# USD 305 Kindergarten Math Curriculum

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U
SD 305
Kindergarten

Math

Curriculum

In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and operating on whole nu
mbers, initially with sets
of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to num
ber than to other topics.

(1) Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as cou
nting objects in a
set; counting out a given number of objects; comparing sets or numerals; and modelin
g 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 Kin
dergarten is encoura
ged, but it is not required.) Students choose, combine, and apply
ing
quantitative questions, including quickly recognizing the cardinalities of small se
ts of objects, counting and pro
ducing 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) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatia
l relations) and vo
cabulary. They iden
tify,
name, and describe basic two
-
dimensional shapes, such as squa
res, triangles, circles, rectan
gles, 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, cy
linders, and spheres. They use basic shapes and
spatial reasoning to model objects in their environment and to construct more complex shapes.

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MATHEMATICS STANDARDS ARTICULATED BY GRADE LEVEL

Standards for Mathematical Practice

Standards

Explanations

and Examples

Students are expected
to:

Mathematical Practices

are
level document in the 2nd
column to reflect the need
to connect the mathematical
practices to mathematical
content in instruction.

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

In Kindergarten, students begin to build the understanding that doing mathematics
involves solving problems and discussing how they solved them.
Students explain to
themselves the meaning of a problem and look for
ways to solve it. Younger students
may use concrete objects or pictures to help them conceptualize and solve problems.

They may check their thinking by asking themselves, “Does this make sense?” or they

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skills as they participate in mathematical discussions involving questions like “How did
you get that?” and “Why is that true?”

to others’ thinkin

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Standards for Mathematical Practice

Standards

Explanations

and Examples

Students are expected
to:

Mathematical Practices

are
level document in the 2nd
column to reflect the need
to connect the mathematical
practices to mathematical
content in instruction.

K.MP.5. Use
appropriate tools
strategically.

Younger students begin to
consider the available tools (including estimation) when
solving a mathematical problem and decide when certain tools might be helpful. For
instance, kindergarteners may decide that it might be advantageous to use linking cubes
to represent two quantities
and then compare the two representations side
-
by
-
side.

K.MP.6. Attend to
precision.

As kindergarteners

begin to develop their mathematical communication skills, they try to
use clear and precise language in their discussions with others and in their own
reasoning.

K.MP.7. Look for and
make use of structure.

Younger students begin to discern a pattern or structure. For instance,
students recognize
the pattern that exists in the teen numbers; every teen number is written with a 1
(representing one ten)
and ends with the digit that is first stated. They also recognize that
3 + 2 = 5 and 2 + 3 = 5.

K.MP.8. Look for and
express regularity in
repeated reasoning.

In the early grades, students notice repetitive actions in counting and computation, etc.
For e
xample, they may notice
that the next number in a counting sequence is one more.
When counting by tens, the next number in the sequence is “ten more” (or one more
group of ten). In addition, students continually check their work by asking themselves
,
“Does

this make sense?”

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This is the Kindergarten

Curriculum for Mathematics. The focus of this document is to provide instructional strategies and resources related to the
standards.

Subject

Domain

Standard #

Domain

Cluster

Standard

M

K

CC

1

Counting and
Cardinality

Know number names and
the count sequence.

Count to 100 by ones and by tens.

M

K

CC

2

Counting and
Cardinality

Know number names and
the count sequence.

Count forward beginning from a given number within the known sequence
(instead of having to begin at 1).

M

K

CC

3

Counting and
Cardinality

Know number names and
the count sequence.

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).

M

K

CC

4

Counting and
Cardinality

Count to tell the number of
objects.

Understand the relationship between numbers and

quantities; connect counting
to cardinality.

M

K

CC

4a

Counting and
Cardinality

Count to tell the number of
objects.

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.

M

K

CC

4b

Counting and
Cardinality

Count to tell the number of
objects.

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

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Subject

Domain

Standard #

Domain

Cluster

Standard

M

K

CC

4c

Counting and
Cardinality

Count to tell the number of
objects.

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

M

K

CC

5

Counting and
Cardinality

Count to tell the number of
objects.

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
.

M

K

CC

6

Counting and
Cardinality

Compare numbers.

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. (Include groups with up to ten objects.)

M

K

CC

7

Counting and
Cardinality

Compare numbers.

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

M

K

OA

1

Operations
and Algebraic
Thinking

to, and underst
and
subtraction as taking apart
and taking from.

Represent addition and subtraction with objects, fingers, mental images,
drawings

(drawings need not show details, but should show the mathematics
in the problem)
, sounds

(e.g., cla
ps)
, acting out situations, verbal explanations,
expressions, or equations.

M

K

OA

2

Operations
and Algebraic
Thinking

to, and understand
subtraction as taking apart
and taking from.

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

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Subject

Domain

Standard #

Domain

Cluster

Standard

M

K

OA

3

Operations
and Algebraic
Thinking

to, and understand
subtraction as taking apart
and taking from.

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).

M

K

OA

4

Operations
and Algebraic
Thinking

to, and understand
subtraction as taking apart
and taking from.

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.

M

K

OA

5

Operations
and Algebraic
Thinking

to, and understand
subtraction as taking apart
and taking from.

Fluently add and subtract within 5.

M

K

NBT

1

Number and
Operations in
Base 10

Work with
numbers 11
-
19
to gain foundations for place
value.

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 eq
uation (such as 18 = 10 + 8); understand
that these numbers are composed of ten ones and one, two, three, four, five,
six, seven, eight, or nine ones.

M

K

MD

1

Measurements
and Data

Describe and compare
measurable attributes.

Describe measurable attributes of objects, such as length or weight. Describe
several measurable attributes of a single object.

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Subject

Domain

Standard #

Domain

Cluster

Standard

M

K

MD

2

Measurements
and Data

Describe and compare
measurable attributes.

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.

M

K

MD

3

Measurements
and Data

Classify objects and count
the number of objects in
each category.

Classify objects into given categories; count the numbers of objects in each
category and sort the categories by

count.

(Limit category counts to be less
than or equal to 10.)

M

K

G

1

Geometry

Identify and describe shapes
(squares, circles, triangles,
rectangles, hexagons, cubes,
cones, cylinders, and
spheres).

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.

M

K

G

2

Geometry

Identify and describe shapes
(squares, circles, triangles,
rectangles, hexagons, cubes,
cones, cylinders, and
spheres).

Correctly name shapes regardless of their orientations or overall size.

M

K

G

3

Geometry

Identify and describe shapes
(squares, circles, triangles,
rectangles, hexagons, cubes,
cones, cylinders, and
spheres).

Identify shapes as two
-
dimensional (lying in a plane, “flat”) or three
-
dimensional (“solid”).

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Subject

Domain

Standard #

Domain

Cluster

Standard

M

K

G

4

Geometry

Analyze, compare, create,
and compose shapes.

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 o
f equal length).

M

K

G

5

Geometry

Analyze, compare, create,
and compose shapes.

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

and drawing shapes.

M

K

G

6

Geometry

Analyze, compare, create,
and compose shapes.

Compose simple shapes to form larger shapes.
For example, "can you join
these two triangles with full sides touching to make a rectangle?”

(H
OME)

Counting and Car
dinality

M.
K
.
CC

Know number names and the count sequence.

Instructional Strategies for Cluster

The Counting and Cardinality domain in Kindergarten contains standard statements that are connected to one another. Examine t
he three samples in
this domain at the same time to obtain a more holistic view of the content.

Provide settings that connect math
ematical language and symbols to the everyday lives of kindergarteners. Support students’ ability to make meaning
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“How do you know?

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Counting on or counting from a given number conflicts with the learned strategy of counting from the beginning. In order to b
e successful in
counting on, students must understand cardinality. Students often merge or separate two groups of objects and then
re
-
count from the beginning to
determine the final number of objects represented. For these students, counting is still a rote skill or the benefits of coun
ting on have not been
realized. Games that require students to add on to a previous count to reach a

goal number encourage developing this concept. Frequent and brief
opportunities utilizing counting on and counting back are recommended. These concepts emerge over time and cannot be forced.

Like counting to 100 by either ones or tens, writing numbers f
rom 0 to 20 is a rote process. Initially, students mimic the actual formation of the
written numerals while also assigning it a name. Over time, children create the understanding that number symbols signify the

meaning of counting.
Numerals are used to com
municate across cultures and through time a certain meaning. Numbers have meaning when children can see mental images
of the number symbols and use those images with which to think. Practice count words and written numerals paired with picture
s, representa
tions of
objects, and objects that represent quantities within the context of life experiences for kindergarteners. For example, dot c
ards, dominoes and number
cubes all create different mental images for relating quantity to number words and numerals.

One way students can learn the left to right orientation of numbers is to use a finger to write numbers in air (sky writing).

Children will see
mathematics as something that is alive and that they are involved.

Common Misconceptions

Some students might not see zero as a number. Ask students to write 0 and say
zero
to represent the number of items left when all items have been
taken away. Avoid using the word
none
to represent this situation.

(HOME)

M.
K.CC
.1

Count to 100 by ones and by tens.

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

All year

Bloom’s Level:

knowledge

What does this standard mean that a student will know and be able to do?

This
stand
ard

calls for students to rote count by starting at one and count to 100. When students count by tens they are only expected to m
aster
counting on the decade (0, 10, 20, 30, 40 …). This objective does not require recognition of numerals. It is focused
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Mathematical Practices:

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in r
epeated reasoning.

Vocabulary:

count

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

What does a number represent?

How many are there?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

The emphasis of this standard is on the counting sequence.

When counting by ones, students need to understand that the next number in the sequence is one more. When counting by tens, t
he next number in
the sequence is “ten more” (or one more group of ten)
.

Instruction on the counting sequence should be
scaffold

(e.g., 1
-
10, then 1
-
20, etc.).

Counting should be reinforced throughout the day, not in isolation.

Examples:

Count the number of chairs of the students who are absent.

Count the number of stairs,
shoes, etc.

Counting groups of ten such as “fingers in the classroom” (ten fingers per student).

When counting orally, students should recognize the patterns that exist from 1 to 100. They should also recognize the pattern
s that exist when
counting by 10
s.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Oral assessment

Instructional Resources/Tools:

Board games that require counting

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg School
Division, 2005
-
2006

Mathematics Learning in Early Childhood
: Paths Toward Excellence and Equity

(HOME)

M.K.CC.2

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Count forward beginning from a given number within the known sequence (instead of having
to begin at 1).

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

All year

Bloom’s Level:

knowledge

What does this standard mean that a student will know and be able to do?

This standard includes
numbers 0 to 100. This asks for students to begin a rote forward counting sequence from a number other than 1. Thus, given
the number 4, the student would count, “4, 5, 6 …” This objective does not require recognition of numerals. It is focused on
the rote

number
sequence.

I can count to 10.

I can count to 100.

I can count on from a number other than 1 up to 100.

Mathematical Practices:

K.MP.7. Look for and make use of structure.

Vocabulary:

count

Essential Questions:

(
What provocative questions will
foster inquiry, understanding, and transfer learning?
)

What does a number represent?

How many are there?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

The emphasis of this standard is on the counting sequence to 100. Students should be able to
count forward from any number, 1
-
99.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Oral assessment

Instructional Resources/Tools:

Board games that require counting

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Mathematics Learning in Early Childhood
: Paths Toward Excellence and Equity

(HOME)

M.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).

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:

Quarter 4:

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Teaching Time:

10 minutes/day

Bloom’s Level:

Understanding

What does this standard mean that a
student will know and be able to do?

This standard
addresses the writing of numbers and using the written numerals (0
-
20) to describe the amount of a set of objects. Due to varied
development of fine motor and visual development, a reversal of numerals is anticipated for a majority of the students. While

r
eversals should be
pointed out to students, the emphasis is on the use of numerals to represent quantities rather than the correct handwriting f
ormation of the actual
numeral itself.

This standard
asks for students to represent a set of objects with a wr
itten numeral. The number of objects being recorded should not be greater than
20. Students can record the quantity of a set by writing the numeral. Students can also create a set of objects based on the
numeral presented.

Mathematical Practices:

K.MP.2.
Reason abstractly and quantitatively.

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in repeated reasoning.

Vocabulary:

count, number, match

Essential Questions:

(
What provocative questions will foster inquiry, und
erstanding, and transfer learning?
)

What does a number represent? How many are there?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Students should be given multiple opportunities to count objects and recognize that a number represents a specific

quantity. Once this is established,
students begin to read and write numerals (numerals are the symbols for the quantities). The emphasis should first be on quan
tity and then connecting
quantities to the written symbols.

A sample unit sequence might inclu
de:

1.

Counting up to 20 objects in many settings and situations over several weeks.

2.

Beginning to recognize, identify, and read the written numerals, and match the numerals to given sets of objects.

3.

Writing the numerals to represent counted objects.

Since the

teen numbers are not written as they are said, teaching the teen numbers as one group of ten and extra ones is foundational t
o
understanding both the concept and the symbol that represents each teen number. For example, when focusing on the number “14,
” s
tudents
should count out fourteen objects using one
-
to
-
one correspondence and then use those objects to make one group of ten and four extra ones.
Students should connect the representation to the symbol “14.”

Assessments:

(
What will be acceptable evidenc
e the
student has achieved the desired results?
)

Instructional Resources/Tools:

Board games that require counting

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Independently writing numbers to 20

Counting objects and writing the correct number

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Mathematics Learning in Early Childhood
: Paths Toward Excellence and Equity

(HOME)

Counting and Cardinality M.K.CC

Count to tell the number of objects.

Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as countin
g objects in a set;
counting out a given number of objects and comparing sets or numerals.

Instructional Strategies for Cluster

On
e of the first major concepts in a student’s mathematical development is cardinality. Cardinality, knowing that the number wo
rd said tells the
quantity you have and that the number you end on when counting represents the entire amount counted. The big idea

is that number means amount
and, no matter how you arrange and rearrange the items, the amount is the same. Until this concept is developed, counting is
merely a routine
procedure done when a number is needed. To determine if students have the cardinality

rule, listen to their responses when you discuss counting
mple, ask, “How many are here?”

The student counts correctly and says that there are seven. Then ask, “Are there seven?”.

Students may count or hesitate if they have not developed cardinality. Students with cardinality may emphasize the last count

or explain that there
are seven because they counted them. These students can now use counting to find a matching set.

Students

develop the understanding of counting and cardinality from experience. Almost any activity or game that engages children in c
ounting and
comparing quantities, such as board games, will encourage the development of cardinality. Frequent opportunities to us
e and discuss counting as a
means of solving problems relevant to kindergarteners is more beneficial than repeating the same routine day after day. For e
questions that can be answered by counting up to 20 items before they change and a
s they change locations throughout the school building.

As students develop meaning for numerals, they also compare numerals to the quantities they represent. The models that can re
present numbers, such
as dot cards and dominoes, become tools for such co
mparisons. Students can concretely, pictorially or mentally look for similarities and differences
in the representations of numbers. They begin to “see” the relationship of one more, one less, two more and two less, thus la
nding on the concept that
success
ive numbers name quantities that are one larger. In order to encourage this idea, children need discussion and reflection of
pairs of numbers
from 1 to 10. Activities that utilize anchors of 5 and 10 are helpful in securing understanding of the relationshi
ps between numbers. This flexibility
with numbers will build students’ ability to break numbers into parts.

Provide a variety of experiences in which students connect count words or number words to the numerals that represent the qua
ntities. Students wil
l
arrive at an understanding of a number when they acquire cardinality and can connect a number with the numerals and the numbe
r word for the
quantity they all represent.

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Common Misconceptions

Some students might think that the count word used to tag an i
tem is permanently connected to that item. So when the item is used again for counting
and should be tagged with a different count word, the student uses the original count word. For example, a student counts fou
r geometric figures:
triangle, square, circl
e and rectangle with the count words: one, two, three, four. If these items are rearranged as rectangle, triangle, circle and

square
and counted, the student says these count words: four, one, three, two.

(HOME)

M.K.CC.4

Understand the relationship between numbers and quantities; connect counting to cardinality.

Quarter Taught:

Quarter 1:

X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

2 times a week for about 15 mins.

Bloom’s Level:

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Le癥氠l

What does this standard mean that a student will know and be able to do?

This standard
asks students to count a set of objects and see sets and numerals in relationship to one another, rather than as isolated num
bers or sets.
These
connections are higher
-
level skills that require students to analyze, to reason about, and to explain relationships between numbers and sets of
objects. This standard should first be addressed using numbers 1
-
5 with teachers building to the numbers 1
-
10 la
ter in the year. The expectation is
that students are comfortable with these skills with the numbers 1
-
10 by the end of Kindergarten.

Mathematical Practices:

K.MP.2. Reason abstractly and quantitatively.

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in repeated re atoning.

Vocabulary:

number, group, set, greater than, less than, equal,

more, fewer, same, most, least

Essential Questions:

(
What provocative questions will foster inquiry, under
standing, and
transfer learning?

o

Can students identify the numeric symbol and word form of a given amount of objects in a set?

o

Can students accurately name the quantity of a set of objects regardless of their physical arrangement?

o

When given a number, can students draw
the amount of objects that accurately depict that numeric value?

o

Are students able to match the word form or numeric symbol with a set of objects?

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15

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65

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

This standard focuses on one
-
to
-
one correspondence and h
ow cardinality connects with quantity.

For example, when counting three bears, the student should use the counting sequence, “1
-
2
-
3,” to count the bears and recognize that “three”
represents the group of bears, not just the third bear. A student may use an

interactive whiteboard to count objects, cluster the objects, and
state, “This is three”.

Students need opportunities to connect number words (orally) and the quantities they represent.

Students should have opportunities to arrange a set of objects in a v
ariety of ways and be shown that the arrangement of those objects does not
change its quantity, numeric symbol or name in word form.

Students need a variety of games and activities which will provide ample opportunities for matching a set of objects with i
ts corresponding numeric
symbol and its word form.

The use of a variety of manipulatives and technologies is useful as well.

Students should be encouraged to draw their own examples of number sets.

In order to understand that each successive number name refers to a quantity that is one larger, students should have experie
nce counting objects,
placing one more object in the group at a time.

For example, using cubes, the student should count the existi
ng group, and then place another cube in the set. Some students may need to re
-
count from one, but the goal is that they would count on from the existing number of cubes. S/he should continue placing one
more cube at a
time and identify the total number in

order to see that the counting sequence results in a quantity that is one larger each time one more cube is
placed in the group.

A student may use a clicker (electronic response system) to communicate his/her count to the teacher.

Assessments:

(
What wil
l be acceptable evidence
the student has achieved the desired results?
)

One on one demonstrates manipulating objects to
show 2 different numbers. (e
.
g. Roll dice, student
sh
ows number using manipulatives)

Then, states
which is “more, fewer, …”

Instructional Resources/Tools:

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg School
Division, 2005
-
2006

http://teachers.net/lessons/posts/1417.html

*Above can be modified using a variety of manipulatives.

Kim Sutton materials

(HOME)

M.K.CC.4a

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.

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

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16

of
65

Teaching Time:

2 times a week for 15 mins.

Bloom’s Level:

Application

Level 3

What does this standard mean that a student will know and be able to do?

This standard
reflects the ideas that students

implement correct counting procedures by pointing to one object at a time (one
-
to
-
one correspondence)
using one counting word for every object (one
-
to
-
one tagging/synchrony), while keeping track of objects that have and have not been counted.. This
is the

foundation of counting.

Vocabulary:

c
ounting by 1’s,
one to one number touch, object

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

What would happen if I said a number and did not touch the

object?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Students need opportunities to connect number words (orally) and the quantities they represent.

Students should have opportunities to arrange a set of objects in a variety of ways and be shown
that the arrangement of those objects does
not change its quantity, numeric symbol or name in word form.

Students need a variety of games and activities which will provide ample opportunities for matching a set of objects with its

corresponding
numeric sym
bol and its word form.

The use of a variety of manipulatives and technologies is useful as well.

Students should be encouraged to draw their own examples of number sets.

Assessments:

(
What will be acceptable evidence
the student has achieved the desired
results?
)

One on one, student counts aloud while touching
objects

Instructional Resources/Tools:

Dot Card and Ten Frame
Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg School
Division, 2005
-
2006

http://teachers.net/lessons/posts/1315.html

Kim Sutton materials

(HOME)

M.K.CC.4b

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

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17

of
65

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:

Teaching Time:

Two times a week for 15
-
20 mins.

Bloo

’s Level:

Application

What does this standard mean that a student will know and be able to do?

This standard
calls for students to answer the question “How many are there?” by counting objects in a set and understanding that the last
number
stated when counting a set (…8, 9,
10
) represents the total amount of objects: “There are
10
bears in this pile.” (
Cardinali
ty
). It also requires students
to understand that the same set counted three different times will end up being the same amount each time. Thus, a purpose of

keeping track of
objects is developed. Therefore, a student who moves each object as it is counted
recognizes that there is a need to keep track in order to figure out
the amount of objects present. While it appears that this standard calls for students to have conservation of number, (regard
less of the arrangement of
objects, the quantity remains the s
ame), conservation of number is a developmental milestone of which some Kindergarten children will not have
mastered. The goal of this

objective is for students to be able to count a set of objects; regardless of the formation those objects are placed.

V
ocabulary:

counting, groups, sets, How many are there?, total, in all, number of object

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

o

What is a number?”

o

“Why is it important to count objects?”

o

“How many would there be if we added one more object?”

o

Can students identify the numeric symbol and word form of a given amount of objects in a set?

o

Can students accurately name the quantity of a set of objects reg
ardless of their physical arrangement?

o

When given a number, can students draw the amount of objects that accurately depict that numeric value?

o

Are students able to match the word form or numeric symbol with a set of objects?

Instructional
/Learning

Activit
ies
:

(
W.H.E.R.E.T.O.
)

Students need opportunities to connect number words (orally) and the quantities they represent.

Students should have opportunities to arrange a set of objects in a variety of ways and be shown that the arrangement of thos
e objects doe
s
not change its quantity, numeric symbol or name in word form.

Students need a variety of games and activities which will provide ample opportunities for matching a set of objects with its

corresponding
numeric symbol and its word form.

The use of a variety of manipulatives and technologies is useful as well.

Students should be encouraged to draw their own examples of number sets.

Assessments:

(
What will be acceptable evidence
Instructional

Resources/Tools:

Page
18

of
65

the student has achieved the desired results?
)

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg School
Division, 2005
-
2006

http://teachers.net/lessons/posts/1289.html

Kim Sutton materials

(HOME)

M.K.CC.4c

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

Quarter Taught:

Quarter 1:

X

Quarter 4:
X

Teaching Time:

Two times a week for 15
-
20 mins.

Bloom’s Level:

䍲e慴楮g

What does this standard mean that a student will know and be able to do?

This standard
represents the concept of “one more” while counting a set of objects. Students are to make the connection that if a set of ob
jects was
increased by one more object then the number name for that set is to be increased by one as well. Students are asked to u
nderstand this concept with
and without objects. For example, after counting a set of 8 objects, students should be able to answer the question, “How man
y would there be if we
added one more object?”; and answer a similar question when not using objects, b
y asking hypothetically, “What if we have 5 cubes and added one
more. How many cubes would there be then?” This concept should be first taught with numbers 1
-
5 before building to numbers 1
-
10. Students
should be expected to be comfortable with this skill w
ith numbers to 10 by the end of Kindergarten.

Vocabulary:

counting, objects, groups, one more

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

“How many would there be if we added one more
object?”; and answer a similar question when not using objects, by asking hypothetically, “What if
we have 5 cubes and added one more. How many cubes would there be then?”
=
=
䥮獴suc瑩潮慬
⽌e慲n楮g
=
䅣瑩癩瑩ts
W
=
(

F
=
http://www.teacherspayteachers.com/Product/Counting
-
Practice
-
Game
-
Using
-
Common
-
Core
-
Standards
-
for
-
Promethean

ou to

Assessments:

(
What will be acceptable evidence the student has achieved the desired results?
)

One on One setting

Complete the assessment
page found on share portal or click on the link below

Instructional Resources/Tools:

Dot Card and Ten Frame Activities

(pp.
1
-
6, 12
-
17) Numeracy Project, Winnipeg
Page
19

of
65

https://shareportal.usd305.com/elementary/kindergarten/Shared%20Documents/Standard%20MKC
C.4c.d
ocx

School Div
ision, 2005
-
2006

http://teachers.net/lessons/posts/3981.ht
ml

(Dice Toss)

(HOME)

M.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.

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

25
minutes a week

Bloom’s Level:

remember楮g

What does this standard mean that a student will know and be able to do?

This standard
asks students to count a set of objects and see sets and numerals in relationship to one another, rather than as
isolated numbers or sets.
These connections are higher
-
level skills that require students to analyze, to reason about, and to explain relationships between numbers and sets of
objects. This standard should first be addressed using numbers 1
-
5 with teachers

building to the numbers 1
-
10 later in the year. The expectation is
that students are comfortable with these skills with the numbers 1
-
10 by the end of Kindergarten.

Mathematical Practices:

K.MP.2. Reason abstractly and quantitatively.

K.MP.7. Look for
and make use of structure.

K.MP.8. Look for and express regularity in repeated reasoning.

Vocabulary:

count, number, numeral, match, how many

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

What does this number represent?

How many are there?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Students should develop counting strategies to help them organize the counting process to avoid re
-
counting or skipping objects.

Examples:

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20

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65

If items

are placed in a circle, the student may mark or identify the starting object.

If items are in a scattered configuration, the student may move the objects into an organized pattern.

Some students may choose to use grouping strategies such as placing object
s in twos, fives, or tens (note: this is not a kindergarten
expectation).

Counting up to

20 objects should be reinforced when collecting data to create charts and graphs.

A student may use a clicker (electronic response system) to communicate his/her count

to the teacher.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Clickers

Interactive white boards

Using manipulatives, students will count a group of
objects. After being mixed around, students can still
i
dentify how many.

Instructional Resources/Tools:

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Clickers and interactive boards

(HOME)

Counting and Cardinality M.K.CC

Compare numbers.

Instructional Strategies for Cluster

As children develop meaning for numerals, they also compare these numerals to the quantities represented and their number wor
ds. The modeling
numbers with manipulatives such as dot cards and five
-

and ten
-
frames become tools for such comparisons. Children
can look for similarities and
differences in these different representations of numbers. They begin to “see” the relationship of one more, one less, two mo
re and two less, thus
landing on the concept that successive numbers name quantities where one is lar
ger. In order to encourage this idea, children need discussion and
reflection of pairs of numbers from 1 to 10. Activities that utilize anchors of 5 and 10 are helpful in securing understandin
g of the relationships
between numbers. This flexibility with nu
mbers will greatly impact children’s ability to break numbers into parts.

Children demonstrate their understanding of the meaning of numbers when they can justify why their answer represents a quanti
ty just counted. This
justification could merely be the

expression that the number said is the total because it was just counted, or a “proof” by demonstrating a one to
-
one
match, by counting again or other similar means (concretely or pictorially) that makes sense. An ultimate level of understand
ing is reache
d when
children can compare two numbers from 1 to10 represented as written numerals without counting.

Students need to explain their reasoning when they determine whether a number is greater than, less than, or equal to another

number. Teachers need
Page
21

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65

to a
sk probing questions such as “How do you know?” to elicit their thinking. For students, these comparisons increase in difficu
lty, from greater
than to less than to equal. It is easier for students to identify differences than to find similarities.

Common
Misconceptions

(HOME)

M.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.
1

1
Include
groups with up to ten objects.

Quarter Taught:

Quarter 1:

X

Quarter 3:

Te慣h楮朠g業e:

㈵⁭楮u瑥猠愠seek

Bloom’s Level:

remember楮g

What does this standard mean that a student will know and be able to do?

This standard
expects mastery of up to ten objects. Students can use matching strategies (Student 1), counting strategies or equal shares (
Student 3) to
determine whether one group
is greater than
,
less than
, or
equal to
the number of objects in another g
roup (Student 2).

Mathematical Practices:

K.MP.2. Reason abstractly and quantitatively.

Page
22

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65

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in repeated reasoning.

Vocabulary:

greater than, less than, equal, larger,
smaller

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

What does a number represent? How many are there?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Students should develop a strong

sense of the relationship between quantities and numerals before they begin comparing numbers.

Other strategies:

Matching: Students use one
-
to
-
one correspondence, repeatedly matching one object from one set with one object from the other set to
determine

which set has more objects.

Counting: Students count the objects in each set, and then identify which set has more, less, or an equal number of objects.

Observation: Students may use observation to compare two quantities (e.g., by looking at two sets of o
bjects, they may be able to tell which
set has more or less without counting).

Observations in comparing two quantities can be accomplished through daily routines of collecting and organizing data in disp
lays. Students
create object graphs and pictographs

using data relevant to their lives (e.g., favorite ice cream, eye color, pets, etc.). Graphs may be
constructed by groups of students as well as by individual students.

Benchmark Numbers: This would be the appropriate time to introduce the use of 0, 5 and

10 as benchmark numbers to help students further
develop their sense of quantity as well as their ability to compare numbers.

○ Students

state whether the number of objects in a set is more, less, or equal to a set that has 0, 5, or 10 objects.

Assessmen
ts:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Clickers/interactive white boards

Teacher observation

Instructional Resources/Tools:

Board games

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Manipulatives

Graphing pap
er

(HOME)

Page
23

of
65

M.K.CC.7

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

Quarter Taught:

Quarter 1:

Quarter 2:
X

Quarter 3:

Quarter 4:

Teaching Time:

25 minutes a week

Bloom’s Level:

remembering

What does this standard mean that a student will know and be able to do?

This standard
calls for students to apply their understanding of numerals 1
-
10 to compare one from another. Thus, looking at the numerals 8 and 10,
a student must be

able to recognize that the numeral 10 represents a larger amount than the numeral 8. Students should begin this standard by h
aving
ample experiences with sets of objects (K.CC.3 and K.CC.6) before completing this standard with just numerals. Based on earl
y childhood research,
students should not be expected to be comfortable with this skill until the end of Kindergarten.

Mathematical Practices:

K.MP.2. Reason abstractly and quantitatively.

Vocabulary:

greater than, less than, equal, larger, smaller

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

What does a number represent?

How many are there?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Given two numerals, students should
determine which is greater or less than the other.

Using a number line, students will find the two numbers and identify which is greater/less than.

Students can draw a picture to check and see if their number is greater.

Using dice, students can compare an
d find the number that is greater or less than.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observation

Clickers/Interactive
w
hiteboards

Instructional Resources/Tools:

Board games

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Dice

Number line

(HOME)

Operations and Algebraic Thinking M.K.OA

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24

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65

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

All standards in this cluster should only include numbers through 10

Students will model simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2

= 7 and 7

2 = 5.
n 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 r
ecognizing the
cardinali
ties of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, o
r counting the
number of objects that remain in a set after some are taken away.

Instructional Strategies for Cluster

Provide cont
extual situations for addition and subtraction that relate to the everyday lives of kindergarteners. A variety of situations
can be found in
children’s literature books. Students then model the addition and subtraction using a variety of representations su
ch as drawings, sounds, acting out
situations, verbal explanations and numerical expressions. Manipulatives, like two
-
color counters, clothespins on hangers, connecting cubes and
stickers can also be used for modeling these operations. Kindergarten student
s should see addition and subtraction equations written by the teacher.
Although students might struggle at first, teachers should encourage them to try writing the equations. Students’ writing of
equations in Kindergarten
is encouraged, but it is not requ
ired.

Create written addition or subtraction problems with sums and differences less than or equal to 10 using the numbers 0 to 10
and Table 1 on page 88
of the
Common Core State Standards (CCSS) for Mathematics

for guidance. It is important to use a pro
blem context that is relevant to
kindergarteners. After the teacher reads the problem, students choose their own method to model the problem and find a soluti
on. Students discuss
their solution strategies while the teacher represents the situation with an
equation written under the problem. The equation should be written by
listing the numbers and symbols for the unknown quantities in the order that follows the meaning of the situation. The teache
r and students should
use the words
equal
and
is the same as
interchangeably.

Have students decompose numbers less than or equal to 5 during a variety of experiences to promote their fluency with sums an
d differences less
than or equal to 5 that result from using the numbers 0 to 5. For example, ask students to us
e different models to decompose 5 and record their work
with drawings or equations. Next, have students decompose 6, 7, 8, 9, and 10 in a similar fashion. As they come to understand

the role and meaning
of arithmetic operations in number systems, students
gain computational fluency, using efficient and accurate methods for computing.

The teacher can use backmapping and scaffolding to teach students who show a need for more help with counting. For instance,
a tower of 5 using 2 green a
nd 3 blue linking cubes while you discuss composing and decomposing 5. Have them identify and compare other ways to
make a tower of 5. Repeat the activity for towers of 7 and 9. Help students use counting as they explore ways to compose 7 an
d 9.

Common Mi
sconceptions

Students may over
-
generalize the vocabulary in word problems and think that certain words indicate solution strategies that must be used to fin
d an
answer. They might think that the word
more
always means to add and the words
take away
or
left

always means to subtract. When students use the
words
take away
to refer to subtraction and its symbol, teachers need to repeat students’ ideas using the words
minus
or
subtract
. For example,
students use addition to solve this Take from/Start Unknown pro
blem: Seth took the 8 stickers he no longer wanted and gave them to Anna. Now
Seth has 11 stickers
left
. How many stickers did Seth have to begin with?

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25

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65

If students progress from working with manipulatives to writing numerical expressions and equations, t
hey skip using pictorial thinking. Students
will then be more likely to use finger counting and rote memorization for work with addition and subtraction. Counting forwar
d builds to the concept
tion. However, counting is an inefficient strategy. Teachers need to provide
instructional experiences so that students progress from the concrete level, to the pictorial level, then to the abstract lev
el when learning mathematical
concepts.

(HOME)

M.K.OA.1

Represent addition and subtraction with objects, fingers, mental images, drawings
2
, sounds (e.g., claps), acting out situations, verbal explanations,
expressions, or equations.

2
Drawings need not show details, but
should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)

Quarter Taught:

Quarter 1:

Te慣h楮朠g業e:

Bloom’s Level:

What does this standard

mean that a student will know and be able to do?

Students understand the concepts of addition and subtraction by manipulating concrete objects, drawing depictions of objects,

making sounds to
represent objects (such as with rhythm sticks), expressing
situations verbally, or acting out story problems.

Mathematical Practices:

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

K.MP.2. Reason abstractly and quantitatively.

K.MP.4. Model with mathematics.

K.MP.5. Use appropriate tools
strategically.

Vocabulary:

joining, separating, addition, subtraction, equal, equation, same amount as, mental picture, less, more

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

Page
26

of
65

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

“Stranger in the Woods”

by Dana Islas

Originally featured in Math Solutions Online Newsletter, Issue 38

This lesson is anchored to the award
-
winning book, Stranger in the Woods by Carl Sams and Jean Stoick. The book features photographs of a series
of forest animals coming forward to explore a snowman (the stranger) that has appeared in the woods. After read
ing the story, children will be asked
to solve three key questions:

• Which animals visited the snowman?

• How many visited the snowman in our problem?

• How could you represent the animals to share your thinking with others?

Students use a combination of
pictures, words, and numbers to describe their problem. Samples of student work are included along with the teacher’s
chart for their formative assessment. (Source: Math Solutions)

“Comparing Connecting Cubes”

by Grace M. Burton

In this five
-
part unit, students use connecting cubes and counting stories, such as Ten Black Dots by Donald Crews, to build an understand
ing of
subtraction. There are five different presentations to

explore subtraction. Lesson One uses counting back/counting on to help children generate
simple subtraction models. In Lesson Two children write subtraction problems and model them with cubes. The results are recor
ded in a table so that
the two sets can b
e compared. Hopping on a number line is the central activity for Lesson Three as children find another way to understand
subtraction. Lesson Four uses cubes and a balance beam to illustrate concepts of more and less. Lesson Five focuses on fact f
amilies. A

brief
bibliography of counting stories that work well in a kindergarten classroom is included. (Source: Illuminations, Comparing Co
nnecting Cubes)

Dragon Feet

Excerpt

posted on website for www.mathsolutions.com Math for All: Differentiating Instruction, Grades K

2 by Linda Dacey and Rebeka Eston
Salemi

This lesson uses the picture book, Dragon Feet by Marjorie Jackson as the anchor for this math lesson. The teacher use
s this story about two children
who are celebrating the Chinese Lunar New Year by being part of a dragon costume. The story prompts students to figure out qu
estions such as how
many children are in the costume and how many feet the dragon has. Through a co
mbination of acting out the story and drawings, the children
explore the math involved in this story and the concepts of one
-
to
-
one correspondence and one
-
to
-
two correspondence of objects. Additionally, the
author provides teaching strategies to build diff
erentiation into the lessons to meet her students varied developmental needs. (Source: Math
Solutions)

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Performance assessment

Instructional Resources/Tools:

“Growing Mathematical Ideas in Kindergarten”

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27

of
65

student exploration and provides a
sample counting activity in which children compare
the lengths of their names using connecting cubes and letters. (Source: Math Solutions)

(HOME)

M.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

Quarter Taught:

Quarter 1:

Te慣h楮朠g業e:

Bloom’s Level:

What does this standard mean that a student will know and be able to do?

Students will solve problems in a story format with a specific emphasis on using objects or drawings to determine the solutio
n using numbers 1
-
10.

Teachers need to be aware
of the three types of problems for addition and subtraction: Result Unknown, Change Unknown, and Start Unknown.

*Result Unknown: There are 3 students on the playground. 4 more students show up. How many students are there now?

*Change Unknown:

There are 3 students on the playground. Some more students show up. There are now 7 students. How many students came?

*Start Unknown: There are some students on the playground. 4 more students come. There are now 7 students. How many stude
nts were
on the
playground at the beginning?

Mathematical Practices:

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

K.MP.2. Reason abstractly and quantitatively.

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

K.MP.4.
Model with mathematics.

K.MP.5. Use appropriate tools strategically.

Vocabulary:

joining, separating, addition, subtraction, equal, equation, same amount as, mental picture, less, more

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Essential Questions:

(
What provocative questions will foster
inquiry, understanding, and transfer learning?
)

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

o

Teach the concept of fact families.

o

When adding, students can look for ways to “make 10” in order to make computing sums and differences easier.

o

Activities that allow for number exploration is critical. One such way to do this is to give students a set of objects that t
otal no more than
10. Have students find multiple ways to separate those objects into two sets and then rationalizing that although

the original set is split,
they still have the same amount that they started with. From here, they can make addition sentences to represent their findin
gs.

o

The use of manipulatives and technologies has a strong influence on their understanding of this sk
ill
.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Performance assessment

Instructional Resources/Tools:

Colored cubes

Students can use a Part
-
Part
-
Whole Mat and objects to model problem situations and
find solutions. This mat is divided into three sections and the labels for the sections in
order are Part, Part, and Whole.

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Common Core State Standards for Mathematics:

situations

Table 1 on page 88 in the
Common Core State Standards for School Mathematics
illustrates 12 addition and subtraction problem situations.

ORC # 1129 From the National Council of Teachers of Mathematics:
with sets

This lesson builds on the previous two lessons in the unit
Do It with Dominoes
and
encourages students to explore another model for addition, the set model.

ORC # 4
319 From the National Council of Teachers of Mathematics:

In the unit
(lessons 2, 4, 5, and 7) students explore models of subtraction
(counting, sets, balanced e
quations, and inverse of addition) and the relation between
addition and subtraction using links. Students also write story problems in which
subtraction is required.

ORC # 4269 From the National Council of Teachers of Mathematics:
More and More
Buttons

In this lesson, students use buttons to create, model, and record addition
sentences. In

this lesson, students use buttons to create, model, and record addition sentences.

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29

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(HOME)

M.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 ea
ch decomposition by a
drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Quarter Taught:

Quarter 1:

Te慣h楮朠g業e:

Bloom’s Level:

What does this standard mean that a student will know and be able to do?

This objective asks students to realize that a set of objects can be broken in multiple ways. When breaking apart a set, stu
dents use the understanding
that a smaller set of objects exists within that a larger set.

Every whole number can be broken in to
parts.

Mathematical Practices:

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

K.MP.2. Reason abstractly and quantitatively.

K.MP.4. Model with mathematics.

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express
regularity in repeated reasoning.

Vocabulary:

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

**Teach the concept of fact families.

**When

adding, students can look for ways to “make 10” in order to make computing sums and differences easier.

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30

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65

**Activities that allow for number exploration is critical. One such way to do this is to give students a set of objects that

total no more than 10. Ha
ve
students find multiple ways to separate those objects into two sets and then rationalizing that although the original set is
split, they still have the same
amount that they started with. From here, they can make addition sentences to represent their fi
ndings.

**The use of manipulatives and technologies has a strong influence on their understanding of this skill.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Performance assessment

Instructional
Resources/Tools:

Colored cubes

Students can use a Part
-
Part
-
Whole Mat and objects to model problem situations and
find solutions. This mat is divided into three sections and the labels for the sections in
order are Part, Part, and Whole.

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Common Core State Standa
rds for Mathematics:

situations

Table 1 on page 88 in the
Common Core State Standards for School Mathematics
illustrates 12 addition and subtraction problem situations.

ORC # 1129 From the National Council of Teachers of M
athematics:
with sets

This lesson builds on the previous two lessons in the unit
Do It with Dominoes
and
encourages students to explore another model for addition, the

set model.

ORC # 4319 From the National Council of Teachers of Mathematics:

In the unit
(lessons 2, 4, 5, and 7) students explore models of subtraction
(counti
ng, sets, balanced equations, and inverse of addition) and the relation between
addition and subtraction using links. Students also write story problems in which
subtraction is required.

ORC # 4269 From the National Council of Teachers of Mathematics:
More and More
Buttons

In this lesson, students use buttons to create, model, and record addition sentences.

(HOME)

M.K.OA.4

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65

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.

Quarter Taught:

Quarter 1:

Quarter 2:

Quarter 3:
X

Quarter 4:

Teaching Time:

150
-
180 minutes weekly including center time

Bloom’s Level:

Level 3
-
Application/Level 4
-
Analysis

What does this standard mean that a student will know and be able to do?

This standard
builds upon the understanding that a number can be decomposed into parts (K.OA.3). Once students have had experiences breakin
g
apart ten into various combinations, this asks students to find a missing part of 10.

Example:

“A full case of juice boxes has

10 boxes. There are only 6 boxes in this case. How many juice boxes are missing?

Mathematical Practices:

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

K.MP.2. Reason abstractly and quantitatively.

K.MP.4. Model with mathematics.

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in repeated reasoning.

Vocabulary:

total, add, equals, missing, number sentence

Introduce and use: addend, sum, equation, combination

Essential Questions:

(
What
provocative questions will foster inquiry, understanding, and transfer learning?
)

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32

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65

What are the different combinations to make 10?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

The number pairs that total ten are foundational for students’ ability
to work fluently within base
-
ten numbers and operations. Different models, such
as ten
-
frames, cubes, two
-
color counters, etc., assist students in visualizing these number pairs for ten.

Example 1:

Students place three objects on a ten frame and then dete
rmine how many more are needed to “make a ten.”

Students may use electronic versions of ten frames to develop this skill.

Example 2:

The student snaps ten cubes together to make a “train.”

Student breaks the “train” into two parts. S/he counts how many

are in each part and record the associated equation (10 = ___ + ___).

Student breaks the “train into two parts. S/he counts how many are in one part and determines how many are in the other part
without directly
counting that part. Then s/he records the a
ssociated equation (if the counted part has 4 cubes, the equation would be 10 = 4 + ___).

Student covers up part of the train, without counting the covered part. S/he counts the cubes that are showing and determines

how many are
covered up. Then s/he recor
ds the associated equation (if the counted part has 7 cubes, the equation would be 10 = 7 + ___).

Example 3:

The student tosses ten two
-
color counters on the table and records how many of each color are facing up.

Assessments:

(
What will be acceptable
evidence the
student has achieved the desired results?
)

Students will be able to find the different combinations
from 1
-
9 to find the sum of ten.

Instructional Resources/Tools:

Colored cubes

Students can use a Part
-
Part
-
Whole Mat and objects to model problem situations and
find solutions. This mat is divided into three sections and the labels for the sections in
order are Part, Part, and Whole.

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Common Core State Standards for Mathematics: Common

situations

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33

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65

Table 1 on page 88 in the
Common Core State Standards for School Mathematics
illustrates 12 addition and subtraction problem situations.

ORC # 1129 From the National Council of Teachers of Mathematics:
with sets

This lesson builds on the previous two lessons in the unit
Do It with Dominoes
and
encourages students to explore another model for addition, the set model.

ORC # 4319 Fro
m the National Council of Teachers of Mathematics:

In the unit
(lessons 2, 4, 5, and 7) students explore models of subtraction
(counting, sets, balanced equation
s, and inverse of addition) and the relation between
addition and subtraction using links. Students also write story problems in which
subtraction is required.

ORC # 4269 From the National Council of Teachers of Mathematics:
More and More
Buttons

In this lesson, students use buttons to create, model, and record addition sentences.

(HOME)

M.K.OA.5

Fluently add and subtract within 5.

Quarter Taught:

Quarter 1:

X

Quarter 4:

Te慣h楮朠g業e:

ㄵ1
-
ㄸ〠楮c汵d楮朠gen瑥r⁴ me

Bloom’s Level:

Le癥氠l
-
䅰A汩c慴楯a

What does this standard mean that a student will know and be able to do?

This standard
uses the word fluently, which means accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility (us
ing
strategies such as the distributive property). Fluency is developed by working with many different kinds of objects over an e
xt
ended amount of time.
This objective does not require students to instantly know the answer. Traditional flash cards or timed tests have not been p
roven as effective
instructional strategies for developing fluency.

Mathematical Practices:

K.MP.2. Reason
abstractly and quantitatively.

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in repeated reasoning.

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Vocabulary:

Introduce and use: addend, sum, equation, subtrahend, difference

Essential
Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

Why is it important to add and subtract quickly?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

This standard focuses on students being able to add
and subtract numbers within 5. Adding and subtracting fluently refers to knowledge of
procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and effic
iently.

Strategies students may use to attai
n fluency include:

Counting on (e.g., for 3+2, students will state, “3,” and then count on two more, “4, 5,” and state the solution is “5”)

Counting back (e.g., for 4
-
3, students will state, “4,” and then count back three, “3, 2, 1” and state the solution
is “1”)

Counting up to subtract (e.g., for 5
-
3, students will say, “3,” and then count up until they get to 5, keeping track of how many they counted
up, stating that the solution is “2”)

Using doubles (e.g., for 2+3, students may say, “I know that 2+2 is
4, and 1 more is 5”)

Using commutative property (e.g., students may say, “I know that 2+1=3, so 1+2=3”)

Using fact families (e.g., students may say, “I know that 2+3=5, so 5
-
3=2”)

Students may use electronic versions of five frames to develop fluency of t
hese facts.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Students will be able to fluently add and subtract within 5.

Instructional Resources/Tools:

Colored cubes

Students can use a Part
-
Part
-
Whole Mat and objects to model problem situations and
find solutions. This mat is divided into three sections and the labels for the sections in
order are Part, Part, and Whole.

Dot Card and Ten Frame Activities

(pp. 1
-
6, 12
-
17) Numeracy Project, Winnipeg
School Division, 2005
-
2006

Common Core State Standards for Mathematics:

ction
situations

Table 1 on page 88 in the
Common Core State Standards for School Mathematics
illustrates 12 addition and subtraction problem situations.

ORC # 1129 From the National Council of Teachers of Mathematics:
with sets

This lesson builds on the previous two lessons in the unit
Do It with Dominoes
and
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35

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65

encourages students to explore another model for addition, the set model.

ORC # 4319 From the National Counc
il of Teachers of Mathematics:

In the unit
(lessons 2, 4, 5, and 7) students explore models of subtraction
(counting, sets, balanced equations, and inverse of addition) and the relation between
addition and subtraction using links. Students also write story problems in which
subtraction is required
.

ORC # 4269 From the National Council of Teachers of Mathematics:
More and More
Buttons

In this lesson, students use buttons to create, model, and record addition sentences.

(HOME)

Number and Operations in Base Ten

Work with numbers 11

ㄹ⁴1⁧慩 ⁦潵nd慴楯a猠f潲⁰污捥⁶慬 e⸠

Instructional Strategies for Cluster

Kindergarteners need to understand the idea of
a ten
so they can develop the strategy of adding onto 10 to add within 20 in Grade 1. Students need to
construct their own base
-
ten ideas about quantities and their symbols by connecting to counting by ones. They should use a variety of manipulatives
to model an
d connect equivalent representations for the numbers 11 to19. For instance, to represent 13, students can count by ones and s
how 13
beans. They can anchor to five and show one group of 5 beans and 8 beans or anchor to ten and show one group of 10 beans and

3 beans. Students
need to eventually see
a ten
as different from 10 ones.

After the students are familiar with counting up to 19 objects by ones, have them explore different ways to group the objects

that will make counting
easier. Have them estimate be
fore they count and group. Discuss their groupings and lead students to conclude that grouping by ten is desirable.
10
ones make 1 ten
makes students wonder how something that means a lot of things can be one thing. They do not see that there are 10 single

objects
represented on the item for ten in pregrouped materials, such as the rod in base
-
ten blocks. Students then attach words to materials and groups
without knowing what they represent. Eventually they need to see the rod as
a ten
that they did not gro
up themselves. Students need to first use
groupable materials to represent numbers 11 to 19 because a group of ten such as a bundle of 10 straws or a cup of 10 beans m
akes more sense than
a
ten
in pregrouped materials.

Kindergarteners should use proporti
onal base
-
ten models, where a group of ten is physically 10 times larger than the model for a one.
Nonproportional models such as an abacus and money should not be used at this grade level.

Students should impose their base
-
ten concepts on a model made f
rom groupable and pregroupable materials (see Resources/Tools). Students can
transition from groupable to pregroupable materials by leaving a group of ten intact to be reused as a pregrouped item. When
using pregrouped
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materials, students should reflect on

the ten
-
to
-
one relationships in the materials, such as the “tenness” of the rod in base
-
ten blocks. After many
experiences with pregrouped materials, students can use dots and a stick (one tally mark) to record singles and a ten.

Encourage students to u
se base
-
ten language to describe quantities between 11 and 19. At the beginning, students do not need to use
ones
for the
singles. Some of the base
-
ten language that is acceptable for describing quantities such as18 includes
one ten and eight, a bundle and

eight
,
a rod
and 8 singles
and
ten and eight more.
Write the horizontal equation 18 = 10 + 8 and connect it to base
-
ten language. Encourage, but do not require,
students to write equations to represent quantities.

Common Misconceptions

Students have difficulty with
ten
as a singular word that means 10 things. For many students, the understanding that a group of 10 things can be
replaced by a single object and they both represent 10 is confusing. Help students develop the sense of 10 by f
irst using groupable materials then
replacing the group with an object or representing 10. Watch for and address the issue of attaching words to materials and gr
oups without knowing
what they represent. If this misconception is not addressed early on it ca
n cause additional issues when working with numbers 11
-
19 and beyond

(HOME)

M.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 reco
rd each
compositi
on 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.

Quarter Taught:

Quarter 1:

X

Quarter 4:
X

Teaching Time:

3
rd

and 4
th

quarters

Bloom’s Level:

What does this standard mean that a student will know and be able to do?

This standard is the first time that students move beyond the number 10 with representations,
such as objects (manipulatives) or drawings. The spirit
of this standard is that students separate out a set of 11
-
19 objects into a group of ten objects with leftovers. This ability is a pre
-
when they need to understand the complex
concept that a group of 10 objects is also one ten (unitizing). Ample experiences with ten frames will help
solidify this concept. Research states that students are not ready to unitize until the end of first grade. Therefore, this w
ork in Kindergarten lay
s the
foundation of composing tens and recognizing leftovers.

Example:

Teacher: “Please count out 15 chips.”

Student: Student counts 15 counters (chips or cubes).

Teacher: “Do you think there is enough to make a group of ten chips? Do you think there
might be some chips leftover?”

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Teacher: “Use your counters to find out.”

Student: Student can either fill a ten frame or make a stick of ten connecting cubes. They answer, “There is enough to make a

group of ten.”

Teacher:

How many leftovers do you have?

Student: Students say, “I have 5 left over.”

Teacher: How could we use words and/or numbers to show this?

Student: Students might say “Ten and five is the same amount as 15”, “15 = 10 + 5”

Mathematical Practices:

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

K.MP.2. Reason abstractly and quantitatively.

K.MP.4. Model with mathematics.

K.MP.7. Look for and make use of structure.

K.MP.8. Look for and express regularity in repeated reasoning.

Voca
bulary:

compose, decompose, equation, base ten

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

How does a digit’s position affect its value?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Special attention needs to be paid to this set of numbers as they do not follow a consistent pattern in the verbal counting s
equence.

Eleven and twelve are special number words.

“Teen” means one “ten” plus ones.

The verbal counting sequence

for teen numbers is backwards

we say the ones digit before the tens digit. For example “27” reads tens to
ones (twenty
-
seven), but 17 reads ones to tens (seven
-
teen).

In order for students to interpret the meaning of written teen numbers, they should re
ad the number as well as describe the quantity. For
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example, for 15, the students should read “fifteen” and state that it is one group of ten
and

five ones and record that 15 = 10 + 5.

Teaching the teen numbers as one group of ten and extra ones is
foundational to understanding both the concept and the symbol that represent each
teen number. For example, when focusing on the number “14,” students should count out fourteen objects using one
-
to
-
one correspondence and then
use those objects to make one
group of ten ones and four additional ones. Students should connect the representation to the symbol “14.” Students
should recognize the pattern that exists in the teen numbers; every teen number is written with a 1 (representing one ten) an
d ends with the

digit that
is first stated.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Visual assessment with the instructor

Instructional Resources/Tools:

Groupable models

Dried beans and small cups for holding group
s of 10 dried beans

Pregrouped materials

Strips (ten connected squares) and squares (singles)

Base
-
ten blocks

Dried beans and bean sticks (10 dried beans glued on a craft stick)

Five
-
frame and Ten
-
frame

Place
-
value mat with ten
-
frames

(HOME)

Measurement and Data

Describe and compare measurable attributes.

Instructional Strategies for Cluster

It is critical for students to be able to identify and describe measureable attributes of objects. An object has different at
tributes that can be measured,
like the height and weight of a can of food. When students compare shapes directly, the attribute be
comes the focus. For example, when comparing
the volume of two different boxes, ask students to discuss and justify their answers to these questions: Which box will hold
the most? Which box will
hold least? Will they hold the same amount? Students can deci
de to fill one box with dried beans then pour the beans into the other box to determine

Have students work in pairs to compare their arm spans. As they stand back
-
to
-
back with outstretched arms, compare the lengths of thei
r spans, then
determine who has the smallest arm span. Ask students to explain their reasoning. Then ask students to suggest other measurea
ble attributes of their
bodies that they could directly compare, such as their height or the length of their feet.

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Connect to other subject areas. For example, suppose that the students have been collecting rocks for classroom observation a
nd they wanted to know
if they have collected typical or unusual rocks. Ask students to discuss the measurable attributes of rocks.

Lead them to first comparing the weights
of the rocks. Have the class chose a rock that seems to be a “typical” rock. Provide the categories:
Lighter Than Our Typical Rock
and
Heavier Than
Our Typical Rock
. Students can take turns holding a different rock

from the collection and directly comparing its weight to the weight of the typical
rock and placing it in the appropriate category. Some rocks will be left over because they have about the same weight as the
typical rock. As a class,
they count the number

of rocks in each category and use these counts to order the categories and discuss whether they collected “typical” rocks.

Common Misconceptions

(HOME)

M.K.
MD
.1

Describe measurable attributes of objects, such as
length
or
weight
. Describe several measurable attributes of a single object.

Quarter Taught:

Quarter 1:

X

Quarter 3:

Te慣h楮朠g業e:

⡑(慲瑥r⁴ 漠o楬氠le⁤e癯瑥d⁴漠摩rec琠楮獴suc瑩潮⤠慮)⁴桥渠

pr慣瑩te⁡ d m慩a瑥n慮ce

Bloom’s Level:

䍯mprehen獩潮

What does this standard mean that a student will know and be able to do?

This standard
calls for students to describe measurable attributes of objects, such as
length
,
weight
, size. For example, a student may describe a shoe
as “This shoe is
he慶a
! It’s also really

.” This standard focuses on using descriptive words and does not mean that students should sort objects
based on attributes. Sorting appears later in the Kinder
garten standards.

Mathematical Practices:

K.MP.7. Look for and make use of structure.

Vocabulary:

length, weight, long, short, heavy, light, size

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer
learning?
)

What words would you use to describe an objects size?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

In order to describe attributes such as length and weight, students must have many opportunities to informally explore these
attributes.

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Students should compare objects verbally and then focus on specific attributes when making verbal comparisons for K.MD.2. The
y may
identify measurable attributes such as length, width, height, and weight. For example, when describing a soda can, a student
how tall, how wide, how heavy, or how much liquid can fit inside. These are all measurable attributes. Non
-
measurable attributes include:
words on the object, colors, pictures, etc.

Use Manipulatives

that allow for direct hands
-
on exploration of measuring length/weight of objects.

Use Technology that allow for direct hands
-
on exploration of measuring length/weight of objects.

An interactive whiteboard or document camera may be used to model objects w
ith measurable attributes.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher Observation

Oral Assessment

Instructional Resources/Tools:

Two
-

and three
-
dimensional real
-
world objects

Dried beans

Rice

Pan Balance

Manipulatives

ORC # 4330 From the National Council of Teachers of Mathematics:
The Weight of
Things

This lesson introduces and provides practice with the measurable attribute of weight.

(HOME)

M.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.

Quarter Taught:

Quarter 1:
X

Quarter 2:

Te慣h楮朠g業e:

Bloom’s Level:

䅮慬祳楳

What does this
standard mean that a student will know and be able to do?

This standard
asks for direct comparisons of objects. Direct comparisons are made when objects are put next to each other, such as two chil
dren, two
books, two pencils. For example, a student may l
ine up two blocks and say, “This block is a lot longer than this one.” Students are not comparing
objects that cannot be moved and lined up next to each other.

Through ample experiences with comparing different objects, children should recognize that obje
cts should be matched up at the end of objects to get
accurate measurements. Since this understanding requires conservation of length, a developmental milestone for young children
, children need
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multiple experiences to move beyond the idea that ….

“Someti
mes this block is
longer than
this one and sometimes it

s
shorter
(depending on how I lay them side by side) and that

s okay.” “This block
is always longer than this block (with each end lined up appropriately).”

Before conservation of length: The striped

block is longer than the plain block when they are lined up like this.

But when I move the blocks around, sometimes the plain block is longer than the striped block.

After conservation of length: I have to line up the blocks to measure them. The plain block is always longer than the striped

block.

Mathematical Practices:

K.MP.6. Attend to precision.

K.MP.7. Look for and make use of structure.

Vocabulary:

short,

-
er,
-
est, long,
-
er,
-
est, heavy,
-
er,
-
est, light,
-
er,
-
est, tall,
-
er,
-
est, big,
-
er,
-
est, small,
-
er,
-
est, more of, less of, same, equal,
difference, size

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and
transfer learning?
)

Why do we need to compare the size of objects in the real world?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

When making direct comparisons for length, students must attend to the “starting point” of each object. For example,
the ends need to be lined up at
the same point, or students need to compensate when the starting points are not lined up (conservation of length includes und
erstanding that if an
object is moved, its length does not change; an important concept when compar
ing the lengths of two objects).

Language plays an important role in this standard as students describe the similarities and differences of measurable attribu
tes of objects (e.g., shorter
than, taller than, lighter than, the same as, etc.).

Use Manipulati
ves that allow for direct hands
-
on exploration of measuring length/weight of objects.

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An interactive whiteboard or document camera may be used to compare objects with measurable attributes.

Use Technology that allow for direct hands
-
on exploration of
measuring length/weight of objects

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Daily Written Work

Teacher Observation

Oral Discussion

Center Activities

Draw Pictures

Instructional
Resources/Tools:

Two
-

and three
-
dimensional real
-
world objects

Dried beans

Rice

Pan Balance

Manipulatives

PBS Kids
-
Measuring up game with Clifford

ORC # 4330 From the National Council of Teachers of Mathematics:
The Weight of
Things

This lesson introduces and provides practice with the measurable attribute of weight.

(HOME)

Measurement and Data

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Instructional Strategies for Cluster

Provide categories for students to use to sort a collection of objects. Each category can relat
e to only one attribute, like
Red
and
Not Red
or
Hexagon
and
Not Hexagon,
and contain up to 10 objects. Students count how many objects are in each category and then order the categories by the numbe
r of
objects they contain.

scussion about the attributes of shapes. Then have students sort a collection of two
-
dimensional and three
-
dimensional
shapes by their attributes. Provide categories like
Circles
and
Not Circles
or
Flat
and
Not Flat
. Have students count the objects in each

category and
order the categories by the number of objects they contain.

Have students infer the classification of objects by guessing the rule for a sort. First, the teacher uses one attribute to s
ort objects into two loops or
regions without labels. T
hen the students determine how the objects were sorted, suggest labels for the two categories and explain their reasoning.

Common Misconceptions

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(HOME)

M.K.MD.3

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

(Limit category counts to be less than or
equal to 10)

Quarter Taught:

Quarter 1:

X

Quarter 3:

Te慣h楮朠g業e:

㈰⁭楮u瑥猠s慩a礠y潲′⁷eeks

Bloom’s Level:

What does this standard mean that a student will know and be able to do?

This standard
asks students to identify similarities and differences between objects (e.g., size, color, shape) and use the identified attr
ibutes to
sort
a
collection of objects. Once the objects are sorted, the student counts the amount in each set. Once each set is co
unted, then the student is asked to sort
(or group) each of the sets by the amount in each set.

For example, when given a collection of buttons, the student separates the buttons into different piles based on color (all t
he blue buttons are in one
pile, a
ll the orange buttons are in a different pile, etc.). Then the student counts the number of buttons in each pile: blue (5), g
reen (4), orange (3),
purple (4). Finally, the student organizes the groups by the quantity in each group (Orange buttons (3), Gree
n buttons next (4), Purple buttons with
the green buttons because purple also had (4), Blue buttons last (5).

This objective helps to build a foundation for data collection in future grades. In later grade, students will transfer these

skills to creating
and
analyzing various graphical representations.

Mathematical Practices:

K.MP.2. Reason abstractly and quantitatively.

K.MP.7. Look for and make use of structure.

Vocabulary:

sort, attribute, alike, different, characteristic, amount, equal, groups

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and transfer learning?
)

How can attributes be used to sort, classify, and compare objects into sets?

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Possible objects
to sort include buttons, shells, shapes, beans, etc. After sorting and counting, it is important for students to:

explain how they sorted the objects;

label each set with a category;

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answer a variety of counting questions that ask, “How many …”; and

compa
re sorted groups using words such as, “most”, “least”, “alike” and “different”.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observation, one to one checking. Students can
verbalize how they sorted

and how many in the group.

Instructional Resources/Tools:

Attribute blocks

Yarn for loops

Large paper to draw loops

A variety of objects to sort

(HOME)

Geometry

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

This entire cluster asks students to understand that certain attributes define what a shape is called (number of sides, numbe
r of angl
es, etc.) and other
attributes do not (color, size, orientation). Then, using geometric attributes, the student identifies and describes particul
ar shapes listed above.
Throughout the year, Kindergarten students move from informal language to describe what

shapes look like (e.g., “That looks like an ice cream
cone!”) to more formal mathematical language (e.g., “That is a triangle. All of its sides are the same length”). In Kindergar
ten, students need ample
experiences exploring various forms of the shapes (
e.g., size: big and small; types: triangles, equilateral, isosceles, scalene; orientation: rotated
slightly to the left, „upside down‟) using geometric vocabulary to describe the different shapes. In addition, students need
numerous experiences
comparing o
ne shape to another, rather than focusing on one shape at a time. This type of experience solidifies the understanding of the

various
attributes and how those attributes are different
-

or similar
-

from one shape to another.

Students in Kindergarten typic
ally recognize figures by appearance alone, often by comparing them to a known example of a shape, such as the
triangle on the left. For example, students in Kindergarten typically recognize that the figure on the left as a triangle, bu
t claim that the fig
ure on the
right is not a triangle, since it does not have a flat bottom. The properties of a figure are not recognized or known. Studen
ts make decisions on
identifying and describing shapes based on perception, not reasoning.

Instructional Strategie
s for Cluster

Develop spatial sense by connecting geometric shapes to students’ everyday lives. Initiate natural conversations about shapes

in the environment.
Have students identify and name two
-

and three
-
dimensional shapes in and outside of the classroo
m and describe their relative position.

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Ask students to find rectangles in the classroom and describe the relative positions of the rectangles they see, e.g.
This rectangle (a poster) is over the
sphere (globe)
. Teachers can use a digital camera to recor
d these relationships.

Hide shapes around the room. Have students say where they found the shape using positional words, e.g
. I found a triangle UNDER the chair
.

Have students create drawings involving shapes and positional words:
Draw a window ON the
door
or
Draw an apple UNDER a tree
. Some students
may be able to follow two
-

or three
-
step instructions to create their drawings.

Use a shape in different orientations and sizes along with non
-
examples of the shape so students can learn to focus on defining attributes of the
shape.

Manipulatives used for shape identification actually have three dimensions. However, Kindergartners n
eed to think of these shapes as two
-
dimensional or “flat” and typical three
-
dimensional shapes as “solid.” Students will identify two
-
dimensional shapes that form surfaces on three
-
dimensional objects. Students need to focus on noticing two and three dimen
sions, not on the words
two
-
dimensional
and
three
-
dimensional.

Common Misconceptions

Students many times use incorrect terminology when describing shapes. For example students may say a
cube
is a
square
or that a
sphere
is a
circle
.
The use of the two
-
dimensional shape that appears to be part of a three
-
dimensional shape to name the three
-
dimensional shape is a common
misconception. Work with students to help them understand that the two
-
dimensional shape is a part of the object but

it has a different name.

(HOME)

M.K.
G.1

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms s
uch as
above
,
below
,
beside
,
in front of
,
behind
,
and
next to
.

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

Throughout the school year

Bloom’s Level:

1
-
㐠4epend楮朠潮⁴桥⁩n獴suc瑩潮

What does this standard mean that a student will know and be able to do?

This standard
expects students to use positional words (such as those italicized above) to describe objects in the environment. Kindergarte
n students
need to focus first on location and position of two
-
and
-
three
-
dimensional objects in their classroom prior

to describing location and position of two
-
and
-
three
-
dimension representations on paper.

Mathematical Practices:

K.MP.7. Look for and make use of structure.

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Vocabulary:

below, beside, in front of, behind, and next to

Essential Questions:

(
What
provocative questions will foster inquiry, understanding, and transfer learning?
)

Are students able to identify basic shapes? Can students describe the attributes of shapes? Are students able to identify t
he shapes within the
environment? Can students
correctly use positional words in relationship to the shapes?

-
3_common_core_math_standards/kindergart
en.aspx

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Examples of environments in which students would be encouraged to identify shapes would include nature, buildings, and the cl
assroom using
positional words in their descriptions.

Teachers should

work with children and pose four mathematical questions: Which way? How far? Where? And what objects? To answer these
questions, children develop a variety of important skills contributing to their spatial thinking.

Examples:

Teacher holds up an object s
uch as an ice cream cone, a number cube, ball, etc. and asks students to identify the shape. Teacher holds up a
can of soup and asks,” What shape is this can?” Students respond “cylinder!”

Teacher places an object next to, behind, above, below, beside, or
in front of another object and asks positional questions. Where is the water
bottle? (water bottle is placed behind a book) Students say “The water bottle is behind the book.”

Students should have multiple opportunities to identify shapes; these may be
displayed as photographs, or pictures using the document camera or
interactive whiteboard.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observation

Instructional Resources/Tools:

Common two
-

and th
ree
-
dimensional items

Digital camera

Pattern blocks

Die cut shapes

Three
-
dimensional models

Assorted shapes

Tangrams

ORC # 4459 From the International Reading Association and the National Council of
Teachers of English:
Going on a Shape Hunt
: Integrating Math and Literacy

In this unit, students are introduced to the idea of shapes through a read
-
aloud session
with an appropriate book. They then u
se models to learn the names of shapes, work
together and individually to locate shapes in their real
-
world environment.

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ORC # 3336 From the National Council of Teachers of Mathematics:
Investigating
Shapes (Triangles)

Students will identify and construct triangles using multiple representations in this unit.

ORC # 423 From the National Council of Teachers of Mathemat
ics:
I’ve Seen That
Shape Before

Students will learn the names of solid geometric shapes and explore their properties at
various centers or during multiple lessons.

(HOME)

M.K.G.2

Correctly name shapes regardless of their orientations or overall size.

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

Taught in the first quarter, but continued review throughout
the year

Bloom’s Level:

1
-
4

What does this standard mean that a student will know and be able to do?

This standard
addresses students‟ identification of shapes based on known examples. Students at this level do not yet recognize triangles t
hat are
turned upside down as triangles, since they don’t “look like” triangles. Students need ample experiences looking at and man
ipulating shapes with
various typical and atypical orientations. Through these experiences, students will begin to move beyond what a shape “looks
like” to identifying
particular geometric attributes that define a shape.

Mathematical Practices:

K.MP.7.
Look for and make use of structure.

Vocabulary:

sides, corners, triangles, squares, rectangles, circles, hexagons, cube, cone, cylinder, sphere, vertices, faces, round, curv
ed, flat

Essential Questions:

(
What provocative questions will foster inquiry,
understanding, and transfer learning?
)

Can students identify a specific shape within a large group even if the shape is sized differently? Can students communicate

a specific attribute?

-
3_common_core_math_standards/kindergarten.aspx

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

S
tudents should be exposed to many types of triangles in many different orientations in order to eliminate the misconception t
hat a triangle is always
right
-
side
-
up and equilateral.

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Students should also be exposed to many shapes in many different sizes.

Examples:

Teacher makes pairs of paper shapes that are different sizes. Each student is given one shape and the objective is to find th
e partner who has
the same shape.

Teacher brings in a variety of spheres (tennis ball, basketball, globe, ping pong ball, etc) to demonstrate that size doesn’t

change the name of
a shape.

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observation

Instructional Resources/Tools:

Common two
-

and three
-
dimensional items

Digital camera

Pattern blocks

Die cut shapes

Three
-
dimensional models

Asso
rted shapes

Tangrams

ORC # 4459 From the International Reading Association and the National Council of
Teachers of English:
Going on a Shape Hunt
: Integrating Math and Literacy

In this unit, students are introduced to the idea of shapes through

-
aloud session
with an appropriate book. They then use models to learn the names of shapes, work
together and individually to locate shapes in their real
-
world environment.

ORC # 3336 From the National Council of Teachers of Mathematics:
Investigating
Shapes (Triangles)

Students will identify and construct triangles using multiple representations in this unit.

ORC # 423 From the National Council of Teachers of Mathematics:
I’ve Seen That
Shape Before

Students will learn the names of solid geometric shapes and explore their properties at
various centers or during multiple lessons.

(
HOME)

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49

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65

M.K.G.3

Identify shapes as two
-
dimensional (lying in a plane, “
flat
”) or three dimensional (“
solid
”).

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

Two
-
dimensional first quarter, three
dimensional second quarter, then continued review

Bloom’s Level:

1
-
4

What does this standard mean that a student will know and be able to do?

This standard
asks students to identify flat objects (2 dimensional) and solid objects (3 dimensional). This standard can be done by having

students
sort flat and solid objects, or by having students describe the appearance or thickness of shapes.

Mathematical
Practices:

K.MP.7. Look for and make use of structure.

Vocabulary:

flat, solid, sides, corners, triangles, squares, rectangles, circles, hexagons, cube, cone, cylinder, sphere, vertices, faces
, round,
curved, figures

Essential Questions:

(
What
provocative questions will foster inquiry, understanding, and transfer learning?
)

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Student should be able to differentiate between two dimensional and three dimensional shapes.

Student names a picture of

a shape as two dimensional because it is flat and can be measured in only two ways (length and width).

Student names an object as three dimensional because it is not flat (it is a solid object/shape) and can be measured in three

different ways
(length, wi
dth, height/depth).

Have the students hold a paper triangle and cone together and compare the difference. Repeat with other similar 2D and 3D sh
apes.

Students explore classroom objects to find 2D and 3D shapes.

Teachers need to model the correct names f
or 2D and 3D shapes.

-
3_common_core_math_standards/kindergarten.aspx

Assessments:

(
What will be
acceptable evidence the
student has achieved the desired results?
)

Teacher observation

Instructional Resources/Tools:

Common two
-

and three
-
dimensional items

Digital camera

Pattern blocks

Die cut shapes

Three
-
dimensional models

Assorted shapes

Tangrams

ORC # 4459 From the International Reading Association and the

National Council of
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50

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65

Teachers of English:
Going on a Shape Hunt
: Integrating Math and Literacy

In this unit, students are introduced to the idea of shapes through a read
-
aloud session
with an appropriate book. They then use models to learn the names of shapes, work
together and individually to locate shapes in their real
-
world environment.

ORC # 33
36 From the National Council of Teachers of Mathematics:
Investigating
Shapes (Triangles)

Students will identify and construct triangles using multiple representations in this unit.

ORC # 423 From the National Council of Teachers of Mathematics:
I’ve Seen That
Shape Before

Students will learn the names of solid geometric shapes and explore their properties at
various centers or during multiple lessons.

(HOME)

Geometry

Analyze, compare, create, and compose shapes.

This entire cluster asks students to understand that certain attributes define what a shape is called (number of sides, numbe
r of angles, etc.) and other
attributes do not (color, size, orientation). Then, using geometric attributes, the student identifies

and describes particular shapes listed above.
Throughout the year, Kindergarten students move from informal language to describe what shapes look like (e.g., “That looks l
ike an ice cream
cone!”) to more formal mathematical language (e.g., “That is a tria
ngle. All of its sides are the same length”). In Kindergarten, students need ample
experiences exploring various forms of the shapes (e.g., size: big and small; types: triangles, equilateral, isosceles, scale
ne; orientation: rotated
slightly to the left, „
upside down‟) using geometric vocabulary to describe the different shapes. In addition, students need numerous experiences
comparing one shape to another, rather than focusing on one shape at a time. This type of experience solidifies the understan
ding of
the various
attributes and how those attributes are different
-

or similar
-

from one shape to another.

Students in Kindergarten typically recognize figures by appearance alone, often by comparing them to a known example of a sha
pe, such as the
triangle on

the left. For example, students in Kindergarten typically recognize that the figure on the left as a triangle, but claim that

the figure on the
right is not a triangle, since it does not have a flat bottom. The properties of a figure are not recognized or

known. Students make decisions on
identifying and describing shapes based on perception, not reasoning.

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51

of
65

Instructional Strategies for Cluster

Use shapes collected from students to begin the investigation into basic properties and characteristics of t
wo
-

and three
-
dimensional shapes. Have
students analyze and compare each shape with other objects in the classroom and describe the similarities and differences bet
students to describe the shapes while the teacher records key descripti
ve words in common student language. Students need to use the word
flat
to
describe two
-
dimensional shapes and the word
solid
to describe three
-
dimensional shapes.

Use the sides, faces and vertices of shapes to practice counting and reinforce the concept of one
-
to
-
one correspondence.

The teacher and students orally describe and name the shapes found on a Shape Hunt. Students draw a shape and build it using
material
s regularly
kept in the classroom such as construction paper, clay, wooden sticks or straws.

Students can use a variety of manipulatives and real
-
world objects to build larger shapes with these and other smaller shapes: squares, circles,
triangles, recta
ngles, hexagons, cubes, cones, cylinders, and spheres. Kindergarteners can manipulate cardboard shapes, paper plates, pattern

blocks,
tiles, canned food, and other common items.

Have students compose (build) a larger shape using only smaller shapes that
have the same size and shape. The sides of the smaller shapes should
touch and there should be no gaps or overlaps within the larger shape. For example, use one
-
inch squares to build a larger square with no gaps or
overlaps. Have students also use differen
t shapes to form a larger shape where the sides of the smaller shapes are touching and there are no gaps or
overlaps. Ask students to describe the larger shape and the shapes that formed it.

Common Misconceptions

One of the most common misconceptions in geometry is the belief that orientation is tied to shape. A student may see the firs
t of the figures below as
a triangle, but claim to not know the name of the second.

Students need to have many experiences wit
h shapes in different orientations. For example, in the
Just Two Triangles
students to form larger triangles with the two triangles in different orientations.

Another misconception is confusing the name of a two
-
dimensional
shape with a related three
-
dimensional shape or the shape of its face. For
example, students might call a
cube
a
square
because the student sees the face of the cube.

(HOME)

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52

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65

M.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 o
f
equal
length).

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

2D shapes in the first quarter, 3D shapes in the second quarter

Bloom’s Level:

1
-
4

What does this standard mean that a student will
know and be able to do?

This standard
asks students to note similarities and differences between and among 2
-
D and 3
-
D shapes using informal language. These experiences
-
dimensional shapes are composed of 2
-
dim
ensional shapes (e.g.., The base and the top of a cylinder
is a circle; a circle is formed when tracing a sphere).

Mathematical Practices:

K.MP.6. Attend to precision.

K.MP.7. Look for and make use of structure.

Vocabulary:

flat, solid, sides, corners, triangles, squares, rectangles, circles, hexagons, cube, cone, cylinder, sphere, vertices, faces
, round,
curved, figures, equal, length

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and

transfer learning?
)

Are the students able to describe the attributes for both 2D and 3D shapes using the appropriate vocabulary? Can students di
stinguish between 2D
and 3D shapes? Are students able to correctly able to identify the parts of both 2D and

3D shapes? Given characteristics can the students identify
the shape?

-
3_common_core_math_standards/k
indergarten.aspx

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Students analyze and compare two
-

and three
-
dimensional shapes by observations. Their visual thinking enables them to determine if things are alike
or different based on the appearance
of the shape. Students sort objects based on appearance. Even in early explorations of geometric properties, they
are introduced to how categories of shapes are subsumed within other categories. For instance, they will recognize that a squ
are is a special
type of
rectangle.

Students should be exposed to triangles, rectangles, and hexagons whose sides are not all congruent. They first begin to desc
ribe these shapes using
everyday language and then refine their vocabulary to include sides and
vertices/corners. Opportunities to work with pictorial representations,
concrete objects, as well as technology helps student develop their understanding and descriptive vocabulary for both two
-

and three
-

dimensional
shapes

Students need opportunities to

sort shapes based on properties like size.

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Students need practice identifying how shapes are alike and different.

Using patterning blocks or tiles students can get exposure to 2D shapes.

-
3_common_core_math_standards/kindergarten.asp

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observation

Instructional Resources/Tools:

Pattern blocks

Tangrams

Colored tiles

Cubes

Three
-
dimensional models

Cans of food

Carpet squares or rectangles

Paper plates

Balls

Boxes that are cubes

Floor tiles

Straws

Wooden sticks

Clay

Construction paper

ORC # 4258 From NCTM:
Building with triangles: what can you build with two
triangles?

The first lesson in this unit includes the
Just Two Triangles
activity worksheet where
students a
re asked to form different larger shapes with two triangles.

(HOME)

M.K.G.5

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

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

2D shapes in the first quarter, 3D shapes in the second quarter

Bloom’s Level:

1
-
6

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65

What does this standard mean that a student will know and be able to do?

This standard
to apply their understanding of geometric attributes of shapes in order to create given shapes. For example, a student may
roll a clump of play
-
doh into a sphere or use their finger to draw a triangle in the sand table, recalling various attributes in orde
r to create that
particular shape.

Mathematical Practices:

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

K.MP.4. Model with mathematics.

K.MP.7. Look for and make use of structure.

Vocabulary:

flat, solid, sides, corners, triangles, squares, rectangles, circles, hexagons, cube, cone, cylinder, sphere, vertices, faces
, round,
curved, figures, equal, length

Essential Questions:

(
What provocative questions will foster inquiry, understanding, and

transfer learning?
)

Given the materials can make a desired shape? Can students figure out the specific materials needed to construct a shape? A
re the students able to
communicate how to construct specific shapes?

-
3_common_core_math_standards/kindergarten.aspx

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Because two
-
dimensional shapes are flat and three
-
dimensional shapes are solid, students should draw two
-
dimensional shapes and build three
-
dimensional shapes. Shapes may be built using materials such as clay, toothpicks, marshmallows, gumdrops, straws, et
c.

Students need opportunities to handle and feel and describe 2D and 3D shapes.

-
3_common_core_math_standards/kindergarten.aspx

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observations

Instructional Resources/Tools:

Pattern blocks

Tangrams

Colored tiles

Cubes

Three
-
dimensional models

Cans of food

Carpet squares or rectangles

Paper plates

Balls

Page
55

of
65

Boxes that are cubes

Floor tiles

Straws

Wooden sticks

Clay

Construction paper

ORC # 4258 From NCTM:
Building with triangles: what can you build with two
triangles?

The first lesson in this unit includes the
Just Two Triangles
activity worksheet where
students are asked to form different larger shapes with two triangles.

(HOME)

M.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?”

Quarter Taught:

Quarter 1:
X

Quarter 2:
X

Quarter 3:
X

Quarter 4:
X

Teaching Time:

While teaching
shapes, review throughout the year

Bloom’s Level:

1
-
6

What does this standard mean that a student will know and be able to do?

This standard
moves beyond identifying and classifying simple shapes to manipulating two or more shapes to create a new shape. This concept

begins to develop as students‟ first move, rotate, flip, and arrange puzzle pieces. Next, students use their experiences with

puzz
les to move given
shapes to make a design (e.g., “Use the 7 tangram pieces to make a fox.”). Finally, using these previous foundational experie
nces, students
manipulate simple shapes to make a new shape.

Mathematical Practices:

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

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

K.MP.4. Model with mathematics.

MP.7. Look for and make use of structure.

Vocabulary:

flat, solid, sides, corners, triangles, squares, rectangles, circles, hexagons, cube, cone, cylinder, sphere, vertices, faces
, round,
curved, figures, equal, length, join, separate

Page
56

of
65

Essential Questions:

(
What provocative questions will foster inquiry, un
derstanding, and transfer learning?
)

Are the students able to recognize the smaller shapes found within larger ones? Are students able to manipulate shapes and p
attern blocks to form
new shapes?

-
3_common_core_math_standards/kindergarten.aspx

Instructional
/Learning

Activities
:

(
W.H.E.R.E.T.O.
)

Students use pattern blocks, tiles, or paper shapes and technology to make new two
-

and three
-
dimensional shapes. Their investigations allow them
to determine what kinds of shapes they can join to create new shapes. They answer questions such as “What shap
es can you use to make a square,
rectangle, circle, triangle? …etc.”

Students may use a document camera to display shapes they have composed from other shapes. They may also use an interactive w
hiteboard to copy
shapes and compose new shapes. They should
describe and name the new shape.

Students can practice with concepts by working with puzzles in which the outline is covered with shapes such as tangram piece
s.

-
3_common_core_math_standards/kindergarten.aspx

Assessments:

(
What will be acceptable evidence the
student has achieved the desired results?
)

Teacher observation

Instructional Resources/Tools:

Pattern blocks

Tangrams

Colored tiles

Cubes

Three
-
dimensional models

Cans of food

Carpet squares or rectangles

Paper plates

Balls

Boxes that are cubes

Floor tiles

Straws

Wooden sticks

Clay

Construction paper

ORC # 4258 From NCTM:
Building with triangles: what can you build with two
Page
57

of
65

triangles?

The first lesson in this unit includes the
Just Two Triangles
activity worksheet where
students are asked to form different larger shapes with two triangles.

(HOME)

Page
58

of
65

Glossary

Addition and subtraction within 5, 10, 20, 100, or 1000
. Addition or subtraction of two whole numbers with whole
number answers, and with sum or minuend in the range 0
-
5,
0
-
10, 0
-
20, or 0
-
100, respectively. Example: 8 + 2 = 10 is an addition within 10, 14

5 = 9 is a subtraction within 20, and 55

18 = 37 is a subtraction within 100.

. Two numbers

whose sum is 0 are additive inverses of one another. Example: 3/4 and

3/4 are additive inverses of one another because 3/4 + (

3/4) = (

3/4) +
3/4 = 0.

. See Table 3 in this Glossary.

Associative property of multiplic
ation.
See Table 3 in this Glossary.

Bivariate data.
Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team.

Box plot
. A method of visually displaying a distribution of data values by using the m
edian, quartiles, and extremes of the data set. A box shows the middle 50% of the data.
29

Commutative property
. See Table 3 in this Glossary.

Complex fraction
. A fraction
A
/
B
where
A
and/or
B
are fractions (
B
nonzero).

Computation algorithm
. A set of
predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out

computation strategy
.

Computation strategy
. Purposeful manipulations that may be chosen for specific problem
s, may not have a fixed order, and may be aimed at converting one problem into another.
computation algorithm.

Congruent
.

Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, refl
ections, and translations).

Counting on
. A strategy for finding the number of objects in a group without having to count every me
mber of the group. For example, if a stack of books is known to have 8
books and 3 more books are added to the top, it is not necessary to count the stack all over again; one can find the total by

counting on

pointing to the top book and saying

eight,

fo
llowing this with

nine, ten, eleven. There are eleven books now.

Dot plot.
See:
line plot.

Dilation
. A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distan
ces from the center by a com
mon scale factor.

Expanded form
. A multidigit number is expressed in expanded form when it is written as a sum of single
-
digit multiples of powers of ten. For example, 643 = 600 + 40 + 3.

Expected value.
For a random variable, the weighted average of its

possible values, with weights given by their respective probabilities.

First quartile
. For a data set with median
M
, the first quartile is the median of the data values less than
M
. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the fir
st
quartile is 6.
30

median, third quartile, interquartile range.

Fraction
. A number expressible in the form
a
/
b
where
a
is a whole number and
b
is a positive whole number. (The word
fraction
in these standards always refers to a nonnegative
numb
er.)
rational number.

Identity property of 0
. See Table 3 in this Glossary.

Independently combined probability models
. Two probability models are said to be combined independently if the probability of each ordered pair in the combined model
eq
uals the product of the original probabilities of the two individual out
-
comes in the ordered pair.

Integer
. A number expressible in the form
a
or

a
for some whole number
a
.

Interquartile Range
. A measure of variation in a set of numerical data, the int
erquartile range is the distance between the first and third quartiles of the data set. Example: For the
data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the interquartile range is 15

6 = 9.
first quartile, third quartile.

Line plot
.

A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number
line. Also known as a dot plot.
31

Mean
. A measure of center in a set of numerical data, computed by adding the values in a list a
nd then dividing by the number of values in the list.
32

Example: For the data set {1,
3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21.

Mean absolute deviation
. A measure of variation in a set of numerical data, computed by adding the distances between e
ach data value and the mean, then dividing by the number
of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation is 20.

2
9

A
d
a
pt
e
d

f
rom

Wi
s
c
on
s
i
n

D
e
p
a
r
t
m
e
n
t

of

Pu
bl
i
c

I
n
s
t
r
u
c
t
i
o
n
,

ht
t
p
:
/
/
d
p
i.
w
i
.
g
o
v
/
s
t
a
nd
a
rd
s
/
m
a
t
h
g
l
o
s
.
h
t
m
l
,

a
cc
e
ssed

M
a
rch
2
,

2
0
1
0
.

3
0

M
a
n
y d
iff
er
e
n
t

m
e
t
h
o
d
s
f
o
r
c
o
m
p
ut
i
n
g
q
u
a
r
t
i
l
e
s
a
re
i
n

use.

T
h
e
m
e
t
h
o
d

de
fi
n
e
d

h
ere
i
s som
e
t
i
m
e
s c
a
ll
e
d

t
h
e
M
oo
re
a
n
d

M
c
Ca
be
m
e
t
h
o
d
.

See

L
a
n
g
f
o
r
d
,

E.
,

Qu
a
r
t
i
l
e
s

i
n

E
l
e
m
e
n
t
a
ry

S
t
a
t
i
s
t
i
c
s
,

J
o
u
rn
a
l
o
f
St
a
t
is
t
i
c
s

E
d
uc
a
t
i
o
n

V
o
lume

1
4
,

N
u
mb
e
r 3 (
2
00
6
),

3
1

A
d
a
pt
e
d

f
rom

Wi
s
c
on
s
i
n

D
e
p
a
r
t
m
e
n
t

of

Pu
bl
i
c

I
n
s
t
r
u
c
t
i
o
n
,

o
p
.

c
i
t
.

3
2

To

be
m
o
re pr
e
c
i
s
e
,

t
hi
s de
fi
n
es

t
h
e

a
ri
t
h
m
e
t
ic

m
e
an
.

Page
59

of
65

Median
. A
measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted

version of the list

or the mean of the two
central values, if the list contains an even number of values. Example: For the data se
t {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11.

Midline.
In the graph of a trigonometric function, the horizontal line half
-
way between its maximum and minimum values.

Multiplication and division within 100
. Multiplication or division of two wh
ole numbers with whole number answers, and with product or dividend in the range 0
-
100. Example:
72 ÷ 8 = 9.

Multiplicative inverses
. Two numbers whose product is 1 are multiplicative inverses of one another. Example:
3
/
4
and
4
/
3
are multiplicative invers
es of one another because
3
/
4
x
4
/
3
=
4
/
3
x
3
/
4
= 1.

Number line diagram.
A diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram for measure
ment quantities, the
interval from 0 to 1 on the diagram represents the unit of measure for the quantity.

Percent rate of change.
A rate of change expressed as a percent. Example: if a population grows from 50 to 55 in a year, it grows by 5/50 = 10% per y
ear.

Probability distribution.
The set of possible values of a random variable with a probability assigned to each.

Properties of

operations
. See Table 3 in this Glossary.

Properties of equality
. See Table 4 in this Glossary.

Properties of inequality
. See Table 5 in this Glossary.

Properties of operations
. See Table 3 in this Glossary.

Probability
. A number between 0 and 1 used
to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from
a group
of people, tossing a ball at a target, testing for a medical condition).

Probability model.
A probability model is used to

assign probabilities to outcomes of a chance process by examining the nature of the process. The set of all outcomes is calle
d
the sample space, and their probabilities sum to 1.
uniform probability model.

Random variable.
An assignment of a nu
merical value to each outcome in a sample space.

Rational expression.
A quotient of two polynomials with a non
-
zero denominator.

Rational number
. A number expressible in the form
a
/
b
or

a
/
b
for some fraction
a
/
b
. The rational numbers include the intege
rs.

Rectilinear figure.
A polygon all angles of which are right angles.

Rigid motion
. A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Ri
gid motions are here assumed to preserve
dist
ances and angle measures.

Repeating decimal
. The decimal form of a rational number.
terminating decimal.

Sample space
. In a probability model for a random process, a list of the individual outcomes that are to be considered.

Scatter plot
. A
graph in the coordinate plane representing a set of bivariate data. For example, the heights and weights of a group of people

could be displayed on a scatter plot.
33

Similarity transformation
. A rigid motion followed by a dilation.

Tape diagram
. A drawin
g that looks like a segment of tape, used to illustrate number relationships. Also known as a strip dia
-
gram, bar model, fraction strip, or length model.

Terminating decimal.
A decimal is called terminating if its repeating digit is 0.

Third quartile
. Fo
r a data set with median
M
, the third quartile is the median of the data values greater than
M
. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the
third quartile is 15.
median, first quartile, interquartile range.

Transitivity
principle for indirect measurement.
If the length of object A is greater than the length of object B, and the length of object B is greater than the length of ob
ject C,
then the length of object A is greater than the length of object C. This principle appl
ies to measurement of other quantities as well.

Uniform probability model
. A probability model which assigns equal probability to all outcomes.
probability model.

Vector.
A quantity with magnitude and direction in the plane or in space, defined

by an ordered pair or triple of real numbers.

Visual fraction model.
A tape diagram, number line diagram, or area model.

Whole numbers
. The numbers 0, 1, 2, 3, ….

33

Adapted from Wisconsin Department of Public Instruction,
op. cit
.

Page
60

of
65

Table 1.

34

Result Unknown

Change Unknown

Start Unknown

Two bunnies sat on the grass.
Three more bunnies hopped
there. How many bunnies are on
the grass now?

2 + 3 = ?

Two bunnies were sitting on the
grass. Som
e more bunnies
hopped there. Then there were
five bunnies. How many bunnies
hopped over to the first two?

2 + ? = 5

Some bunnies were sitting on
the grass. Three more bunnies
hopped there. Then there were
five bunnies. How many bunnies
were on the grass be
fore?

? + 3 = 5

Take From

Five apples were on the table. I
ate two apples. How many
apples are on the table now?

5

2 = ?

Five apples were on the table. I
ate some apples. Then there
were three apples. How many
apples did I eat?

5

? = 3

Some apples were on the table. I
ate two apples. Then there were
three apples. How many apples
were on the table before?

?

2 = 3

Total Unknown

35

Put Together/

Take Apart
36

Three red apples and two green
apples are on the table. How
many apples are on the table?

3 + 2 = ?

Five apples are on the table.
Three are red and the rest are
green. How many apples are
green?

3 + ? = 5, 5

3 = ?

Grandma has five flowers. How
many c
an she put in her red vase
and how many in her blue vase?

5 = 0 + 5, 5 = 5 + 0

5 = 1 + 4, 5 = 4 + 1

5 = 2 + 3, 5 = 3 + 2

Difference Unknown

Bigger Unknown

Smaller Unknown

Compare
37

(“How many more?” version):

Lucy has two apples. Julie has
five apples. How many more
apples does Julie have than
Lucy?

(“How many fewer?” version):

Lucy has two apples. Julie has
five apples. How many fewer
apples does Lucy have than
Julie?

2 + ? = 5, 5

2 = ?

(Version with “more”):

Julie has three more apples than
Lucy. Lucy has two apples. How
many apples does Julie have?

(Version with “fewer”):

Lucy has 3 fewer apples than
Julie. Lucy has two apples. How
many apples does Julie have?

2 + 3 = ?, 3 + 2 =
?

(Version with “more”):

Julie has three more apples than
Lucy. Julie has five apples. How
many apples does Lucy have?

(Version with “fewer”):

Lucy has 3 fewer apples than
Julie. Julie has five apples. How
many apples does Lucy have?

5

3 = ?, ? + 3 =
5

3
4

A
d
a
pt
e
d

f
rom

B
o
x

2
-
4

o
f

M
a
t
h
em
a
t
i
cs

L
e
a
rn
i
n
g
i
n

Ea
rly
Chi
ld
h
o
o
d
,

N
a
t
i
o
n
a
l Res
ea
rch
C
o
u
n
c
i
l (
2
0
0
9
,

p
p
.

3
2
,

3
3
)
.

3
5

T
h
ese
t
a
k
e
a
pa
rt
s
it
u
a
t
i
o
n
s c
a
n

be used

t
o

s
h
o
w

a
ll

t
h
e d
e
c
o
m
po
s
i
t
i
on
s

o
f

a

g
i
ven
n
u
mb
e
r.

T
h
e
a
ss
o
c
ia
t
ed

e
q
u
a
t
i
on
s
,

w
hi
ch

ha
ve
t
h
e
t
o
t
a
l on

t
h
e l
ef
t

of

t
h
e
e
q
u
a
l

s
i
g
n
,

h
elp

c
hi
ldr
e
n

u
nd
e
r
s
t
a
n
d

t
ha
t

t
h
e =

s
i
gn

d
o
es

n
o
t

a
lw
a
ys m
ea
n

m
a
k
e
s

o
r
r
e
sul
t
s

i
n
b
u
t

a
lw
a
ys

do
es

m
ea
n

is

t
he
s
a
m
e nu
mb
er

as
.

3
6

E
i
t
h
er
a
dd
e
n
d

c
a
n

be

un
k
n
o
w
n
,

s
o

t
h
ere

a
re
t
h
r
e
e v
a
r
ia
t
i
on
s of

t
h
ese
p
r
o
bl
e
m

s
it
u
a
t
i
on
s
.

B
o
t
h

A
d
d
e
n
d
s
U
n
k
n
o
w
n

i
s a

pr
o
d
u
c
t
i
v
e
e
x
t
e
n
s
i
o
n

of

t
hi
s b
a
s
i
c

s
i
t
u
a
t
i
o
n

e
s
p
ec
ia
lly
f
o
r s
ma
ll

n
u
mb
e
rs

l
e
ss
t
ha
n

or
e
q
u
a
l
t
o

1
0
.

3
7

Fo
r
t
h
e
Bi
gger

Un
k
n
o
w
n

o
r S
ma
ll
e
r U
n
k
no
w
n

s
it
u
a
t
i
o
n
s,

on
e vers
i
o
n

d
i
r
e
c
t
s

t
h
e
c
o
rr
e
ct

op
er
a
t
i
o
n

(
t
h
e vers
i
o
n

us
i
n
g

mo
re

f
o
r
t
h
e b
i
gger
u
n
k
n
o
w
n

a
n
d

us
i
n
g

le
s
s

f
o
r
t
h
e s
ma
ll
e
r u
n
k
n
o
w
n
)
.

The
o
t
h
er v
e
r
s
i
on
s
a
re
m
o
re
d
iffi
c
u
l
t
.

Page
61

of
65

Table 2.

Common multiplication and division situations
.
38

Unknown Product

Group Size Unknown

(“How many in each group?” Division)

Number of Groups Unknown

(“How many groups?” Division)

3
x
6
=
?

3
x
? = 18 and 18
÷
3 = ?

?
x
6 = 18 and 18
÷
6
=
?

Equal Groups

There are 3 bags with 6 plums
in each bag. How many plums
are there in all?

Measurement example
. You
need 3 lengths of string, each
6 inches long. How much
string will you need
altogether?

If 18 plums are shared equally into 3
bags, then how many plums will be in
each bag?

Measurement example
. You have 18
inches of string, which you will cut into

3
equal pieces. How long will each piece
of string be?

If 18 plums are to be packed 6 to
a bag, then how many bags are
needed?

Measurement example
. You
have 18 inches of string, which
you will cut into pieces that are
6 inches long. How many pieces
of
string will you have?

Arrays,
39

Area
40

There are 3 rows of apples
with 6 apples in each row.
How many apples are there?

Area example
. What is the
area of a 3 cm by 6 cm
rectangle?

If 18 apples are arranged into 3 equal
rows, how many apples will be i
n each
row?

Area example
. A rectangle has area 18
square centimeters. If one side is 3 cm
long, how long is a side next to it?

If 18 apples are arranged into
equal rows of 6 apples, how
many rows will there be?

Area example
. A rectangle has
area 18 square centimeters. If
one side is 6 cm long, how long
is a side next to it?

Compare

A blue hat costs \$6. A red hat
costs 3 times as much as the
blue hat. How much does the
red hat cost?

Measurement example
. A
rubber band is 6 cm long.
How long will the rubber
band be when it is stretched to
be 3 times as long?

A red hat costs \$18 and that is 3 times as
much as a blue hat costs. How much does
a blue hat cost?

Measurement example
. A rubber band is
stretche
d to be 18 cm long and that is 3
times as long as it was at first. How long
was the rubber band at first?

A red hat costs \$18 and a blue
hat costs \$6. How many times as
much does the red hat cost as the
blue hat?

Measurement example
. A rubber
band was 6

cm long at first. Now
it is stretched to be 18 cm long.
How many times as long is the
rubber band now as it was at
first?

General

a x b = ?

a x ? = p and p ÷ a = ?

? x b = p and p ÷ b = ?

38

The first examples in each cell are examples of discrete things. These are easier for students and should be given before the

measurement examples.

39

The language in the array examples shows the easiest form of array problems. A harder form is to use th
e terms rows and columns: The apples in the

grocery window are in 3 rows and 6 columns. How many apples are in
there? Both forms are valuable.

40

Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so arra
y problems include these especially important measurement situations.

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TABLE 3
.
The properties of operations
. Here
a
,
b
and
c
stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system,
the real

number system, and the complex number system.

Associative property of multiplication

Commutative property of
multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication

(a + b) + c = a + (b + c)

a + b = b + a

a + 0 = 0 + a = a

For every
a

there exists

a

so that
a + (

a)
= (

a) + a = 0.

(a x b) x c = a x (b x c)

a x b = b x a

a x 1 = 1 x a = a

For every
a ≠ 0

1/a

so that
a x 1/a = 1/a x a = 1.

a x (b + c) = a x b + a x c

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TABLE 4
.
The properties of equality
. Here
a
,
b
and
c
stand for arbitrary

numbers in the rational, real, or complex number systems.

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Subtraction property of equality

Multiplication property of
equality

Division property of equality

Substitution property of equality

a = a

If
a = b
, then
b = a.

If
a = b
and
b = c
, then
a = c.

If
a = b
, then
a + c = b + c.

If
a = b
, then
a

=
c
=
b

=
c.

If
a = b
, then
a
x
c
=
b
x
c.

If
a = b
and
c
≠ 0, then
a
÷
c
=
b
÷
c.

If
a
=
b
, then
b
may be substituted for
a

in any expression containing
a
.

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TABLE 5
.
The properties of inequality
. Here
a
,
b
and
c
stand for arbitrary numbers in the rational or real number systems.

Exactly one of the following is true:
a < b, a = b, a > b
.

If
a
>
b
and
b
>
c
then
a
>
c
.

If
a > b
, then
b < a
.

If
a > b
, then

a <

b
.

If
a > b
, then
a ± c > b ± c
.

If
a
>
b
and
c
> 0, then
a
x
c
>
b
x
c.

If
a
>
b
and
c
< 0, then
a
x
c
<
b
x
c.

If
a
>
b
and
c
> 0, then
a
x
c
>
b
÷
c.

If
a
>
b
and
c
< 0, then
a
÷
c
<
b
÷
c.

(HOME)

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Appendix A

Instructional/Learning Activities: W.H.E.R.E.T.O.

What learning experiences and instruction will enable students to achieve the desired results?

How will the design
-

W

= Help the students know
W
here the unit is going and
W
hat is expected? Help the teacher know
W
here the students are coming from (prior
knowledge, interests)?

H
=
H
ook all students and
H
old their interest?

E
=
E
quip students, help them
E
xperience the key ideas and
E
xplore the issues?

R

= Provide opportunities to
R
ethink and
R
evise their understandings and work?

E

= Allow students to
E
valuate their work and its implications?

T

= Be
T
ailored (personalized) to

the different needs, interests, and abilities of learners?

O

= Be
O
rganized to maximize initial and sustained engagement as well as effective learning? Chunking is evident.

**Begin by unpacking individual standards with the plan to move to more
thematic units of instruction that teach multiple standards in the
future.