Math PROBES - Amazon S3 · Cheryl Tobey Math Probes appear in each module of Reveal Math, a core...

Formative Assessment Math Probes written by Cheryl R. Tobey provide you with insight into student thinking.

Deliver insightful instruction today!

Uncover misconceptions to deepen understanding

Math PROBES

Grades 6–12

Cheryl Tobey Math Probes appear in each module of Reveal Math, a core mathematics program. Enjoy these samples and use them in your classroom today.

Learn more about the full Reveal Math program at RevealMath.com

Cheryl R. Tobey is an Author and Educational Consultant. She has been project director for Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students and a mathematics specialist for Differentiated Professional Development: Building Mathematics Knowledge for Teaching Struggling Students. Both of these project areas are funded by the National Science Foundation. Cheryl’s work, including published books, is primarily in the areas of formative assessment and professional development.

MATHPROBES

CHERYL TOBEY

163b Module 3 • Compute with Multi-Digit Numbers and Fractions

Program: MSM Component: MOPDF Pass

Vendor: Lumina Grade: TE_M3

Analyze the ProbeReview the probe prior to assigning it to your students.

In this probe, students estimate the size of each quotient, without calculating. Exercises involve division with decimals.

Targeted Concept Reason about the value of numbers and understand the effects of division given those values.

Targeted Misconceptions• Students may apply the whole number concept of “division makes quantities smaller”. • Students may have inaccuracies about the value of decimal numbers.

Assign the probe after Lesson 2.

Collect and Assess Student Work

the student selects... the student likely...

1. c, d, e 2. c, d, e 3. d, e 4. e

overgeneralizes from operations with whole numbers, and reasons the rule “division makes smaller” applies to all numbers.

Example: The student chooses a combination of these answers, because the quotient is greater than the dividend.

Various patterns of incorrect responses

applies incorrect reasoning about either the size of the decimal or the effect of the operation; and/or incorrectly applies an algorithm.

Example: The student chooses c for Exercise 1 by incorrectly reasoning that dividing by one half is the same as dividing a quantity half and option c is the closest answer to half of 31.

Take ActionAfter the Probe Design a plan to address any possible misconceptions. You may wish to assign the following resources.

• Decimals• Lesson 2, Example 5

Revisit the probe at the end of the module to be sure your students no longer carry these misconceptions.

Correct Answers: 1. b; 2. a; 3. d; 4. c

Formative Assessment Math ProbeEstimate Quotients

ThenIf

MATHPROBES

CHERYL TOBEY

Module Resource

0163A-0164_MSM_TE_V1_C1M3_MO_899720.indd 2 11/27/17 5:59 PM

Reveal Math K-12 Contributing Author

Grade 6 Probe 1

TE page

163b Module 3 • Compute with Multi-Digit Numbers and Fractions




In this probe, students estimate the size of each quotient, without calculating. Exercises involve division with decimals.

Targeted Concept Reason about the value of numbers and understand the effects of division given those values.

Targeted Misconceptions• Students may apply the whole number concept of “division makes quantities smaller”. • Students may have inaccuracies about the value of decimal numbers.




1. c, d, e 2. c, d, e 3. d, e 4. e

overgeneralizes from operations with whole numbers, and reasons the rule “division makes smaller” applies to all numbers.

Example: The student chooses a combination of these answers, because the quotient is greater than the dividend.

Various patterns of incorrect responses

applies incorrect reasoning about either the size of the decimal or the effect of the operation; and/or incorrectly applies an algorithm.

Example: The student chooses c for Exercise 1 by incorrectly reasoning that dividing by one half is the same as dividing a quantity half and option c is the closest answer to half of 31.


• Decimals• Lesson 2, Example 5


Correct Answers: 1. b; 2. a; 3. d; 4. c

Formative Assessment Math ProbeEstimate Quotients

ThenIf

MATHPROBES

CHERYL TOBEY

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135b Module 3 • Operations with Integers


In this probe, students will determine whether each simplified expression is positive or not, without actually calculating.

Targeted Concept Determining whether the result is positive or negative involves understanding the effects of integer addition, subtraction, multiplication and division.

Targeted Misconceptions• Students may overgeneralize or mix up the rules governing rational number operations.• Students may think that when opposites are multiplied or divided, the result is zero.• Explanations are ambiguous. For example, students may indicate “two negatives is

always a positive” without specifying for which operations.




Yes for a, c, e, f, g, or h misapplied the rule of “two negatives make a positive” to all operations.

e. Yes distributed the negative to the first integer inside the parentheses, but not to the second.

h. Yes thinks a negative divided by a negative results in a positive answer, without simplifying each part of the problem first.

“Two negatives cancel each other out.”

overgeneralized in their explanations.


• Whole Numbers, Integers• Lesson 1, Examples 1–7• Lesson 2, Examples 1–5• Lesson 3, Examples 1–6• Lesson 4, Examples 1–4


Correct Answers: a. No; b. No; c. Yes; d. Yes; e. No; f. Yes; g. Yes; h. No

Formative Assessment Math ProbeOperations with Integers

ThenIf

MATHPROBES

CHERYL TOBEY

Module Resource


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135b Module 3 • Operations with Integers


In this probe, students will determine whether each simplified expression is positive or not, without actually calculating.

Targeted Concept Determining whether the result is positive or negative involves understanding the effects of integer addition, subtraction, multiplication and division.

Targeted Misconceptions• Students may overgeneralize or mix up the rules governing rational number operations.• Students may think that when opposites are multiplied or divided, the result is zero.• Explanations are ambiguous. For example, students may indicate “two negatives is

always a positive” without specifying for which operations.




Yes for a, c, e, f, g, or h misapplied the rule of “two negatives make a positive” to all operations.

e. Yes distributed the negative to the first integer inside the parentheses, but not to the second.

h. Yes thinks a negative divided by a negative results in a positive answer, without simplifying each part of the problem first.

“Two negatives cancel each other out.”

overgeneralized in their explanations.


• Whole Numbers, Integers• Lesson 1, Examples 1–7• Lesson 2, Examples 1–5• Lesson 3, Examples 1–6• Lesson 4, Examples 1–4


Correct Answers: a. No; b. No; c. Yes; d. Yes; e. No; f. Yes; g. Yes; h. No

Formative Assessment Math ProbeOperations with Integers

ThenIf

MATHPROBES

CHERYL TOBEY

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Cheryl Tobey Math Probe � Graphical Representations © McGraw-Hill Education

Cheryl Tobey Math Probe Graphical Representations Alex runs up a hill at a steady pace and continues to run around at the top of the flat hill at the same pace before running down the hill at an increased pace. Determine whether each graph could describe his running pattern.

Circle yes or no. Explain your choice.

1.

yes no

2.

yes no

3.

yes no

4.

yes no

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Module Resource

MATHPROBES

CHERYL TOBEY Formative Assessment Math ProbeMultiplying Binomials

543b Module 10 • Polynomials

Collect and Assess Student Answers


• Polynomial Expressions• Lesson 10-4, Learn, Examples 1-3

Revisit the probe at the end of the module to be sure that your students no longer carry these misconceptions.

If the student selects these responses...

Then the student likely...

Student 1: yes incorrectly distributes the power treating the binomial like a monomial. They are following the pattern of (xy)2 = x2y2.

Student 2: yes is overgeneralizing about the meaning of the parentheses using multiplication to distribute the power, following the pattern of 2(x + y) = 2x + 2y.

Student 3: yes believes that their answer needs to be simplified to one term and combines all numbers and variables.

Student 4: no Is not recognizing the equivalent form of the trinomial due to one of the misconceptions described above. These students often do not understand where the linear (middle) term comes from.


In this probe, students will determine whether four students expanded (x + y)2 correctly and explain their choices.

Targeted Concepts Squaring a binomial means multiplying the binomial by itself.

Targeted Misconceptions• Students may not understand that squaring a binomial means that the binomial is

multiplied by itself and instead multiply by 2.• Students may incorrectly expand a squared binomial by overgeneralizing from work with

squaring monomials or multiplying a monomial with a binomial.• Students may not recognize a trinomial as the result of expanding a squared binomial.Use the Probe after Lesson 10-4. Correct Answers:

1. no 2. yes 3. no 4. no

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Cheryl Tobey Math Probe � Multiplying Binomials © McGraw-Hill Education

Cheryl Tobey Math Probe Multiplying Binomials Four students were asked to find (x + y)2. Which student do you think did it correctly?


Student 1:

(x + y)2 = x2 + y2

yes no

Student 2:

(x + y)2 = 2x + 2y

yes no

Student 3:

(x + y)2 = 2xy

yes no

Student 4:

(x + y)2 = x2 + 2xy + y2

yes no

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Module Resource

Correct Answers: 1. yes; 2. not enough information; 3. no; 4. yes; 5. not enough information; 6. not enough information

Collect and Assess Student Answers


• Isosceles and Equilateral Triangles• Lesson 5-6, Learn, Examples 1-2

Revisit the Probe at the end of the module to be sure that your students no longer carry these misconceptions.


In this probe, students will determine whether pairs of triangles are congruent and explain.

Targeted Concepts Understand the information needed to prove triangle congruence.

Targeted Misconceptions• Students struggle to connect corresponding sides of triangles with different orientation.• When using the Side-Angle-Side Postulate, students do not check to make sure that the

angle is the included angle.• When using AAS, students do not check that the congruent sides are corresponding.• Students sometimes see congruence marks, but do not consider whether the segments

of one triangle are congruent to segments of another triangle.• Students transfer the AA Similarity Postulate to congruent triangles.• Students only consider using the Hypotenuse-Leg Theorem and not the Side-Angle-Side

Postulate in right triangles and/or not consider using HL as it follows SSA in nonright triangles.

Use the Probe after Lesson 5-6.

If the student selects these responses…

Then the student likely...

1. no or not enough information

4. no or not enough information

does not recognize that HL and/or SAS can be used with right triangles. In item 1, students overgeneralize that SSA cannot be used with nonright triangles and only look for HL in Item 4.

2. yes or no does not recognize that the congruency marks compare segments of individual triangles.

3. yes or not enough information

does not notice that the congruent sides are not corresponding sides.

5. yes does not recognize that both angles are not included angles.

5. no is not considering that the other missing side could also be 15.

6. yes is confusing similarity with congruence.

6. no Is not considering that the corresponding sides could be congruent.

MATHPROBES

CHERYL TOBEY Formative Assessment Math ProbeCongruent Triangles

Program: Reveal Math

Vendor: AptaraPDF PASS

Component: Module 5 Opener

Grade: 9-12, TE

289b Module 5 • Triangles and Congruence

289a_289b_HSM_NA_T_G1M05_MO_899750.indd Page 2 3/14/18 8:44 AM f-0296 /110/GO02370/Reveal_Math_HSM/NA/Geometry/TE/2017/Vol1/007_899750_X_P1/Application ...

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NAME DATE ___________ PERIOD ___________

Cheryl Tobey Math Probe Congruent Triangles © McGraw‐Hill Education

Cheryl Tobey Math Probe Congruent Triangles

Determine whether the pairs of triangles are congruent.

Circle yes, no, or not enough information. Explain your choice.

1.

yes

no

not enough information

2.

yes

no


3.

yes

no


4.

yes

no


5.

yes

no


6.

yes

no


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Cheryl Tobey Math Probe Solving Exponential Equations © McGraw‐Hill Education

Cheryl Tobey Math Probe Solving Exponential Equations Three students were discussing how to solve 42x + 1 = 8x + 6. Which student do you think is correct?


Sadie’s Solution:

42x + 1 = 8x + 6

2x + 1 = x + 6

x = 5 yes no

Stefan’s Solution:

42x + 1 = 8x + 6

42x + 1 = 2 4x + 6

2x + 1 = 2(x + 6)

2x + 1 = 2x + 12

1 ≠ 12

Therefore, no solution. yes no

Wen’s Solution:

42x + 1 = 8x + 6

(22)2x + 1 = (23)x + 6

4x + 2 = 3x + 18

x = 16 yes no

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Course 1 • Volume 1

Interactive Student Edition

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Reveal Math empowers educators to uncover the mathematician in every student through powerful explorations, rich discourse, and timely and insightful differentiation opportunities.

Learn more at RevealMath.com

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Reveal the Full Potential in Every Student.


Course 2 • Volume 1

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Course 3 • Pre-Algebra Volume 1


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CONTENTS

MODULE 1 EXPRESSIONS

MODULE 2 EQUATIONS IN ONE VARIABLE

MODULE 3 RELATIONS AND FUNCTIONS

MODULE 4 LINEAR AND NONLINEAR FUNCTIONS

MODULE 5 CREATING LINEAR EQUATIONS

MODULE 6 LINEAR INEQUALITIES

MODULE 7 SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES

MODULE 8 EXPONENTS AND ROOTS

MODULE 9 EXPONENTIAL FUNCTIONS

MODULE 10 POLYNOMIALS

MODULE 11 QUADRATIC FUNCTIONS

MODULE 12 STATISTICS

Make Math Meaningful to Reveal Every Student’s Full Potential

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CONTENTS

MODULE 1 TOOLS OF GEOMETRY

MODULE 2 ANGLES AND GEOMETRIC FIGURES

MODULE 3 LOGICAL ARGUMENTS AND LINE RELATIONSHIPS

MODULE 4 RIGID TRANSFORMATIONS AND SYMMETRY

MODULE 5 TRIANGLES AND CONGRUENCE

MODULE 6 RELATIONSHIPS IN TRIANGLES

MODULE 7 QUADRILATERALS

MODULE 8 SIMILARITY

MODULE 9 RIGHT TRIANGLES AND TRIGONOMETRY

MODULE 10 CIRCLES

MODULE 11 MEASUREMENT

MODULE 12 PROBABILITY

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CONTENTS

MODULE 1 RELATIONS AND FUNCTIONS

MODULE 2 LINEAR EQUATIONS, INEQUALITIES, AND SYSTEMS

MODULE 3 QUADRATIC FUNCTIONS

MODULE 4 POLYNOMIALS AND POLYNOMIAL FUNCTIONS

MODULE 5 POLYNOMIAL EQUATIONS

MODULE 6 INVERSES AND RADICAL FUNCTIONS

MODULE 7 EXPONENTIAL FUNCTIONS

MODULE 8 LOGARITHMIC FUNCTIONS

MODULE 9 RATIONAL FUNCTIONS

MODULE 10 INFERENTIAL STATISTICS

MODULE 11 TRIGONOMETRIC FUNCTIONS

MODULE 12 TRIGONOMETRIC IDENTITIES AND EQUATIONS


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