Grade 3 - Unit 9: Algebraic Reasoning-All Operations
5E Lesson Plan Math
Grade Level: 3
Subject Area: Math
Lesson Title: Algebraic Reasoning-All
Unit Number: 9
Lesson Length:
Operations
13 Days
Lesson Overview:
This unit bundles student expectations that address representing and solving one- and twostep addition, subtraction, multiplication, and division problems; representing real-world
relationships using number pairs in tables; and summarizing a data set using a frequency
table, dot plot, pictograph, or bar graph. According to the Texas Education Agency,
mathematical process standards including application, a problem-solving model, tools and
techniques, communication, representations, relationships, and justifications should be
integrated (when applicable) with content knowledge and skills so that students are prepared
to use mathematics in everyday life, society, and the workplace.
Prior to this unit, in Unit 03 and Unit 07, students applied addition, subtraction, multiplication,
and division to solve one- and two-step problems. In Unit 04, students summarized a set of
data using frequency tables, dot plots, pictographs, and bar graphs. Students have no prior
experience representing numerical relationships using number pairs in tables according to the
standards.
During this unit, students gain fluency, efficiency, and accuracy while solving one- and twostep problems involving addition and subtraction within 1,000 and multiplication and division
within 100. Students build on previous understandings of strategies based on place value,
properties of operations, and pictorial representations to reason through and solve real-world
problem situations. Students explain their reasoning and solution strategies using
expressions, equations, and precise mathematical language. Through repeated exposure and
practice, students solidify their understanding of the standard algorithm to solve problems
involving multiplication of a two-digit number by a one-digit number and develop fluency using
standard algorithms to solve addition and subtraction problems within 1,000. Students
experience various real-world situations that involve various operations, including
decomposing composite figures to determine area using the additive property of area (the
sum of the areas of each non-overlapping region of a composite figure equals the area of the
original figure) to determine to area of the original figure. Real-world numerical relationships
are presented using input-output tables. Students explore number pairs in tables to determine
additive and multiplicative patterns that exist and represent the pattern (or process) using
equations and expressions. Students also revisit summarizing a set of data using a frequency
table, dot plot, pictograph, or bar graph. Students use these data representations to solve
one- or two-step problems involving the categorical data represented.
Unit Objectives:
Students will…
 Build on previous understandings of strategies based on place value, properties of
operations, and pictorial representations to reason through and solve real-world
problem situations.
 Explain their reasoning and solution strategies using expressions, equations, and
precise mathematical language.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
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Solidify their understanding of the standard algorithm to solve problems involving
multiplication of a two-digit number by a one-digit number and develop fluency using
standard algorithms to solve addition and subtraction problems within 1,000.
Experience various real-world situations that involve various operations, including
decomposing composite figures to determine area using the additive property of area
(the sum of the areas of each non-overlapping region of a composite figure equals the
area of the original figure) to determine to area of the original figure. Real-world
numerical relationships are presented using input-output tables.
Explore number pairs in tables to determine additive and multiplicative patterns that
exist and represent the pattern (or process) using equations and expressions.
Summarize a set of data using a frequency table, dot plot, pictograph, or bar graph.
Students use these data representations to solve one- or two-step problems involving
the categorical data represented.
Standards addressed:
Processing Standard
Supporting Standard
Readiness Standard
TEKS:
3.1A Apply mathematics to problems arising in everyday life, society, and the
workplace.
3.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution and
evaluating the problem-solving process and the reasonableness of the solution
3.1C Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation, and
number sense as appropriate, to solve problems
3.1D Communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language as
appropriate
3.1E Create and use representations to organize, record, and communicate
mathematical ideas
3.1F Analyze mathematical relationships to connect and communicate mathematical
ideas
3.1G Display, explain, and justify mathematical ideas and arguments using precise
mathematical language in written or oral communication.
3.4A Solve with fluency one-step and two-step problems involving addition and
subtraction within 1,000 using strategies based on place value, properties of
operations, and the relationship between addition and subtraction.
3.4G Use strategies and algorithms, including the standard algorithm, to multiply a twodigit number by a one-digit number. Strategies may include mental math, partial
products, and the commutative, associative, and distributive properties.
3.4K Solve one-step and two-step problems involving multiplication and division within
100 using strategies based on objects; pictorial models, including arrays, area models,
and equal groups; properties of operations; or recall of facts.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
3.5A Represent one- and two-step problems involving addition and subtraction of whole
numbers to 1,000 using pictorial models, number lines, and equations.
3.5B Represent and solve one- and two-step multiplication and division problems within
100 using arrays, strip diagrams, and equations.
3.5D Determine the unknown whole number in a multiplication or division equation
relating three whole numbers when the unknown is either a missing factor or product.
3.5E Represent real-world relationships using number pairs in a table and verbal
descriptions.
3.6D Decompose composite figures formed by rectangles into non-overlapping
rectangles to determine the area of the original figure using the additive property of
area.
3.8A Summarize a data set with multiple categories using a frequency table, dot plot,
pictograph, or bar graph with scaled intervals.
3.8B Solve one- and two-step problems using categorical data represented with a
frequency table, dot plot, pictograph, or bar graph with scaled intervals.
ELPS:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.2D monitor understanding of spoken language during classroom instruction and
interactions and seek clarification as needed
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types,
and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new
English words and build academic language proficiency
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more
English is acquired
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught
topic-related vocabulary and other prereading activities to enhance comprehension of written
text
ELPS.c.4F use visual and contextual support and support from peers and teachers to read
grade-appropriate content area text, enhance and confirm understanding, and develop
vocabulary, grasp of language structures, and background knowledge needed to comprehend
increasingly challenging language
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level
vocabulary
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and
connecting words to combine phrases, clauses, and sentences in increasingly accurate ways
as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content
area writing needs as more English is acquired.
Misconceptions:
•Some students may think numerical patterns in tables only apply to horizontal number pairs
rather than recognizing numerical patterns that exist in a vertical column of numbers.
•Some students may think numerical patterns in tables are always multiplicative rather than
recognizing additive patterns that may also exist.
•Some students may think the term “multiplicative pattern” means always multiplying the input
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
column (left) by a common factor to equal the output column (right) rather than recognizing
that a multiplicative pattern may also describe a situation in which the input column is divided
by a common divisor to equal the output column.
•Some students may think the term “additive pattern” means always adding the input column
(left) to a common addend to equal the output column (right) rather than recognizing that an
additive pattern may also describe a situation in which a common subtrahend is subtracted
from the input column to equal the output column.
Vocabulary:
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Addend – a number being added or joined together with another number(s)
Additive numerical pattern – a pattern that occurs when a constant non-zero value is
added to an input value to determine the output value
Area – the measurement attribute that describes the number of unit squares (or square
units) a figure or region covers
Bar graph – a graphical representation to organize data that uses solid bars that do not
touch each other to show the frequency (number of times) that each category occurs
Categorical data – data that represents the attributes of a group of people, events, or
objects
Composite figure – a figure that is composed of two or more two-dimensional figures
Counting (natural) numbers – the set of positive numbers that begins at one and
increases by increments of one each time {1, 2, 3, ..., n}
Data – information that is collected about people, events, or objects
Decompose figures – to break a geometric figure into two or more smaller geometric
figures
Difference – the remaining amount after the subtrahend has been subtracted from the
minuend
Dividend – the number that is being divided
Divisor – the number the dividend is being divided by
Dot plot – a graphical representation to organize data that uses dots (or Xs) to show
the frequency (number of times) that each category occurs
Equation – a mathematical statement composed of equivalent expressions separated
by an equal sign number(s), an unknown(s), and/or an operator(s)
Fact families – related number sentences using the same set of numbers
Factor – a number multiplied by another number to find a product
Fluency – efficient application of procedures with accuracy
Frequency table – a table to organize data that lists categories and the frequency
(number of times) that each category occurs
Input-output table – a table which represents how the application of a rule on a value,
input, results in a different value, output
Minuend – a number from which another number will be subtracted
Multiplicative numerical pattern – a pattern that occurs when a constant non-zero
value is multiplied by an input value to determine the output value
Pictograph – a graphical representation to organize data that uses a picture or symbol,
where each picture or symbol may represent one or more than one unit of data, to show
the frequency (number of times) that each category occurs
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
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Product – the total when two or more factors are multiplied
Quotient – the size or measure of each group or the number of groups when the
dividend is divided by the divisor
Subtrahend – a number to be subtracted from a minuend
Sum – the total when two or more addends are joined
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
Related Vocabulary:
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Algorithm
Area model
Array
Category
Column
List of Materials:
 Grocery circulars/ads
 Scissors
 Glue
 Graph paper
 Student math journals
 Sticky notes
 Construction
 Chart paper (Anchor
charts)
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Horizontal
Partial products
Row
Scaled interval
Square unit
Dice or Playing cards
Dominos
Multiplication/Division
handout
http://www.enchantedl
earning.com/dominoe
s/blank/1.shtml
Make a Sum Handout
K5_strip_diagrams_wor
kshop by F. Schwope
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Graph paper
Graphing my Math
Problem Solving
Model (Template)
Rulers
Markers/color pencils
www.math-aids.com
Copy of “Magic Pot” or
Internet access
Handout: Multi-Step
Word Problem #1
Handout: Multi-Step
Word Problem #2
Handout: Multi-Step
Word Problem
Assessment
INSTRUCTIONAL SEQUENCE
Algebraic Reasoning-All Operations
Phase: Engage the Learner
Day 1 Activity:
Materials: You Tube access, math journals, and chart paper.
Prior to watching the “Magic Pot,” pose the following questions to the class:
 How would you like to have two (double) of everything?
 Would you like two of any family members?
 What if there were two of you, how would this affect your life? Answer will vary
but some will realize the family must double everything such as (groceries,
clothing, games, etc…)
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Using You Tube, Play “The Magic Pot” by story cove
https://www.youtube.com/watch?v=cYRo0o1sZ9o
After watching the video, students will create a T-chart in their math journals as the teacher
creates one chart paper (later display in the class).
 Write on the left side (Into the Magic Pot) and on the right side (Out of the Magic Pot).
 Have students recall from the story the items dropped into the pot and the
corresponding “output”. Guide and model for the class and if needed, replay the video.
 Have the students fill in 1 axe as the input and 2 as the output. Then 1 hat on the input
and 2 as the output. Continue until all items dropped into the “Magic Pot” are
documented.
Ask the following:
What pattern do you see with the output? (The items are doubling).
Did you see or hear from the story an item dropped into the “Magic Pot” and it didn’t
double? (No).
How would you feel if you dropped a dollar bill into the pot but, it only gave you $0.25 or
if it gave you $5? (Mad about the quarter but happy with the $5).
Do you think this is fair?
Then explain to the class that with every input/output model, the output will reflect a
REPEATABLE pattern. Thus, if you do anything to one input, then it must be done to the
others to create a relational pattern.
What rule or repeatable pattern do you see with “The Magic Pot?” (Doubling, multiplies by
2, or skip counting the input value.)
Using www.math-aids.com the teacher will create input/output worksheets for
independent/partner work, centers, or homework.
What’s the teacher doing?
What are the students doing?
 Read prior to the lesson and be able
 Listen and watch “The Magic Pot”
to answer all content related questions
 Provide thought-out responses based
 Model properly
on the content being discussed
 Ask “open-ended questions” and be
 Fill in the T-chart accurately and neatly
able to relate the concepts to self and
 Identify and document the proper rule
others
 Ask for clarifications as needed
 Distribute the necessary paperwork to
 Stay focused and try to make
each student
connections
 Monitor for student understanding
Algebraic Reasoning-All Operations
Phase: Explore the Concepts
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Day 2 Activity:
Materials: Grocery circular/ad (see attachment if you do not have one), scissors, glue,
construction paper.
Review Input/Output table and rules from yesterday. More importantly, it is important for
students to express the (rule) as an algebraic equation/expression. Yesterday, in the story
The Magic Pot, the “Wood Cutter’s” rule was (n x 2) where n is the variable (input) and the x2
is the constant (repeatable) producing the output.
Have students locate a ratio using a grocery circular/ad. The ratio should give the amount of
items and the price (BBQ ribs-2 for $10). Make sure students do not select items that are a
one-to-one outcome (bottled water- 10 for $10). The students will:
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Select 2 items from the circular
Fold the construction paper (vertically) and glue the items on top of each column
Construct an input/output chart (using a T-chart) that continues the pattern in their
coupon located below the cut-out item example: coupon (BBQ ribs- 1 for $5, 2 for $10…
up to 10 for $50)
Find relationships between the quantity of items and the total costs.
Develop a ‘rule’ for their table.
Write an explanation for the pattern(s) they see.
What’s the teacher doing?
 Model properly
 Ask “open-ended questions” and be
able to relate the concepts to self and
others
 Distribute the necessary paperwork
to each student
 Monitor for student understanding
What are the students doing?
 Provide thought-out responses based
on the content being discussed
 Fill in the necessary information onto
their charts accurately and neatly
 Ask for clarifications as needed
 Stay focused and try to make
connections
Algebraic Reasoning-All Operations
Engage the Learner
Day 3 Activity:
Materials: Chart paper, math journals, and sticky notes
As the students enter the classroom, hand them a sticky note and ask them to think about their
favorite math shape and write it on the sticky note. Ask the following questions:
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How can data be presented to you? (Written, orally, charts, etc…)
If you had a choice between reading a paragraph of data or a bar graph of data,
which do you prefer? (Graphs would be the likely answer)
Why? (Less words, easier to read, colorful, interesting, etc…)
When have you ever seen a bar graph? (Answer varies)
Does anyone know certain parts of the bar graph? (Title, quantities
(spacing/intervals), items, and key/legend.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Have the students document their responses in their math journals. Inform the students they
will become a Data Analyst today (a person who studies data).
The teacher labels the chart paper as shown below:
Number of Students
Class Favorite Colors
10
9
8
7
6
5
4
3
2
1
x
x
x
Blue
x
x
x
x
x
x
Red
x
x
x
x
x
Green
x
x
x
Yellow
x
Orange
Ask the following questions while the students are looking at the chart “Class Favorite Colors” :
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Is it easy to read?
Do you see any of the reasons listed above why it is easy to read?
How many students like Red? (6)
What is the least favorite color? (Orange)
Which two colors are most popular? (Red, Green)
Which two colors are like equally? (Blue, Yellow)
How many more students like red than Orange? (6-1=5 because comparison)
How many times greater is Yellow than Orange? (1 x 3=3)
Is Blue and Yellow equal to the number of students who liked Red? (Yes, because
3 + 3 = 6 and 6 is the total for Red. Thus, 6 = 6)
What is the total number of students who participated in this survey? (18, if you
add 3 + 6 + 5 + 3 + 1 = 18)
Using another chart paper, label the bottom of the chart paper with the various shape names
(Circle, Triangle, Square, Rectangle, etc…). Have the students place their sticker onto the
chart paper according to the proper column (starting from the bottom moving upwards.) Once
everyone had the opportunity to place their sticker, have the students do the following:
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Using the newly created bar graph of shapes, document the chart in their math journals
Have them write 5-8 questions based on the Shape Graph (allow students to refer to
the types of questions from the Color Graphs
Have the students exchange their journals with their elbow partners and solve each
other’s problems
Allow students to communicate their answers with one another using appropriate
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
academic language
What’s the teacher doing?
 Model properly
 Ask “open-ended questions” and be
able to relate the concepts to self and
others
 Distribute the necessary paperwork
to each student
 Monitor for student understanding
What are the students doing?
 Provide thought-out responses based
on the content being discussed
 Fill in the necessary information onto
their charts accurately and neatly
 Ask for clarifications as needed
 Model good communication skills when
speaking/listening to their partners
 Stay focused and try to make
connections
Algebraic Reasoning-All Operations
Explore and Elaborate
Day 4 & 5 Activity:
Materials: Cards (Ace-9), Graph my Math (handout), color pencil/markers/crayons
Review Bar Graphs from yesterday. Ask if anyone had seen a Bar Graph after our class
discussions. Have them share with the class. If nobody shares, have an example ready. You
can use the following: As I was driving home, I realized in my car as I was listening to the
radio, the louder I turned the volume up, a Bar Graph appeared to get higher. This should
begin the class willingness to share and think a bit harder. The more a student is able to
articulate the learned concepts, the better their comprehension/understanding of that concept.
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Graphing My Math (Adding & Subtracting)
Skills: Adding, Subtracting, Collecting and Organizing Data, and Solving One - and
Two-step problems
Materials: Cards (Ace=1) through 9, take out face cards and jokers, and hand-out
Players: 1-2
Rule: Each player has a blank document sheet. The goal is to fill in a column (Full
10) before the other player. Player number ONE turns over two cards. This player
decides whether to add or subtract the numbers. The player records his/her answer in
the appropriate column. Player TWO will then draw two cards and repeat the
procedure until one player fills in a column (10 combinations of one number).
Example: Player ONE draws (3 and 6), he can find the sum (9) or difference (3) and record in
the proper column (but only one column). Player TWO draws (A and 6), she can find the sum
(7) or difference (5) and record in the proper column (but only one column). The blind card
draw will dictate which column they may use.
While monitoring the class, ask the students the following:
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What were you hoping to accomplish as you were drawing the cards? (Hoping
the sum or difference between the cards would be one that I had already started.)
Did you pay more attention to the cards your opponent drew and why? (Make
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
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sure they did the math computation correctly and that they didn’t get to add onto
their existing column faster than me.)
Based on the chart created, do you think you can create some similar questions
like we did yesterday with the “Color and Shape Graphs?” (Sure, let me show
you.)
What’s the teacher doing?
 Model properly
 Ask “open-ended questions” and be
able to relate the concepts to self and
others
 Distribute the necessary paperwork
to each student
 Monitor for student understanding
What are the students doing?
 Provide thought-out responses based
on the content being discussed
 Fill in the necessary information onto
their handouts accurately and neatly
 Ask for clarifications as needed
 Model good communication skills when
speaking/listening to their partners
 Stay focused and try to make
connections
Algebraic Reasoning-All Operations
Engage & Explain the Concepts
Day 6 & 7 Activity:
Materials: K-5_strip_diagrams_workshop by F. Schwope (Teacher resource), math journals,
problem solving template, Handout: Make a Sum
Warm-up: Play the game Make a Sum.
Prior to teaching this section, it is extremely important to be knowledgeable about Strip
Diagrams (up to the Multiplication/Division section of the packet). Please refer to the attached
Teacher resource.
Strip Diagrams provide children a pictorial (diagram) of a given problem to reduce uneasiness
and discomfort. It is a “thinking tool” a picture form to help guide students to transform words
into a practical and appropriate numerical operation/process.
Pose the following questions:
How many have ever been on a See-Saw before? (This should bring about
conversations)
What happens when you weigh more than the other person? (I go down and the other
person goes up)
What must be done for the See-Saw to be balanced? (Same weight distribution is
required)
How do you think this relates to MATH? (Blank looks possible)
Do you know that you have been solving Equations but did not realize it? (Really…How)
Let me show you!
Ask the students the following questions:
 How many can solve 1 + 3 =? (Many will say 4)
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
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How did you get your answers? (Counted up 1 from 3)
Can you solve 2 + Z = 5? (3 because I counted from 2 until I got to 5)
Do you know what the Z represents in this equation (number sentence)? (Missing
part, Variable, the Unknown)
Have students document the following examples into their Math Journals.
Now using a Strip Diagram model how 1 + 3 = 4
1 part red plus 3 part grey = 4 Total parts
Another example
97
66
?
Whole – Part = Another Part
97 – 66 = 31
As the questions become more difficult, it will benefit the students to utilize this
strategy.
Example:
Alicia has $6 more than Bobby. If Bobby has $10, how much did they have altogether?
The example below might be how one student solved the problem.
Student’s work
Using strip diagrams to solve the problem:
(filled in during discussion)
Example:
Step 1: 10 +6=16
Step 2: 10+16= 26 Alicia
money:
Alicia and Bobby
10
6
had $26 altogether
26
10
Bobby’s
money:
Solution: Alicia has $10+ $6=$16 And Bobby has $10
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
So altogether, they have $26. Because 10+10+6=26
After discussing strip diagrams, the students will complete the quick assessment
Handout: Multi-Step Word Problems #1.
What’s the teacher doing?
 Model properly
 Ask “open-ended questions” and
be able to relate the concepts to
self and others
 Distribute the necessary
paperwork to each student
 Monitor for student
understanding
What are the students doing?
 Provide thought-out responses based on the
content being discussed
 Fill in the necessary information into their
math journals accurately and neatly
 Ask for clarifications as needed
 Model good communication skills when
speaking/listening to their partners
 Stay focused and try to make connections
Algebraic Reasoning-All Operations
Explain the Concepts
Day 8 & 9 Activity:
Materials: Graph paper, math journals, marker, and ruler.
Write the vocabulary words and definitions on the board or chart paper: Area-the
measurement attribute that describes the number of unit squares a figure or region covers.
Using the board, projector, or chart paper, create an array (example: 6 x 7). Ask the following
questions:
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How many rows would this problem have? (6)
How many units within each row? (7)
How many total square units? (42)
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Today, we will take a more in depth look at this array and see what will happen if we take
away some square units. Have the students create a 6 x 7 array using the graph paper.
Provide a marker and ruler. Once they finished tracing the perimeter (outer edge), with their
color pencils, have them color in the following example:
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How many square units are colored in? (28)
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How did you get 28 square units? (I counted them)
Is there another way to find how many units the Red represents? (Pause…)
I will show you another way to find the value of square units the Red represents algebraically.
Will you help me? (Sure)…Here we go!
Ask the following:
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What was the area (square units for our original 6 x7)? (42)
Can anyone see an array only in the Red section? (I do…2 x 2=4)
Does anyone else see another one? (5 x 4 = 20)
Is there another one that we have not accounted for? (4 x 1=4)
If we add all the products of each smaller squares, do you think we will get 28?
o 4 + 20 + 4 = 28
What is the difference from the original (6x7) array and the parts of the Red? (4228=14)
Project the following figure on the overhead. Have the students create this model on the
bottom of the graph paper. See if students can find the area of this shape. Remind students
to extend the lines as needed to create smaller rectangles or squares. Once complete, have
the students come to the overhead and let them share with the class the different lines they
extended, the product of each smaller squares/rectangles, and the computation accuracy.
The design (size/shape of the smaller squares/rectangles will vary but, the product will be the
same 72 square feet).
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Additionally, create your own problems and have the kids create their own and partner them
up with another student. Allow partner to create/solve the other persons model.
Students will complete an assessment over multiplication and division word problems called:
Multi-Step Word Problems #2.
What’s the teacher doing?
 Model properly
 Ask “open-ended questions” and
be able to relate the concepts to
self and others
 Distribute the necessary
paperwork to each student
 Monitor for student
understanding
What are the students doing?
 Provide thought-out responses based on the
content being discussed
 Fill in the attached worksheets accurately
and neatly
 Ask for clarifications as needed
 Stay focused and try to make connections
Algebraic Reasoning-All Operations
Explain, Explore, and Elaborate the Concepts
Day 10, 11, & 12 Activities:
Materials: Problem Solving Template, Problem Solving Folder (addition, subtraction, division,
multiplication), Quiz Folder (addition, subtraction, division, multiplication), and notes from math
journals (reference)
Distribute the Problem Solving template (Blank form). Allow time for students to read and
absorb the necessary content. Afterwards, have them communicate what they believe each
component requires in their own words.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Day 10
Communicate the two top boxes (Understand-U, and Procedure/process P). This template is
intended for students to organize their information from top left to the right (clockwise
direction).
Ask the following questions:

What would happen if you came to school and there were no rules? (Students
could do as they pleased, no learning would take place, chaos/disorder, etc…)
Do you believe this would be the best thing for students, not having rules? (With
some disgruntled sounds, “NO”).

We all have rules to follow to keep us safe and to provide structure in life. With structure, we
still make mistakes but we tend to have others help us and guide us to reduce the changes of
additional mistakes.
The problem solving template (strategy) provide/affords you the opportunity to organize your
thoughts and keep you on track. If you make a mistake, the progression of your work using
the template will allow the teacher to help you quickly and efficiently.
Understand (U): Allows you the opportunity to pull out factual information and lets you know
what each number/vocabulary word means. Additionally, after you pull out the information,
you will need to re-write the question you must be able to answer. Can you solve a problem
without understanding what the information is and what the problem is asking you to do? (No)
Procedure (P): Allows the student to use the information from the Understand (U) and
formulate several mathematical expressions using correct academic language.
EXAMPLE:
Jenny equally distributed a bag of 36 marbles to 4 friends. How many marbles will each
friend receive?
Understand (U):





Procedure/Process (P)
Jenny equally distributed = gave away
same amount to every friend
If you give away, the amount you
have left is decreasing (only division
or subtraction can occur)
Started with 36 total marbles
4 friends will receive the marbles
Each friend will get how many
marbles?
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+, -, x,



÷
36 ÷ 4 = ?
Divide 36 by 4
Quotient of 36 and 4
Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Distribute one Problem Solving Questions from the Problem Solving Folder. Have students
work collectively in groups or pairs to discuss the Understand (U) Box only. Teachers will
monitor for understanding. Model using the overhead projector and record student responses.
Then move to the Procedure (P) Box and discuss the findings from students. Reinforce proper
academic language. If time permits, distribute another word problem from the folder and allow
students to fill in only the U and P Boxes.
Day 11
Solve (S): Allows students to perform the computation of the problem using any method from
the Procedure Box. The checking of the work by utilizing another method is paramount. This
is the reason why we must expose the students to multiple strategies and the
reciprocal/inverse functions of any mathematical process.
Evaluate (E): Allows students to pause and REFLECT to determine if they answered the
questions and to see if the answer is a REASONABLE one. This is done by taking a bit from
each of the (U,P,S) sections and writing a sentence which demonstrate the understanding of
the problem.
Using the same example from yesterday, refer to the above example:
Evaluate (E)

Solve/Check (S)
In order to find the number of marbles
Jenny distributed among her four
friends, you must divide 36 by 4 to get
9 marbles.
36 / 4 = 9 (knowing the 4’s or 9’s facts)
9+9+9+9=36 (4 groups of 9)
Have students to work on the other worksheets from the Problem Solving Folder. Allow for
students to work with others and monitor by asking relevant questions to get insight into the
thinking of your students.
Day 12
Center Day! Separate the class into 4 groupings. Using the Quiz Folder, students will work
independently or with partners within the groupings to complete all the worksheets (what is not
complete, they may take home and complete). 20 minute rotations or longer if possible!
Group 1: Worksheets from Quiz Folder: Addition and Subtraction
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
Group 2: Worksheets from Quiz Folder: Multiplication and Division or
Group 3: Domino Multiplication/Division. Goal is to connect and get rid of all the dominoes
first. If the player is unable to connect the matching domino, they must draw from the stack of
dominos and wait for their next turn.
http://www.enchantedlearning.com/dominoes/blank/1.shtml






(Each member of the group will draw 5 dominoes each.
Teacher decides who goes first.
Each member receives a Multiplication/Division Relationship with Dominos sheet
(possibly copy front/back).
Each member will document the dominoes of all players as played.
Each will write the algebraic equations (fact families) for each domino played.
Finally, each student will create an array for the given domino factors for each student
as they are played.)
Group 4: Teacher Pull-out. Working with students to address misconceptions and provide
individualized instructions.
Extension: If time permits the students can complete Handout: Multi-Step Word Problems
Assessment.
What’s the teacher doing?
What are the students doing?
 Model properly
 Provide thought-out responses based on the
content being discussed
 Ask “open-ended questions” and
be able to relate the concepts to
 Fill in the attached worksheets accurately
self and others
and neatly
 Distribute the necessary
 Ask for clarifications as needed
paperwork to each student
 Stay focused and try to make connections
 Monitor for student
understanding
Algebraic Reasoning-All Operations
Evaluate Students’ Understanding of the
Concept
Day 13
Activity:
The students will work on the performance assessment from the IFD.
Performance Assessment #1:
Analyze the problem situation(s) described below. Organize and record your work for each of
the following tasks. Using precise mathematical language, justify and explain each solution
process.
1) Clara bought 96 seeds for her vegetable garden. She bought an equal number of carrot
seeds, squash seeds, and cucumber seeds. She plans to put 8 carrot seeds in each row of her
garden. How many rows of carrot seeds can Clara plant?
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
a) Estimate the solution to the problem.
b) Create a pictorial model that could be used to represent the problem situation.
c) Determine the actual solution to the problem.
d) Describe the strategies used to solve the problem.
e) Explain how your estimation could be used to justify your actual answer.
2) A builder needed to purchase 997 bricks. One local brick supplier had 502 bricks in stock
and another supplier had 399 bricks in stock. How many more bricks will the brick supplier
need to find to have the exact number of bricks needed to complete his project?
a) Estimate the solution to the problem.
b) Write an equation that could be used to represent the problem situation.
c) Determine the actual solution to the problem.
d) Describe the strategies used to solve the problem.
e) Explain how your estimation could be used to justify your actual answer.
3) Kristina is planting a flowerbed. The flowerbed will be a total of 14 feet long and 7 feet wide.
Each flower planted will require 1 square foot of space in the flowerbed. Kristina has planted
24 lilies so far. She still needs to plant some chrysanthemums and daisies as shown in the
model.
a) Estimate the total area of the garden that still needs to be planted with chrysanthemums
and daisies.
b) Write equations that could be used to represent the area of the garden that still needs to be
planted with chrysanthemums and daisies.
c) Determine the actual area of the garden that still needs to be planted with chrysanthemums
and daisies.
d) Describe the strategies used to determine area of the garden that still needs to be planted
with chrysanthemums and daisies.
e) Explain how your estimation in part a could be used to justify your actual answer in part c.
Performance Assessment #2:
Analyze the problem situation(s) described below. Organize and record your work for each of
the following tasks. Using precise mathematical language, justify and explain each solution
process.
1) Ms. Krause wants to make confetti eggs for her daughter’s birthday party. The local grocery
store is having a sale on half-dozen cartons of eggs. Mrs. Krause created a table to track the
number of confetti eggs she would be able to make based on the number of cartons of eggs
she purchased.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
a) Write a verbal description of the relationship between the number of cartons and the
number of eggs represented in the table.
b) Write an equation with the unknown indicated that could be used to determine the number
of eggs in any given number of cartons.
c) Use the relationship within each number pair to complete the table.
d) Use a standard algorithm to determine how many confetti eggs Mrs. Krause could make if
she purchased 34 half-dozen cartons.
2) Toni has started walking every evening to get in shape. She purchased a pedometer to
record the number of steps she walked each week. She created a table to track the number of
steps she walked each week for five weeks and noticed a pattern in the table. Toni set a goal
to increase the number of steps she walked each week based on the pattern in the table.
a) Write a verbal description of the relationship between the week number and the number of
steps represented in the table.
b) Use the pattern to extend the table to determine which week Toni will have walked 200
steps more than she walked in week 1.
Performance Assessment #3:
Analyze the problem situation(s) described below. Organize and record your work for each of
the following tasks. Using precise mathematical language, justify and explain each solution
process.
1) Mrs. Green created a frequency table to record her students’ scores on last week’s math
assessment.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
a) Create a dot plot to represent the data from the frequency table.
b) Use the dot plot to answer the following questions:



How many students in Mrs. Green’s class took the assessment?
How many of Mrs. Green’s students scored below an 80?
How many fewer students in Mrs. Green’s class scored 80 or below than those who
scored above 80?
c) Use the frequency table and/or dot plot you created to complete the bar graph.
d) Analyze the dot plot and bar graph. Describe how the graphs are the same and how they
are different. Explain how the two graphs can look different yet summarize the same data.
e) Mr. Black’s class took the same math assessment as Mrs. Green’s class. Complete the
frequency table for Mr. Black’s scores using the following information:


Mr. Black’s class and Mrs. Green’s class have the same number of students.
In Mr. Black’s class, the same number of students scored 75, 85, and 95.
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Grade 3 - Unit 9: Algebraic Reasoning-All Operations
f) Using the data in the frequency table, create a pictograph to represent the math assessment
scores for Mr. Black’s class.
g) Use the pictograph to answer the following questions:



How many students in Mr. Black’s class took the assessment?
How many of Mr. Black’s students scored below an 80?
How many fewer students in Mr. Black’s class scored 80 or below than those
who scored above 80?
What’s the teacher doing?

What are the students doing?
Actively monitor as students work
to demonstrate mastery





Provide thought-out responses based on the
content being assessed
Apply the learned concepts by providing
accuracy of work
Provide neat work
Ask for clarifications as needed
Stay focused and try to make connections
Page 21 of 21