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Fall 2013
Putting the Mathematical
Practices Into Action
Welcome“Who’s in the Room”
Norms
• Listen as an Ally
• Value Differences
• Maintain Professionalism
• Participate Actively
http://maccss.ncdpi.wikispaces.net/Home
Materials Found on Mathematics Wiki
Session Outcome
• Understand the Standards for Mathematical Practice
• Explore strategies for implementing the Standards effectively
What does it mean to be Mathematically Proficient?
Turn and Talk
• Are students who can remember formulas or memorize algorithms truly mathematically proficient, or are there other skills that are necessary?
• Is the correct answer the ultimate goal of mathematics, or do we expect a greater level of competence?
For the first time, mathematical processes are elevated to essential expectations, changing our view of math to encompass more than just content.
The goal now is to apply, communicate, make connections, and reason about math content rather than simply compute.
What areThe Standards for
Mathematical Practice?
What has been your biggest challenge with the implementation of The Standards for
Mathematical Practice?
Overarching habits of mind of a productive Mathematical Thinker
Exploring Standard 1
Make Sense of Problems and Persevere in Solving them.
Understanding the Standard
• What do we do each day in our classroom to build mathematical thinkers?
• What do we do to keep our students actively engaged in solving problems?
• How do we help our students develop positive attitudes and demonstrate perseverance during problem solving?
How Do We Get There?Brainstorming Strategies…
1. Choose an Operation
2. Draw a Picture
3. Find a Pattern
4. Make a Table
5. Guess and Check
6. Make an Organized List
7. Use Logical Reasoning
8. Work Backward Page 11
The Holiday Tree
The Partin family counted the different types of ornaments on the town’s holiday tree. Here is the list of what they saw.
stars – 24gingerbread men – 14snowflakes – 12reindeer – 18candy canes – 6
Six of the reindeer had red noses. What fraction of the reindeer had red noses? Tell how you would get the answer.
Page 14
The Holiday Tree
The Partin family counted the different types of ornaments on the town’s holiday tree. Here is the list of what they saw.
stars – 24gingerbread men – 14snowflakes – 12reindeer – 18candy canes – 6
What fraction of the ornaments were snowflakes?
What fraction of the ornaments were edible?
If 6 of the stars were silver, what fractions of the stars were not silver?
Page 14
What questions could you ask?
Look on page 19 for some additional suggestions.
What questions could you ask?
Shipley AquariumAdmission Cost
Adults - $8.00Children (ages 3 and over) - $6.50Children (ages 2 and under) – Free
Look on page 19 for some additional suggestions.
Traditional Problems vs. Rich Problems• We can ask questions that stifle learning by
prompting a quick number response.
– What is the answer to number 3 on your worksheet?
– What is 5 x 4?
• We can ask questions that promote discussion, thinking, and perseverance.
Sort the math questions.
Check your arrangement on page 22
page 26
Reflecting on strengthening student problem solving experience…1. Do I routinely provide opportunity for my students
to share their solutions and processes with partners, groups, and the whole class?
2. Do I show my students that I value process (how they did it) rather than simply the correct answer?
3. Do I pose problems that require perseverance? Do I use thoughtful questions to guide and encourage students as they struggle with problems?
Exploring Standard 2
Reason abstractly and quantitatively.
Understanding the Standard
• What can we do in our classrooms each day to help students build a strong understanding of numbers (quantities)?
• How do we help students convert problems to abstract representations?
• What can we do to help students understand what numbers stand for in a given situation?
How Do We Get There?
• Number Webs
• Headline Stories– Post It– Book It– Reverse It– Match It– Question It
• Pinch Cards
Let’s Create a Number Web!
Select a number.
Number Web
Number Webs encourage flexibility with numbers. page 34
Headline Stories
• Headlines sum up a story.
• Equations are like newspaper headlines—short and to the point.
• Equations are connected to word problems the same way a headline is connected to a news story.
• Using headlines can help you see students understandings and misunderstandings.
Headline Stories
page 36
Headline Stories
page 36
Headline Stories
Headline: 52÷4 =
Headline stories can be as easy or as difficult as you make them!
Students might be asked to write problems about equations that include fractions, decimals, percents, or variables.
Let’s look at some variations found on page 39.
Pinch Cards
Pinch cards are an all-pupil response technique.
There were 6 soccer teams in the league and 12 players on each team. How many players were in the league?
The 4 members of the High Rollers Bowling Team scored 120, 136, 128, and 162. What was the team’s mean score?
page 41
Avoiding Key Words
• Key words are misleading.
• Many problems have no key words.
• The key word strategy sends a terribly wrong message about doing mathematics.
A sense making strategy will always work.
Van de Walle & Lovin (2006)
Read and Discuss page 33
• What is contextualization and decontextualization?
• Why is it important?
• Discuss at your table.
Contextualize and Decontextualize
120 students and 5 chaperones went on the field trip. Each bus held 35 people. How many buses were needed?
Decontextualize: consider the data, the action of the problem, and create an equation to represent the problem in an abstract way
Contextualize: refer back to the context of the problem to determine if the answer makes sense
• Understanding the units and quantities within a problem is an important factor in making sense of the numbers within the problem.
• Labeling answers forces students to refer back to the context of the problem.
page 40
Exploring Standard 3
Construct viable arguments and critique the reasoning
of others.
Understanding the Standard
• What do we do in the classroom to get students to justify their answer and defend their process for finding the answer?
• How do we help students understand math skills and concepts so they can construct viable arguments?
• How do we help students consider and judge the reasonableness of other answers and strategies?
How Do We Get There?
• Eliminate It
• Agree or Disagree?
• My 2 Cents
Eliminate It!
• As a group, decide on the concept that should be eliminated with reasoning or math data to back up your decision.
• There may be more than one way to eliminate an item!
• Create your own.
Eliminate It
page 50
Agree or Disagree?
• 75% is more than 2/3.
• Tell why you agree or disagree.
Agree or Disagree?
Agree or Disagree?
Agree or Disagree?
• Jim has 12 pencils and Annie has 8. Jim has more than Annie.
• 7 + 3 and 4 + 6 are the only ways to make 10.
• 9 is an even number.
• 6 tens and 3 ones is the same as 5 tens and 13 ones.
• 3 jars of peanut butter for $7.50 is a better deal than 4 jars of peanut butter for $10.20.
page 53
My 2 Cents
Constructing Arguments
• Read page 44-46
• What is the difference between an assertion and an argument?
• Be prepared to share your thinking.
Assertion vs. Argument
• Assertion: a statement of what students want us to believe without support or reasoning. – The answer is correct “because it is,” “because I know
it,” or “because I followed the steps.”
• Argument: a statement that is backed up with facts, data, or mathematical reasons
• Constructing viable arguments is not possible for students who lack an understanding of math skills and concepts.
Page 58
Exploring Standard 4
Model with mathematics.
Understanding the Standard
• As teachers, we model with mathematics routinely in our classrooms. Should students be able to model? Why?
• How do students modeling mathematics look?
• How does student modeling of mathematics affect instruction?
How Do We Get There?
• Model It
• Part-Part-Whole mats (addition & subtraction)
• Bar Diagrams (multiplication & division)
• Bar Diagrams (solving equations)
How would you model…
• 123 + 57
• 1 – 1 3
• 3.4 + 5.07
Part-Part-Whole Mat
• There were 2 yellow lollipops and 3 red lollipops. How many lollipops were there?
Part-Part-Whole Mat
• There were 6 children. 3 were boys. How many were girls?
Part-Part-Whole Mat
• There were 5 cupcakes. Jan ate some. There were 2 left. How many did she eat?
Bar Diagram
• There are 3 boxes with 6 toys in each box. How many toys are there?
6 6 6
Bar Diagram
• 18 toys are packed equally into 3 boxes. How many toys are in each box?
18
Bar Diagram
• 18 toys are packed 6 to a box. How many boxes are needed?
18
Bar Diagram
• 18 toys are packed 6 to a box. How many boxes are needed?
18
6 6 6
Page 74
Assessment Tips
• Tell me what your model represents.
• Why did you choose this model?
• Did creating a model help you any way? If so, how?
• Did you get any insights by looking at your model?
• Is there another way you might model this problem or idea? How?
Page 75
Exploring Standard 5
Use appropriate tools strategically.
Understanding the Standard
• What are tools used by our students?
• Why is it important to use tools?
• Tools enhance our students’ mathematical power by assisting them as they perform tasks.
• The ability to select appropriate tools is an important reasoning skill.
Which tool is more efficient?
There is often more than one tool that will work for a task, but some tools are more efficient than others.
Paper & PencilMental MathCalculator
Solve using your assigned tool!1. 5 x 6
2. 23 x 15
3. Estimate the cost of 2 pies @ $3.75 each Cereal @ $3.20 each Milk @ $1.79 gal Bananas @ 59 cents/lb
4. 236 x 0 x 341
5. What comes next 3, 7, 15, 31, ___
6. A local TV store had a sale on TV’s. They sold 7 for $1,699.95 each. They made a profit of $169.00 on each TV. What did the store pay for the 7 TVs?
A. $1,183.00 C. $13,082.65B. $10,716.65 D. $11,899.65
How Do We Get There?
• Use tools appropriately
• Number Lines (It’s Close to…)
• Rulers (broken ruler, magnified inch)
• Mental Math– Number Partners– In My Head?
• Students benefit from opportunity to select a tool that makes sense for the math task and to evaluate which tool is most efficient for that task.
• Not only do our students need to be able to select appropriate tools, they must be able to effectively use those tools. (page 80-81)
Number Lines
Rulers
Page 84
Folding Paper
• Fold the strip in half. Open it, and mark ½ at the center fold.
• Refold the strip in half and fold it in half again. Label 1/4, 2/4, 3/4 on the three folds.
Questions
• Are the sections equal in size?
• Do the fraction labels make sense? Why?
• Where is 0? Why?
• Where is 1? Why?
• Why are 1/2 and 2/4 on the same fold?
Folding Paper
• Refold the paper and then fold it in half one more time.
• Open the paper, place a mark on each fold and indicate what each of the new marks represent.
Folding Paper
• Why is there more than one fraction on some folds?
• Does it make sense that those fractions are on the same fold? Why?
• Which of those fractions is easiest to understand? Would you say 1/2 or 2/4 or 4/8? Why?
Number Partners (Mental Math)
• Find a Number Partner that Makes 10
5 4 9 3 6 1 7 2 5 8
Page 86
What are some modifications for this task?
In My Head? (Mental Math)
Do I use paper & pencil or do it in my head?
–734 x 82
–63 x 4
–1/4 + 2/8
–930 ÷ 3Page 86-87
Students need to identify tools that increase their efficiency with math tasks.
Page 88-89
Exploring Standard 6
Attend to precision.
Understanding the Standard
• Why is precision important in mathematics?
• What does it mean to be precise?
• What can we do in the classroom to promote precise communication in mathematics?
How Do We Get There?
• Estimate and Exact
• Vocabulary
– Word Webs
– Word Walls
– Sort and Label
– Mystery Words
– Translate the Symbol
– Word Boxes
Estimate and Exact
• Buying bags of candy to put in party treat bags
• Measuring the dimensions of the doorway to install a screen door
• Buying pizzas for a class party
• Buying carpeting for a living room floor
Estimation Skills
• Will the sum of 8 + 7 be greater than or less than 20? Why?
• Is the difference of 81 and 29 closer to 40, 50, or 60? Why?
• Is the sum of 1/3 + 4/8 greater than or less than 2? Why?
• How would you estimate the product of 2.4 and 63? Will the product be between 2 x 60 and 3 x 60? Why or why not?
Word Webs
Select a word or phrase.
Sort and Label
Sort and Label
• sum, minus, join, compare, subtract, add, take apart, plus
• pint, foot, measuring cup, ounce, inch, scale, yard, pound, ruler
• square, trapezoid, hexagon, rectangle, rhombus, triangle, pentagon
• expression, equation, addition, operation, inequality, comparison, variable, division
Mystery Words
area perimeter
volume length
Translate the Symbol
• 4 dollars and 10 cents is greater than 4 dollars and 5 cents
• One-fourth of 16 is 4
• Doubling a number then adding six more
Translate the Symbol
• 12 = 7 + 5
• 4 + x = 6
• 3 x 4 > 2 x 5
Which Is More Challenging?
• 4 dollars and 10 cents is greater than 4 dollars and 5 cents
• One-fourth of 16 is 4
• Doubling a number then adding six more
• 12 = 7 + 5
• 4 + x = 6
• 3 x 4 > 2 x 5
Word Boxes
word
Definition
Real life example
Picture
Other words
Page 102-103
Exploring Standard 7
Look for and make use of structure.
Understanding the Standard
• How do we help students discover patterns in the number system?
• What can we do to help students make sense of mathematics through the use of structure?
Can you see the pattern?
1/2 = .50 1/3 = .33 1/5 = .20
1/4 = .25 1/6 = .167 1/10 = .10
1/8 = .125 1/12 = .083 1/20 = .05
1/16 = .0625 1/24 = .0467 1/40 = .025
page 107
Properties, Patterns, and Functions
There were 10 children at the party. How many were boys and how many were girls?
Boys Girls
0 10
1 9
2 8
3 7
4 6
5 5
6 4
7 3
8 2
9 1
10 0
Perimeter Patterns
If there was a row of 50 connected equilateral triangles, what would the perimeter measure?
Number Flexibility
There was 1 ½ cupcakes left on the plate and Liam and Molly decided they would eat them. How much might each person have eaten? Be ready to justify your answers.
How Do We Get There?
• Pattern Cover Up
• Pattern in the Hundreds Chart or Multiplication Chart
• Ratio Tables
• Number Lines
Pattern Cover Up
2 5 9 14 20
Hundreds Chart
On a hundred chart, can students explain the vertical and horizontal pattern? Do they see diagonal patterns?
Multiplication Chart
Can students explain vertical and horizontal patterns? Do they see diagonal patterns? Can they find patterns that explore equivalent fractions or proportions?
Ratio Table
Margo was painting flowers on the classroom mural. Every flower had 3 leaves. How many leaves were on the mural after 7 flowers had been painted.
Number of flowers 1 2 3 4 5 6 7
Number of leaves 3 6 9 12 15 18 21
Ratio Table
• Each chicken has two legs. How many legs are on 4 chickens?
• 2 jars of peanut butter cost $4.50. How much do 8 jars cost?
• To make one dozen ice-cream sandwiches, Katie used ¾ gallon of ice cream. How much ice cream did she need for 60 ice cream sandwiches?
Number Line
page 117-118
Exploring Standard 8
Look for and express regularity in repeated
reasoning.
Understanding the Standard
• Why is it important for students to recognize repetition and reason why it is happening?
• When you think about repeated reasoning, what do you think about?
How Do We Get There?
• Exploring Repetition
• Finding Shortcuts
• Organizing and Displaying Data to Discover Rules
• Classroom Investigations
Exploring Repetition
2 + 1 = 3
5 + 1 = 6
3 + 1 = 4
6 + 1 = 7
4 + 1 = 5
What do you notice?What conclusion can you draw?
Exploring Repetition
Investigations to Find Shortcuts
• Read first paragraph under heading Investigations to Find Shortcuts - page 124
• Turn and Talk – about your experience with “Memorization”
Investigations to Find Shortcuts
• What do you notice?• Do you see any pattern?• Did you notice anything
interesting about the solution?
Organizing and Displaying Data to Discover Rules
Alice jumps rope faster then anyone in her class. She can jump 8 times in 4 seconds. How long will it take her to jump 40 times? Justify your answer.
Page 127
Orchestrating Classroom Investigations to Discover Shortcuts1. Opportunities for all students to gather data
with partners or teams.
2. Creating compilations of class data.
3. Observing compiled data and discussing insights.
Page 128
Page 130
Our “To Do” List • Introduce all teachers to the practice standards.
• Provide examples to illustrate the standards.
• Encourage ongoing reflection about the standards.
– Professional development
– Faculty meetings
– Book study or PLCs
– Grade level teams
– Math coaches
Let’s Do Some Math!
Practice the Mathematical Practices
Three different veterinarians each help a total of 63 dogs and cats in a week, but each veterinarian helps a different number of dogs and cats. How many dogs and cats could each veterinarian have helped?
Use tangrams to create the Cat…
What is the fractional value of your cat’s tail?
How can you find the fractional value of each piece?
What questions do you have?
Updates
• NCCTM Conference: Oct 31-Nov 1
• K-2 Mid-Year Assessments
• 3-5 Formative Tasks
• Vocabulary Document
DPI Mathematics Section
Kitty RutherfordElementary Mathematics [email protected]
Denise SchulzElementary Mathematics [email protected]
Johannah MaynorSecondary Mathematics [email protected]
VacantSecondary Mathematics Consultant919-807-3934
Jennifer CurtisK – 12 Mathematics Section [email protected]
Susan HartMathematics Program [email protected]
