Effective Instructional Practices in Mathematics grades 6 12

Effective Instructional Practices in Mathematics grades 6 -12 November 3, 2017 9 am – 3 pm presented by JUDITH T. BRENDEL Ed. M. FEA consultant

DIFFERENTIATE why? • For students to experience multiple opportunities to work within the same topic. • To keep all students actively engaged and interested. • Meet the different learning style preferences of your students. Give them different types of activities as they learn, practice and assess their understanding. • Meet the different levels of understanding of all the students in your class. • Give students experiences outside-their-comfort-zone; so they learn to persevere and be successful.

What will you do? -Participate in engaging activities that focus on Math Practices; CCSS and differentiating tasks; -Debrief activities -Select a topic/standards for your students -Create a similar activity for your students Use √off-list Use templates formatting Use hardcopy & online resources for content

FEA Website http: //www. njpsa Scroll to right … to FEA PROGRAMS Scroll down … to PROFESSIONAL LEARNING RESOURCE LINK http: //njpsa. org/professional-learning-resourcelinks/ Later today, the Power. Point and some resources from today’s workshop will be listed here under Nov. 3, 2017 “Effective Instructional Practices in Mathematics” Grades 6 -12. ”

The 8 MATHEMATICAL ”PRACTICES” (process) STANDARDS 1. Make sense of problems; persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

Which one Doesn’t Belong and Why? RAISE the level of EACH student’s thinking THINK out-of-the-box MONITOR each student’s understanding CHALLENGE ALL! A In FOLDER B C D

Which one Doesn’t Belong and Why? A. 5 < 6 C. 2 + 4 > 1+ 5 B. 6 > 2+1 D. √ 4 = 12 + √ 9 See ‘A’ above: Compare 2 two-digit numbers … record results of comparisons with symbols > = < (use of symbols new to grade-1. ) ABCD monitor each

Which one Doesn’t Belong and Why? A. B. C. D. ABCD monitor each

Which is the most difficult to solve? A) 2 x = - 14 2 X = - 14 2 2 X= -7 B) 14 = 2 - x +X +X X + 14 = 2 - 14 -14 N W O X = -12 R U O Y. E e T u A g a e l l CRE o c a h t i C) x + 2 k = 14 D) x/2 = 14 w Wor - 2 -2 X = 12 BLANK FORM in FOLDER (2)x/2 = - 14(2) x = - 28

Math Practices #1 and #2 1. Make sense of problems and persevere in solving them. Make sense of problems and persevere in solving • them. “Mathematically proficient students start by explaining to the meaning a problem and forto • themselves “Mathematically proficientofstudents start bylooking explaining entry pointsthe to meaning its solution. ” themselves of a problem and looking for entry points to its solution. ” • “They can understand the approaches of others to complex problems and identifyof others to • solving “They can understand the approaches correspondences betweenand different solving complex problems identifyapproaches. ” correspondences between different approaches. ” 2. Reason abstractly and quantitatively. “… knowing and flexibly different of operations andand Reasonusing abstractly andproperties quantitatively. “… knowing objects. …” different properties of operations and objects flexibly using. …”

MOVIE CLIP Lucy and Ethel in the CANDY FACTORY Google: You tube Lucy Ethel Chocolate Factory

USE VIDEOS “THE CANDY FACTORY” math worksheet (Handout to be distributed) 1. What SKILLS do you see here? 2. What GRADE-LEVELS or COURSES? Math. Bits. com (worksheet)

GET THEM MOVING MAKING DECISIONS BEING SUCCESSFUL SEE EACH OTHER THINKING

CHANGE PROCESS!!! 21 st Century Assembly Line Move, have fun, see each other thinking a. Organize thoughts; show process; b. Follow rules; defend/critique each others work Arithmetic 8 -10 steps (yellow) Algebra about 8 steps (white) with Answer Keys Geom. /Alg. II 8 steps Posted around the room

CHANGE PROCESS!!! 21 st Century Assembly Line PASS IT ON Move, have fun, see each other thinking a. Organize thoughts; show process; b. Follow rules; defend/critique each others work Geometry about 7 steps (gr. 4/5) WORK TOGETHER * CREATE YOUR OWN Fractions 8 steps (note step #1) Pre-Algebra (more fractions) (white) Blank FORM in FOLDER

MP. 4: Model with mathematics Math Practices #4 Model with mathematics. “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society and the workplace…. . ”

2 D – to – 3 D You already know so much! Yes, CONCRETE! Yes, MANIPULATIVES in MS HS!

USE non-math FOCUS ON CONCEPT before the math FUN with PUZZLE PIECES (work in pairs or groups of 3) 1 4 2 3 ENVELOPES of PUZZLE PIECES and Answer Keys

CAROUSEL*ACTIVITIES Why do these? SAME THEME different TYPES of tasks SAME THEME different LEVELS of difficulty INDIVIDUAL, PAIRED or TEAM activity Students HELP & TEACH each other POSTED ABOUT ROOM: 1. Turquoise sheets: PARCC like format Students move about the room (w/clip board, handout, pencil) 2. Fun Shapes: Grades 8 -9 to complete multiple tasks. 3. Yellow Index cards: Geometry • 4. Posted about room are sheets, cards or post-its with tasks Algebra at 3 levels • Needed tools are at each site FROM DECK: • PICK Answer keys are available to students Green Index cards: Pick two to do.

CAROUSEL WORK TOGETHER * CREATE / PLAN YOUR OWN

MP. 3: Construct viable arguments and critique the reasoning of others Math Practices #3 Construct viable arguments and critique the reasoning of others. • They justify their conclusions, communicate them to others, and respond to the arguments of others. … • Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

MATCH your PARTNER • • Set CD GH (MS/HS) Set A & B (Expressions) Set A & B (Equations) Set C & D (Inequalities) • Worksheet form Blank form in Folder WORK TOGETHER * CREATE YOUR OWN

Math Practices #6 Attend to precision. • Mathematically proficient students communicate precisely to others. Use correct math terminology.

Frayer Model from Ad. Lit. org Classroom Strategy Library Definition Fact Word / Concept Examples Non-examples

Frayer Model from Ad. Lit. org Classroom Strategy Library Definition Fact Word / Concept CREATE YOUR OWN Form in your folder. Examples Non-examples

A VOCAB LADDER ACTIVITY PAIRS PLAY a few rounds: (select a page; Directions: fold vertically) - Students work in pairs (fold the sheet) - Teacher decides who goes first (closest birthday? ) - First student: begins at the bottom and reads one definition slowly - Second student: gives the vocabulary word to match - If correct – then the second student reads … - If incorrect – the first student continues (If a student is incorrect … must begin at step-1 next time. Repetition = continued practice. ) Add challenge: student #1 reads word, student #2 gives definition Forms: Grade 5, 6, Alg. II, Geom , Parabolas

Match Parts with whole Activity: Student pairs or groups of 3 (compete) Envelopes contain: - Fractions and Exponents - Geometry: characteristics and shapes - Algebra-II: - Blank form -Formulas and uses -Vocabulary words and definitions -Students work together and teach other.

MATCH the GRAPH GEOMETRY ACTIVITY: Battle-ship-like game (Student pairs) Play games to practice; improve organization; improve sequencing, problem-solving, use of specific vocabulary, …. Match the Graph Separate Handouts straight edge, graph paper • 1. Geometric Shapes/polygons: Geometry 2. Match graphed Equations (MS or HS) 3. Match graphed parabolas (HS) 4. Create own for “your” student needs

MATCH the GRAPH ALGEBRA ACTIVITY: Battle-ship-like game (Student pairs) Play games to practice; improve organization; improve sequencing, problem-solving, vocabulary, following directions, …. Match the Graph • Geometric Shapes • Linear Equations Give directions with coordinates points, slope and/or x- or y-intercepts • Algebra-I or II: Parabolas

Equation Tic Tac Toe 1 2 4 7 3 5 8 6 9 Math + Fun = more learning grade-5 samples to see format w/Math Practices, Standards and Answers

CHANGE the QUESTIONS How do PARCC questions look? A. …. WRITE B. . . DRAW C. …. . D. …. EXPLAINCOMPARE SKETCH CONSTRUCT BUILD WHY? PROVE YES? NO? NOT TRUE?

More then one right answer MORE RIGOR ACTIVITY Student pairs GEOMETRY (probability? ) - Same perimeter different areas - Same area different perimeters

Activity: FIND THE AREA: Draw 3 -4 different rectangles that have a perimeter of 30. Record the area of each. (Use whole numbers only. ) • Which shapes have the largest & smallest area? • What do you observe? AREA with PERIMETER

Activity: FIND THE PERIMETER: Draw 4 -5 different rectangles that each have an area of 36. Record the perimeter of each. (Use whole numbers only. ) • Which shape has the largest or smallest area? • Which has the largest or smallest perimeter? PERIMETER with AREA

ERROR ANALYSIS Teacher selects examples focusing on most common errors. 1. Students solve the example correctly 2. Students discuss with partner 3. Students also explain the error that had been made This can be done individually or in pairs (This had been used successfully in a collaborative ‘Basic Algebra’ class. )

CHANGE the QUESTION How should I write my answer? I S L A IM C E D e. r o t If I us a l u lc a c y m can use at work? Will th WHAT makes SENSE? Shoul d. I ESTIM ATE ? ? ? Sh wo ould FR rk AC wit I TI h ON S

HOW should I write my answer? As an IMPROPER FRACTION? ( 13/2 ) A MIXED NUMBER? ( 5 -1/2 ) A DECIMAL NUMBER? (4. 32 ) Should I ROUND UP? (4. 8 is about 5) Should I ROUND DOWN? (4. 8 can round down to 4) Should I use π on my calculator or 3. 14 ? and WHY ? Presenter to distribute HANDOUTS TO USE NOW

If you have, DISTRIBUTE Sleeves and Markers or Individual White Boards SEE EACH ONE THINKING

ORIGINAL NEW QUESTION same example 1 Add 3/8 and 4/8 Is the sum of these two fractions closer to ½ or closer to 1 whole? Explain or show you know. 2 4 x 2/3 = Estimate your answer. The product is (a) > 4 b) < 4 (c) < 1 (d) < 2 3 Find the sum 5/12 6/12 7/12 REWRITE this example using only two fractions so you will get the same answer when they are added. 4 What is the total? How can you tell by just looking at this example that 5 and 5/7, and your answer will be greater than 10? EXPLAIN in 4 and 6/7 WORDS. Change the Question not the Textbook See #2 above: NEW grade-5: Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; … (5. NF. 5. b)

ORIGINAL NEW QUESTION same example 5 2/3 + ¾ = ? A. 5/7 B. 17/24 C. 5/12 Choices ‘A’ and ‘B’ are incorrect. What do you think someone might have thought if they selected one of these choices? 6 0. 25, 50% and 2/3 Create two different examples using these numbers. Solve them. 7 5/12 + 6/12 + 7/12 What could you add to these three fractions so the sum would be A) A mixed number? Add ___ new sum = _____ B) A whole number? Add ___ new sum = _____ WORK TOGETHER C) A negative integer? Add ___ new sum = ____ D) Zero? Add ____ new sum = ____ 8 1/20 1/10 1/5 and 1/8 * Mark says there is a very easy way to put these CREATE fractions in order without making common denominators. Explain. YOUR OWN Change the Question not the Textbook

WHICH 2 are the easiest and which 2 are the most difficult to solve? Why? 1 Solve for n: 2 n – 4 = 12 6 Solve for x: 3 x = 22 + 2 2 Solve for x: 10 = 3 x + 4 7 Solve for w: √ 25 = w 3 Solve for a: 1/3 a = 10 8 Solve for a: 2900 = 100 a 4 Solve for x: 2 (x +1) = 12 9 Evaluate if x = 4: x 2 + x – 16 5 Solve for w: 3 (2 - w) = 12 10 Evaluate if y = 3: 9 – y + y 3 Change the Homework Question

WHICH 2 are the easiest and which 2 are the most difficult to solve? Why? 1 Solve for n: n=8 2 n – 4 = 12 6 Solve for x: 3 x = 22 + 2 x=2 2 Solve for x: 2=x 10 = 3 x + 4 7 Solve for w: √ 25 = w 5=w 3 Solve for a: a = 30 1/3 a = 10 8 Solve for a: 2900 = 100 x x=29 4 Solve for x: x=5 2 (x +1) = 12 9 Evaluate if x = 4: x 2 + x – 16 4 5 Solve for w: w=6 3 (2 - w) = 12 10 Evaluate if y = 3: 9 – y + y 3 33 Change the Homework Question

(grades 4 -5) Make 2 groups: WHICH are the easiest and which are the most difficult to solve? Why? 1 Find the sum: 15. 2 + 12. 7 6 Which is larger? 2/3 or 7/8 2 Find the sum: 12. 8 + 6. 29 7 Which is larger? 5/9 or 7/8? 3 Find difference: 16. 04 – 1. 8 8 Which is closer to ½? 3/8 or 6/10 4 Find difference: 16. 98 – 6. 25 9 How much time between 2: 05 pm and 8: 20 pm? 5 Which is larger? 6/9 or 2/3? 10 How much time between 10: 05 am and 5: 20 pm? Change the Homework Question

(grades 4 -5 ) Possible Answers Easier More Difficult 1 15. 2 (easy to line-up) 12. 7 27. 9 (no regrouping) 2 12. 8 6. 29 (line up decimals) 19. 09 (regroup) 4 16. 98 – 6. 25 10. 73 (no regrouping) 3 16. 04 -1. 08 (need to regroup twice) 14. 96 5 Which is larger? 6/9 or 2/3? (the same 6/9 reduces to 2/3) 7 larger? 5/9 or 7/8? Ans. 7/8 Need common denom. to compare 5/9=40/72 7/8=63/72 6 Which is larger? 2/3 or 7/8? 7/8. It is only 1/8 away from 1; 2/3 is 1/3 away; 1/3 is larger than 1/8 8 Which closer to ½? 3/8 or 6/10 9 8: 20 pm -2: 05 pm is 6 hours 15 min (no regrouping) 10 time between 10: 05 am & 5: 20 pm 10 am to 12 pm = 2, 12 pm to 5 pm = 5 2 hr+5 hr=7 hr (add am + pm hrs) 20 -5=15 min. Answer: 7 hr. 15 min

1 Separate these 10 examples into 2 groups and explain why you separated them that way. Solve for n: 2 n – 4 = -12 6 Solve for x: 3 x = 22 + 2 2 Solve for x: 10 = 3 x + 4 7 Solve for w: √ 25 = w 3 8 4 Solve for a: 1/3 a = 10 Solve for x: 2 (x +1)= -17 9 Solve for a: 2900 = 100 x Evaluate if x=4: x 2 + x – 16 5 Solve for w: 3(2 – w)=12 10 Evaluate if y = 3: 9 – y + y 3 Change the Homework Question Easy Content, New Process

Separate these 10 examples into 2 groups and explain why you separated them that way. 1 Solve for n: n = -8 2 n – 4 = -12 6 Solve for x: 3 x = 22 + 2 x=2 2 Solve for x: 2= x 10 = 3 x + 4 7 Solve for w: √ 25 = w + 32 -4 = w 3 Solve for a: a = 30 1/3 a = 10 8 Solve for a: 2900 = 100 x x =29 4 Solve for x: 2 (x +1) = -17 x = -19/2 = -9. 5 9 Evaluate if x = 4: x 2 + x – 16 4 5 Solve for w: w=6 10 Evaluate if y = 3: 9 – y + y 3 33 3 (2 - w) = 12 Change the Homework Question Easy Content, New Process

As The Crow Flies Activity CCSS takes what seems so usual, just a bit further. Your house School Friend’s House

As The Crow Flies Activity CCSS takes what seems so usual, just a bit further. Your house School Friend’s House

As The Crow Flies Activity CCSS takes what seems so usual, just a bit further. Your house School Friend’s House

U. S. A. MAP MAKE THE TASK Individually meaningful. Add grid lines • • • PLOT TRIP PLOT PLANE FLIGHT DETERMINE SLOPE ESTIMATE DISTANCE ESTIMATE AREAS What questions could you ask?

LAB STATIONS for any grade or course 1. What are they? 2. Why have them? Think of lab stations appropriate for one of your classes (topic, standards, format, levels … tools, manipulatives, technology, … skills, applications, . . . individual or pairs. . . ) K. I. S. S. for this activity!

LAB STATIONS grades 3 -6 TOPIC: ESTIMATION Forms for selected MS attendees Station # • 8 - Coiled rope: estimate length • 2 - Overflow the spoon • 3 - Toothpick count • 4 - Jellybean mix-up • 5 - Unifix cubes

LAB STATIONS (any grade or subject) Work alone, in pairs, or teams of three Each student submits own lab report • • • • Discover Create Experiment Compare/contrast Solve (skill based) Apply to real-life situation Apply to self Relate to another subject, another genre Use manipulatives/tools Build, construct Write, explain, critique Graph, draw, model, photo, speak, record, …. Survey, simulations Online: research, interactivity, self-assess online, communicate Teach others, compete, collaborate DIF FE REN TAS TIAT E KS

Increase Rigor • Take the usual … this time see if they really understand. • Use manipulatives (algebra-tiles, cardboard shapes or even online virtual manipulatives) National library of Virtual Manipulatives • CCSS takes what seems to be usual, just a bit further. http: //nlvm. usu. edu/en/nav/vlibrary. html

Mini-Projects • • Give them choices (4 -MATT example) Online Research Work in pairs Collect data Organize data Analyze data Come to conclusions Explain conclusions

STUDENTS CHOOSE DRAW …. WRITE …. DISCOVER …. BUILD …. GRAPH …. MEASURE? . . .

Share the Wealth: COLLABORATE across SUBJECTS R–A–F–T

*ACTIVITY Sample RAFT Strips blank form in folder

EXAMPLE

Ideas for Cubing in Math…

CUBING FRACTIONS TWO NUMBERS: - Coordinates of a point - Measurement of an angle - Point of intersection

CUBING (general)

PORTFOLIO - Congratulations: Dear Assistant, Printing day is Tuesday and I need your help! The Arts section of our local paper is completing a page on all subjects covered in 9 th grade. I need you to cover the math piece for me … Write a poem, song, rap, or story about order of operations. Just follow the attached rubric and you should be fine. Be creative and watch your spelling, but have fun! Human Growth and Development: Teachers know and understand how student learning is influenced by individual experiences, talents and prior learning …. Portfolio: A Creative Piece Special Needs: Teachers engage in activities to: Apply knowledge of students’ abilities, … talents … to positively impact student learning.

DESIGN a SKI SLOPE A WEBQUEST for grade 6 -8 students o Introduction o The Task o The Process o Evaluation o Conclusions Handout hardcopy sample

Ski Indoors o Notes to the teacher: (2 -3 week project) o Benchmark lessons (whole-class, concept-based skills, such as: • 1. concept of slope • 2. concept of relationship among numbers (rise/run; 3 sides of a right triangle). o Assignments – individual for content mastery o Mini-Lessons – small group, short lessons on a specific skills such as: …. o How-To-Sheets – step-by-step directions for skill development o Web Resources o Teacher Notes WEB QUEST! IDE portal sample

RUN FOREST RUN • http: //www. idecorp. com/? page_id=117 See video: Learner-Active, Technology. Infused Classrooms • A WEBQUEST for ALGEBRA students o Introduction o The Task o The Process o Evaluation o Conclusion

Where do I find similar resources? Webquests: Quest. Garden http: //questgarden. com [Register for a FREE 30 -day TRIAL] Kathy Schrock’s Guide to Everything http: //www. schrockguide. net [Search] for [Rubrics]

How to Make Sure a Butterfly Doesn’t Fly

How do you get a butterfly? First, there is the egg which hatches into a caterpillar. The caterpillar eats and grows. At the right time, it makes a cocoon out of its own body. While in the cocoon, the caterpillar changes into a butterfly.

When the butterfly is ready, it starts to break through the cocoon. First a hole appears. Then the butterfly struggles to come out through the hole. This can take a few hours. If you try to “help” the butterfly by cutting the cocoon, the butterfly will come out easily but it will never fly. Your “help” has destroyed the butterfly.

The butterfly can fly because it has to struggle to come out. The ‘pushing’ forces lots of enzymes from the body to the wing tips. This strengthens the muscles, and reduces the body weight. In this way, the butterfly will be able to fly the moment it comes out of the cocoon. Otherwise it will simply fall to the ground, crawl around with a swollen body and shrunken wings, and soon die.

If the butterfly is not left to struggle to come out of the cocoon, it will never fly. We can learn an important lesson from the butterfly. If we do not have struggles and challenges in our work, we will never grow strong and capable. If life has no difficulties, we will become weak and helpless. -- Lim Siong Guan, Former Secretary, Singapore’s Ministry of Education

RESOURCES REFLECT, DECIDE, PLAN FOLDER • Differentiate Instruction: “Questions that Encourage Thinking and Increase Student Understanding” • Differentiated Instruction: Checklist _ Do I ?

www. NCTM. ORG Classroom Resources (activity, instructions, explorations) • Adjustable spinner w/interactivities (K-12) http: //www. nctm. org/Classroom-Resources/Interactives/Adjustable-Spinner/ • Geometric Solids (solid, faces, edges, vertices) opens to net/closes back to solid http: //www. nctm. org/Classroom. Resources/Interactives/Geometric-Solids/ • Trigonometric Graphing http: //www. nctm. org/Classroom. Resources/Interactives/Trigonometric-Graphing/

High School http: //www. nctm. org/Classroom-Resources/Browse. All/? ps=20&cp=2&tx=2681 Find one to use next week! • Movie Lines: Apply knowledge of linear equations and graphs in an authentic situation. • Sickle Cell Anemia Inheritance Explore the concept of genetics and inheritance using probability. • Graphs from the Unit Circle Make the connection between trigonometric rations and graphs of sine and cosine functions. • Sample Interactive Tasks (from NAEP) Scroll down to select grade 12. http: //www. nationsreportcard. gov/science_2009/ict_tasks. asp

Where do I find resources? Illuminations from NCTM website (http: //www. nctm. org) Dana Center Toolbox http: //www. ccsstoolbox. org - Resources for Implementation - PARCC Prototype Project Engage. NY http: //www. engageny. org/subject/math PARCC http: //parcc. pearson. com/practice-tests/math/ Achieve the Core http: //achievethecore. org Regents Prep www. regentsprep. org Review, Practice worksheets, Quiz

Exit Thoughts 1. ___ A D Students work in pairs: Students give Carousel cards posted about room. specific verbal directions; use correct math terminology A 2. ____ Students work with different levels of the same standard; students teach other B 21 st Century: Do one step, move right or pass right F 3. ____ Student pairs practice vocabulary words many times over C Tic-Tac-Toe 4. ____ E Students work in pairs; …. Correct errors … Teach other … create examples D Battle Ship Game 5. ____ B Students focus on process and the way they “show work” and “see each other thinking” without talking E Match Your Partner’s Answer F The Ladder Game

“Questions to Engage Students’ Thinking” 4 -page packet “Check Off List” To see what you already do; and might do

What/How will you use? • • • • Which one doesn’t belong and why? 21 st C. move right; pass right Circle Fold Match Your Partner’s Answer (create own) – pairs Match Cards (2 -3 student teams) Carousel (different levels)-whole class/indiv. /teams Battleship (2 -3 students) Vocabulary Ladder (student pairs) RAFT model Thayer organization model What will you use next Cubing week? Share with a Webquest colleague? Change the Question How should you write your answer?

JUDITH T. BRENDEL, ED. M. • edleaderk 12@hushmail. com • jbrendel 2112@gmail. com KEEP IN TOUCH