Grade 1
Grade 1: Unitizing and Skip-counting by 2s, 5s, and 10s
Multiplication and Division Horizontal Content Strand
Texas Essential Knowledge and Skills (TEKS):
(1.1) Number, operation, and quantitative reasoning. The student uses whole numbers to describe
and compare quantities. The student is expected to:
(A) Compare and order whole numbers up to 99 (less than, greater than, or equal to) using sets
of concrete objects and pictorial models;
(B) Create sets of tens and ones using concrete objects to describe, compare, and order whole
numbers;
(C) Read and write numbers to 99 to describe sets of concrete objects.
(1.4) Patterns, relationships, and algebraic thinking. The student uses repeating patterns and
additive patterns to make predictions. The student is expected to identify, describe, and extend
concrete and pictorial patterns in order to make predictions and solve problems
(1.5) Patterns, relationships, and algebraic thinking. The student recognizes patterns in numbers
and operations. The student is expected to:
(A) Use patterns to skip count by twos, fives, and tens;
(B) Find patterns in numbers, including odd and even;
(1.7) Measurement. The student directly compares the attributes of length, area, weight/mass,
capacity, and temperature. The student uses comparative language to solve problems and
answer questions. The student selects and uses nonstandard units to describe length. The
student is expected to:
(A) Estimate and measure length using nonstandard units such as paper clips or sides of color
tiles;
(B) Compare and order two or more concrete objects according to length (from longest to
shortest);
(1.11) Underlying Processes and Tools. The student applies Grade 1 mathematics to solve problems
connected to everyday experiences and activities in and outside of school. The student is
expected to:
(B) Solve problems with guidance that incorporates the processes of understanding the
problem, making a plan, carrying out the plan, and evaluating the solution for
reasonableness;
(C) Select or develop an appropriate problem-solving plan or strategy including drawing a
picture, looking for a pattern, systematic guessing and checking, or acting it out in order to
solve a problem;
(D) Use tools such as real objects, manipulatives, and technology to solve problems.
TEKS Connections to Other Grade Level Strands:
•
•
•
Number and Operations (Multiplication and Division)
Algebra (Patterns)
Measurement
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
380
Grade 1
Purpose:
This lesson introduces first graders to two important knowledge and skills—skip-counting and
unitizing. By internalizing these concepts, children can begin to apply more efficient quantification
strategies involving repeated addition and eventually formal multiplication for solving word problems.
The unitizing nature of skip-counting lays the foundation for multiplication by helping a child to
organize a set of objects into clusters of equal groups, thereby allowing him or her to land predictably
on designated numbers and pass over others. By the end of first grade, children should be able to use
skip-counting beyond a mere rote level, incorporating it as a strategy, in addition to one-to-one
correspondence, to quantify rationally large sets of objects.
Suggested Vocabulary:
equal groups
length
measure
repeated addition
skip-counting
total
Materials:
•
•
•
•
•
Baggies each containing all ten of the basic proportional fraction manipulatives such as
Cuisenaire® rods - per pairs of students
Chart Paper
Linking Cubes such as Unifix® cubes
Class-size 100s Chart with transparent squares for marking numbers
Markers and paper per student
Advanced Preparation:
• Handout 1-2: Riddle Cards
For Each Student:
• Markers and paper
• Handout 1-3: Riddle Recording Sheet
• Handout 1-1: Individual student 100s Chart
For Each Pair of Students:
• Baggies each containing all ten of the basic proportional fraction manipulatives such as
Cuisenaire® rods
Suggested Pacing:
•
3 days
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
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Grade 1
Procedures
Engage:
1. Distribute to pairs of students baggies containing a
set of each of the ten different proportional fraction
manipulatives such as Cuisenaire® rods.
These bars are called Cuisenaire® rods. I am
going to let you explore them with your partner
for about 5 minutes. Study them carefully and
be ready to share your discoveries to the whole
group.
2. As the students explore, observe and listen to their
conversations. Pay attention to students who try to
assign numerical values to each rod. Make note of
the strategies these students use to figure out what
each rod is worth.
3. After the five minutes of free exploration, call on
volunteers to share their observations and
discoveries. If the rods are new to students, this
exploration will take more than five minutes.
4. If during this initial discussion the students fail to
explain or express inquiries on how to figure out
the value of each rod, guide the class in thinking
about these ideas through the following questions:
• How could you order these rods from
shortest to longest/ tallest?
Responses may vary. Possible responses
include: White is the shortest, followed by red,
then light green, etc.
• Does the way you order the rods by length
help you figure out the numerical value of
each rod?
Responses may vary. Possible responses
include: If there are ten rods, then white is
equal to 1, red is equal to 2, etc.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
Teacher Notes
Materials
There are ten basic bar units within a set
of proportional fraction manipulatives
such as Cuisenaire® rods.
They are, from shortest to longest:
1. White (Tan)
2. Red
3. Light green
4. Violet
5. Yellow
6. Dark green
7. Black
8. Brown
9. Blue
10. Orange
Informal Assessment
As students are exploring (Refer to Step
2), look for the following attempts
initiated by students to figure out of the
size and numerical value of each rod:
• Do the students order the rods from
shortest to longest and then iterate a
shorter bar against a longer bar to
figure out how many of the former
will equal the length of the latter?
TEKS Connection Unit iteration is a
measurement concept referred to in
TEKS 1.7A
• Do students immediately figure out
that the shortest rod is worth one
unit by comparing it with the other
10 rods? Do they then determine that
the numerical value of each bar must
correspond to its ordinal position
relative to its size when the rods are
ordered from shortest to longest?
• If after figuring out the value of
shorter rods, do students relate this
understanding and apply number
facts (e.g., 2 + 2 = 4, so 2 reds = 1
purple) to determine the value of
longer rods?
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Grade 1
•
If you claim that the longest rod (the orange
one) is worth 10, could you use any of the
other rods to help you prove your
statement?
Responses may vary. Possible responses
include: I knew just by looking [comparing]
that it takes 2 white rods to make one red.
Then, I saw that the red rod is half of the
purple rod, so I knew that 2+2 = 4. Then, I put
a white rod (1) with a purple rod (4) and saw
that it was the same size as a yellow rod. So
then I knew that a yellow rod is equal to
5 because I know that 1+ 4 = 5. Then, I saw
that a yellow rod is half of an orange rod. So
I knew that an orange rod is worth 10 because
5 + 5 = 10.
5. Explain to the class that they will now measure
how long objects in the classroom are using the red
rod as a non-standard unit. Use this scenario to set
the stage for solving the following word problem:
Since there are not enough red rods for every
student to measure with during tomorrow’s
math lesson, you will each make your own red
rods using interlocking cubes. Since a red rod is
worth two units, you will need two white
interlocking cubes for every rod you make.
If each student is supposed to make 9 red rods,
how many white interlocking cubes will you
need?
Responses may vary.
6. Remind the class that they must first solve the
problem described in Step 5 then record their
strategy on paper using numbers, pictures, or
words.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
Strategy for ALL Learners
So that all students are able to make
sense of the strategies explained by
classmates during the sharing session
outlined in Step 4, illustrate and label
each student’s response on chart paper.
Figure A
Jessica’s strategy:
1+1 = 2
+ =
2+2 = 4
+
=
1+4 = 5 +
=
So, 5+5 = 10
Problem-Solving Strategy
To ensure that the students make sense
of and comprehend the word problem
described in Step #5, call on various
students to summarize in their own
words what you said.
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Grade 1
Explore:
Research
1. Distribute sets of interlocking cubes to pairs of
students. Every child will need markers and a sheet Fennema, Carpenter, Levi, Franke, &
Empson (1999) have conducted
of paper for recording his or her work.
considerable research on the informal
2. As children are working, move about from student strategies children use to solve word
problems. Below are three different
to student, asking questions to probe for
levels of quantification strategies that
understanding, redirecting misconceptions, and
taking note of different problem-solving strategies. Fennema, et al., have observed across
If students are having difficulty getting started, ask several student populations: (The
examples below relate to strategies for
the following questions:
solving multiplicative word problems.)
• When you make one red rod, how many
•
•
•
cubes do you have?
Two
When you make two red rods, how many
cubes do you have?
Four
What about when you have three rods?
I will have six cubes.
What is happening to the amount each time
you make more rods? Do you notice a
pattern?
Responses may vary. Possible responses
include: It is getting bigger, or it keeps going
up two more each time I make one more.
3. When students finish, ask them to demonstrate
how they figured out how many cubes they would
need to make 9 red rod units. Make note of 3
different problem –solving levels. (For a
description of these levels, refer to the teacher
notes.)
Direct Modeling: The student models
the structure of the problem, making
stacks of each group with counters, then
counts all of the objects one-by-one in
every group to figure out the total.
Counting Strategies: Instead of
counting each object one-by-one, the
student skip counts, enumerating the
accumulating total or each group to
determine how many there are
altogether.
Numerical Reasoning: The student
uses derived number facts or repeated
addition (e.g., “I know that 2 + 2 = 4
and then I made another group of 4 so
that I could add 4 + 4 to get 8. I counted
10 over here, and so 10 + 8 = 18. There
are 18 altogether.”)
4. Allow students who finish sooner than others to
connect the red rods they made into one train of 18
connecting cubes. Then, have the children make
size comparisons with their train, determining
which objects are about as long, longer than, or
shorter than a series of 9 red rod cube units.
Instruct students to record their findings on a sheet
of paper.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
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Grade 1
Explain:
1. Gather the class together to debrief students’
problem-solving strategies and discoveries.
2. Select students to share who solved the problem
using one of the quantification strategies outlined
in the teacher notes from the Explore section.
3. As these students share, summarize and label their
strategies on chart paper. Make sure that each
selected student demonstrates his or her counting
strategy. See the Pacing explanation in the teacher
notes column.
4. As a whole group, compare and contrast the
different strategies that were shared:
• How are these different problem-solving
solutions alike?
Responses may vary. Possible responses
include: Everyone said they needed 18 cubes to
make 9 red rods or they all made groups.
• How is each of these students’ approach
different from the other? Did everyone use
counting to figure out the total?
Responses may vary. Possible responses
include: [Student’s name] counted all of his
cubes, but [Student’s name] only counted the
last numbers in each rod.
5. Discuss key mathematical vocabulary. For
example, if some students point out that they
added in order to figure out the total, introduce the
terms repeated addition and equal groups. Lead the
students to this understanding through questioning:
▪ How is adding together one group of 2,
another group of 2, and another group of 2
different from adding a group of 4, 5, and 3?
Responses may vary. Possible responses
include: In the first type of adding, all of the
groups are equal, in the second example, all of
the groups have different numbers.
Then, make the targeted vocabulary explicit:
When you join together equal groups, you are
using repeated addition.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
Scheduling
Make sure that the students have been
physically active before the sharing
session so that they are ready to
concentrate and listen to classmates’
strategies.
Pacing
In order to go more in depth into key
mathematical ideas, select only
3 different strategies to share. Allow
those students who were not selected to
share to vote on which of the
3 strategies is closest to the approach
that they used. (See Figure B below:)
Whose strategy is most like the one you used
to solve the “How many cubes to make 9
red rods” problem?
Stephen’s
Tonya’s
Lupe’s
Counting all
Skip-counting
Number facts
Lisa
Tyrone
Moesha
Denise
Lyle
Sunny
Harold
Anthony
Ariel
Vocabulary
To make the concept of repeated
addition more concrete, relate it to
students’ background and familiarity
with repeating patterns—ABB ABB
ABB. Point out how the unit of the
pattern (the part of the arrangement that
repeats) continues over and over, similar
to the situation modeled in the word
problem in which equal sets of 2 were
repeatedly joined together.
Materials
A 100s Chart is a powerful visual model
for illustrating counting patterns and
representing place value. As children go
from left to right horizontally on the
chart, they see the repetitive cycle of the
digits 1 – 9 in the ones place. This same
pattern emerges in the tens place,
moving vertically from top to bottom.
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Grade 1
Elaborate:
1. Display a class size 100s Chart. Connect the skipcounting and grouping that some of the students
demonstrated in the Explain section to how the
100s chart can be used as a tool to visualize and
keep track of this type of counting and quantifying.
2. Begin by placing a transparent marker or square
over the numbers on the chart where each of the
nine red rods would stop. Engage the students in
this activity by asking them which numbers you
should cover:
• On which number do I land when I count
the first rod?
2
• The second rod?
4
• The third rod?
6…etc.
• What do you notice about the 100s chart
when I place a marker over every number
we land on after counting each of the rods?
Responses may vary. Possible responses
include: It makes an AB pattern, or it skips
every other number, or it lands on all of the
even numbers.
Research
Clements, Samara, and DiBiase (2002)
site evidence gathered by a panel of
researchers indicating that most
typically developing 6- and 7- year-olds
are just learning to skip count
meaningfully when quantifying sets of
objects and can begin to do with
groupings of 2, 5, and 10. By age 7 and
8, children begin to develop greater
facility skip-counting by less regular
numbers, such as 3 and 4.
3. Begin to introduce other skip-counting rules, such
as counting by 5s and 10s. Continue to use the
rods, connecting cubes, and the 100s chart to
illustrate these counting methods. For example,
show a yellow rod and an orange rod.
• What is the value of the yellow rod?
5
• The orange rod?
10
• How many squares would I take up on the
100s chart to represent 3 yellow rods?
15
4. Call on a student volunteer to demonstrate how to
count out 5 yellow connecting cubes for each of
the 3 yellow rods. Then, count and mark every 5th
numeral on the 100s chart to show a counting by
5s pattern. Follow the same steps to illustrate skipcounting by 10s using the orange rod.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
386
Grade 1
5. Discuss the patterns that unfold on the 100s chart
as you mark the numbers the students land on
when they count by 5s and 10s. (See Figure C in
the teacher notes section.)
Evaluate:
1. Display the stack of Riddle cards (Refer to
Handout 1-3: Riddle). Read Card A: I am made
up of 5 yellow Cuisenaire rods. How many cubes
do you need to make me?
Figure C
Handout 1-3: Riddle
2. Gather suggestions from students about how to
solve the riddle. Remind them of the work they did
during the Explore section solving the 9 red rods
problem.
3. Demonstrate how to record your counting on an
individual 100s chart Handout 1-1: Student 100s
Chart Recording Sheet by encircling groups of 5,
starting at 1, then continuing until there are
5 groups of 5 squares circled, ending at 25.
Pacing
The activity outlined in the Evaluate
section would work best as a follow-up
lesson scheduled during a different
block of time or, preferably, on the
following school day.
4. Model how to record the number of rods, the
number of cubes needed to make each rod, and the
total number of cubes in the indicated columns on
Handout 1-3: Riddle.
5. Distribute the Student 100s Chart and Riddle
handout. Students will also need sets of connecting
cubes.
6. As a formal assessment, have the students solve
their riddles independently.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
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Grade 1
7. Differentiate as needed based on your observations
of the students’ counting. Refer to the Teacher
Notes section for ideas on how to incorporate
appropriate scaffolding.
8. When students finish, have them put together their
rods into one train and use that train to measure
objects in the classroom that are the same length.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
Scaffolding
The Riddle Cards are differentiated to
accommodate children’s counting range.
Select a card that will meet the
instructional level of each student.
Based on the students’ performance, try
the following interventions and
challenges:
• Below: If some children are having
difficulty keeping track of the
number of groups they need to
make, in addition to the number of
cubes they need to count into each
group, provide 5 or 10 frames to
help these students organize their
count.
• Developing: If some children are
still relying on one-to-one
correspondence to count all of the
cubes, point to and recite the last
number of each group encircled on
the 100s chart while the student
attempts to correspond his or her
enumeration of the set of objects to
your lead.
• Achieving: If some children are
skip-counting with ease and
flexibility, provide these students
with an extension activity using a
calculator. Have them program a
skip-counting command—e.g., [+2],
[+5], [+10]—into the calculator and
then continue pressing the [=] button
until they reach their target number.
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Grade 1
Handout 1-1: Individual Student 100s Chart
________________ ’s
100’s chart
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Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
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Grade 1
Handout 1-2: Riddle Cards
I am made up of 5 yellow
rods. How many cubes do
you need to make me?
I am made up of 3 red rods.
How many cubes do you
need to make me?
(Card A)
(Card B)
I am made up of 4 orange
rods. How many cubes do
you need to make me?
I am made up of 4 yellow
rods. How many cubes do
you need to make me?
(Card C)
(Card D)
I am made up of 11 red rods. I am made up of 6 orange
How many cubes do you
rods. How many cubes do
need to make me?
you need to make me?
(Card E)
(Card F)
I am made up of 7 yellow
rods. How many cubes do
you need to make me?
I am made up of 5 red rods.
How many cubes do you
need to make me?
(Card G)
(Card H)
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
390
Grade 1
I am made up of 9 orange
rods. How many cubes do
you need to make me?
I am made up of 8 yellow
rods. How many cubes do
you need to make me?
(Card I)
(Card J)
I am made up of 6 red rods.
How many cubes do you
need to make me?
I am made up of 8 orange
rods. How many cubes do
you need to make me?
(Card K)
(Card L)
I am made up of 4 yellow
rods. How many cubes do
you need to make me?
I am made up of 12 red rods.
How many cubes do you
need to make me?
(Card M)
(Card N)
I am made up of 5 orange
rods. How many cubes do
you need to make me?
I am made up of 10 yellow
rods. How many cubes do
you need to make me?
(Card O)
(Card P)
I am made up of 7 red rods.
How many cubes do you
need to make me?
I am made up of 4 orange
rods. How many cubes do
you need to make me?
(Card Q)
(Card R)
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
391
Grade 1
I am made up of 6 yellow
rods. How many cubes do
you need to make me?
I am made up of 4 red rods.
How many cubes do you
need to make me?
(Card S)
(Card T)
I am made up of 7 orange
rods. How many cubes do
you need to make me?
I am made up of 9 yellow
rods. How many cubes do
you need to make me?
(Card U)
(Card V)
I am made up of 8 red rods.
How many cubes do you
need to make me?
I am made up of 2 orange
rods. How many cubes do
you need to make me?
(Card W)
(Card X)
I am made up of 2 yellow
rods. How many cubes do
you need to make me?
I am made up of 20 red rods.
How many cubes do you
need to make me?
(Card Y)
(Card Z)
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
392
Grade 1
Handout 1-3: Riddle
Name: _______________________________________________
Card #: _______ Riddle: _______________________________
_______________________________
Color of Rod
Number of Rods
Total Cubes
Number of Cubes: _____
Show how you solved the riddle and counted the cubes. Use numbers, pictures,
and words.
Mathematics TEKS Connections: Grades K-2
Transition 2: Multiplication & Division – Grade 1
393