Common Core State Standards

Mathematical Practices

Jeremy Centeno

You would never hear someone say:

I am not very good at reading.

I can’t read.

You do hear people say:

I am not good at math.I can’t do math.

When I was a kid math was my worst subject

What is the difference?

The difference between the USA and other higher performing nations is that a culture of learning math is established from the beginning of a students career in school.

Students are informed and taught everyone can do math.

What are the following?

• Cryptanalyst $137,780/yr• Computational Biologist $150,000/yr• Mathematical Physicist $166,400/yr• Actuary $160,000/yr

• The way we taught students in the past simply does not prepare them for the higher demands of college and careers today and in the future. Your school and schools throughout the country are working to improve teaching and learning to ensure that all children will graduate high school with the skills they need to be successful. In mathematics, this means three major changes.

• Teachers will concentrate on teaching a more focused set of major math concepts and skills. This will allow students time to master key math concepts and skills in a more organized way throughout the year and from one grade to the next. It will also call for teachers to use rich and challenging math content and to engage students in solving real-world problems in order to inspire greater interest in mathematics.

Class Building/Corners

• Look at the posters in the room• Pick which is your favorite math concept to teach:– Operations: Addition/Subtraction/Multiplication/Division– Place Value: Skip counting/Base Ten/ Greater Than/ Less

Than/ Equal To– Measurement– GeometryThink about the following Question:– Why is this concept your favorite concept to teach?Timed Rally Robin

Corners Continued

• Pick your corner by least favorite concept to teachThink about the following Question:

Why do you feel this concept is your least favorite concept to teach?

Timed Rally Robin:

Team Building/4 Corners Name Tag

• Take a Piece of notebook paper and fold it so that it stands on its own

• Write Your Name in the Center– In the Upper Left hand corner draw a shape

that represents you– In the Upper Right hand corner pick an

operation that represents you– In the Lower Left Corner pick your favorite

temperature– In the Lower Right Hand corner write your

favorite number

Team Interview• Person Number 1 will ask person number 2 to

answer upper left hand corner out loud for the group

• Person Number 2 will be standing and Answer Question

• Person Number 3 will make a connection • Person Number 4 will praise• Person 1 turns Mat and Person Number 2

becomes person number 1 and repeats the same process (Continue until I say stop)

Reflection/Talking Chips

• Think about the activities for class building and team building

• How would you use these activities with students in a math class? Why is it necessary to team build and class build in order to have engaging activities in your classroom?

• When the signal word is given anyone can speak but they must put a chip in as they talk.

• All chips must be down on the table if there is time pick up chips and continue conversation

Mathematical Practice #1Problems and Perseverance

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. … —CCSS

CCSS Mathematical Practice #1

• I can try many times to understand and solve a math problem.

Key Points

• Life Happens (Not just a Chapter in a Book)

• Puzzler’s Disposition (A good puzzle creates a drive to succeed)

• Toolkit (Need to solve)• Hard problems broken into simpler

problems

CCSS Practice #1

Practice: #1 Make Sense of problems and perserver in solving them

Three Main Points:Make a PlanSelf Monitor and ExplainDemonstrate understanding by corresponding

Example: Student Friendly Definition:I can make a plan, explain my answers, and show how I did it; then I can try another way

My Number Bond Name TagJeremy Centeno 6 73 3 73 103 + 10 =

3 + 10 = 13

Choose Three Ways

• Each person will get a math problem• On your graphic organizer you will chose three

ways to solve your problem and show how you solved it

• You will work on your own first• When you hear the signal word begin

Numbered Head Together

• Person number one reads their problem• Everyone Answers their problem without

looking at each others board• As each person gets done stand up• When all people are standing discuss how

each of you solved the problem• Agree and praise/ Or Coach and Praise• Repeat with person number 2

Video

• https://www.teachingchannel.org/videos/problem-solving-math?fd=1

• http://www.insidemathematics.org/index.php/classroom-video-visits/public-lessons-word-problem-clues/438-word-problem-clues-lesson-5

Mathematical Practice #2Reason Abstractly and

QuantitativelyMathematically proficient students make sense of

quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. —CCSS

CCSS Mathematical Practice #2

• I can think about the math problem in my head first

Practice: #2: Reason Abstractly and Quantitatively

Three Main Points:Makes sense of quantities and their relationship to the problemBring complementary abilities togetherUse reasoning that entails creating a coherent representation

Example: Student Friendly Definition:I can think about the math problem in my head first

CCSS Practice #2

Math StringsMental Math

• The number of fingers on two human hands• Subtract the number of toes on one human foot• Multiply it by the number doughnuts in a half

dozen• Divide by the number of eyes on a human face• Add to it the number of hearts in a human body• The answer is? 16

Math Task Cards/Word Problem Activities

• Taking a look at Task Cards• Taking a Look at Activities

Decontextualizing/Recontextualizing

• How many buses are needed for 99 children if each bus fits 44 students?

• Find the Answer• How did I come about my answer?

(recontexualize)

Video

• https://www.teachingchannel.org/videos/third-grade-mental-math

• http://insidemathematics.org/index.php/classroom-video-visits/public-lessons-proportions-a-ratios/199-proportions-a-ratios-problem-1

Mathematical Practice #3Construct Arguments and Critique

ReasoningMathematically proficient students … justify their

conclusions, communicate them to others, and respond to the arguments of others. They … distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. … Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. … 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. —CCSS

CCSS Mathematical Practice #3

• I can make a plan, called a strategy, to solve the problem and discuss other students’ strategies to make sense.

Key Points

• Children like to talk but explanation is difficult

• Focus on “Shared Context”• Interest Based• Student should not critique from desk• Key is to “Show as well as tell”• Give Depth not one process problems• Challenging problems worked on

together creates a natural pull to reason

Practice: #3: Construct viable arguments and critique the reasoning of others

Three Main Points:Understand and use stated assumptions and definitionsConstruct arguments using objects, drawings, and actionsListen, read, and critique to find out what makes sense

Example: Student Friendly Definition:I can make a plan.I can tell my partner how I did it and listen to how they did it too!Talk about it!

CCSS Practice #3

Find the Fiction

• My number is 100.1.I can be broken into 4 parts equally2.I represent a millennium3.My quantity in pennies is equal to a dollar

Find the Fiction

• On your board write the number of the statement that is fiction and write the word fiction next to that number (DO NOT SHOW ANYONE)

• Example: 4 Fiction • When you hear the signal word discuss with your

group one at a time your answer. Come to a consensus

• Answer: 2 is the Fiction • Praise: Expert Thinking

Partners take turns, One solving a problem while the other coaches.

100% Engagement! 0 % Engagement

Teacher Coach

Page 6.32

How Many Ways Task

• How many ways can I make 28 cents?• How many ways can I make 28 cents

without quarters?• How many ways can I make a 5 inch

tower with 1 inch cubes using 1 white cube and 4 blue cubes?

• Now try with 3 white and 2 blue cubes.

Consideration

• Young students should not developmentally be held accountable for critiquing another students argument.

• http://insidemathematics.org/index.php/classroom-video-visits/public-lessons-proportions-a-ratios/204-proportions-a-ratios-problem-3-part-c

Mathematical Practice #4Model Mathematics

• Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life…. In early grades, this might be as simple as writing an addition equation to describe a situation.… Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation…. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs…. They…reflect on whether the results make sense…. —CCSS

CCSS Mathematical Practice #4

• I can use math symbols and numbers to solve the problem.

Practice: #4 Model with mathematics

Three Main Points:Reflect on whether the results make senseApply the math they know to solve problem in everyday lifeMap relationships using tools

Example: Student Friendly Definition:I can use symbols and numbers to solve problems

CCSS Practice #4

Children are Curious about the World

• Use of real life interest• Play: Experiment, Tinker, and Push

Buttons to see what happens• Curiosity: Size, Shape, Fit, Quantity,

and Number• To catch a ball what does one need to

figure?• Math is not a collection of skills that

are demonstrated

What does this Array tell us?

Why Arrays?

Peter Penguins Clues

• As a team use your mats to see who is person 1,2,3, or 4

• Person 1: Pass out clue cards• Person 2: Colors in Answer Sheet when

done• Person 3: Checks all Answers• Person 4: Praises if correct• Turn board for next game

• http://www.insidemathematics.org/index.php/classroom-video-visits/public-lessons-proportions-a-ratios/202-proportions-a-ratios-problem-3-part-a

Mathematical Practice #5Use Tools Appropriately

• Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet…. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. —CCSS

CCSS Mathematical Practice #5

• I can use math tools, pictures, drawings, and objects to solve problems

Practice: #5 Use appropriate tools strategically

Three Main Points:Be familiar with and consider all available toolsUse Technology tools to deepen understandingIdentify and Use other math resources

Example: Student Friendly Definition:I can use all available tools and technology appropriately when solving math problems

CCSS Practice #5

Key Notes

• Make a list of standard tools in classrooms

• Pencil and Paper• Area Model of Multiplication• Many Choice Options for Students• Student made decisions on tools• Different problems to create choice of

tools• Proscribed or Prescribed tool activities

Number Lines

Blind Sequence• Person Number 1 distributes one card face down to each

person• Each person puts their initials on the back to identify their own

card• Each person looks at their card and describes it to the group• Once they have completed that keeping cards face down the

group will sequence the cards• Once the group feels they have properly sequenced they will

turn cards over• If correct praise if incorrect discuss how the cards should be

placed then praise• If a person forgets what they have they may look at their card

only

• http://www.insidemathematics.org/index.php/classroom-video-visits/public-lessons-numerical-patterning/223-numerical-patterning-problem-2

• Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently…. In the elementary grades, students give carefully formulated explanations to each other…. —CCSS

Mathematical Practice #6Attend to Precision

CCSS Mathematical Practice #6

• I can check to see if my strategy or calculations are correct.

Practice: #6 Attend to Precision

Three Main Points:Communicate precisely and use clear definitionsState the meaning of the symbolSpecify units of measure

Example: Student Friendly Definition:I can carefully explain to my partner how I came across my answer and why I think it is correct

CCSS Practice #6

Key Points

• Focus on:– Precision of Communication– Speech and Writing– Specify units in numerical answers,

graphs, and diagrams

Vocabulary

• Use in context• Children build their own definitions• Refine definitions• Make Precise• Holistic Imagery • Draw a triangle• Meticulous use of vocabulary and

symbols• Clarity of Communication

Draw a Triangle

Match Mine

• Assign roles of sender and reciever• Sender Creates the arrangement• Sender Directs Reciever• Partners Check• Praise and Plan• Switch Roles

• http://insidemathematics.org/index.php/classroom-video-visits/public-lessons-numerical-patterning/221-numerical-patterning-introduction-part-c

Mathematical Practice #7Look/Make Use of Structure

• Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property…. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects…. —CCSS

CCSS Mathematical Practice #7

• I can use what I already know about math to solve the problem.

Practice: #7 Look for and make use of structures

Three Main Points:Find a patternUse what you knowSolve the problem!!!

Example: Student Friendly Definition:I can use what I already know to solve a problem using patterns

CCSS Practice #7

Key Points• Children look for structure• Mathematics is a structured language– Children learn orally– Two Sheep plus three sheep makes five

sheep– Two hundred plus three hundred makes five

hundred– Two wugs plus three wugs makes five wugs

no matter what a wug might be– So two eighths plus three eighths makes five

eighths

Word ProblemTake a Deep Breath!

• Mr. Centeno had a fruit fly problem• On day 1 there were two fruit flies.• On day 2 there were four fruit flies.• On day 3 there were six fruit flies.• How many fruit flies would Mr. Centeno have

on day 5?• Answer: 10• What was the pattern? +2

Line Ups

• Students each receive a card• When the signal word is given students must line up

according to sequence• They may or may not talk during the line up• If they talk they must discuss why and where they

should go• Once done look at line up and determine if correct• Praise when done

• http://insidemathematics.org/index.php/classroom-video-visits/public-lesson-number-operations/179-multiplication-a-divison-problem-3-part-b

• Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal…. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. —CCSS

Mathematical Practice #8Repeated Reasoning

CCSS Mathematical Practice #8

• I can use a strategy I used to solve another math problem.

Practice: #8 Look for and express regularity in repeated reasoning

Three Main Points:Notice if calculations are repeated by looking for methodsAttend to detailOften evaluate results (check understanding)

Example: Student Friendly Definition:I can look for shortcuts, use steps to solve problems, and see if it makes sense

CCSS Practice #8

Key Points

• Students should draw conjectures• Look for patterns in results• Look for patterns in what is being

done

Repeated Pattern ActivityDirections. Multiply the middle number by

itself. Multiply the outer numbers to each other. Compare the products

5,6,73,4,56,7,8What conjecture can you come up with?What is 29x31 and why?What would the Algebraic formula look like?

Expressing Regularity

Start with 28 and Respond to me:+10 Repeat

Start with 28 and Respond to me:+9 Repeat

What conjecture can you make?Start with 100 and Respond to me:

+10+99

Quiz Quiz Trade• Students stand up, hand up, and pair up with

a partner• Partner A quizzes partner B• Partner B answers• Partner A praises partner B if he or she

answers correctly/Coach if not• Switch roles• Trade Cards• Repeat

Video

• http://www.youtube.com/watch?v=kZxNldBEU6o

Resource

• http://www.k-5mathteachingresources.com/1st-grade-number-activities.html

Contact Me

Jeremy Centenocentej@bay.k12.fl.us

Website:https://sites.google.com/a/

bay.k12.fl.us/jeremy-centeno/