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PK – 2 Mathematics Common Practice Aligned to the Expectations of the CCSSM

October 3, 2014 Session Description: The CCSSM Content Standards and Standards for Mathematical Practice frame the teaching and learning of mathematics in the classroom. How can these be effectively taught by teachers and learned by students? What are the expectations as shown in SBAC? In this session, participants will:

• Identify the priority and supporting clusters. • Explore high leverage instructional strategies. • Analyze and design a CCSS-‐M lesson.

Learning Targets:

• I can use high leverage instructional strategies to improve student learning.

• I can design lessons focused on students learning the CCSSM – the content standards and the standards for mathematical practice

Agenda

Welcome and Introductions

What do we expect students to learn? Ø Content Standards: Identify the priority and supporting standards Ø Standards for Mathematical Practice: Instructional practices and strategies that develop student

mathematical thinking and learning ♦ How do students make sense of math in their world? How do they recognize

important information in a problem? ♦ Which questions can we use to encourage reasoning? ♦ How can high cognitive tasks be used to further student learning?

How will we know students have learned it?

Ø Classroom assessments and links to SBAC Ø Create lessons that provide opportunities for formative feedback

Starting … Getting There … Got It!

Starting … Getting There … Got It!

1

Classroom Prac+ce Aligned to the CCSSM

Sarah Schuhl SMc Currriculum

Teaching and Assessing the CCSSM § What have students already learned this year related to the CCSSM §  Content Standards? §  Standards for Mathema7cal Prac7ce?

§ What tasks or ac7vi7es have you used to support students’ learning?

Reasoning and Explaining “A boy bought some things at the store and gave the merchant a two-‐dollar bill; he received in change five coins, no two of them of the same value. What was the amount of his purchase? Find six correct answers.” Explain your reasoning. -‐-‐Gillan, S.Y. (1909) Problems without Figures, p8.

2

What Do We Expect Students To Learn?

Priority Clusters & Suppor+ng Clusters §  Read the Priority and Suppor7ng Clusters.

§  Highlight the accompanying content standards.

§ Which of the Priority standards are your students currently learning?

§  Choose a Suppor7ng standard and explain how it can be taught while students are learning a Priority standard.

Standards for Mathema+cal Prac+ce

1.  Make sense of problems and persevere in solving them.

6. Attend to precision.

2.  Reason abstractly and quantitatively.

3.  Construct viable arguments and critique the reasoning of others.

4.  Model with mathematics.

5.  Use appropriate tools strategically.

7.  Look for and make use of structure.

8.  Look for and express regularity in repeated reasoning.

3

CCSSM (SBAC) Priority Clusters K – 2

Kindergarten

Grade 1

Grade 2

Counting and Cardinality

Know num

ber nam

es and the count

sequence.

Count to tell the number of objects.

Compare num

bers.

Operations and Algebraic Thinking

Understand addition as putting together

and adding to, and understand

subtraction as taking apart and taking

from

. Num

ber and Operations in Base Ten

Work with num

bers 11-‐19 to gain

foundations for place value.


Represent and solve problems involving

addition and subtraction.

Understand and apply properties of

operations and the relationship between


Add and subtract within 20.

Work with addition and subtraction

equations.

Num


Extending the counting sequence.

Understand place value.

Use place value understanding and

properties of operations to add and

subtract.

Measurement and Data

Measure lengths indirectly and by

iterating length units.


Represent and solve problems involving


Add and subtract within 20.

Num


Understand place value.

Use place value understanding and

properties of operations to add and

subtract.


Measure and estimate lengths in

standard units.

Relate addition and subtraction to

length.

4

CCSSM (SBAC) Supporting Clusters K – 2

Kindergarten

Grade 1

Grade 2


Classify objects and count the num

ber of

objects in categories.

Describe and compare measureable

attributes.

Geom

etry

Identify and describe shapes.

Analyze, compare, create, and compose

shapes.


Represent and interpret data.

Tell and write time.

Geom

etry

Reason with shapes and their attributes.


Work with equal groups of objects to

gain foundations for m

ultiplication.


Work with time and money.

Represent and interpret data.

Geom

etry

Reason with shapes and their attributes.

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Counting and CardinalityKnow number names and the count sequence.K.CC.1 Count to 100 by ones and by tens.K.CC.2 Count forward beginning from a given number within the known

sequence (instead of having to begin at 1).K.CC.3 Write numbers from 0 to 20. Represent a number of objects with a

written numeral 0-20 (with 0 representing a count of no objects).

Count to tell the number of objects.K.CC.4 Understand the relationship between numbers and quantities;

connect counting to cardinality. a. When counting objects, say the number names in the standard

order, pairing each object with one and only one number name and each number name with one and only one object.

b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

c. Understand that each successive number name refers to a quantity that is one larger.

K.CC.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.

Compare numbers.K.CC.6 Identify whether the number of objects in one group is greater than,

less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. (Note: Include groups with up to ten objects.)

K.CC.7 Compare two numbers between 1 and 10 presented as written numerals.

operations and algebraiC thinKingunderstand addition as putting together and adding to, and understand subtraction as taking apart and taking from.K.oa.1 Represent addition and subtraction with objects, fingers, mental

images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. (Note: Drawings need not show details, but should show the mathematics in the problem – this applies wherever drawings are mentioned in the Standards.)

K.oa.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

K.oa.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

K.oa.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

K.oa.5 Fluently add and subtract within 5.

number and operations in base tenWork with numbers 11 – 19 to gain foundations for place value.K.nbt.1 Compose and decompose numbers from 11 to 19 into ten ones and

some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 +8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

measurement and datadescribe and compare measurable attributes.K.md.1 Describe measurable attributes of objects, such as length or weight.

Describe several measurable attributes of a single object.K.md.2 Directly compare two objects with a measurable attribute in

common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.K.md.3 Classify objects into given categories; count the numbers of objects

in each category and sort the categories by count. (Note: Limit category counts to be less than or equal to 10.)

geometryidentify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).K.g.1 Describe objects in the environment using names of shapes, and

describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

K.g.2 Correctly name shapes regardless of their orientations or overall size.K.g.3 Identify shapes as two-dimensional (lying in a plane, “flat”) or three-

dimensional (“solid”).

analyze, compare, create, and compose shapes.K.g.4 Analyze and compare two- and three-dimensional shapes, in different

sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/ “corners”) and other attributes (e.g., having sides of equal length).

K.g.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

K.g.6 Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”

Kindergarten – Standards 1. representing, relating and operating on whole numbers, initially with

sets of objects – Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.

2. describing shapes and space – Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and

vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

mathematiCal praCtiCes1. make sense of problems and 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.

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operations and algebraiC thinKingrepresent and solve problems involving addition and subtraction.1.oa.1 Use addition and subtraction within 20 to solve word problems

involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (Note: See Glossary, Table 1.)

1.oa.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

understand and apply properties of operations and the relationship between addition and subtraction.1.oa.3 Apply properties of operations as strategies to add and subtract.

(Note: Students need not use formal terms for these properties.) Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

1.oa.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

add and subtract within 20.1.oa.5 Relate counting to addition and subtraction (e.g., by counting on 2 to

add 2).1.oa.6 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).

Work with addition and subtraction equations.1.oa.7 Understand the meaning of the equal sign, and determine if

equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

1.oa.8 Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = o – 3, 6 + 6 = o.

number and operations in base tenextend the counting sequence.1.nbt.1 Count to 120, starting at any number less than 120. In this range,

read and write numerals and represent a number of objects with a written numeral.

understand place value.1.nbt.2 Understand that the two digits of a two-digit number represent

amounts of tens and ones. Understand the following as special cases: a. 10 can be thought of as a bundle of ten ones – called a “ten.” b. The numbers from 11 to 19 are composed of a ten and one, two,

three, four, five, six, seven, eight, or nine ones. c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two,

three, four, five, six, seven, eight, or nine tens (and 0 ones).1.nbt.3 Compare two two-digit numbers based on meanings of the tens and

ones digits, recording the results of comparisons with the symbols >, =, and <.

use place value understanding and properties of operations to add and subtract.1.nbt.4 Add within 100, including adding a two-digit number and a one-digit

number, and adding a two-digit number and a multiple of 10, 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 and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

First Grade – Standards

1. developing understanding of addition, subtraction, and strategies for addition and subtraction within 20 – Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.

2. developing understanding of whole number relationship and place value, including grouping in tens and ones – Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. The compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes. They think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and some ones). Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes.

3. developing understanding of linear measurement and measuring lengths as iterating length units – Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement. (Note: students should apply the principle of transitivity of measurement to make direct comparisons, but they need not use this technical term.)

4. reasoning about attributes of, and composing and decomposing geometric shapes – Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine shapes, they recognize them from different perspectives and orientations, describe their geometric attributes, and determine how they are alike and different, to develop the background for measurement and for initial understandings of properties such as congruence and symmetry.


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1.nbt.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

1.nbt.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), 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 and explain the reasoning used.

measurement and datameasure lengths indirectly and by iterating length units.1.md.1 Order three objects by length; compare the lengths of two objects

indirectly by using a third object.1.md.2 Express the length of an object as a whole number of length units, by

laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

tell and write time.1.md.3 Tell and write time in hours and half-hours using analog and digital clocks.

represent and interpret data.1.md.4 Organize, represent, and interpret data with up to three categories;

ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

geometryreason with shapes and their attributes.1.g.1 Distinguish between defining attributes (e.g., triangles are closed and

three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

1.g.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Note: Students do not need to learn formal names such as “right rectangular prism.”)

1.g.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

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operations and algebraiC thinKingrepresent and solve problems involving addition and subtraction.2.oa.1 Use addition and subtraction within 100 to solve one- and two-step

word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (Note: See Glossary, Table 1.)

add and subtract within 20.2.oa.2 Fluently add and subtract within 20 using mental strategies. (Note:

See standard 1.OA.6 for a list of mental strategies). By end of Grade 2, know from memory all sums of two one-digit numbers.

Work with equal groups of objects to gain foundations for multiplication.2.oa.3 Determine whether a group of objects (up to 20) has an odd or even

number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

2.oa.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

number and operations in base tenunderstand place value.2.nbt.1 Understand that the three digits of a three-digit number represent

amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens – called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one,

two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2.nbt.2 Count within 1000; skip-count by 5s, 10s, and 100s.2.nbt.3 Read and write numbers to 1000 using base-ten numerals, number

names, and expanded form.2.nbt.4 Compare two three-digit numbers based on meanings of the

hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

use place value understanding and properties of operations to add and subtract.2.nbt.5 Fluently add and subtract within 100 using strategies based on

place value, properties of operations, and/or the relationship between addition and subtraction.

2.nbt.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.

2.nbt.7 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

2.nbt.8 Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.

2.nbt.9 Explain why addition and subtraction strategies work, using place value and the properties of operations. (Note: Explanations may be supported by drawings or objects.)

measurement and datameasure and estimate lengths in standard units.2.md.1 Measure the length of an object by selecting and using appropriate

tools such as rulers, yardsticks, meter sticks, and measuring tapes.2.md.2 Measure the length of an object twice, using length units of

different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

2.md.3 Estimate lengths using units of inches, feet, centimeters, and meters.2.md.4 Measure to determine how much longer one object is than another,

expressing the length difference in terms of a standard length unit.

relate addition and subtraction to length.2.md.5 Use addition and subtraction within 100 to solve word problems

involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

2.md.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

Second Grade – Standards 1. extending understanding of base-ten notation – Students extend their

understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones).

2. building fluency with addition and subtraction – Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.

3. using standard units of measure – Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the understanding that linear measure

involves iteration of units. They recognize that the smaller the unit, the more iterations they need to cover a given length.

4. describing and analyzing shapes – Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding attributes of two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.


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Work with time and money.2.md.7 Tell and write time from analog and digital clocks to the nearest five

minutes, using a.m. and p.m.2.md.8 Solve word problems involving dollar bills, quarters, dimes, nickels,

and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

represent and interpret data.2.md.9 Generate measurement data by measuring lengths of several

objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

2.md.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put together, take-apart, and compare problems using information presented in a bar graph. (Note: See Glossary, Table 1.)

geometryreason with shapes and their attributes.2.g.1 Recognize and draw shapes having specified attributes, such as

a given number of angles or a given number of equal faces. (Note: Sizes are compared directly or visually, not compared by measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

2.g.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

2.g.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

10

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are

so

me

oth

er p

rob

lem

s th

at a

re s

imila

r to

th

is o

ne?

H

ow

mig

ht

you

use

on

e o

f yo

ur

pre

vio

us

pro

ble

ms

to

hel

p y

ou

beg

in?

H

ow

els

e m

igh

t yo

u o

rgan

ize.

..re

pre

sen

t...

sh

ow

...?

2. Reason abstractly and quantitatively

Faci

litat

e o

pp

ort

un

itie

s fo

r st

ud

en

ts

to d

iscu

ss o

r u

se r

epre

sen

tati

on

s to

m

ake

sen

se o

f q

uan

titi

es a

nd

th

eir

rela

tio

nsh

ips.

Enco

ura

ge t

he

flex

ible

use

of

pro

per

ties

of

op

erat

ion

s, o

bje

cts,

an

d

solu

tio

n s

trat

egie

s w

hen

so

lvin

g p

rob

lem

s.

Pro

vid

e o

pp

ort

un

itie

s fo

r st

ud

ents

to

d

eco

nte

xtu

aliz

e (a

bst

ract

a s

itu

atio

n)

and

/or

con

text

ual

ize

(id

enti

fy

refe

ren

ts f

or

sym

bo

ls in

volv

ed)

the

mat

hem

atic

s th

ey a

re le

arn

ing.

Mak

e se

nse

of

qu

anti

ties

an

d t

hei

r re

lati

on

ship

s.

Are

ab

le t

o d

eco

nte

xtu

aliz

e (r

epre

sen

t a

situ

atio

n s

ymb

olic

ally

an

d m

anip

ula

te

the

sym

bo

ls)

and

co

nte

xtu

aliz

e (m

ake

mea

nin

g o

f th

e sy

mb

ols

in a

pro

ble

m)

qu

anti

tati

ve r

ela

tio

nsh

ips.

Un

der

stan

d t

he

mea

nin

g o

f q

uan

titi

es

and

are

fle

xib

le in

th

e u

se o

f o

per

atio

ns

and

th

eir

pro

per

tie

s.

Cre

ate

a lo

gica

l re

pre

sen

tati

on

of

the

pro

ble

m.

Att

end

s to

th

e m

ean

ing

of

qu

anti

ties

, n

ot

just

ho

w t

o c

om

pu

te

Wh

at d

o t

he

nu

mb

ers

use

d in

th

e p

rob

lem

re

pre

sen

t?

Wh

at is

th

e re

lati

on

ship

of

the

qu

anti

ties

?

Ho

w is

rel

ate

d t

o

?

Wh

at is

th

e re

lati

on

ship

bet

wee

n a

nd

?

Wh

at d

oes

mea

n t

o y

ou

? (e

.g. s

ymb

ol,

qu

anti

ty,

dia

gram

)

Wh

at p

rop

erti

es m

igh

t w

e u

se t

o f

ind

a s

olu

tio

n?

H

ow

did

yo

u d

ecid

e in

th

is t

ask

that

yo

u n

eed

ed t

o

use

...?

C

ou

ld w

e h

ave

use

d a

no

ther

op

erat

ion

or

pro

per

ty

to s

olv

e th

is t

ask?

Wh

y o

r w

hy

no

t?

11

Te

ach

er A

ctio

ns

M

ath

em

atic

al P

ract

ice

s (S

tud

en

t p

ers

pec

tive

)

Qu

esti

on

s to

De

velo

p M

ath

em

atic

al T

hin

kin

g

3. Construct viable arguments and

critique the reasoning of others.

Pro

vid

e an

d o

rch

estr

ate

op

po

rtu

nit

ies

for

stu

den

ts t

o li

ste

n t

o

the

solu

tio

n s

trat

egie

s o

f o

ther

s,

dis

cuss

alt

ern

ativ

e so

luti

on

s, a

nd

d

efen

d t

hei

r id

eas.

Ask

hig

her

-ord

er q

ues

tio

ns

that

en

cou

rage

stu

den

ts t

o d

efen

d t

hei

r id

eas.

Pro

vid

e p

rom

pts

th

at e

nco

ura

ge

stu

den

ts t

o t

hin

k cr

itic

ally

ab

ou

t th

e m

ath

emat

ics

they

are

lear

nin

g.

An

alyz

e p

rob

lem

s an

d u

se s

tate

d

mat

hem

atic

al a

ssu

mp

tio

ns,

def

init

ion

s,

and

est

ablis

hed

res

ult

s in

co

nst

ruct

ing

argu

men

ts.

Just

ify

con

clu

sio

ns

wit

h m

ath

emat

ical

id

eas.

List

en t

o t

he

argu

men

ts o

f o

ther

s an

d

ask

use

ful q

ues

tio

ns

to d

eter

min

e if

an

ar

gum

ent

mak

es

sen

se.

Ask

cla

rify

ing

qu

esti

on

s o

r su

gges

t id

eas

to im

pro

ve/r

evis

e th

e ar

gum

ent.

Co

mp

are

two

arg

um

ents

an

d d

ete

rmin

e co

rrec

t o

r fl

awe

d lo

gic.

Wh

at m

ath

emat

ical

evi

den

ce w

ou

ld s

up

po

rt y

ou

r so

luti

on

? H

ow

can

we

be

sure

th

at..

.? /

Ho

w c

ou

ld

you

pro

ve t

hat

...?

Will

it s

till

wo

rk if

...?

Wh

at w

ere

you

co

nsi

der

ing

wh

en..

.?

Ho

w d

id y

ou

dec

ide

to t

ry t

hat

str

ateg

y?

Ho

w d

id y

ou

te

st w

het

her

yo

ur

app

roac

h w

ork

ed

? H

ow

did

yo

u d

ecid

e w

hat

th

e p

rob

lem

was

ask

ing

you

to

fin

d?

(Wh

at w

as u

nkn

ow

n?)

D

id y

ou

try

a m

eth

od

th

at d

id n

ot

wo

rk?

Wh

y d

idn

’t

it w

ork

? W

ou

ld it

eve

r w

ork

? W

hy

or

wh

y n

ot?

W

hat

is t

he

sam

e an

d w

hat

is d

iffe

ren

t ab

ou

t...?

H

ow

co

uld

yo

u d

emo

nst

rate

a c

ou

nte

r-ex

amp

le?

4. Model with mathematics.

Use

mat

hem

atic

al m

od

els

app

rop

riat

e f

or

the

focu

s o

f th

e le

sso

n.

Enco

ura

ge s

tud

ent

use

of

dev

elo

pm

enta

lly a

nd

co

nte

nt-

app

rop

riat

e m

ath

emat

ical

mo

del

s (e

.g.,

var

iab

les,

eq

uat

ion

s, c

oo

rdin

ate

grid

s).

Rem

ind

stu

den

ts t

hat

a m

ath

emat

ical

m

od

el u

sed

to

rep

rese

nt

a p

rob

lem

’s

solu

tio

n is

a w

ork

in p

rogr

ess,

an

d

may

be

revi

sed

as

nee

ded

Un

der

stan

d t

his

is a

way

to

rea

son

q

uan

tita

tive

ly a

nd

ab

stra

ctly

(ab

le t

o d

eco

nte

xtu

aliz

e an

d

con

text

ual

ize)

.

Ap

ply

th

e m

ath

th

ey k

no

w t

o s

olv

e p

rob

lem

s in

eve

ryd

ay li

fe.

Are

ab

le t

o s

imp

lify

a co

mp

lex

pro

ble

m

and

iden

tify

imp

ort

ant

qu

anti

ties

to

loo

k at

re

lati

on

ship

s.

Rep

rese

nt

mat

hem

atic

s to

des

crib

e a

situ

atio

n e

ith

er w

ith

an

eq

uat

ion

or

a d

iagr

am a

nd

inte

rpre

t th

e re

sult

s o

f a

mat

hem

atic

al s

itu

atio

n.

Ref

lect

on

wh

eth

er t

he

resu

lts

mak

e

sen

se, p

oss

ibly

imp

rovi

ng

or

revi

sin

g th

e m

od

el.

Ask

th

emse

lves

, “H

ow

can

I re

pre

sen

t th

is m

ath

emat

ical

ly?”

Wh

at n

um

ber

mo

del

co

uld

yo

u c

on

stru

ct t

o

rep

rese

nt

the

pro

ble

m?

Wh

at a

re s

om

e w

ays

to r

epre

sen

t th

e q

uan

titi

es?

Wh

at’s

an

eq

uat

ion

or

exp

ress

ion

th

at m

atch

es t

he

dia

gram

...,

nu

mb

er li

ne.

., c

har

t...

, tab

le..

? W

her

e d

id y

ou

see

on

e o

f th

e q

uan

titi

es in

th

e ta

sk

in y

ou

r eq

uat

ion

or

exp

ress

ion

? W

ou

ld it

hel

p t

o c

reat

e a

dia

gram

, gra

ph

, tab

le..

.?

Wh

at a

re s

om

e w

ays

to v

isu

ally

re

pre

sen

t...

? W

hat

fo

rmu

la m

igh

t ap

ply

in t

his

sit

uat

ion

?

12

Te

ach

er A

ctio

ns

M

ath

em

atic

al P

ract

ice

s (S

tud

en

t p

ers

pec

tive

)

Qu

esti

on

s to

De

velo

p M

ath

em

atic

al T

hin

kin

g

5. Use appropriate tools

strategically.

Use

ap

pro

pri

ate

ph

ysic

al a

nd

/or

dig

ital

to

ols

to

rep

rese

nt,

exp

lore

, an

d d

eep

en s

tud

ent

un

der

stan

din

g.

Hel

p s

tud

ents

mak

e s

ou

nd

dec

isio

ns

con

cern

ing

the

use

of

spec

ific

to

ols

ap

pro

pri

ate

fo

r th

e gr

ade

-lev

el a

nd

co

nte

nt

focu

s o

f th

e le

sso

n.

Pro

vid

e ac

cess

to

mat

eria

ls, m

od

els,

to

ols

, an

d/o

r te

chn

olo

gy‐

bas

ed

reso

urc

es t

hat

ass

ist

stu

den

ts in

m

akin

g co

nje

ctu

res

nec

ess

ary

for

solv

ing

pro

ble

ms.

Use

ava

ilab

le t

oo

ls r

eco

gniz

ing

the

stre

ngt

hs

and

lim

itat

ion

s o

f e

ach

.

Use

est

imat

ion

an

d o

ther

mat

hem

atic

al

kno

wle

dge

to

det

ect

po

ssib

le e

rro

rs.

Iden

tify

rel

evan

t ex

tern

al m

ath

emat

ical

re

sou

rces

to

po

se a

nd

so

lve

pro

ble

ms.

Use

tec

hn

olo

gica

l to

ols

to

dee

pen

th

eir

un

der

stan

din

g o

f m

ath

emat

ics.

Wh

at m

ath

emat

ical

to

ols

co

uld

we

use

to

vis

ual

ize

and

re

pre

sen

t th

e si

tuat

ion

? W

hat

info

rmat

ion

do

yo

u h

ave?

W

hat

do

yo

u k

no

w t

hat

is n

ot

stat

ed

in t

he

pro

ble

m?

Wh

at a

pp

roac

h a

re y

ou

co

nsi

der

ing

tryi

ng

firs

t?

Wh

at e

stim

ate

did

yo

u m

ake

fo

r th

e so

luti

on

? In

th

is s

itu

atio

n w

ou

ld it

be

hel

pfu

l to

use

...a

gra

ph

...,

n

um

ber

lin

e...

, ru

ler.

.., d

iagr

am..

., c

alcu

lato

r...

, m

anip

ula

tive

? W

hy

was

it h

elp

ful t

o u

se..

.?

Wh

at c

an u

sin

g a

sho

w u

s, t

hat

_m

ay n

ot?

In

wh

at s

itu

atio

ns

mig

ht

it b

e m

ore

info

rmat

ive

or

hel

pfu

l to

use

...?

6. Attend to precision.

Emp

has

ize

the

imp

ort

ance

of

pre

cise

co

mm

un

icat

ion

by

en

cou

ragi

ng

stu

den

ts t

o f

ocu

s o

n c

lari

ty o

f th

e d

efin

itio

ns,

no

tati

on

, an

d v

oca

bu

lary

to

co

nve

y th

eir

reas

on

ing.

Enco

ura

ge a

ccu

racy

an

d e

ffic

ien

cy in

co

mp

uta

tio

n a

nd

pro

ble

m-b

ased

so

luti

on

s, e

xpre

ssin

g n

um

eric

al

answ

ers

, dat

a, a

nd

/or

mea

sure

men

ts

wit

h a

deg

ree

of

pre

cisi

on

ap

pro

pri

ate

fo

r th

e co

nte

xt o

f th

e p

rob

lem

.

Co

mm

un

icat

e p

reci

sely

wit

h o

ther

s an

d

try

to u

se c

lear

mat

hem

atic

al la

ngu

age

wh

en d

iscu

ssin

g th

eir

reas

on

ing.

Un

der

stan

d m

ean

ings

of

sym

bo

ls u

sed

in

mat

hem

atic

s an

d c

an la

bel

qu

anti

ties

ap

pro

pri

ate

ly.

Exp

ress

nu

mer

ical

an

swer

s w

ith

a

deg

ree

of

pre

cisi

on

ap

pro

pri

ate

fo

r th

e p

rob

lem

co

nte

xt.

Cal

cula

te e

ffic

ien

tly

and

acc

ura

tely

.

Wh

at m

ath

emat

ical

ter

ms

app

ly in

th

is s

itu

atio

n?

Ho

w d

id y

ou

kn

ow

yo

ur

solu

tio

n w

as r

eas

on

able

? Ex

pla

in h

ow

yo

u m

igh

t sh

ow

th

at y

ou

r so

luti

on

an

swe

rs t

he

pro

ble

m.

Is t

her

e a

mo

re e

ffic

ien

t st

rate

gy?

Ho

w a

re y

ou

sh

ow

ing

the

mea

nin

g o

f th

e q

uan

titi

es?

Wh

at s

ymb

ols

or

mat

hem

atic

al n

ota

tio

ns

are

imp

ort

ant

in t

his

pro

ble

m?

Wh

at m

ath

emat

ical

lan

guag

e...

,def

init

ion

s...

, p

rop

erti

es

can

yo

u u

se t

o e

xpla

in..

.?

Ho

w c

ou

ld y

ou

te

st y

ou

r so

luti

on

to

see

if it

an

swe

rs

the

pro

ble

m?

13

Te

ach

er A

ctio

ns

M

ath

em

atic

al P

ract

ice

s (S

tud

en

t p

ers

pec

tive

)

Qu

esti

on

s to

De

velo

p M

ath

em

atic

al T

hin

kin

g 7. Look for and make use of structure.

Enga

ge s

tud

ents

in d

iscu

ssio

ns

emp

has

izin

g re

lati

on

ship

s b

etw

een

p

arti

cula

r to

pic

s w

ith

in a

co

nte

nt

do

mai

n o

r ac

ross

co

nte

nt

do

mai

ns.

Rec

ogn

ize

that

th

e q

uan

tita

tive

re

lati

on

ship

s m

od

eled

by

op

erat

ion

s an

d t

hei

r p

rop

erti

es r

emai

n

imp

ort

ant

rega

rdle

ss o

f th

e o

per

atio

nal

fo

cus

of

a le

sso

n.

Pro

vid

e ac

tivi

ties

in w

hic

h s

tud

ents

d

emo

nst

rate

th

eir

flex

ibili

ty in

re

pre

sen

tin

g m

ath

emat

ics

in a

n

um

ber

of

way

s, e

.g.,

76

= (

7 x

10

) +

6; d

iscu

ssin

g ty

pes

of

qu

adri

late

rals

, an

d s

o o

n.

Ap

ply

gen

eral

mat

hem

atic

al r

ule

s to

sp

ecif

ic s

itu

atio

ns.

Loo

k fo

r th

e o

vera

ll st

ruct

ure

an

d

pat

tern

s in

mat

hem

atic

s.

See

com

plic

ate

d t

hin

gs a

s si

ngl

e o

bje

cts

or

as b

ein

g co

mp

ose

d o

f se

vera

l ob

ject

s.

Wh

at o

bse

rvat

ion

s d

o y

ou

mak

e ab

ou

t...

? W

hat

do

yo

u n

oti

ce w

hen

...?

Wh

at p

arts

of

the

pro

ble

m m

igh

t yo

u

elim

inat

e...

,sim

plif

y...

? W

hat

pat

tern

s d

o y

ou

fin

d in

...?

H

ow

do

yo

u k

no

w if

so

met

hin

g is

a p

atte

rn?

Wh

at id

eas

that

we

hav

e le

arn

ed b

efo

re w

ere

use

ful

in s

olv

ing

this

pro

ble

m?

Wh

at a

re s

om

e o

ther

pro

ble

ms

that

are

sim

ilar

to

this

on

e?

Ho

w d

oes

th

is r

elat

e to

...?

In

wh

at w

ays

do

es t

his

pro

ble

m c

on

nec

t to

oth

er

mat

hem

atic

al c

on

cep

ts?

8. Look for and express regularity in repeated reasoning.

En

gage

stu

den

ts in

dis

cuss

ion

rel

ate

d

to r

epea

ted

rea

son

ing

that

may

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14

Standard for Mathematical Practice Assume you are teaching a unit on addition and subtraction. Create a problem that assesses content while also assessing your assigned standard for mathematical practice. Mathematics Problem: (Standard for Mathematical Practice: _______)

How will you know students are learning the standard for mathematical practice? What will it look like and sound like in the classroom?

15

Ques+oning in Class §  How do you inten7onally use ques7ons in class?

§ Why and how do you use wait 7me?

Ques+oning in Class §  Ques7ons to seldom ask:

§  Are there any ques7ons? §  How many of you understood that?

§  Ques7ons/Prompts to ask: § What ques7ons do you have? §  Tell me more about your thinking. §  Think about the answer and write down your thoughts. I will check your work and have 3 students share.

Effec+ve Classroom Communica+on §  How do students express their

ideas, ques7ons, insights, and difficul7es?

§  Where are the most significant conversa7ons taking place (student to teacher, student to student, teacher to student)?

Do students see each other as reliable and valuable

resources?

16

5 Questions Mathematics can be used to quantify the world. After each picture is shown, generate mathematical questions you first think of and then identify what you would need to know to answer each question. This helps students when problem solving answer: What do I know? What do I need to know?

Picture 1 Question What do you need to know?



17

Beginning Repertoire of Teacher Questions

1. Initial eliciting of students’ thinking—

Does anyone have a solution they would like to share?

Please raise your hand when you are ready to share your solution.

What did you come up with? What are you thinking?

Be ready to explain the solution you got.

Please explain to the rest of the class how you got your answer, ____.

How did you begin working on this problem?

What have you found so far?

Would anyone be willing to explain their solution?

Can you point to a part of this problem that was difficult?

What are some ideas you had?

Raise your hand if you have a different idea.

Did anyone approach the problem in a different way?

What do you already know about ____?

2. Probing students’ answers to— Figure out what a student means or is thinking when you don’t understand what

they are saying Check whether right answers are supported by correct understanding Probe wrong answers to understand student thinking Explain what you have done so far? What else is there to do?

How do you know?

Why did you ____?

How did you get ____?

Could you use [materials] to show how that works?

What led you to that idea?

Walk us through your steps. Where did you begin?

Please give an example.

Would you please repeat what you said about that?

Say a little more about your idea.

So is what you’re saying ____?

When you say ____, do you mean ___?

Could you explain a little more about what you are thinking? Can you explain that in a different way?

What do you notice when _____?

18

3. To help when students get stuck—

How would you describe the problem in your own words?

What do you know that is not stated in the problem?

What facts do you have?

Could you try it with simpler numbers? Fewer numbers? Using a number line?

What about putting things in order?

Would it help to create a diagram? Make a table? Draw a picture?

Can you guess and check?

What did other members of your group try?

What do you already know that could help you figure that out?

4. Focusing students to listen and respond to others’ ideas—

What do other people think?

What do other people think about what ____ said? Do you agree or disagree with the idea?

Would someone be willing to add on to what ____ said?

What do you think ____ means by that?

How does what ____ said go along with what you were thinking?

How could you explain what ____ said in a different way?

Can you repeat what ____ just said in your own words?

Why do you think ____ did it that way?

Why is it okay for ____ to do that?

Who can explain this using ____’s idea?

Does anyone have the same answer but a different way to explain it?

Can you convince us that your answer makes sense?

Can anybody see what method ____ might have used to come up with that solution?

How do you think ____ got his/her solution?

5. Supporting students to make connections (e.g., between a model and a mathematical

idea or a specific notation)—

How is ____’s method similar to (or different from) ____’s method?

How does [one representation] correspond to [another representation]?

Can you think of another problem that is similar to this one?

How does that match what you wrote on the board?

Can you explain your representation?

Can you use the [representation] to explain what you are thinking?

How is this similar to what we learned about _____?

How is this related to [a particular problem students already solved or something students already learned]?

19

How does that relate to what ____ said?

How can we make a [picture, graph, model, chart] of this solution?

What part of the problem/solution does this [pointing to a particular part of representation] represent?

6. To guide students and encourage mathematical reflection and reasoning (e.g., make conjectures, state definitions, generalize, prove)

Can you explain the method you used?

Does this method always work?

Why does that work in this case?

When do you think that would be true?

Do you notice any patterns?

What do these solutions have in common?

Can this method be used for other problems?

What do we mean when we say _____ in math class?

What math terms help us to talk about that? Did you learn any new words today?

What do you mean by ___? Can you give a definition?

Does this match our reasoning? How?

Have we found all the possible answers?

How do you know it works in all cases?

What about [counterexample]?

How would you describe ____’s method?

Can you represent the solution in another way?

Using this problem as an example, what can you say about problems like this in general?

What are the main ideas that you learned about today?

7. Extending students’ current thinking, and assessing how far they can be stretched—

Can you think of another way to solve this problem?

Can you use this same method to solve _____?

What would happen if the numbers were changed to _____?

What if the problem was like this instead: [give slight variation of problem]?

If someone said [wrong answer], how would you respond?

If we notice/know ______ then what does that mean for _______?

Can you predict the next one?

Can you think of another problem that could be solved with this method?

20

Using High Cogni+ve Tasks

Low and High Level Tasks

•  Low-‐Level Tasks – memoriza7on –  procedures without connec7ons to meaning

•  High-‐Level Tasks –  procedures with connec7ons to meaning –  doing mathema7cs

Which type of tasks are most oLen used in your classroom?

ShiLs in Classroom Prac+ce

What are the shi^s in classroom prac7ce when the CCSS content standards and Mathema+cal Prac+ces are merged? Read ShiLs in Classroom Prac+ce. Where is your math classroom on each con+nuum?

21

The Task Analysis Guide (Source: This table is provided verbatim from Smith & Stein, “Selecting and Creating Mathematical Tasks: From Research to Practice,” Mathematics Teaching in the Middle School, February 1998, 3(5), p. 348)

Lower Level Higher Level Memorization Task x� Involve reproducing previously learned facts, rules,

formulae, or definitions to memory.

x� Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use procedure.

x� Are not ambiguousȄsuch tasks involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated.

x� Have no connection to the concepts or meaning that underlie the facts, rules, formulae, or definitions being learned or reproduced.

Procedures With Connections Tasks x� Focus students’ attention on the use of procedures for

the purpose of developing deeper levels of understanding of mathematical concepts and ideas.

x� Suggest pathways to follow (explicitly or implicitly) that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

x� Usually are represented in multiple ways (e.g., visual diagrams, manipulatives, symbols, problem situations). Require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with the conceptual ideas that underlie the procedures in order to successfully complete the task and develop understanding.

Procedures Without Connections Tasks x� Are algorithmic. Use of the procedure is either

specifically called for or its use is evident based on prior instruction, experience, or placement of the task.

x� Require limited cognition demand for successful completion. There is little ambiguity about what needs to be done and how to do it.

x� Have no connection to the concepts or meaning that underlie the procedure being used.

x� Are focused on producing correct answers rather than developing mathematical understanding.

x� Require no explanations, or explanations that focus solely on describing the procedure that was used.

Doing Mathematics Tasks x� Require complex and no algorithmic thinking (i.e.,

there is not a predictable, wellǦrehearsed approach or pathway explicitly suggested by the task, task instructions, or workedǦout example).

x� Require students to explore and understand the nature of mathematical concepts, processes, or relationships.

x� Demand selfǦmonitoring or selfǦregulation of one’s own cognitive processes.

x� Require students to access relevant knowledge and experiences and make appropriate use of them in working through her task.

x� Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions.

x� Require considerable cognitive effort and may involve some level of anxiety for the student due to unpredictable nature of the solution process required.

(Task Analysis Guide, page 1 of 1)

Beyond the Common Core: A Handbook for Mathematics in a PLC at Work™ (Grades K–5, Grades 6–8, High School, and Leader’s Guide)

© Solution Tree Press (forthcoming). solution-tree.comDo not duplicate.

10

22

School Store: Grade 1

Carl earned 15 tickets to spend at the school store. The numbers of tickets needed to buy each item are in the chart.

Item Number of Tickets Needed Sticker 1 Pencil 3 Bookmark 5

1. Carl wants to buy at least one of each item. He needs to use all of his tickets.

How many of each item might he buy? Find at least two different ways Carl can spend his tickets.

2. Carl thinks he can buy two of each item. Is he correct? Use numbers, words or

pictures to explain your answer.

23

Shifts in Classroom Practice

Shift 1: From same instruction toward differentiated instruction.

Shift 2: From students working individually toward community of learners.

Shift 3: From mathematical authority coming from the teacher or textbook toward mathematical authority coming from sound student reasoning.

Shift 4: From teacher demonstrating ‘how to’ toward teacher communicating ‘expectations’ for learning.

Shift 5: From content taught in isolation toward content connected to prior knowledge.

Shift 6: From focus on correct answer toward focus on explanation and understanding.

Shift 7: From mathematics-made-easy for students toward engaging students in productive struggle.

Differentiated instruction but same learning outcomes for all students.

Same instruction for all students.

Community of learners where students hear, share, and judge reasonableness of strategies and solutions.

Students work individually on tasks and seek feedback from teacher on reasonableness of strategies and solutions.

Correctness of solution is based on reasoning about the accuracy of the solution strategy.

Correctness of solutions is determined by seeking input from teacher or textbook.

Teacher facilitates high-‐level performance by sharing learning goals and expectations for products that demonstrate learning.

Teacher demonstrates the way in which to solve a problem and helps students in solving the problem in that way.

Discussions and classroom routines focus on student explanations that address why an answer is (or isn’t) correct.

Discussions and classroom routines focus on student explanation of how they solved a task and if it is correct.

Content presented in ways where explicit attention is given to making connections among mathematical ideas.

Content presented independent of its connections to what has been previously learned.

Teacher poses tasks and challenges students to persevere and attempt multiple approaches to solving problems.

Mathematics is presented in small chunks and help is provided so that students reach solutions quickly and without higher level thinking.

Used with permission. From “Coaching Tools for Supporting the Common Core State Standards for Mathematical Practice,” presented at the NCSM Annual Conference Philadelphia, PA,

Monday, April 23, 2012, by Maggie B. McGatha and Jennifer M. Bay-Williams.

Shifts in Classroom Practice

Shift 1: From same instruction toward differentiated instruction.

Shift 2: From students working individually toward community of learners.

Shift 3: From mathematical authority coming from the teacher or textbook toward mathematical authority coming from sound student reasoning.

Shift 4: From teacher demonstrating ‘how to’ toward teacher communicating ‘expectations’ for learning.

Shift 5: From content taught in isolation toward content connected to prior knowledge.

Shift 6: From focus on correct answer toward focus on explanation and understanding.

Shift 7: From mathematics-made-easy for students toward engaging students in productive struggle.

Differentiated instruction but same learning outcomes for all students.

Same instruction for all students.

Community of learners where students hear, share, and judge reasonableness of strategies and solutions.

Students work individually on tasks and seek feedback from teacher on reasonableness of strategies and solutions.

Correctness of solution is based on reasoning about the accuracy of the solution strategy.

Correctness of solutions is determined by seeking input from teacher or textbook.

Teacher facilitates high-‐level performance by sharing learning goals and expectations for products that demonstrate learning.

Teacher demonstrates the way in which to solve a problem and helps students in solving the problem in that way.

Discussions and classroom routines focus on student explanations that address why an answer is (or isn’t) correct.

Discussions and classroom routines focus on student explanation of how they solved a task and if it is correct.

Content presented in ways where explicit attention is given to making connections among mathematical ideas.

Content presented independent of its connections to what has been previously learned.

Teacher poses tasks and challenges students to persevere and attempt multiple approaches to solving problems.

Mathematics is presented in small chunks and help is provided so that students reach solutions quickly and without higher level thinking.

Used with permission. From “Coaching Tools for Supporting the Common Core State Standards for Mathematical Practice,” presented at the NCSM Annual Conference Philadelphia, PA,

Monday, April 23, 2012, by Maggie B. McGatha and Jennifer M. Bay-Williams.

11824

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25

How do we know students have learned the CCSSM?

CCSS and Assessments

•  Smarter Balanced will be given once in the last 12 weeks of school for Grades 3, 4, 5, 6, 7, 8, 11

•  Finalized Achievement Level Descriptors will be released at the end of October 2014

•  Two assessments: – CAT (Computer Adap7ve Test) – Performance Task

Four Claims Used in DRAFT SBAC Test Specifica+ons

Claim #1 Concepts & Procedures

Claim #2 Problem Solving -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Claim #4 Modeling & Data

Analysis

Claim #3 Communica+ng

Reasoning

Students can explain and apply mathema+cal concepts and interpret and carry out mathema+cal procedures with precision and fluency.

Students can clearly and precisely construct viable arguments to support their own reasoning and to cri+que the reasoning of others.

Students can solve a range of complex well-‐posed problems in pure and applied mathema+cs, making produc+ve use of knowledge and problem solving strategies. -‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐ Students can analyze complex, real-‐world scenarios and can construct and use mathema+cal models to interpret and solve problems.

40%

60%

26

Pu]ng it All Together §  Design a lesson using the Lesson Planning Tool and a Digging Into Math Lesson. §  How will you emphasize a mathema7cal prac7ce? § What are your assessing and advancing ques7ons?

§  How will the lesson begin and end? § What are students doing during the lesson?

§  Find/Create a high cogni7ve task to use in the first unit with students. §  How will you also teach a mathema7cal prac7ce?

27

Mathematics Guidelines for Inclusion of

Measurement Conversions and Formulas

1

Conversions or formulas that are allowable with a particular item type will be given either in the item

stem or in the stimulus for a performance task.

Measurement Conversions

LIST A: The measurement conversions listed here are not included in directions for any item type at any grade level in either the Smarter Balanced summative or interim assessments.

1 foot = 12 inches

1 yard = 3 feet

1 meter = 100 centimeters

1 kilometer = 1000 meters

1 kilogram = 1000 grams

1 liter = 1000 milliliters

1 hour = 60 minutes

1 minute = 60 seconds

1 week = 7 days

LIST B: The measurement conversions listed here are given only in performance task stimuli, and not

in computer adaptive testing (CAT) items. The first grade level at which tasks are eligible for the given

conversion is listed in brackets, and the conversion is then allowable in performance tasks at subsequent grades as appropriate to the task. Item authors must still attend to the language as

written in the standards for a particular grade level (e.g., attending to limits within the standard such as expressing a larger unit in terms of a smaller unit).

1 cup = 8 fluid ounces [Grade 4]

1 mile = 5280 feet [Grade 5]

1 mile = 1760 yards [Grade 5]

1 pound = 16 ounces [Grade 4]

1 pint = 2 cups [Grade 4]

1 quart = 2 pints [Grade 4]

1 gallon = 4 quarts [Grade 4]

All measurement conversions not appearing in List A or List B may be given in CAT item stems and performance task stimuli as appropriate for a particular item or task with respect to the standards

for the item or task’s intended grade level.

Formulas

LIST C: The formulas listed here are not included for any item type at any grade level in the Smarter

Balanced summative or interim assessments.

Perimeter or area of a rectangle

Perimeter or area of a parallelogram

Perimeter or area of a trapezoid

Perimeter or area of a square

Perimeter or area of a triangle

Circumference or area of a circle

Arc length

Area of a sector of a circle

Sum of interior angles of a polygon

Measure of one interior angle of a regular polygon

28

Mathematics Guidelines for Inclusion of

Measurement Conversions and Formulas

2

Surface area of a rectangular prism

Surface area of a cube

Surface area of right pyramids with a rectangular or triangular base

Volume of a right rectangular prism

Volume of a cube

Quadratic formula

Pythagorean Theorem

LIST D: The formulas listed here are given only in performance task stimuli, and not in computer

adaptive testing items. The first grade level at which tasks are eligible for the given formula is listed in brackets.

Volume of a cone [HS1]

Volume of a sphere [HS]

Volume of a cylinder [HS]

All formulas not appearing in List C or List D may be given in CAT item stems and performance task stimuli as appropriate for a particular item or task with respect to the standards for the item or task’s

intended grade level.

Angle Measurement Facts and Ratios

LIST E: The angle measurement facts and ratios listed here are not included for any item type at any grade level in the Smarter Balanced summative or interim assessments.

Number of degrees in a circle

Sum of interior angles of a triangle

Side length ratios in a 45-45-90 triangle

Side length ratios in a 30-60-90 triangle

Trigonometric ratios for sin, cos, and tan

All angle measurement facts and ratios not appearing in List E may be given in CAT item stems and

performance task stimuli as appropriate for a particular item or task with respect to the standards for the item or task’s intended grade level.

1 Since applications of volume formulas are not considered major work in grade 8, performance tasks that measure more complex applications of such formulas are reserved for high school.

29

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide Visit go.solution-tree.com/commoncore to download this page.

Figure 2.12: CCSS Mathematical Practices Lesson-Planning Tool

Unit: Date: Lesson:

Learning target: As a result of today’s class, students will be able to

Formative assessment: How will students be expected to demonstrate mastery of the learning target during

Probing Questions for Differentiation on Mathematical Tasks

Assessing Questions

(Create questions to scaffold instruction for students who are “stuck” during the lesson or the lesson tasks.)

Advancing Questions

(Create questions to further learning for students who are ready to advance beyond the learning target.)

Targeted Standard for Mathematical Practice:

Which Mathematical Practice will be targeted for pro�ciency development during this lesson?

Tasks

(Tasks can vary from lesson to lesson.)

What Will the Teacher Be Doing?

(How will the teacher present and then monitor student response to the task?)

What Will the Students Be Doing?

(How will students be actively engaged in each part of the lesson?)

Beginning-of-Class Routines

connect to students’ prior knowledge, or how is it based on analysis of homework?

page 1 of 2

R E PRO D UCI B LE



Unit: Date: Lesson:


Formative assessment: How will students be expected to demonstrate mastery of the learning target during


Assessing Questions


Advancing Questions



Which Mathematical Practice will be targeted for pro�ciency development during this lesson?

Tasks







connect to students’ prior knowledge, or how is it based on analysis of homework?

page 1 of 2

REPRODUCIBLEREPRODUCIBLEREPRODUCIBLE

11630

hannah.kirchner

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Students will work in pairs, but each will write equations on a whiteboard to show the teacher and other students in the class. Students will write share and discuss their observations.

hannah.kirchner

Text Box

Add./Sub. 1/27/14 Introduction to Fact Families (Grade 1)

hannah.kirchner

Text Box

Write related addition and subtraction equations.

hannah.kirchner

Text Box

Show me 7 on a ten frame (or using cubes or a number line). What are two numbers that add to 7? Show me 7 on a ten frame (or using cubes or a number line). Show me 7 - 3. What is the difference? If a student struggles to write, allow them to explain verbally while practicing writing.

hannah.kirchner

Text Box

How many different fact families can you make that have sums of 7? Write a fact family using the numbers 5, 6, and 11. What other number could be in a fact family with 4 and 8? How many different equations are in a fact family with 4, 4, and 8?

hannah.kirchner

Text Box

MP 6: Attend to precision. (Students will write equations with correct symbols and numerals.) MP 7 Look for and make use of structure. (Students will recognize the structures of addition and subtraction and the relationship between the two.)

hannah.kirchner

Text Box

At the carpet, ask the students to find as many sums to 7 as possible.

hannah.kirchner

Text Box

At the carpet with a partner ask: What is a family? Facilitate a discussion and note relationships in families and relationships between addition and subtraction in fact families we will see today. Prompt students to write sums to 7 on white boards. Teacher writes student sums on board horizontally.

hannah.kirchner

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Describe a family to a partner. Students write sums to 7 on white boards and then share their sums.

R E PRO DUCI B LE


Tasks






Task 1

How will the students be engaged in understanding the learning target?

Task 2

How will the task develop student sense making and reasoning?

Task 3

How will the task require student conjectures and communication?

Closure

How will student questions and re�ections be elicited in the summary of the lesson? How will students’ understanding of the learning target be determined?

page 2 of 2

R E PRO DUCI B LE


Tasks






Task 1


Task 2


Task 3


Closure

How will student questions and re�ections be elicited in the summary of the lesson? How will students’ understanding of the learning target be determined?

page 2 of 2

REPRODUCIBLEREPRODUCIBLEREPRODUCIBLE

11731

hannah.kirchner

Text Box

Teacher observes, questions and guides students as they write. Teacher has student share one that matches an equation on the board. Put a star above the equation. (e.g., 4 + 3 = 7) Teacher has another student share a sum. (e.g., 3 + 4 = 7) Ask: What do you notice? Repeat above actions and list equations beneath the sums. Ask: What do you notice?

hannah.kirchner

Text Box

Give students a flap card with a sum of 7 that differs from 3 + 4. Not all students have the same addends and sum shown.

hannah.kirchner

Text Box

Monitor pairs as they write a fact family using their flap card. First ask for the addition equations and then the subtraction equations. Record answers on the board. Put a circle around each fact family to show the families with sums of 7.

hannah.kirchner

Text Box

Show students the flap card with 3 black dots and 4 white dots and a 7 on the outside flap. Ask: Write an addition equation to match this card (students have used flap cards before). Use the flap card to have students write subtraction equations starting with 7.

hannah.kirchner

Text Box

Students work in pairs and each write a sum on their whiteboards. Share answer when selected. Students notice the commutative property they have already been learning. Students write and share differences. They notice all of the equations in the family use the same numbers in different orders.

hannah.kirchner

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Students write addition and subtraction equations to match their flap cards.

hannah.kirchner

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Ask students to describe what they think a fact family is. Ask: I have 2 stickers. Mandy has some stickers. Together we have 6 stickers. How many stickers does Mandy have?

hannah.kirchner

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Facilitate a class discussion. Have students listening to one another and speaking to the group. Have pairs write an equation to show their thinking to the word problem. Record the addition: 2 + ? = 6 [Record 2 + 4 = 6] Record the subtraction: 6 - 2 = ? [Record 6 - 2 = 4]

hannah.kirchner

Text Box

Students discuss and create a definition of a fact family. Students solve the problem and in the discussion notice they can solve the problem using addition or subtraction and that the equations are part of a fact family.

hannah.kirchner

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Have students use their thinking to write a fact family using 2, 3, and 5 on a notecard.

hannah.kirchner

Text Box

Watch what students write. Facilitate a discussion that includes the answer.

hannah.kirchner

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Write the fact family on a card to use a reference tomorrow. Discuss how they knew the fact family.

R E PRO DUCI B LE

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Unit: Date: Lesson:


Formative assessment: How will students be expected to demonstrate mastery of the learning target during in-class checks for understanding?


Assessing Questions


Advancing Questions



Which Mathematical Practice will be targeted for proficiency development during this lesson?

Tasks







How does the warm-up activity connect to students’ prior knowledge, or how is it based on analysis of homework?

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R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Tasks






Task 1


Task 2


Task 3


Closure

How will student questions and reflections be elicited in the summary of the lesson? How will students’ understanding of the learning target be determined?

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Kindergarten Task

http://www.k-5mathteachingresources.com

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11 12 13

14 15 1635

17 18 19

20Print the cards on cardstock. Laminate, cut out and store in a zip-lock bag. Students can:- match dot cards with numeral cards- order cards from least to greatest/greatest to least- place cards facedown in two rows and play Memory with a partner .

You may wish to consider:- printing the dot and numeral cards on different colored card. Alternatively, put a large on theback of the dot cards so that students can differentiate between dot and numeral cards when thecards are facedown.- placing blank paper and markers in the center for students who wish to record their work.


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What basic fact could you use to find the

product of 30 x 4? Find the product.

3. A bus driver picked up passengers at the first

stop. 32 more people got on the bus at the second

stop. If there were a total of 56 people on the

bus, how many people got on at the first stop?

3. Peter read 10 pages of his book before lunch.

After lunch he read some more. If Peter read

19 pages in all, how many pages did he read

after lunch?

2. Tess picked 7 flowers on Monday morning.

In the afternoon she picked some more flowers.

If Tess picked 15 flowers in all, how many did she

pick in the afternoon?

1. Peter had 6 marbles. Mike gave Peter

some more marbles and then he had 13 in all. How

many marbles did Mike give Peter?

4. Jess gave out 8 cookies to girls in her class.

She then gave out cookies to boys in her

class. If Jess gave out 14 cookies in all, how many

cookies did she give out to boys?

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Grade 1 Task


7. Jack ate 12 grapes at morning snack time and

more grapes at lunch time. If Jack ate 20 grapes

in all, how many grapes did he eat at lunch time?

6. A bakery sold 11 birthday cakes before midday.

After midday more birthday cakes were sold.

If 16 birthday cakes were sold in all, how many

were sold after midday?

5. Dad baked cakes for the Bake Sale. He baked

9 cakes before breakfast and more cakes

after breakfast. If dad baked 15 cakes in all,

how many cakes did he bake after breakfast?

8. On Monday 5 caterpillars hatched from eggs. On

Tuesday some more caterpillars hatched. If 12

caterpillars hatched in all, how many caterpillars

hatched on Tuesday?

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There were 46 balls on the playground. 20 were soccer balls and 15 were basketballs. The rest were tennis balls. How many were tennis balls?

At the park I saw 32 animals. I saw 12 dogs, 15 squirrels, and some frogs. How many frogs did I see?

Dad caught 22 fish in the morning. He threw 5 back because they were too small. He caught 12 more in the afternoon. How many fish did dad have then?

30 children lined up to jump rope. 9 children joined them. 4 children left to get a drink of water. How many children were left in the line?

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Grade 2 Task


I drew a pattern with 32 triangles and 34 squares. I erased 4 shapes. How many shapes were left?

I read 25 pages of my book yesterday and 41 pages today. I need to read 14 more pages to finish my book. How many pages are in my book?

Ben has 22 red and 22 blue marbles. Tim has 19 marbles. How many more marbles does Ben have than Tim?

I spent 25 cents on gum, 9 cents on a pencil, and 50 cents on a drink. How much money did I spend in all?

©K-5MathTeachingResources.com 40


3 – 2 – 1 Reflection List 3 things to remember when planning a lesson. List 2 questions you will use during instruction. List 1 instructional shift you will work to incorporate during instruction.

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PK#–#2#Mathematics# - malesd.k12.or.us Math... · objects in a set; counting out a given number of ... 5 = o – 3, 6 + 6 = o. number and ... digits of a two-digit number represent