School District of Philadelphia, Office of Multilingual Curriculum and Programs (OMCP) Language of Mathematics

Index

Index ...... 1

Course Description ...... 2

Core Materials ...... 3 Language Central for Math: ...... 3 Student texts: ...... 3

Optional Materials for September / October ...... 3 Option 1 ...... 3 Option 2 ...... 5

Additional Resources ...... 5 Mathematics and the English Language ...... 7 Mathematics Operations Terminology ...... 9 Mathematics Syntax (Word Order) ...... 10

Sample Unit Curriculum Map for Language Central for Math ...... 11

Lesson Planning Resources ...... 12 QTEL Principles ...... 13 QTEL Unit / Lesson Planning: Three Moments (components) in a Lesson and Sample Tasks ...... 16

WIDA ...... 16 Scaffolds Supports (WIDA) ...... 17

Placement Test ...... 18 Answer Key ...... 18

2 Course Description

Language of Mathematics is an mathematics credit bearing course. The class MUST be taught by a teacher with 7-12 mathematics certification. In addition, the teacher should have either ESL certification or have participate in Quality Teaching for English Learner (QTEL) professional development. The course is designed for newcomer high school students who either have limited and/or interrupted formal schooling or will benefit from a mathematics course to prepare them for Algebra I. In addition to learning and/or reviewing mathematical concepts and skills, students will expand their knowledge of the English academic language of mathematics. An assessment is at the end of this document to determine if students should be rostered in the course.

Goals of the course include (1) building on students’ “funds of knowledge” / prior knowledge and experiences in mathematics, (2) engaging students in meaningful mathematical language practices (reading, writing, speaking, listening) with appropriate scaffolds, (3) learning in context the disciplinary vocabulary of mathematics, and (4) supporting students’ understanding, application and appreciation of mathematical concepts and skills, and (5) preparing students for advanced high school mathematics courses.

When working with English Learners:

1. Focus on students’ mathematical reasoning, not accuracy in using language. 2. Focus on ALL students participating in mathematical discussions on important mathematical concepts and reasoning rather than pronunciation, vocabulary or low-level linguistic skills. 3. Recognize and support students to engage with the complexity of language in math classrooms. Student learn content / concepts and language simultaneously. Students learn to describe patterns, make generalization and use representation to support their claims. 4. Treat everyday language and experiences as resources, not as obstacles. 5. Uncover the mathematics in what students say and do. 6. Draw on multiple resources available in classrooms such as objects, drawings, graphs and gestures as well as home languages and experiences outside of school.

(Judith Moschkovich in Mathematics, the Common Core, and Language: Recommendations for Mathematics Instruction for ELs Aligned with the Common Core)

3 Core Materials

Language Central for Math:

Student texts: o 6th: Number and Operations, Algebra, Geometry and Measurement, Data Analysis and Probability (24 lessons)

o 7th: Number and Operations, Algebra, Geometry and Measurement, Data Analysis and Probability (24 lessons)

o 8th: Number and Operations, Algebra, Geometry and Measurement, Data Analysis and Probability (24 lessons)

Teacher’s Edition: (Grades 6 – 8)

o Language Proficiency Chart (pgs. T12 – T13) o Building Lesson Plans (T14 – T23) o Strategies for Teaching English Learners (T24 – T35) o Scope and Sequence (pgs. T38 – T39)

Students may need additional preparation BEFORE using Language Central for Math.

Optional Materials for September / October

Option 1

Bridges to Academic Success

Mathematics Unit 1 http://bridges-sifeproject.com/classes/math-unit-1/

Essential Question: How do we use math to describe the world around us? 6 sets or units with 5 lessons each (30 lessons) • Structure of the number system • Place value • Symbolic notation

4 • Four central operations (addition, subtraction, multiplication, division) • Solving simple world problems with whole numbers • Concrete measurement (area, perimeter) • Final performance task: Designing the Farm (apply knowledge of area, perimeter and operations with whole numbers)

Materials: • Unit Map • Performance Task • Unit Guide • Lesson Plans • Glossaries • Flash Cards • Bilingual Translations (Bengla, Haitian Creole, Karen, Spanish, Somali)

Mathematics Unit 2 http://bridges-sifeproject.com/classes/math-unit-2/

Essential Questions: How do we represent parts of a whole using words, numbers and symbols? What strategies and we use to solve problems with parts of a whole? 7 sets or units with 3 to nine lessons in each (35 lessons) • Concepts behind rational numbers • Fractions • Decimals • Percents • Construct number line, area models, ratio tables • Ratio and proportion

Materials: • Unit Map • Performance Task • Unit Guide • Lesson Plans • Glossaries • Flash Cards • Bilingual Translations and Bilingual Unit Previews (Bengla, Haitian Creole, Karen, Spanish, Somali)

5 Option 2

Longman Mathematics (Out of print)

Text is out of print. Each school will receive 1 – 2 copies. Materials will have to be photocopied.

Content in Longman Mathematics that may be used before using Language of Mathematics includes: Unit 1: Reading and Saying Cardinal Numbers (pgs. 3 – 12) Reading and Saying Ordinal Numbers (pgs. 13 – 24) Rounding Off Numbers (pgs. 25 – 36) Unit 2: Addition (pgs. 39 – 50) Subtraction (pgs. 51 – 66) Multiplication (pgs. 67 – 86) Division (pgs. 87-110)

Additional Resources

A Framework for Re-envisioning Mathematics Instruction with English Language Learners (Council of Great City Schools, Dec. 2016) https://tinyurl.com/y77qevsu

Bilingual Mathematics Word-to-Word Dictionaries http://steinhardt.nyu.edu/metrocenter/resources/glossaries

• By course (e.g. middle school math, Algebra, Geometry, etc.) • By language (Albanian, Arabic, Bengali, Burmese, Chinese (traditional), Dutch, French, Greek, Haitian, Hindi, Japanese, Karen, Korean, Nepali, Portuguese, Punjabi, Romanian, Russian, Spanish, Tagalog, Tibetan, Ukrainian, Urdu, Uzbek, Vietnamese)

Mathematical Notation Comparisons Between U.S. and Latin American Countries https://tinyurl.com/y7qytav7

Mathematics, the Common Core, and Language: Recommendations for Mathematics Instruction for ELs Aligned with the Common Core Judith Moschkovich https://tinyurl.com/hf2cebm https://www.youtube.com/watch?v=gUfpnIbq4TA

6 Principles for Mathematics Instruction for ELs Understanding Language, Literacy and Learning in the Content Areas Judith Moschkovich https://tinyurl.com/hbxnw2g

Supporting Language and Content Learning in Math (newcomers) https://www.teachingchannel.org/videos/math-for-newcomers-ousd

Tasks (strategies) to Support Reading Math Problems https://tinyurl.com/llkzk8w Additional Task Teaching Resources: http://ell.stanford.edu/teaching_resources/math

Teaching Mathematics to English Language Learners National Council of Teachers of Mathematics Position Statement https://tinyurl.com/ybfu6zvx

Tutorial: Language Central for Math https://mypearsontraining.com/products/language-central-math-2011 Language Central for Math Program Overview – Tutorial (If the tutorial does not work with Google Chrome, try Safari or Firefox.)

Mathematic related homophones (and near homophones) that English Learners may confuse when listening to a discussion, lecture, etc. angle / ankle flower / flour pair / pear area / era give / keep (for Arabic quotient / quota before / four / for speakers) remainder / remain a / remain column / calm in / an the divisor / advisor know / no son / sun eight / ate many / money sum / some exchange / change meet / meat tens / tense factor / factory one / won two / to vary / very weight / way

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Mathematics and the English Language

Category of Difficulty Examples Mathematics and everyday English Right angles versus right answer share some words but they have Right angle versus right hand different meanings in the two Reflection as flipping over a line versus reflection as contexts. thinking about something Foot as 12 inches verses the foot on a leg

Number: prime, power, factor Algebra: origin, function, domain, radical, imaginary Geometry: volume, leg, right Statistics/probability: mode, event, combination Discrete mathematics: tree

Mathematics and everyday English Difference as the answer to a subtraction problem verses share some words and have difference as a general comparison comparable meanings but the mathematical meaning is more Number: divide, equivalent, even, difference precise. Algebra: continuous, limit, amplitude, slope Geometry: similar, reflection Statistics: probability: average Discrete mathematics: array, edge, and, or

Some mathematical terms are found Number: quotient, decimal, denominator, algorithm only in mathematics. Algebra: asymptote, integer, hyperbola Geometry: quadrilateral, parallelogram, isosceles, hypotenuse Statistics/ probability: outlier, permutation Discrete mathematics: contrapositive

Some words have more than one Round as a circle versus to round a number to the tenths mathematical meaning. place Square as a shape versus square as a number times itself

Number: inverse, round Algebra: square, range, base, inverse, degree Geometry: square, round, dimensions, median, base, degree, vertex Statistics/probability: median, range Discrete mathematics: dimensions, inverse, vertex

8 Some words shared with science have Variable in mathematics is a letter that represents different technical meanings. possible numerical values, but variable clouds in science are a weather condition.

Number: divide, density Algebra: solution, radical, variable Geometry: prism, degree, image, radian Statistics/probability: simulation, experiment Discrete mathematics: matrix, element, cell, tree

Mathematics and everyday English Number: sum or some share some terms as homonyms. Algebra: sine or sign, cosine or cosign Geometry: pi or pie, dual or duel, plane or plain, arc or ark Statistics / probability: leaf, as in stem-and-leaf, or leave Discrete mathematics: complement or compliment, graph or graft Some mathematical words are related, Number: factor and multiple, hundreds and hundredths, but students may confuse their numerator and denominator distinct meanings. Algebra: equation and expression, solve and simplify Geometry: theorem and theory Statistics/ probability: dependent events and independent events Discrete mathematics: converse, inverse, and contrapositive

Some mathematical phrases must be Number: at most, at least learned and understood in their Algebra: one-to-one entirety. Geometry: if-then, if-and-only-if Statistics/probability: stems-and-leaf Discrete mathematics: if-then, if-and-only-if

Modifiers may change mathematical Number: value or absolute value, prime or relatively meaning in important ways. prime Algebra: root or square root, inverse or inverse function Geometry: polygon or regular polygon, bisector or perpendicular bisector Statistics/probability: number or random number, probability or conditional probability Discrete mathematics: sequence or arithmetic sequence, reasoning or circular reasoning

9 A single English word may translate In Spanish, the table at which we eat is mesa, but a into Spanish or another language in mathematical table is a tabla. two different ways. In Spanish, round (redondear), as in “round off,” or round (redondo), as in “circular.” English spelling and usage have Four has a u, but forty does not. many irregularities. Faction denominators, such as sixth, fifth, fourth, and third are like the ordinal numbers, but rather than second, the next fraction is half.

Some mathematical concepts are Skip count by threes versus tell the multiples of 3 verbalized in more than one way. One-quarter versus one-fourth

Students may adopt an informal term Diamond for rhombus as if it is a mathematical term. Corner for vertex

Adapted from Rubenstein, R.N. & Thompson, D.R. (2002) and Thompson, D. & Rubenstein, RH. (2000)

Mathematics Operations Terminology

Addition + Subtraction - Notes Combine(d) Minus Student must learn what words mean in a Increased (by) Less than particular mathematical expression. Examples: (1) More than Less 3 multiplied by 10 is vastly different than 3 Total of Fewer than increased by 10, (2) divided by and divided into Sum Difference will give entirely different results. Add Decreased And Take away • When students learn mathematical symbols such Added to Subtract from as +, ×, ÷, >, <, they must learn to relate them to Together (1) mathematical processes or operations, and (2) Plus translate them into everyday concepts.

Multiplication x Division ÷ Notes Multiplied Divided by • If students have had math education in their Product of Into home countries, they may find differences in Times Per symbolic use confusing. Of Quotient of 300.000 versus 300,000 Percent (divided by 100) 1,73 versus 1.73 Out of Ratio of

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Mathematics Syntax (Word Order)

Structures that are frequently used in math and may be difficult for English Learners. greater than/less than as in All numbers greater than 4 n times as much (as) as in Mr. C. earns six times as much as I do. Mr. C. earns $40,000. How much do I earn? as…as as in The tennis ball is as big as the plastic ball.

-er than as in Fernando is three years older than Leslie. Leslie is 25. How old is Fernando?

Numbers used as nouns Twenty is five times a certain number. What is the (rather than adjectives) as in number?

Prepositions Eight divided by four and eight divided into four as in Passive voice When 15 is added to a number, the result is 21. as in Find the number.

Ellipsis (words left out) as in All numbers (that are) greater than four Maria earns six times as much as Peter (earns)

Pronoun reference as in Rachel had 17 toy cars. She gave 11 of them away. How many toy cars does she have now?

Spread your thumb and first finer as far apart as you can. Do this in the air. Don’t use your other hand to help. Trace them on the board.

Article determiners (the, a, as in When 15 is added to a number, the result is 21. an, this, that, these those) Find the number. usually indicate that it is a Twenty is five times a certain number. What is number that has been that number? mentioned before. Adapted from Heinze, K. (2005)

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Sample Unit Curriculum Map for Language Central for Math

Grade 6 / Unit 1: Materials WIDA Can Do Descriptors (Key Uses) and Number and Core Standards Operations Language Central for WIDA ELD Standard 3: The Language of Essential Questions: Math, Grade 6 Math Lesson 1: What terms do you use to Unit 1 – Number and Speaking: Students interact with talk about greatest Operations mathematical grade-level vocabulary, common factor expressions and concepts (GCF), least common Student’s Edition Ask and answer questions / describe multiple (LCM) and Lesson 1: Pages 1 – 4 relationships related to topic using teach factorization? Lesson 2: Pages 5 – 8 modeling, word banks and visual supports Lesson 3: Pages 9 – 12 Lesson 2: What terms Lesson 4: Pages 13 – 16 Writing: Students interact with do you need to know Lesson 5: Pages 17 – 20 mathematical grade-level vocabulary, and use to learn about Lesson 6: Pages 21 – 24 expressions and concepts the order of Lesson 7: Page 25 – 28 List, describe, compare or explain choices operations? Lesson 8: Pages 29 - 32 related to topic using modeling, and word banks with a partner Lesson 3: How do Teacher’s Edition you use pictures, Lesson 1: Pages 1 – 4 Reading: Students interact with numbers, and words Lesson 2: Pages 5 – 8 mathematical grade-level vocabulary, to explain how to Lesson 3: Pages 9 – 12 expressions and concepts solve multiplication Lesson 4: Pages 13 – 16 Identify key language that provides problems? Lesson 5: Pages 17 – 20 information to solve real-life mathematical Lesson 6: Pages 21 – 24 problems using visual and graphic supports Lesson 4: How do Lesson 7: Pages 25 – 28 with a partner you use pictures, Lesson 8: Pages 29 - 32 numbers, and words Listening: Students interact with to explain how you mathematical grade-level vocabulary, solve division expressions and concepts problems? Respond to prompts based on oral descriptions using visual supports with a Lesson 5: What partner terms do you need to use and understand to talk about simplifying PA Core Standards fractions? Lesson 1 and 4: Lesson 6: What CC.2.1.6.E.2: Identify and choose terms do you need to appropriate processes to compute fluently learn in order to talk with multi-digit numbers (add, subtract, about adding, multiply and divide whole numbers, subtracting, decimals, fractions and mixed numbers)

12 multiplying and Lesson 2 – 4, 6: dividing fractions and CC. 2.2.6.B.1: Apply and extend previous mixed numbers? understandings of arithmetic to algebraic expressions. (Identify parts of an expression Lesson 7: What using mathematical terms such as sum, term, terms do you need to product, factor, quotient, coefficient) understand to work Lessons 2 - 4: with fractions, CC.1.6.E.1: Apply and extend previous decimals, and understandings of multiplication and division percents? to divide fractions by fractions. Lessons 3 – 4: Lesson 8: How do CC.2.2.6.B.2: Understand the process of you talk about and solving a one-variable equation or inequality understand fractions and apply to real-world and mathematical and decimals on the problems. number line? Lessons 5 – 7: CC.2.1.6.D.1: Understand ratio concepts and Lesson Structure: use ratio reasoning to solve problems. 1 - Prepare the CC. Analyze proportional relationship and Learner / Build use them to model and solve real-world and Background mathematical problems. 2 - Interaction with Lesson 8: the content/ CC.2.1.6.E.4: Apply and extend previous Model, guided understandings of numbers to the system of discussion, write and rational numbers. share, table talk Lessons 2, 3, 4, 6 and 8: 3 – Extending C.C.2.1.7.E.1: Apply and extend previous understanding understandings of operations with factions to operations with rational numbers.

Lesson Planning Resources

Common Core Standards aligned with Language Central of Math: http://assets.pearsonschool.com/correlations/LC4M_CCSS_Grades_3-8.pdf Standards are listed with corresponding lessons. Use PA Crosswalk to align with PA Core Standards.

Pennsylvania Department of Education English Language Learner Crosswalk (Standards) for Mathematics: http://static.pdesas.org/content/documents/PA_CC_Mat_Crosswalk_9-17- 12.pdf

Pennsylvania Department of Education Educating English Learners Basic Education Circulars (BECS), July 2017 https://tinyurl.com/y86q9svj (22 Pa. Code 4.26)

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Quality Teaching for English Learners (QTEL) – https://qtel.wested.org/

QTEL provides a framework / foundation for planning and instruction for 6th – 12th grade just as Comprehensive Literacy provides a framework / foundation for planning and instruction for K-5.

QTEL Principles From Walqui, A & van Lier, L. (2010) Scaffolding the Academic Success of Adolescent English Language Learners: A Pedagogy of Promise. San Francisco, CA: WestEd; pp. 84, 85, 152, 155,169, 174.

Principles Goals Objectives Sustain 1. Promote deep • Develop central ideas in the discipline first, postponing Academic disciplinary interesting but secondary details Rigor knowledge • Establish interconnections among central ideas of the

disciplines

• Deepen understanding of themes over time • Have students anchor new knowledge to central 2. Engage students concepts to build understanding in generative disciplinary • Have students apply familiar central ideas or concepts and strategies to their emerging understanding of new skills concepts • Invite students to build increasingly complex

explanations of disciplinary concepts and processes

• Have students combine facts and ideas to synthesize, 3. Engage students evaluate, and generalize in generative Have students build arguments, solve problems, and cognitive skills • construct new meanings and understandings (higher order thinking)

Hold High 1. Engage students • Provide students with activities that are robust, but Expectations in tasks that flexible enough to allow multiple entry points: all provide high students, regardless of where they starts, will benefit challenge and from participation high support • Scaffold students’ ability to participate in the activities • Ensure that students are asked to engage in

increasingly more complex tasks • Treat students as if they already possess the abilities you are seeking to develop

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• Conduct metacognitive activities so that students gain 2. Engage students knowledge of how to learn, how to monitor their (and teacher) in progress, and how to self-correct the development • Provide practice in the use of academic tools and of their own activities so that students appropriate them over time expertise • Encourage students to support each other in their development • Encourage students to support each other in building academic stamina 3. Make criteria for • Use rubrics to spell out expected quality of work quality work clear • Encourage students to take risks and to work hard to for all master challenging academic work

Engage 1. Engage students • Invite students to go beyond brief, single responses Students in in sustained and to elaborate, illustrate, and connect to their Quality interactions with interlocutors’ ideas Interactions teacher and peers

2. Focus interactions on the • State explicitly that constructing new understandings construction of is hard work, that is requires listening intently to knowledge interlocutors, making sense of what they are saying, and deciding how to respond, either by agreeing and providing further evidence or by disagreeing and stating why this is the case • Ask students to focus on the coherence of what they are saying (Are they staying with the main ideas? Are they making sense?) and to deepen their understanding by making connections to related ideas

Sustain a 1. Promote • Provide explicit examples, for example, formulaic Language language learning expressions, of how to mark agreement, Focus in meaningful disagreement, and other moves in response to an contexts interlocutor or text.

2. Promote • Focus on social purpose of genre, audience, structure, disciplinary and specific language of disciplinary texts; have language use students practice deconstructing and creating similar texts.

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3. Amplify rather • Give rich and varied examples, looking at difficult than simplify concepts from several angles. communications

4. Address specific • Focus corrective feedback on fluency, complexity, or language issues accuracy, but not at the same time judiciously

Develop 1. Structure • Set long-term goals and benchmarks Quality opportunities to • Use a problem-based approach with increasing Curriculum scaffold learning, interrelated lessons

incorporating the • Use a spiraling progression goals above • Make connections between subject matter and students’ reality • Build on students’ lives and experiences

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QTEL Unit / Lesson Planning: Three Moments (components) in a Lesson and Sample Tasks

Preparing Learners Sample Tasks • Focus attention on concepts to be developed Think-Write-Pair-Share • Activate / build on background knowledge Quick-Write / Round-Robin Anticipatory Guide or Extended Anticipatory Guide • Introduce essential new vocabulary in context Knowledge Rating Scale • Connect lessons to students’ experiences Novel Ideas Only Jig-Saw Project Frayer Model List – Group – Label (vocabulary)

Interacting with Text / Concepts / Content Sample Tasks • Deconstruct text; focus on understanding a Teacher Model chunk and reconnect a chunk to the emerging Double-entry Journal / Triple-Entry Journal whole text Reading with a Focus / Viewing with a Focus Clarifying Bookmark or Partner Clarifying Bookmark • Establish connections between ideas within Reading Aloud in Four Voices text Partner Reading and Discussion • Work collaboratively to discuss, evaluate, Novel Ideas Only predict, check for understanding, summarize, Four Corners etc. Carousel

Extending Understanding Sample Tasks • Re-create text in a new genre or create new Collaborative Mind Mirror / Monologue text to represent new understanding Collaborative Poster Famous Phrases • Apply newly gained knowledge to novel Create, Exchange, Assess situations or use to problem-solve Collaborative (Dialogue) Writing • Connect ideas learned to other ideas and Literary Elements experiences outside the text - compare, Famous Phrases synthesize, evaluate, create, critique, problem Gallery Walk solve, etc.

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WIDA

WIDA is an educational consortium of 39 state education agencies and 200 international schools. ACCESS testing is developed by WIDA. The mission of WIDA is to advance academic language development and academic achievement for children and youth who are culturally and linguistically diverse. WIDA’s Can Do Philosophy is grounded on students’ assets and

17 contributions to the classroom and community. Lastly, WIDA challenges linguistic discrimination, cultural biases and racism in education.

WIDA English Language Development Standards: https://www.wida.us/standards/eld.aspx

WIDA Can Do Descriptors and Key Uses Edition: https://www.wida.us/standards/CAN_DOs/

Scaffolds Supports (WIDA)

Sensory Graphic o Real life objects (Realia) / concrete objects / o Charts / Tables Physical models o Graphs o Manipulative (e.g. measurement tools, o Timelines models of geometric figures, scientific o Number lines instruments, etc.) o Graphic organizers o Pictures / photos o Graphing paper o Visual representations (illustrations, o Number lines diagrams, drawings, etc.) / Cartoons o Timelines o Videos / broadcasts / audio books o Maps o Newspapers / magazines o Rubrics o Gestures / Physical movement o Study guides / Guided Notes o Music / songs / chants o Posters / display Interactive Verbal and Textual o Whole group o Labeling o Small group o Teacher Modeling / Monitoring o Partner (turn-&-talk) o Repetition o Cooperative groups (think/write/pair/share) o Paraphrasing / Summarizing o Triads o Guiding, clarifying, probing questions o Interactive websites / software o Leveled questions (5Ws) o Mentor / coach o Questioning prompts / cues o L1 (home or first language) o Word banks / phrase banks / word walls o Word to Word Dictionary / Picture Dictionary o Sentence starters / sentence frames / discussion o Jigsaw activities frames / formulaic expressions o Cloze paragraphs / sentences o Talk moves (structured academic conversations: re-voicing/clarifying, restating, reasoning, adding on, wait time) o Wait time

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Placement Test Use the placement test to determine if students should be rostered in Language of Math and/or what skills and concepts need to be taught / reviewed.

Answer Key (1) 8 + 7 = (2) 21 + 36 = (3) 9 – 2 =

15 57 7

(4) 83 – 16 = (5) 6 x 3 = (6) 3.2 x .3 =

67 18 .96

(7) 18 ÷ 6 = (8) 112 ÷ 7 = (9) ½ + ⅔ =

3 16 3/6 + 4/6 = 7/6 or 1 1/6

(10) ⅘ - ⅓ = (11) ½ × ⅖ = (12) ¼ ÷ ½ =

12 / 15 – 5/ 15 = 2 / 10 = 1/5 ¼ x 2/1 = 2/4 = 7 / 15 1/2

(13) x + 4 = 11 (14) x – 7 = 13 (15) 3! + 4! = (3^3 + 4^2=)

7 x = 13 + 7 27 + 16 = 43 x = 20

(16) 3x + 2x + x = 0 (17) 2(x-2) = 20 (18) x + 9 = 18 + -2X

6x = 0 (x – 2) = 20/2 x = 9 + -2X x = 0 (x – 2) = 10 3x = 9 x = 12 x = 3

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Math Placement Name ______Date______(1) 8 + 7 = (2) 21 + 36 = (3) 9 – 2 =

(4) 83 – 16 = (5) 6 x 3 = (6) 3.2 x .3 =

(7) 18 ÷ 6 = (8) 112 ÷ 7 = (9) ½ + ⅔ =

(10) ⅘ - ⅓ = (11) ½ × ⅖ = (12) ¼ ÷ ½ =

(13) x + 4 = 11 (14) x – 7 = 13 (15) 3! + 4! = (3^3 + 4^2=)

(16) 3x + 2x + x = 0 (17) 2(x-2) = 20 (18) x + 9 = 18 + -2X

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