Transcript PUSD March Math Leader Workshop, video removed

Welcome Back!
March 3, 2015
Goals for Today
 Follow up and deepen Measurement Menu experiences from
last week
 Re-examine key components of Procedural Fluency and
Mastering Math Facts
 Explore timely site math leadership topics
**LUNCH IS 11:30-12:15**
Back to the Shoe Menu
Focus Question: When the perimeter stays the same and the
shape changes, what happens to the area?
Task:
Consider the area and perimeter of the sole of your shoe.
If you made a square that had the same perimeter as your sole,
how would the two areas compare? Write your predictions.
How can you find out?
Area and Perimeter
What did you discover?
Relevant Content Standards
 CCSS.MATH.CONTENT.3.MD.C.7.D
Recognize area as additive. Find areas of rectilinear figures by decomposing them into
non-overlapping rectangles and adding the areas of the non-overlapping parts, applying
this technique to solve real world problems.
 CCSS.MATH.CONTENT.3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including
finding the perimeter given the side lengths, finding an unknown side length, and
exhibiting rectangles with the same perimeter and different areas or with the same area
and different perimeters.
 CCSS.MATH.CONTENT.4.MD.A.3
Apply the area and perimeter formulas for rectangles in real world and mathematical
problems. For example, find the width of a rectangular room given the area of the flooring and the
length, by viewing the area formula as a multiplication equation with an unknown factor.
Your Half-Size Shoes
What is “half-size?”
Focus Practice: Construct viable arguments and critique the reasoning of others.
Partner Task:
How do the length, width, height, sole area, and volume of the original
shoe compare to the ½ size shoe?
1. Make some predictions.
I predict that the actual shoe’s _____ will be ______ the tape shoe.
I predict the _____ of the tape shoe will be ____ the actual shoe because _____.
2. Then find out. Prove your thinking.
Questions to Consider . . .
 What tools might you use?
 How might you organize your measurement data to help you look for and make use of
structure?
Measurement and Dimensions
Linear (one dimension: length)
Area
(two dimensions: length x width)
Volume
(three dimensions: length x width x height)
How many cubes would you need to make the next biggest cube?
Relevant Content Standards
CCSS.MATH.CONTENT.5.MD.C.3
Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.
CCSS.MATH.CONTENT.5.MD.C.5
Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
Analyze patterns and relationships.
CCSS.MATH.CONTENT.5.OA.B.3
Generate two numerical patterns using two given rules. Identify apparent relationships between
corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns,
and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting
number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and
observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally
why this is so.
CCSS.MATH.CONTENT.5.NF.B.5
Interpret multiplication as scaling (resizing), by:
CCSS.MATH.CONTENT.5.NF.B.5.A
Comparing the size of a product to the size of one factor on the basis of the size of the
other factor, without performing the indicated multiplication.
Performance Assessment
Grade 5 Standard:
CCSS.MATH.CONTENT.5.MD.C.5.B
Apply the formulas V = l × w × h and V = b × h for rectangular
prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and
mathematical problems.
How do they get to the formula?
Grade 3:
CCSS.MATH.CONTENT.3.MD.A.2
Measure and estimate liquid volumes and masses of objects using standard units of grams
(g), kilograms (kg), and liters (l).1 Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using
drawings (such as a beaker with a measurement scale) to represent the problem.2
Grade 4:
CCSS.MATH.CONTENT.4.MD.A.2
Use the four operations to solve word problems involving distances, intervals of time, liquid
volumes, masses of objects, and money, including problems involving simple fractions or
decimals, and problems that require expressing measurements given in a larger unit in terms
of a smaller unit. Represent measurement quantities using diagrams such as number
line diagrams that feature a measurement scale.
Performance Task: Shoe Box Design
Design a Shoe Box: (with a partner)
Design an efficient shoe box (right rectangular prism) for a pair of
your half size shoes (both shoes).
Develop an Argument for Your Shoe Box:
Include relevant measurements, formulas, diagrams and a
description of your solution process on a poster.
Assessing Student Learning
What might we learn about students from observing them and
analyzing their work during this performance task?
Break
Revisiting
Procedural Fluency
Session Goals
 Revisit the meaning of procedural fluency.
 Continue to examine ways to help students become
procedurally fluent.
Refresher:
Procedural Fluency refers to …
(National Research Council (2001), Adding It Up)
 knowledge of procedures,
 knowledge of when and how to use them
appropriately, and
 skill in performing them flexibly, accurately, and
efficiently.
You may remember . . .
Examining Common Errors
1)
86
+ 47
2)
101
- 63
3)
24
x 16
4)
1 2

2 3
How might horizontal versus vertical presentation of
problems affect students’ approaches?
86 + 47 =
86
+ 47
Common Core Progression of Procedural Fluency
Year 1: Concrete Modeling
Year 2: Strategies and Algorithms Based on Place Value
Year 3: Standard Algorithm
Example:
NBT Add within 1000 Progression
2nd: Add and subtract within 1000, using concrete models or
drawings and strategies based on place value, properties of
operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method.
3rd: Fluently add and subtract within 1000 using strategies and
algorithms based on place value, properties of operations,
and/or the relationship between addition and subtraction.
4th: Fluently add and subtract multi-digit whole numbers using
the standard algorithm.
What is the role of estimation in
procedural fluency?
Another Refresher:
Tell Me All You Can
502 – 196 =
The sum will be less than __ because ____________.
The sum will be greater than __ because __________.
The sum will be about ___ because _____________.
The sum will be between ___ and ___ because __________.
What role does mental math play in students’
development of procedural fluency?
Number Talk
99 + 17 =
2nd Grade Number Talk: 99 + 17
No student permission for posting the video
What evidence of procedural fluency do you see?
What does Olivia do to support the development of procedural
fluency?
Assessing Procedural Fluency:
A Free Tool – The Math Reasoning Inventory
Using Number Lines
The Number Line Is a
Recurring Model in the CC Standards
 Used as a tool for modeling operations and computing
 Used as a tool for organizing data
 Used as a tool for solving word problems
Using a Number Line to Conceptualize
Large Numbers and Understand Magnitude
Where would you place 1 million on this number line?
1__________________________________________ 1Billion
A Billion is a Difficult Number to
Comprehend
 A billion seconds ago it was 1984.
 A billion minutes ago The Roman Empire was expanding into
Armenia and Mesopotamia.
 A billion hours ago the most recent North American glacial
period was starting
 A billion days is almost 3 million years.
Using a Number Line to Help Build Procedural Fluency
Open Number Line: Addition
How might you use an open number line to model 999 + 75?
Open Number Line: Subtraction
How might you use an open number line to model 500 – 299 ?
You can also try:
40 – 17 =
1,000 – 295 =
251 – 39 =
About Open Number Lines
 An open number line reinforces the idea that the answer
to a subtraction problem is the difference between two
numbers.
 It supports using the inverse relationship between addition
and subtraction by counting up instead of back.
 It engages students in decomposing numbers and
reasoning.
 It provides a visual model that’s a useful tool.
- Marilyn Burns’ Blog, Feb. 25, 2015
What evidence of procedural fluency do you see?
Kiara, Grade 3
What evidence of procedural fluency do you see?
Kiara, Grade 3
What evidence of procedural fluency do you see?
Kiara, Grade 3
Wearing Two Hats . . .
What new ideas do you have about helping your
students develop procedural fluency?
How might you share this session’s ideas about
procedural fluency with your grade level? With
your staff ?
Lunch
Helping Children Master
the Basic Facts
Ideas in this session are derived from the work of the
following key people:
John Van de Walle
Jo Boaler
Cathy Seeley
Kathy Richardson
Mark Alcorn
Session Goals
 Think about the role of basic math fact mastery in
procedural fluency.
 Examine ways to support students in mastering the
basic math facts.
Why do we notice the same thing every year?
Every teacher knows students who are still counting on their
fingers, making marks in the margins, or simply guessing at
answers.
These students have certainly been given more than adequate
opportunity to drill their facts in past years.
They have not mastered their facts because they have not
developed efficient methods of producing a fact answer. Drill of
inefficient methods does not produce mastery.
--Van de Walle
All Students Can Master Basic Facts
“Basic facts for addition and multiplication refer to combinations
in which both addends or both factors are less than 10 . . .
Subtraction facts and division facts can and should be related to
their corresponding addition and multiplication facts . . . Mastery
of a basic fact means a child can give a quick response (in about 3
seconds) without resorting to non-efficient means, such as
counting. All children are able to master basic facts – including
children with learning disabilities. They simply need to construct
efficient mental tools that will help them . . .”
Van de Walle (2006). Teaching Student-Centered Mathematics.
Research-Based Progression of
Developing Fact Mastery
1. Help children develop a strong understanding of number
relationships of operations.
2. Develop efficient strategies for fact retrieval through
practice.
3. Then provide drill in the use and selection of those
strategies once they have been developed.
Common Core Shift: Rigor
Pursue conceptual understanding, procedural skills
& fluency, and application with equal intensity
Procedural skills and fluency: The standards call for
speed and accuracy in calculation. Students must practice core
functions, such as single-digit multiplication, in order to have
access to more complex concepts and procedures. Fluency
must be addressed in the classroom or through supporting
materials, as some students might require more practice than
others.
--Common Core Documentation
Research-Based Progression of
Developing Fact Mastery
1. Help children develop a strong understanding of number
relationships of operations.
2. Develop efficient strategies for fact retrieval through
practice.
3. Then provide drill in the use and selection of those
strategies once they have been developed.
Helping Students Develop
Efficient Strategies
Two Approaches to Fact Strategies
1. Design and use story problems which are likely to be solved
using a specific strategy.
2. Teach a lesson focused on a special collection of facts for
which a particular strategy is appropriate. Facilitate a way that
allows students to “uncover” the strategy.
If you did not know this fact, what strategy
might you use to solve this problem?
9x8=
9x1=9
9 x 2 = 18
9 x 3 = 27
What patterns do you notice?
How can you use those patterns to help you
develop a strategy for basic multiplication by 9
facts?
How are 9 facts related to 10 facts? How might
this relationship help you develop a strategy for
finding any 9 facts? How might modeling this
relationship with cubes support student thinking?
How is it related to the 9s finger trick?
9 x 4 = 36
9 x 5 = 45
9 x 6 = 54
9 x 7 = 63
9 x 8 = 72
9 x 9 = 81
Making Strategy Decisions: Fact Sort
Directions:
Select one of the strategies we listed.
Sort the math facts on the index cards into two piles:
1. Can be efficiently solved using the strategy.
1. Cannot be efficiently solved using the strategy.
Practice with Strategies
When a student is developing a specific strategy,
what might practice look like?
Practice: problem-based activities in which students are
encouraged to develop (invent, consider, try – but not
master) flexible and useful strategies that are meaningful.
Drill
1. How will you know when a student is ready to move
beyond practicing strategies and move toward drill?
(avoid premature drill)
Drill: Repetitive non-problem-based activity. Appropriate for
students who have a strategy they understand, like, and know
how to use but have not become facile with it. Drilling an inplace strategy builds automaticity.
2. What would those drills look like?
To pursue this idea further, read Van de Walle
during your independent menu time.
Current Research
In recent years brain researchers have found that the students who are most
successful with number problems are those who are using different brain
pathways – one that is numerical and symbolic and the other that involves more
intuitive and spatial reasoning (Park & Brannon, 2013). …
Additionally brain researchers have studied students learning math facts in two
ways – through strategies or memorization. They found that the two
approaches (strategies or memorization) involve two distinct pathways in the
brain and that both pathways are perfectly good for life long use.
Importantly the study also found that those who learned through strategies
achieved ‘superior performance’ over those who memorized, they solved
problems at the same speed, and showed better transfer to new problems.
The brain researchers concluded that automaticity should be reached through
understanding of numerical relations, achieved through thinking about number
strategies (Delazer et al, 2005).
Common Core Automaticity Progression
“Fluency” = Use strategies
“From Memory” = Automaticity
Kindergarten
All work with addition and subtraction involves concrete experiences,
contexts, and/or drawings. (partners of numbers equal to or less than 10; find
the number that makes a ten for any number between 0 and 9)
Fluency within 5 is expected.
First Grade
Add and subtract within 20, demonstrating fluency for addition and
subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 +
6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4
= 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction
(e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but
easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 +
1 = 12 + 1 = 13).
2nd Grade: Fluently add and subtract within 20 using mental strategies.2 By end
of Grade 2, know from memory all sums of two one-digit numbers.
Accountability?
 Beyond timed tests . . .what alternatives might we offer students?
 What online resources have you found helpful in holding students
accountable and providing you with data?
 How can we help students self-assess and set reasonable, incremental
goals for themselves?
Some Online Game Resources for Math Fact Practice and
Accountability/Record Keeping: (not clustered by strategy)
Big Brainz
First in Math
Leadership Reflections
What new thoughts do you have about helping students master
their basic facts?
What is your next step in your own classroom?
How might you share this session’s ideas with your grade level?
With your staff ?
Smarter Balanced
Assessment
For more detailed information about the test design,
you can view a webinar on the SBAC site with all the
Common Core experts explaining their intent.
Table Talk
What do conversations about the SBAC sound like at your school?
Teachers? Students? Principals? Parents?
SBAC Member States
2013
2015
(new map not yet made showing it is down to 18 states)
Next Generation Assessment
Guiding Principles
 Assessments are common across states and aligned to the
Common Core State Standards.
 Students take “performance-based” assessments for
accountability.
 The assessment systems are “computer-based” and “computer
adaptive” for more sophisticated design and quick, reliable
scoring.
 Transparent reporting systems drive effective decision-making.
Leadership Menu
Main Dish:
 Math Central Scavenger Hunt**
Side Dishes:
 Explore the SBAC website
 Take an SBAC Practice Test
 Read some research on automacity
 Read Marilyn Burns’ blog about open #lines (link on
MathCentral)
 Look at the Math Reasoning Inventory (MRI)
Dessert:
 Common Core on Twitter (hashtagcommoncore.com)
**Do This First
Math Central Scavenger Hunt
Visit each of these elements (recommended by Lynne). Be ready to share out anything
new or interesting that you find. When you finish, go onto the Math Leadership page
and see what you find.
Rich Tasks
School-wide Tasks
Learning Targets (including “I
Can” Statements)
Coaches’ Corner (Specifically the
grade level sharing, and the Articles
button which contains two brand
new articles from the latest NCTM
journal “ Effective Textbook Use”
link and “Understanding
Mathematical Practices” link) Both
articles are highly recommended
Math Games
Math Games
Parent Information and Support
Math Expressions Vocabulary
Cards
Marilyn Burn’s Math Blog
TLC Templates
Family Math Nights (in particular
the games link which has grade level
games that can also be used in the
classroom)
Literature Based Lessons
Units of Study (always posting new
units here)
Focus Questions . . .
1.
What new, interesting, useful, and/or exciting things did
you find?
2. In what ways might you help other teachers access and
use these resources?
If you pursue math fact mastery:
 What new ideas do you have about supporting the development of highly
efficient strategies?





What sorts of activities can you offer students to work toward efficient strategies
and/or automaticity?
What role might student goal setting play?
What role might student self-monitoring play?
What role might choice in both practice and accountability play?
What role does time allotted for practice during class play?
By 2:30, be prepared to share 1-2 important ideas your
learned from your independent research.
Questions and Reflections