangle through multiplication of the side lengths.



How to Use Manipulatives•Instruments to Help Students Understand Math Concepts and Skills

•Help to Work with Abstract Ideas at a Concrete LevelKnow common manipulatives, when you might use them, and how you might use them

– Examples: pattern blocks, Cuisenaire rods, base ten blocks, geometric shapes, geoboards, chips, fraction tiles, Unifix cubes

Descriptive to Analytic to Abstract•Van Hiele Levels of Geometric Reasoning

Visualization: Can name and recognize shapes based on characteristics

Analysis: Can identify properties of shapes and uses appropriate vocabulary

Informal Deduction: Understands relationships between and among properties of shapes

Deduction: Proving relationships among geometric shapes

What to Know about Effective Math Instruction

Ways to Assess in Math• Diagnostic assessments• Formative assessments• Summative assessments• Traditional assessments• Performance based assessments

Differentiation in Math Instruction• English Language Learners (ELL)• Students with Exceptionalities• Response to Intervention (RtI)• Multi-Tiered System of Support (MTSS)

Preconceptions and Misconceptions• Error Pattern Analysis

What to Know about the Nature of Math Instruction

Concepts vs. Procedures• Concept: Big Idea or Understanding• Procedure (Skill): Rules, Steps, Routines, Processes, and Algorithms • Move from concept to procedure

Mathematical Fluency• Efficiency: Using the least number of steps to solve a problem

Subitizing: Conceptual or proceduralIterationAccuracyAutomaticity

• Accuracy: Knowing what algorithm to use, Memorization of basic facts, Knowing the relationships between numbers, Place value, Correctness of answer

• Flexibility: Can use more than one strategy to solve a problem• Rate: The number correct within a given period of time


Sample Questions with Mathematical Fluency• Which of the following is the most appropriate sequence of stages for

students’ development of basic automaticity?A. Stage 1: Strategies for remembering math facts

Stage 2: Figuring out math factsStages 3: Developing speed and accuracy with math facts

B. Stage 1: Figuring out math factsStage 2: Strategies for remembering math factsStages 3: Developing speed and accuracy with math facts

C. Stage 1: Developing speed and accuracy with math factsStage 2: Figuring out math factsStages 3: Strategies for remembering math facts

D. Stage 1: Figuring out math factsStage 2: Developing speed and accuracy with math factsStages 3: Strategies for remembering math facts

• A student provided an alternative computation algorithm for 52 = 38. He indicated that 52 + 38 = 50 + 40 = 90. What component of computational fluency is most likely represented by this student’s answer?

A. RateB. AutomaticityC. AccuracyD. Flexibility








• A student provided an alternative computation algorithm for 52 + 38. He indicated that 52 + 38 = 50 + 40 = 90. What component of computational fluency is most likely represented by this student’s answer?









• A student provided an alternative computation algorithm for 52 + 38. He indicated that 52 + 38 = 50 + 40 = 90. What component of computational fluency is most likely represented by this student’s answer?



Content Complexity• Recalls information• Basic application of skills and

concepts• Strategic thinking• Extended thinking

What to Know about the Content of Math

• Selecting and Performing Operations to Solve Problems– Addition and Subtraction

• Types of Problems: Join, Separate, Part Part Whole, and Compare– Question Example: (Join) Dr. O. had 5 geoboards. A student

gave her 4 more. How many geoboards does she now have?– Question Example: (Separate) Dr. O. had 30 equilateral

triangles. She lost 8 of them. How many triangles does she have left?

– Question Example: (Part Part Whole) Dr. O. has a total of 20 right triangles and 10 scalene triangles. How many triangles does she have?

– Question Example: (Compare) Dr. O. has 37 rulers. Dr. Weber has 15 rulers. How many more rulers does Dr. O. have?


• Selecting and Performing Operations to Solve Problems

– Multiplication and Division• Types of Problems: Equal Groups or Repeated Addition,

Area and Arrays, Multiplicative Comparisons, and Combinations

– Question Example: (Equal Groups or Repeated Addition) Dr. O. has 5 buckets of pattern blocks with 100 pieces in each bucket. How many patterns does she have altogether?

– Question Example: (Area and Array) Dr. O. has 50 counting bears. She wants to put them in 5 equal rows. How many counting bears will be in each row?

– Question Example: (Multiplicative Comparisons) Dr. O. has 18 bags of counting chips and Dr. Boote has twice as many bags of counting chips. How many bags of counting chips does Dr. Boote have?

– Question Example: (Combinations) How many combinations of circles and semi circles can be made out of 3 circles and 5 semi circles.


Number Theory Concepts– Factors and Multiples– Prime and Composite Numbers– Whole Numbers– Natural Numbers– Greatest Common Factor– Least Common Multiple– Divisibility Rules

Order of Operations– Example: 20 + 3 (5 - 1) = ? What is the answer?

• How would you approach this problem?– PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction)


Algebraic Thinking– Properties of Equality

• Reflexive• Symmetric• Transitive• Commutative• Associative• Identity• Inverse• Distributive

– Using Properties to Solve Problems• Solve: 7x = 2x + 35 What is x?


Fractions, Ratios, and IntegersIt’s all about Equivalence, Comparisons, and Relations


Measurement, Data Analysis, and Statistics– Range, Mean, Median, Mode, Frequency Tables, Graphs

• Helpful Strategies

1. Always put the data into a frequency table.2. Arrange in ascending order from least to greatest.3. Pay attention to the details of graphs.


Geometric Concepts– All about shapes and their properties

• True or False?

– All squares are parallelograms. True

– Some rectangles are trapezoids. False

– All quadrilaterals are parallelograms. False

– Some squares are rectangles. False



• What did you need to know to be able to answer the questions?

http://home.blarg.net/~math/deflist.html

– Congruence and Similarities • What is the difference between the two?• Sample Question: A mason was tiling a living room. To add to the appearance of the floor, the

designer decided to use similar triangles within the tile.

Given ABC~ DEF, find the length of DF.

A. 2 inchesB. 4 inchesC. 6 inchesD. 8 inches



• What did you need to know to be able to answer the questions?

http://home.blarg.net/~math/deflist.html

– Congruence and Similarities • What is the difference between the two?• Sample Question: A mason was tiling a living room. To add to the appearance of the floor, the

designer decided to use similar triangles within the tile.

Given ABC~ DEF, find the length of DF.

A. 2 inchesB. 4 inchesC. 6 inchesD. 8 inches


Geometric Concepts– Flips, Slides, Rotations


Geometric Measurement– Length, Width, Height, Angles– Perimeter, Area, Volume– Surface Area– Coordinate Planes

Practice Questions

1. Which of the following demonstrates the commutative property of multiplication? (Easy)

a. (3 x 4) x 6 = 3 x (4 x 6)b. 3 x 4 x 6 = 4 x 3 x 6c. 3 x 4 x 6 x 1 = 3 x 4 x 6d. 0 x 3 x 4 x 6 = 0

Practice Questions

1. Which of the following demonstrates the commutative property of multiplication? (Easy)

a. (3 x 4) x 6 = 3 x (4 x 6)b. 3 x 4 x 6 = 4 x 3 x 6c. 3 x 4 x 6 x 1 = 3 x 4 x 6d. 0 x 3 x 4 x 6 = 0

Practice Questions

2. A parallelogram must have which of the following properties? (Easy)

a. All equal sidesb. Two sets of parallel sidesc. All right anglesd. Five sides

Practice Questions

2. A parallelogram must have which of the following properties? (Easy)

a. All equal sidesb. Two sets of parallel sidesc. All right anglesd. Five sides

Practice Questions

3. What is the greatest common factor of 16, 28, and 36? (Rigorous)

a. 2b. 4c. 8d. 16

Practice Questions

3. What is the greatest common factor of 16, 28, and 36? (Rigorous)

a. 2b. 4c. 8d. 16

Resources

• Florida Department of Education FTCE website http://www.fl.nesinc.com/testpage.Asp?Test=060• CliffsNotes FTCE: Elementary K-6http://www.scribd.com/doc/26446600/CliffsNotes-FTCE-Elementary-Education-K-6• Quizlet Flashcards - Mathhttps://quizlet.com/subject/FTCE-k%252D6-math/• Content Module - Mathhttp://www.fl-pda.org/independent/courses/elementary/math/unitObj.htm

Documents

FTCE K-6 Subtest 4: Mathematics. General Exam Knowledge Competency# of Questions 1 Knowledge of Student Thinking and Instructional Practices About 13