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Grade 4
Mathematics
Grade 4
Mathematics
Table of Contents
Unit 1: Data, Graphs, and Numbers ................................................................................1
Unit 2: Place Value, Number Sense and Measurement ................................................15
Unit 3: Understanding Multiplication and Division .....................................................26
Unit 4: The Multiplication Algorithm ............................................................................35
Unit 5: Dividing by 1-Digit Divisors ...............................................................................46
Unit 6: Geometry and Measurement..............................................................................56
Unit 7: Fun with Fractions and Chance .........................................................................71
Unit 8: Algebraic Thinking-Patterns, Counting Techniques, and Probability ..........80
Louisiana Comprehensive Curriculum, Revised 2008
Course Introduction
The Louisiana Department of Education issued the Comprehensive Curriculum in 2005. The
curriculum has been revised based on teacher feedback, an external review by a team of content
experts from outside the state, and input from course writers. As in the first edition, the
Louisiana Comprehensive Curriculum, revised 2008 is aligned with state content standards, as
defined by Grade-Level Expectations (GLEs), and organized into coherent, time-bound units
with sample activities and classroom assessments to guide teaching and learning. The order of
the units ensures that all GLEs to be tested are addressed prior to the administration of iLEAP
assessments.
District Implementation Guidelines
Local districts are responsible for implementation and monitoring of the Louisiana
Comprehensive Curriculum and have been delegated the responsibility to decide if
 units are to be taught in the order presented
 substitutions of equivalent activities are allowed
 GLES can be adequately addressed using fewer activities than presented
 permitted changes are to be made at the district, school, or teacher level
Districts have been requested to inform teachers of decisions made.
Implementation of Activities in the Classroom
Incorporation of activities into lesson plans is critical to the successful implementation of the
Louisiana Comprehensive Curriculum. Lesson plans should be designed to introduce students to
one or more of the activities, to provide background information and follow-up, and to prepare
students for success in mastering the Grade-Level Expectations associated with the activities.
Lesson plans should address individual needs of students and should include processes for reteaching concepts or skills for students who need additional instruction. Appropriate
accommodations must be made for students with disabilities.
New Features
Content Area Literacy Strategies are an integral part of approximately one-third of the activities.
Strategy names are italicized. The link (view literacy strategy descriptions) opens a document
containing detailed descriptions and examples of the literacy strategies. This document can also
be accessed directly at http://www.louisianaschools.net/lde/uploads/11056.doc.
A Materials List is provided for each activity and Blackline Masters (BLMs) are provided to
assist in the delivery of activities or to assess student learning. A separate Blackline Master
document is provided for each course.
The Access Guide to the Comprehensive Curriculum is an online database of
suggested strategies, accommodations, assistive technology, and assessment
options that may provide greater access to the curriculum activities. The Access
Guide will be piloted during the 2008-2009 school year in Grades 4 and 8, with
other grades to be added over time. Click on the Access Guide icon found on the first page of
each unit or by going directly to the url http://mconn.doe.state.la.us/accessguide/default.aspx.
Louisiana Comprehensive Curriculum, Revised 2008
Grade 4
Mathematics
Unit 1: Data, Graphs, and Numbers
Time Frame: Approximately four weeks
Unit Description
Mastery of numbers including counting, writing, comparing, rounding, estimating and
ordering large numbers (to 1,000,000) using place value strategies is achieved. Types of
graphs (bar, pictograph, line plot, line graph) are made and interpreted, comparing and
contrasting the data involved. Number patterns, including input-output situations and
odd and even patterns resulting from operations are examined, the corresponding rules
are stated, and predictions are made concerning next terms.
Student Understandings
Students demonstrate an understanding of place value in comparing large numbers to
1,000,000. They make, use, and interpret data from graphs and tables. They utilize
number patterns to predict missing elements in a pattern and apply probability to real-life
situations.
Guiding Questions
1. Can students show command of basic facts from grade 3?
2. Can students demonstrate an understanding of large numbers to 1,000,000?
3. Can students read, compare, and order large numbers to 1,000,000 using place
value strategies?
4. Can students select and make appropriate graphs for data sets and graph inputoutput patterns?
5. Can students find the mean, median and mode for a small set of numbers
when the answer is a whole number?
6. Can students investigate and determine patterns in operations on odd-even
numbers and generalize?
7. Can students use data sets and graphs to answer real-life questions?
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Louisiana Comprehensive Curriculum, Revised 2008
Unit 1 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
2.
Read, write, compare, and order whole numbers using place value concepts,
standard notation, and models through 1,000,000 (N-1-E) (N-3-E) (A-1-E)
4.
Know all basic facts for multiplication and division through 12 x 12 and 144 ÷
12, and recognize factors of composite numbers less than 50 (N-1-E) (N-6-E)
(N-7-E)
12.
Count money, determine change, and solve simple word problems involving
money amounts using decimal notation (N-6-E) (N-9-E) (M-1-E) (M-5-E)
13.
Determine when and how to estimate and when and how to use mental math,
calculators, or paper/pencil strategies to solve multiplication and division
problems (N-8-E)
14.
Solve real-life problems, including those in which some information is not
given (N-9-E)
Data Analysis, Probability, and Discrete Math
34.
Summarize information and relationships revealed by patterns or trends in a
graph, and use the information to make predictions (D-1-E)
35.
Find and interpret the meaning of mean, mode, and median of a small set of
numbers (using concrete objects) when the answer is a whole number (D-1-E)
36.
Analyze, describe, interpret, and construct various types of charts and graphs
using appropriate titles, axis labels, scales, and legends (D-2-E) (D-1-E)
37.
Determine which type of graph best represents a given set of data (D-1-E), (D2-E)
39.
Use lists, tables, and tree diagrams to generate and record all possible
combinations for 2 sets of 3 or fewer objects (e.g., combinations of pants and
shirts, days and games) and for given experiments (D-3-E) (D-4-E)
40.
Determine the total number of possible outcomes for a given experiment using
lists, tables, and tree diagrams (e.g., spinning a spinner, tossing 2 coins) (D-4E) (D-5-E)
41.
Apply appropriate probabilistic reasoning in real-life contexts using games and
other activities (e.g., examining fair and unfair situations) (D-5-E) (D-6-E)
Patterns, Relations, and Functions
42.
Find and describe patterns resulting from operations involving even and odd
numbers (such as even + even = even) (P-1-E)
43.
Identify missing elements in a number pattern (P-1-E)
44.
Represent the relationship in an input-output situation using a simple equation,
graph, table, or word description (P-2-E)
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Louisiana Comprehensive Curriculum, Revised 2008
Sample Activities
Activity 1: Develop a Sense of Large Numbers (GLEs: 2, 13, 14)
Materials List: clock, calculator, How Much is a Million? by David Schwartz, estimation
materials, (e.g., beans on the overhead, dots that can be drawn in one minute, rice in a jar,
pretzels in a bag, etc.)
Have students count for a given amount of time (one minute for example). Using the
number to which they counted as a benchmark, have them round their number to the
nearest hundred. Using the rounded number, have the students use a calculator or mental
math to predict how long it would take them to count to 1000, to 10,000, to 100,000, to
1,000,000. Then read How Much Is a Million? by David M. Schwartz. Have them
compare their predictions to the book.
 Give the students many varied opportunities to estimate large quantities of objects
throughout this unit using a benchmark (e.g., beans on the overhead, dots that can be
drawn in one minute, rice in a jar, pretzels in a bag, etc.). Note: Show students how
food items can be estimated by using the amount of an individual serving and the
number of servings in a container.
Example:
(Amount in an individual serving) X (Number of servings) = (Estimated amount of items in the
container).
Activity 2: Just a Grain of Rice (GLEs: 2, 13, 14, 42, 43, 44)
Materials List: rice, spoon, jar, calculator, pencil, overhead, One Grain of Rice by Demi,
or The King’s Chessboard by David Birch, Grain of Rice BLM
As an introduction to this book, show the students a spoonful of rice. Have them write
down their estimate of the number of grains of rice in the spoon. Pour the rice on the
overhead. Put the rice into estimated groups of 20 grains. Count by 20’s the groups of
rice. Compare the students’ estimated amounts of rice to the estimated amounts of rice
on the overhead. Repeat this activity using the spoon of rice as the new benchmark to fill
a small 3 ounce paper cup. Were the estimates closer this time? Why or why not?
Read either One Grain of Rice by Demi, or The King’s Chessboard by David Birch. Have
students use a calculator to determine the amount of rice that is received each day. Extend
the activity by having students use a calculator to complete an input output table for each
of the 31 days using the Grain of Rice BLM to discover the growing amounts of grains of
rice discussed in either book. Through this activity, have them identify the odd and even
number patterns that result from adding or multiplying numbers.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Activity 3: If You Made a Million! (GLEs: 2, 4, 12, 14)
Materials List: resources to find costs of items, (newspapers, magazines, or Internet
access), calculator, pencil, paper, If You Made a Million by David M. Schwartz (optional)
Students will create a story chain (view literacy strategy descriptions). Story chains are
especially useful in teaching math concepts, while at the same time promoting writing
and reading. Students can be creative and use information and math from their everyday
life. To create this story chain, put students in groups of four. On a sheet of paper, ask
the first student to write the opening sentence of the Math Chain Story, “If I had a Million
Dollars I would...” (The first student will put what they would do or buy and how much it
would cost.) The second student would use the calculator and subtract that amount. Next,
he/she would continue the story by adding what they would do/buy. The third and fourth
persons would do the same. They will continue in this manner until there is no more
money. The students may use newspapers, magazines, or the computer to find the costs
of different items they would buy “if they made a million.”
Optional: Read and discuss the book, If You Made a Million by David M. Schwartz. (This
book describes the various forms money can take, including coins, paper money and
personal checks and how money can be used to make purchases, pay off loans, or build
interest in a bank.)
Activity 4: Reach the Target Calculator Game (GLEs: 2, 42, 43)
Materials List: calculators, math learning log, pencil
Use the calculator to introduce counting large numbers. Have students enter 10,000
(100,000 or any large number as a starting point) on the calculator, press 1 (or 2 , or
5 , or 10 , etc.) depending on the target number, then press =. Ask them to continue to
hit the equal sign to make the number grow. Choose any large number (the target
number). Tell the students what number to start on and what number to put in to count
by ( 1 , or 2 , or 5 , or 10 , etc.) When everyone is ready, tell them to begin. The
students race to see who reaches the target number first (e.g. the first one to get to 10,
672). Have students record any findings or observations made about these number
patterns in their math learning log, (view literacy strategy descriptions). Each student
should have a notebook specifically for writing math learning logs. Have them describe
any number patterns they notice. For example:
 If you start with an even number and put in + 2, all the numbers are even.
 If you start with an odd number and put in + 2, all the numbers are odd.
 If you start with a number that is a multiple of 5, when you put in + 5, you have
numbers that are multiples of 5 and all the numbers will end in either a 5 or 0.
 If you start with a number that ends in 0 and put in + 10, you will have numbers that
are multiples of 10..
 Numbers that end in 2, 4, 6, 8, 0 or 5 have a factor of 2, 5, or 10; therefore, they will
all be composite numbers except the number 2. (2 is the only even prime number.)
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Activity 5: King of the Hill (GLEs: 2, 4, 42, 43)
Materials List: calculators
A calculator is used to play this game which provides practice in reading and comparing
large numbers. Ask the students to begin by entering into the calculator any hundreds
number (100, 200, 300, etc.) or thousands number (1000, 2000, 3000, etc.) or ten
thousands number, (10,000, 20,000, 30,000, et.) that you have chosen. Have students
press, 1 (or 2 , or 5 , or 10 , etc.), and then press the =. When everyone has
completed the above task, ask students to place their elbows on their desks and their
pointer finger in the air. Say, “Go!” and have the students press the = sign until you say,
“Stop!” Ask one student to read his/her number. The next student challenges the first by
reading his/her number. The student with the larger number remains standing. Continue
with this comparison of numbers until there is only one person standing (The King/Queen
of the Hill). Point out that the numbers that appear on the display bar on the calculator are
multiples of the number you are skip counting by. (Notice the number patterns of
multiples of 2 are even numbers, multiples of 5 end in either a 5 or 0, multiples of 10 end
in 0. Numbers that have 2, 5, or 10 for a multiple will always be a composite number.)
 Play professor know-it-all (view literacy strategy descriptions) to help the
students practice what they have discovered about large numbers. Write a
number on the board. Choose a professor to come up and tell as much as possible
about the number and explain their response (e.g. the teacher writes, 12,540 on
the board. Professor know-it-all reads the number. Then a student in the class
may ask, “Is your number a multiple of any other number? Professor Know-It-All
answers, yes, 12,540 is a multiple of 10, 5, 2. I know that because it ends in a 0.)
Sample questions that might be asked are: Is it a composite/prime number? Is it
larger/smaller than 12,000? Is it an even/odd number? What would the number be
if it were rounded to the nearest hundred/thousand/ten-thousand?)
Activity 6: Number Riddles Using Place Value Strategies to 1,000,000 (GLE: 2)
Materials List: set of Number Riddles Cards BLM for each student, scissors
Give students the Number Riddle cards made from the Number Riddles Cards BLM and
have them separate them by odd and even. Next, have them put the numbers in order
from least to greatest. Make up number riddles utilizing place value strategies for their
number cards or use the ones provided at the end of this activity. The students will use
deductive reasoning to locate the card that answers the riddle.
Example Riddle: I have one less ten thousand than I do thousands. I am greater than
30,000 but less than 45,000. I am an odd number. The student would then hold up the
card with 34, 687 on it.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Louisiana Comprehensive Curriculum, Revised 2008
Even Numbers
Odd Numbers
32,564
34,687
45,238
45,459
.
Using the Number Riddle Cards BLM, the teacher will read the riddles below to the
students and the students will hold up the card with the correct answer.
Riddles and Answer Key
Give one part of the riddle at a time. Have the students turn over any cards that do not
apply to that part of the riddle.
1. My digit in the ten thousands place is one less than my digit in the thousands
2.
3.
4.
5.
6.
place. I am greater than 30,000 but less than 45,000. I am an odd number.
Answer---34,687
My thousands digit and my tens digit sum is 10. I have four less tens than I do
hundreds. I am an even number.
Answer---45,954
I have one more hundred than ones. My tens digit is two times greater than my
thousands digit. I am an even number. Answer---23,564
I have just as many tens as thousands. I am an odd number. If you add the digits
in the ten thousands and thousands place, you will find the number of ones I
have. Answer---45,459
I have one more ones than ten thousands. I am greater than 30,000 but less than
40,000. I am an even number. Answer---32,564
I have the same number of hundreds as I do thousands. I am an odd number. I
am more than ½ of a million. Answer---504,409
Activity 7: Number Grid Puzzles (GLEs: 2, 43)
Materials List: grid paper taped together to create large number chart for recording
numbers, number cards, marker
Center activity---Have students study the patterns of numbers by creating a number chart
of large numbers (for instance, from 10,000 to 11,000). Place landmark numbers
(numbers that are familiar landing places, that make for simple calculations, and to which
other numbers can be related, such as 10, 100, 1000, and their multiples and factors) on
the chart and hang the chart in a center. Have number cards (with a specific number on
each card) available for a student to choose. Using the number landmarks along with
place value strategies, have them locate the placement of that number and write the
number on the chart.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Example: A student chooses a card. It has 4,787 on it. The student, using landmark
numbers and number pattern strategies, then writes that number in the appropriate space
on the number chart. This center activity remains open until all the numbers have been
filled in.
Solution:
4,781
4,785
4,787
4,797
4,800
Activity 8: Spin and Win (GLE: 2)
Materials List: Ten Digit Spinner BLM, Spin and Win BLM, paper, pencil
Students work with a partner or in groups of four using SQPL, student questions for
purposeful learning (view literacy strategy descriptions) in this readiness activity.
Students will discuss this statement that the teacher writes on the board: There is only one
number that will make this statement true: 2,863 < _________ < 8,623. Have the
students turn to their partner or to their group and come up with one question they would
like answered about that statement. The teacher will write the questions on the board.
Next, the students will use prior knowledge about place value to answer the questions
generated by the class.
Next, have the students play Spin and Win to provide additional practice with large
numbers. They will need the Ten Digit Spinner BLM, a pencil, and a paper clip to make a
spinner. 1. Place the point of a pencil through a large paper clip. 2. Place the point of the
pencil on the center of the spinner. 3. Adjust the paper clip so that the end of the paper
clip is on the center of the spinner. 4. “Flick” the paper clip to spin it.
Have them play the game with a partner. Using the Spin and Win BLM and the Ten Digit
Spinner BLM, they will try to build the largest number possible. A turn is signified by
each player spinning, the Ten Digit Spinner BLM six times. After each spin the player
decides where to place the digit he has spun. Once a digit is placed on the paper, it cannot
be moved. After all six digits are written down, the player reads his number. The partner
does the same thing. Together they must use the correct symbols (<, >, =) to record the
comparison of the numbers to declare the winner.
Example:
Spin and Win
Partner 1
Partner 2
865730
<
878621
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Activity 9: Building Mean, Mode, and Median (GLE: 35)
Materials List: linking cubes, index cards, pencil, (Optional zip lock bag or envelope
glued into learning log notebook to store vocabulary cards)
Have students work with a partner using linking cubes to display data. Using the cubes,
have students find the mode, median, and mean of the given data. Have the students work
with small numbers of linking cubes and build to larger numbers. Example: Students are
given a list of numbers (5, 5, 3, 5, 2, 1, 7). The students make the corresponding number
towers using cubes. Looking at the cube towers, students determine the mode (which
number occurred most often). Next, they arrange the towers from least to greatest. (1, 2,
3, 5, 5, 5, 7) to locate the median number (The middle number is 5. Three numbers are in
front of 5 and three numbers are after 5). The last step is to discover the mean. To do this,
have students rearrange the cubes, removing only one cube at a time, until all seven
towers have the same number. (Each tower will have 4 cubes.) Repeat this activity
several times with additional cubes.
Have the students make vocabulary cards (view literacy strategy descriptions) for
mean, mode, and median. Have them write the word in the middle of the card. Put the
definition in one top corner. Write characteristics in the other top corner. Put an
example in a bottom corner and an illustration in the other bottom corner. Cards can be
kept in a zip lock bag or in the front of the math log in an envelope that has been glued
to the inside cover as an easy reference for future activities and for preparing for a test.
Vocabulary card example:
The number that
occurs most often
in a given set of data.
The number
shows up more
than once.
Mode
4,3,6,4
4 is the mode
Game 1---4 runs
Game 2---3 runs
Game 3---6 runs
Game 4---4 runs
4 runs is the mode
Activity 10: Analyzing Data (GLEs: 2, 4, 34, 35, 36, 37)
Materials List: index cards, pencils and/or markers, learning log notebook
Have the students complete a word grid (view literacy strategy descriptions) to
differentiate attributes of graphs and kinds of graphs. Lead a discussion of which
attributes are shown on which graph. Students will place a check in the appropriate
boxes. Further discussion should be held on the purposes of graphs and which graphs
are best used to show certain kinds of data.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Analyzing data should be an ongoing activity throughout the unit. Bring in a variety of
graphs and charts from various media sources and from other content areas. Make sure to
include circle graphs, line plots, line graphs, bar graphs, and pictographs. Ask students to
look for graphs and charts and bring them to class, too. Have students analyze, describe,
and interpret the graphs. Have them determine which type of graph best represents a
given set of data. Have them find the mode, median, and mean of the given data when
possible. Provide concrete objects when needed. Ask them to make predictions based on
trends in a graph. In the process of analyzing the graphs, have students review number
and operation skills mastered in grade 3. These would include writing, ordering,
comparing numbers, and basic number facts.
Need
scale
Need
horizontal
and vertical
axes
Need
number
line
Compares
parts to
whole
Shows
individual
pieces of
data
Shows
trends.
Shows
frequency
Bar graph
Pictograph
Circle
graph
Line
graph
Line plot
Have the students make vocabulary cards (view literacy strategy descriptions) for each
kind of graph. Have them write the word in the middle of the card. Put the definition
in one top corner. Write characteristics in the other top corner. Put an example in a
bottom corner and an illustration in the other bottom corner. Cards can be kept in a zip
lock bag or in the front of their math log in an envelope that has been glued to the
inside cover as an easy reference for use with future activities and for reviewing for a
test.
Activity 11: How Do You Know? (GLEs: 14, 34, 35, 36, 37)
Materials List: paper, pencil and/or markers
Have students work in groups to survey and record data (using a table, chart, list, or line
plot) from fellow classmates to determine answers to questions such as “What is the most
popular TV show among fourth-grade students?” or “How much time do fourth-grade
students spend riding the school bus per school day?” Then have them construct the best
graph for displaying the collected data. (For example, the amount of time students spend
on homework each week could be put into a line graph.) Students should then find the
mean, mode, and median for their graphs. Groups will give oral reports of their findings
to the class. Have them develop questions for other students to answer by looking at the
data. Remind them to use the data to make questions based on predictions.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Additional information for students on using and constructing graphs are available on
LPB Cyberchannel: www.lpb.org/cyberchannel.
Math Mastery: Graphs and Statistics
Lesson 1: Reading & Interpreting Tables (05:00)
Lesson 2: Reading & Interpreting Graphs (05:13)
Lesson 3: Organizing Data with Graphs (04:03)
Lesson 4: Understanding Circle Graphs (03:46)
Lesson 5: Understanding Line Graphs (05:55)
Activity 12: Graphing Classroom Data (GLEs: 34, 36, 37)
Materials List: previously made graph vocabulary cards and graph word grid from
activity 10
This activity may be set up as a center or as an on-going activity. Create a bulletin board
of various graphs collected from Activity 10 and Activity 11. Have students analyze the
graphs according to the terms and rules they have learned. They may refer to their
vocabulary cards and/or their word grid created in activity 10 to check for: title,
horizontal axis, vertical axis, labels, equal increments, etc.
Activity 13: Probability and Graphing (GLEs: 4, 34, 36, 37, 39, 40, 41)
Materials List: dice, coins or number spinners, paper, pencil
Have students use activities involving probability to create graphs, such as:
 flipping one coin (head, tail)
 flipping two coins (head/head, tail/tail, head/tail)
 rolling a die/number cube for even and odd numbers
 rolling a die/number cube for prime or composite numbers
 finding sums of two random number generators (e.g., dice, number cubes, spinners)
 finding the products of two random number generators (e.g., dice, number cubes,
spinners)
Have the students work in groups. They will begin by making a table, list, or tree
diagram to discover all the possible results and combinations of their activity. Next, have
the students predict the probability of each result occurring. Then have them carry out the
activity (flip two coins, find the sum or product of a random number generator) a
specified number of times, record the results and construct the most appropriate graph to
show their collected data. Ask students to compare actual results and the predicted
probable results. Have each group share results with class. Decide if the activity is fair or
unfair. More probability will come in Unit 8.
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Activity 14: Computer Graphing (GLEs: 34, 35, 36, 37)
Materials List: math learning log, computer, pencil, paper, Internet access or a graphing
program
Have students use the computer to graph and analyze real-world data they have collected
over a period of a week or a month. Examples include: daily weather, absences, hours of
personal TV viewing, etc. Have them look for mode, median, mean. Print the student
generated charts, line graphs, pictographs, and circle graphs. Then have students write an
analysis of their data in their math learning log (view literacy strategy descriptions),
based on the graphs they generated. Have them demonstrate comprehension by
explaining what the graphs do and do not reveal about the data. Ask students to make
predictions based on the information from the graphs.
Graphs may be created at Create A Graph:
http://nces.ed.gov/nceskids/createagraph/index.asp
Activity 15: Graphs with a Spin (GLEs: 34, 35, 36, 37)
Materials List: paper spinners (or some other kind of top), toothpicks, construction paper,
paper, pencil
This can be done in a center. Have students construct spinners (tops) (out of interlocking
blocks or with paper and toothpicks, etc.). Once spinners have been made, have each
person take turns spinning their spinner ten times. Each time have them record how many
seconds they spin. Using their data, ask them to construct a line graph to display the
length of time of each spin. Plotting the change in time will help them develop a better
understanding that line graphs are best used to show change. Have them find the mean,
mode, and median of their graphs. (To make a paper and toothpick spinner have the
students cut a 2-inch diameter circle out of construction paper. Next insert the toothpick
into the center of the circle to complete the spinner.)
Activity 16: Student Meteorologists (GLEs: 34, 36, 37)
Materials List: Internet access or newspaper, paper, pencil
This can be done in an on-going center. Ask students, individually or in small groups, to
select a city of their choice. Using the newspaper or the Internet, have them track daily
temperatures for a one-week period. Using the data they collected, have students create
charts of high and low temperatures, calculate the mean temperatures, plot data (high and
low temperatures) on a line graph, and display results. Using the data collected, have
them write a sentence that shows the probability of a weather event happening. Have the
groups present oral reports of their findings or display their findings on a thematic
bulletin board.
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Activity17: Finding Averages (GLEs: 14, 35, 37)
Materials List: paper, pencil or markers, cubes
This can be done in a center. Students explore the concept of “mean” or “average” by
working with sets of data that are of interest to them. Have students collect data from
their classmates about the number of pets owned or the number of siblings, etc. Using
this data, have them answer questions about the average number of pets owned by each
class member, the average number of siblings of each class member, etc. Provide cubes
for the students to use to help them figure out the mean of a given set. Have students
create a graph to show their data. Add these graphs to the bulletin board display.
Activity 18: Explain the Rule (GLEs: 4, 14, 43)
Materials List: Explain the Rule BLM, calculator, pencils
Write these open-ended number sequences on the board.



2, 7, 12, 17, _____, _____, _____
Rule: __________________________________
8, 16, 32, 64, _____, _____, ______
Rule: __________________________________
203, 195, 187, 179, _____, _____, ______
Rule: __________________________________
Have students find the next three numbers in the pattern, describe the rule (function), and
then name the 5th and 6th number in the pattern. Have students determine if the numbers
are increasing (adding or multiplying) or decreasing (subtracting or dividing).
Have the students work in pairs to complete the Explain the Rule BLM. When they have
completed this activity, have students generate their own open-ended number sequence
for a partner to solve by using the calculator. First, have them choose a starting number.
Then they put in “the rule” ( + 5, or  3, or - 4, or  6, etc.). Next, have them write
down the first four numbers of their number sequence. Their partner tries to discover the
rule and writes down the 5th and 6th number in the pattern.
Activity 19: Input-Output Tables (GLEs: 12, 14, 44)
Materials List: Input-Output Tables BLM, pencils
Have students explore the operation and pattern that determines the change from the
input to the output column of a given table and/or situation. Have them complete and
discuss the Input-Output Tables BLM and represent the relationship of the input-output
tables using a simple equation.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Louisiana Comprehensive Curriculum, Revised 2008
Activity 20: Odd, Even (GLEs: 42, 43)
Materials List: calculator, math learning log
Have students use the calculator to observe number patterns; starting with any odd
number, add an even number. Have students record at least five examples of this number
pattern in their math learning logs (view literacy strategy descriptions). Next, have them
record any observations about the sums resulting from this number pattern in their math
learning logs, (e.g. odd + even = even). Then have them repeat the process by adding
any even number to an even number and record at least five examples and a summary
observation about the sums of this number pattern in their math learning logs. Lastly,
have them add any odd number to an odd number, recording examples and observations
of the sums in their math learning logs. Ask them to write or draw a picture that will
explain the results of number patterns for addition of even + even, even + odd, and odd +
odd patterns. Later in the year, we will extend this activity to include multiplication
patterns.
Sample Assessments
General Assessments



Maintain portfolios containing samples of graphs.
Record anecdotal notes on students as they complete tasks.
Give prompts such as the ones that follow, and the student will record his/her
thoughts in a personal math journal.
1. If you were playing Spin and Win, and you had written _ _ 6 _ 2 _ and on
your next spin you got a 7, where would you place the 7? Why did you make
that choice?
2. If you were using the calculator to get to the targeted number of 3,562 in less
than 10 turns, what number would you start with on your calculator and what
number would you skip count by?
3. Given (specific data), what kind of graph would you use to display it and
why?
4. When would you use an input/output table to help you in everyday life? Give
an example.
Activity-Specific Assessments

Activity 7: The student will find given numbers (between 10,000 to 1,000,000) on
a blank number chart using place value strategies and record the given number in
the appropriate spot, on the Number Grid BLM
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Louisiana Comprehensive Curriculum, Revised 2008

Activity 7: Give the Number Grids BLM puzzles to fill in and have the student
explain the strategies used to solve the grids.
Example
Solution
7,652
7,642
7,652
7,643

Activities 9 and 14: Have the student answer specific questions, such as mean,
mode, and median of a small set of numbers; as well as gather inferred data using
student-generated graphs from real-life experiences. Concrete objects will be
available, if needed.

Activities 11 and 12: The student will create bar graphs of a variety of real-world
data they collect and display them on a thematic bulletin board.

Activities 11, 13, 14, 15, and 16: Use this scoring rubric to score the performance
tasks in these Activities. Students may use the same rubric for Activity 12.
Scoring Rubric:
o 4 points---All components (title, horizontal axis, vertical axis, points
plotted correctly, appropriate scale, color, and neatness) of the activity are
completed, labeled, and correctly done.
o 3 points---All components of the activity are completed but not properly
labeled but otherwise correctly done.
o 2 points---All components of the activity are completed, but they are not
properly labeled nor correctly done.
o 1 point---The activity was attempted. Effort was evident; parts of the
activity were correct, but the activity was not completed.
o 0 points---The activity was attempted but little or no effort was evident.
No component of the activity was completed.

Activity 19: Have the student complete input/output tables and state the rule used.
The student will use the relationship between input and output to determine
missing data of a given problem.
Grade 4 MathematicsUnit 1Data, Graphs, and Numbers
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Louisiana Comprehensive Curriculum, Revised 2008
Grade 4
Mathematics
Unit 2: Place Value, Number Sense and Measurement
Time Frame: Approximately four weeks
Unit Description
This unit focuses on understanding numbers through 1,000,000 in standard, written, and
expanded form. It also addresses measurement as it relates to money and time.
Student Understandings
Students read and express numbers in standard, expanded, and written forms through
1,000,000. Students use estimation, mental math, or addition and subtraction computation
to solve real-world problems with multi-digit entries and solutions, along with problems
that include money, elapsed time, and possible outcomes.
Guiding Questions
1. Can students represent and interpret representations for numbers through
1,000,000?
2. Can students work easy problems with money and make change to a given
amount under $50?
3. Can students tell time and work elapsed time problems that do not cross
midnight or noon?
4. Can students write, interpret, and solve number sentences for simple real-life
problems involving + and –?
5. Can students determine the total possible outcomes of a given situation using
a table, graph, or list?
Unit 2 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
1.
Read and write place value in word, standard, and expanded form through
1,000,000 (N-1-E)
2.
Read, write, compare, and order whole numbers using place value concepts,
standard notation, and models through 1,000,000 (N-1-E) (N-3-E) (A-1-E)
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
GLE #
7.
8.
12.
13.
14.
GLE Text and Benchmarks
Give decimal equivalents of halves, fourths, and tenths (N-2-E) (N-1-E)
Use common equivalent reference points for percents (i.e., ¼ , ½, ¾, and 1
whole) (N-2-E)
Count money, determine change, and solve simple word problems involving
money amounts using decimal notation (N-6-E) (N-9-E) (M-1-E) (M-5-E)
Determine when and how to estimate and when and how to use mental math,
calculators, or paper/pencil strategies to solve multiplication and division
problems (N-8-E)
Solve real-life problems, including those in which some information is not
given (N-9-E)
Algebra
15.
Write number sentences or formulas containing a variable to represent real-life
problems (A-1-E)
Measurement
23.
Set up, solve, and interpret elapsed time problems (M-2-E) (M-5-E)
Data Analysis, Probability, and Discrete Math
36.
Analyze, describe, interpret, and construct various types of charts and graphs
using appropriate titles, axis labels, scales, and legends (D-2-E) (D-1-E)
40.
Determine the total number of possible outcomes for a given experiment using
lists, tables, and tree diagrams (e.g., spinning a spinner, tossing 2 coins) (D-4E) (D-5-E)
Sample Activities
Activity 1: Getting to Know a Million (GLEs: 1, 2)
Materials List: base ten blocks or grid paper, calculator, ruler
Ask students what they know about the number one million. They may mention that it
has six zeros, it’s an even number or it’s a lot of money. Help students understand the
number by using various ways to visualize it. One way might be to use the flat that
represents 100 from a set of base ten manipulatives. If none are available, a 10 × 10 grid
can be used. Ask the following questions:
 What number does one row of ten flats represent? (1000)
 What number does a square consisting of ten rows of ten flats represent?
(10,000)
 What number does a row of ten such squares represent? (100,000)
 What number does a square of ten such rows represent? (1,000,000)
Use illustrations to help students visualize the process. As students go through the
example, help them relate to the expanded and written forms of numbers. Use other
examples to illustrate the relative size of large numbers. Ask students to predict how long
it would take a hundred seconds to tick away, and then a thousand, and then a million.
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Show the students the answers using a calculator. Another example would be the lengths
of a hundred inches, a thousand inches, and a million inches.
Activity 2: Read, Write, Compare, and Order (GLEs: 1, 2, 14, 36)
Materials: computer, encyclopedias, or current atlas, learning log notebook, pencil
Provide student groups with a list of several cities in Louisiana. Have students research
(using the computer, encyclopedias, current atlas, etc.) to find the most current population
of each city and have them compare the populations and put them in order from smallest
to largest. Have them record this information in their math learning log (view literacy
strategy descriptions). It will be used later on in the specific assessment activity in which
GLE 36 will be covered. Have students indicate the reason for the ordering by
underlining the appropriate place-value digit. Next, have students write the word names
for each population number and write the numbers in expanded form.
Activity 3: Find Your Partners (GLEs: 1, 2)
Materials List: index cards with numbers written in corresponding standard, word, and
expanded form, Find Your Partners BLM
Use the Find Your Partner BLM to make three sets of cards. On the first set, write
numbers in standard form. On the second set, write the corresponding numbers in written
form. On the third set, write the corresponding numbers in expanded form. Give each
student one of the cards. When everyone has received a card, have them quietly move
around the room to find their partners. When each group has found their number in
written, standard, and expanded forms, have them read their cards aloud. If the class
agrees, have them give a thumbs-up. If they disagree, have the class give a thumbs-down.
Any group receiving a thumbs-down needs to try again to find the correct partners.
32, 604
Thirty-two
thousand, six
hundred four
30,000 + 2,000 + 600 + 4
Activity 4: Number Loops (GLE: 1)
Materials List: paper, pencil, post its
Have the students practice making number loops. This will give them practice with
numbers written in standard and word form. Numbers will always loop back to 4.
Begin by writing any number in standard form. Next, write that same number in word
form. Next, count the letters in the written form. Write that number in standard form.
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Continue the pattern of word form, standard form until the number loops to 4. (This is a
good warm-up activity.)
Sample 1
378,010
Three hundred seventy-eight thousand ten
35
thirty-five
10
ten
3
three
5
five
4

Sample 2
6,907
six thousand nine hundred seven
27
twenty-seven
11
eleven
6
six
3
three
5
five
4
Give each student a post-it sheet. Have them write a number between 35,001 and
75,009 at the top of the sheet using standard form. Next, have them write the
number they chose in word form. Lastly, have them write their number in expanded
form. Have the students place their number on a Venn diagram that has one circle
labeled---even numbers and the other circle labeled numbers greater than 50,000.
Have the class discuss any numbers that they disagree with on the placement.
(Don’t forget to discuss the numbers that belong on the outside of the circles and the
numbers that would be shared by both circles.)
Even Numbers
45,213
forty-five
thousand, two
hundred
thirteen
40,000+5,000+
200+10+3
35,604
thirty-five
thousand, six
hundred four
30,000+5,00
0+600+4
Numbers > 50,000
61,215
sixty–one
thousand,
two hundred
fifteen
60,000+1,00
0+200+10+5
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
Activity 5: Bonus Purchase Bingo (GLEs: 12, 13)
Materials List: Bonus Purchase Bingo BLM, chips, or crayons, calculator, paper
Give each student a sheet of paper. Have them fold it in half, then into fourths, eighths,
and finally into sixteenths to make their bingo sheet. While they are doing that, write
these amounts on the board.
$15.00
$9.00
$24.00
$39.60
$13.50
$18.00
$ 14.00
$2.25
$34.42
$9.75
$3.00
$24.90
$40.00
$17.10
$34.00
$10.00
$14.78
$9.20
Have the students copy them in any space they would like using a crayon. (You may
want to designate one or two spaces as “free.”) Each student creates their own Bingo
card. The teacher will use the Bonus Purchase Bingo BLM to make the game calling
cards needed to play the game.
Have students play Bonus Purchase Bingo using the Bonus Purchase Bingo BLM. The
caller uses money flash cards to call out purchases (e.g., the total cost of two items, each
priced $5.60). Student cards contain whole-dollar estimates of the total cost and/or the
exact total cost. Have students use an appropriate calculation strategy (e.g., mental math,
estimation, paper and pencil, calculator) to determine the total cost, estimated or actual.
The estimated or actual total cost may then be covered with a chip or marked off with a
crayon on the students’ cards. The first student to cover four in a row wins. The “bonus”
comes when a student uses estimation and exact calculations to get both answers.
Activity 6: Money Hunt (GLEs: 12, 13, 40)
Materials List: Pigs Will Be Pigs, Money Hunt BLM, notebook, pencils
Read the story, Pigs Will Be Pigs, by Amy Axelrod, or a similar story about hunting for
money. As you read the story, have the students keep a running total of the amount of
money the pigs find on their money hunt utilizing a modified split-page notetaking (view
literacy strategy descriptions). Model for the students the first few steps of split-page
notetaking.
Example:
Mr. Pig found a dollar in his wallet.
$1.00
Mr. Pig found a two dollar bill in his sock
drawer.
$2.00 + $1.00 = $3.00
Mrs. Pig found two nickels, five pennies,
and one quarter in the beds, under the
carpet, in the night stand, and in her
jewelry box.
10¢ + 5¢ + 25¢ + $3.00 =$3.40
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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When the pigs have enough money to go out to eat, have the students predict what menu
items they will buy with their money.
After the story, have students work in groups to make a story chain (view literacy
strategy descriptions). Give each group a menu. Have them find as many possible
menu combinations for spending their money. The first person chooses any item from
the given menu (Money Hunt BLM) and writes what they would order. Next, they
subtract the cost of their item from the amount they have. The second person then
“purchases” the next item and subtracts the amount of their item from the money that
was left. The third and fourth persons continue this process until all the money is spent
or there is not enough money to buy anything else. Have them share their menu
combinations. Have them make a table, tree diagram, or organized list to determine
how many different combinations were possible.
Activity 7: Pigs Pizza Party (GLE: 40)
Materials List: pencil, paper
The teacher writes on the board:
Pigs Pizza Party
Types of Crusts---Thick, thin, pan
Types of Toppings--- pepperoni, hamburger, cheese
The Pigs can only order 3 kinds of pizza given these choices.
Using the strategy SQPL, student questions for purposeful learning (view literacy
strategy descriptions), ask the students to turn to a partner and think of one good question
they have about the statement on the board. Write the questions generated by the students
on the board. Example: Can the pigs have more than one topping? Next, have the
students make a tree diagram, a table, or an organized list to discover answers to the
questions written on the board.
Activity 8: Presto, Change-o! (GLEs: 12, 40)
Materials List: play coins, learning log notebook
Have students work with a partner to complete a table that shows all the possible
combinations to have the same amount of money. Begin by assigning each group a given
amount of money. Have them construct a table or an organized list to show the different
combinations of coins that will make that amount. Have coins available for students to
use, if needed. When they are sure they have found all the possible combinations, have
them write about the strategy they used in their math learning logs (view literacy strategy
descriptions). Discuss the different strategies that were used to find ALL the possible
combinations of coins.
87¢
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
Half dollars
1
0
Quarters
Dimes
Nickels
Pennies
1
1
0
2
3
0
2
2
2
3
0
7
Activity 9: Organizing Your Money (GLEs: 7, 8)
Materials List: Organizing Your Money BLM, colored pencils, index cards with money
amounts on them, box, learning log notebook
In this activity, have students create a graphic organizer in their math learning log (view
literacy strategy descriptions) connecting fractions to decimals to percents as it relates to
a dollar bill. They will need this graphic organizer as a reference to help study for future
tests. Have them complete the grid dollars on Organizing Your Money BLM. They will
use this dollar to display 101 of the dollar, .10 of the dollar, 10% of the dollar by shading
in the corresponding spaces on the grid dollar. Have them do the same for 14 , .25, 25%;
½, .50, 50%; ¾, .75, 75%; 100/100, 1 whole, 100%.
The professor know-it-all (view literacy strategy descriptions) strategy can be used to
check for understanding. A child is chosen to come to the front of the class. They will
pull a money amount card from a box and read it. (The amount could be written as a
decimal, with words, or using a cent sign.) After the student reads the amount, the
class will ask him a variety of questions. For instance: Can you rewrite your money
amount in another form? What fraction of a dollar is that amount? What percent of a
dollar do you have? Can you describe your amount in coins?
Activity 10: Sunday Shopping (GLEs: 12, 13, 14)
Materials List: Sunday circulars, play money
Provide students with sales circulars from the Sunday newspaper and play money. Have
students work in pairs and switch roles after each purchase. One student is the shopper,
and the other student is the sales clerk. Inform the students of the amount of money they
can spend for their purchase. The shoppers should go through the circulars and decide
what to buy. The sales clerk should provide the appropriate change. Repeat the activity a
few times using different amounts of money. As a variation on this activity, use menus
collected from local restaurants.
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Activity 11: Real-World Scenarios (GLEs: 13, 14, 15)
Materials List: paper, pencil, Real-World Scenarios BLM
Have the students write number sentences or equations for each situation on the Real-World
Scenarios BLM and solve using algebraic thinking.
Activity 12: Elapsed Time (GLEs: 14, 23)
Materials List: learning log notebook, pencils
Ask students to make a table to keep track of elapsed time during their after school
activities in their math learning log (view literacy strategy descriptions). Conduct a
whole-group “brainstorming” session to create a sample table with few activities so that the
students fully understand their assignment. For example, if a favorite TV show begins at 6:30
p.m. and ends at 7:00 p.m., have them record how much time has elapsed. If students are
engaged in after-school activities (e.g., sports, scouting, etc.), have them calculate when
the activity will end, given a start time and the duration of the activity. If practice starts at
3:00 p.m. and ends 1 hour and 45 minutes later, then have them record what time it will
end.
Activity 13: Time’s Running Out (GLE: 23)
Materials List: student clocks, pencils, Pigs on a Blanket by Amy Axelrod, notebook
Using student clocks, have students display the elapsed time as you read the story, Pigs
on a Blanket by Amy Axelrod, or a similar story. Have them use split-page notetaking
(view literacy strategy descriptions) to accurately record the elapsed time. Have the
students record the actual times on one side of their page. On the other side, have them
record the time that has elapsed and the new time. Accurate note-taking will reveal that
time has “run out.”
Example:
It was 11:30 when the pigs decided
Starting time-11:30
to go to the beach.
The children were ready in 10
minutes.
11:30 + 10 minutes = 11:40
Have students work in a group to write their own story of an adventurous day, using a
story chain (view literacy strategy descriptions) to record the elapsed time of their
events. The first student will choose the starting time and the beginning activity.
He/She will then pass the story for the next person to add on a continuous happening
and an amount of time for the event. The group will record the time as it passes for
each event. The group will continue passing the story around as each person adds to
the story. After each new event and time is added, the group will figure out the new
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
time. Have them share their stories with the class as they practice elapsed time of reallife events.
Activity 14: Planning Daily Schedules (GLE: 23)
Materials List: paper, pencils, (computers can also be used)
Ask students to keep a log of their service hours—time spent reading to younger students,
helping in the library, and so on. Logs could be created by making a table in Word or
Excel on the computer where computers are available.) Involve students in planning daily
schedules, field trip timetables, homework, and after-school activity times year round.
Example:
Activity
Beginning Time
Ending Time
Elapsed Time
Reading to a young
child
Helping in the Library
Cleaning my room
Activity 15: Let’s Take a Trip (GLEs: 12, 13, 14, 23)
Materials List: computers, pencils, paper
Provide a demonstration of how to plan a trip by planning a trip for yourself. Using the
computers, have students work in groups to plan a trip to a specific location. Ask them to
compare the fares for airlines, trains, and/or buses and to determine the amount of time
that will elapse during their travel. Have them make a travel log showing what they will
see, what time they will arrive/leave, and estimate how much time will elapse between
their activities. Finally, have them make a table or a list of their travel expenses, not
forgetting souvenirs. If computers are unavailable, provide brochures/travel books from
travel agencies for students to use. Additional assistance may be needed to help groups
determine a transportation mode and costs.
Activity 16: What Time Is It, Anyway? (GLEs: 13, 14, 23)
Materials List: computers, pencils, paper
Have the students research time zones on the Internet and compile a list of states that are
in each time zone and any states that have more than one time zone. Conduct a class
discussion of when time zones may be important to know (e.g., when you are traveling,
when you phone a friend in a different time zone, etc.). Using their favorite television
programs, have them record the viewing times of the programs in different time zones.
As an extension, have them work in a group to write their own elapsed time real-life
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
problems, involving time zones. Have them exchange their problems with another group
to solve. Have groups share their solutions with the class.
Sample Assessments
General Assessments



Maintain portfolios containing student work.
Create teacher-made, written tests that include:
o Matching of standard, written, and expanded forms of numbers.
o Questions about elapsed time utilizing time zones
o Open-ended questions such as: How many combinations of coins can you use
to make $.85?
Give journal prompts such as the ones that follow, and the student will add the
entries to a personal math journal.
o
If you have saved $12.57, and you get $5.50 a week allowance, will you
have enough money in four weeks to buy a video game that costs $35.00?
Why or why not? Explain
o
West Virginia is in the Eastern Time Zone. If a movie is advertised to come
on at 6:00 p.m. in West Virginia, what time will you be able to watch the
movie? Why?
o
Describe a time when you used an estimate instead of an actual amount.
Activity-Specific Assessments

Activity 2: The student groups will give oral group reports on the population of
their Louisiana cities. Together they will create and explain a corresponding
graph displaying their findings. The graphs will later be displayed on a bulletin
board. The teacher will conduct a class discussion of the displayed graphs in
which the student will compare and contrast the given data.
Use the following rubric to grade the Louisiana project:
 Scoring Rubric for Louisiana Project:
4 points---All components of the activity are completed, labeled, and correctly
done. Oral presentation was well delivered.
3 points---All components of the activity are completed but not properly
labeled but otherwise correctly done. Oral presentation was
adequate.
2 points---All components of the activity are completed but they are not
properly labeled nor correctly done. Oral report was not well
delivered but was adequately covered.
1 point---The activity was attempted. Effort was evident; parts of the activity
were correct, but the activity was not completed. Oral report was
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
not adequately covered.
0 points---The activity was attempted but little or no effort was evident. Oral
report was not given. No component of the activity was completed.

Activities 4, 5, 6, and 7: The student will participate in a “Family Night Out”
Project. The student will be asked to estimate how much money it costs for
his/her family to eat dinner at a favorite restaurant. Using donated menus, the
student will “order” a complete meal for each family member. The student will
create a bill for the charges and calculate how much change will be received if the
waiter is given $50.00 (or more).

Activities 11 and 12: The student will explore other elapsed time events, such as
the ones that follow and record the ideas in his/her personal math journal.
o How many movies can someone watch in a half-day? In 12 hours? What
else can take 12 hours to complete?
o How many regulation football games can be played in 12 hours?
o Why are election results not given as soon as the polls close in your state?
Grade 4 MathematicsUnit 2Place Value, Number Sense, and Measurement
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Louisiana Comprehensive Curriculum, Revised 2008
Grade 4
Mathematics
Unit 3: Understanding Multiplication and Division
Time Frame: Approximately three weeks
Unit Description
This unit develops a more complete understanding of the operations of multiplication and
division. Application of these operations as they are used in real-life situations will be
explored.
Student Understandings
Students will use multiplication as repeated addition tied to work with equal groups and
area, and they will work with division tied to sharing and measuring (repeated subtraction
of equal groups) developing a basis for mastery of the basic facts for multiplication and
division through 12x12 and 144÷12 by successfully completing multiplication and
division activities with and without manipulatives. Students will also determine the
appropriateness of using mental mathematics, calculators, or paper/pencil strategies for
calculations.
Guiding Questions
1. Can students model and represent multiplication and division with objects and
verbal situations?
2. Can students show mastery of the basic facts for multiplication and division
through 12×12?
3. Can students solve simple whole number sentences having whole number
solutions related to the facts?
4. Can students illustrate divisibility patterns for 2, 3, 5, and 10?
5. Can students estimate products and quotients?
Grade 4 MathematicsUnit 3Understanding Multiplication and Division
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Louisiana Comprehensive Curriculum, Revised 2008
Unit 3 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
3.
Illustrate with manipulatives when a number is divisible by 2, 3, 5, or 10 (N-1-E)
4.
Know all basic facts for multiplication and division through 12 x 12 and 144 ÷ 12,
and recognize factors of composite numbers less than 50 (N-1-E) (N-6-E) (N-7-E)
10.
Solve multiplication and division number sentences including interpreting
remainders (N-4-E) (A-3-E)
13.
Determine when and how to estimate, and when and how to use mental math,
calculators, or paper/pencil strategies to solve multiplication and division problems
(N-8-E)
14.
Solve real-life problems, including those in which some information is not given
(N-9-E)
Algebra
19.
Solve one-step equations with whole number solutions (A-2-E) (N-4-E)
Patterns, Relations, and Functions
42.
Find and describe patterns resulting from operations involving even and odd
numbers (such as even + even = even) (P-1-E)
Sample Activities
Activity 1: Talking Calculators (GLEs: 3, 4)
Materials List: calculator, pencil, paper
Have students use calculators to find multiples of 2 by pressing +2=. Have them record
the number on the display bar. Students will continue to hit the = for 20 more times, each
time recording the number in the display bar. Have the students look at the numbers they
recorded and make observations. (Possible answers: All the numbers are even. Every
number ends in a 2, 4, 6, 8, or 0. You’re counting by 2’s.) After all observations are
made, have students try larger numbers that are even and divide them by 2 on the
calculator. Help them discover that all even numbers are divisible by 2. Repeat this
activity for +5=, +10=, and +3=. (Someone may realize that a number is divisible by
three if the sums of their digits are divisible by three. If not, the teacher will need to
show the connection. Ex. 1,356; 1+3+5+6=15; so since 15 is divisible by 3; 1,356 is also
divisible by 3.)
Example:
Input for +2
1 (1 time of pressing =
Output (after pressing =)
2
after +2 was put in)
2 (times of pressing =)
4
3
6
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Activity 2: Dividing with Tiles (GLEs: 3, 4, 10)
Materials List: tiles, Tiles BLM, Dividing with Tiles BLM, scissors, learning log
notebook
Students, working in pairs, will use the Dividing with Tiles BLM to begin exploring
number patterns of division. Students will need to cut the Tiles BLM sheet into individual
tiles. Students will sort tiles into groups of twos and record their answers on the table
according to the directions on the Dividing with Tiles BLM. Students should begin to
recognize a pattern of numbers divisible by two. (Numbers that end in 2, 4, 6, 8, or, 0 are
divisible by 2.) This activity will continue with numbers that are divisible by 5 (numbers
that are multiples of five). Have the students sort the tiles into groups of fives. Students
should be able to identify a pattern for numbers that are evenly divisible by 5. (Numbers
that end in 5 or 0 are divisible by 5) Students progress to sorting tiles into groups of tens,
then threes. Students should be able to recognize and explain patterns that emerge for
numbers that are divisible by 2, 3, 5, and 10. Students should record the patterns that they
have discovered in their math learning log (view literacy strategy descriptions). This
information will serve as a reference for future division problems and serve as a key for
unlocking many simple division problems. After the students have recorded their
observations in their math logs, discuss the divisibility rules they have discovered. Check
that all number patterns for divisibility have been found.
(A number is divisible by another number if there is no remainder when you divide.)
 A number is divisible by 2 if it is an even number. (Even numbers end with 2, 4, 6, 8, or
0.)
 A number is divisible by 5 if it ends in 5 or 0.
 A number is divisible by 10 if it ends in 0.
 A number is divisible by 3 if the digits of the number add up to a number that is divisible
by 3.
Repeat this activity on another day, and this time include numbers that are evenly
divisible by a given number, as well as numbers that will have remainders. Conduct a
class discussion on how you are able to predict whether a division problem will have a
remainder. If they cannot figure this out on their own, have them refer back to their math
logs and the divisibility rules.
Activity 3: “Array! Array!” (GLEs: 3, 4, 10 )
Materials List: red beans, One Hundred Hungry Ants, math learning logs, pencils
Sing the song, “The Ants Go Marching” substituting “Array” for “Hooray,” or read the
story One Hundred Hungry Ants by Elinor J. Pinczes. Have 12 students act out how the
arrangement (“array”gement) of the ants changes when they march by 1’s, 2’s, 3’s, and
4’s. Have the students draw corresponding ant arrays in their math learning logs (view
literacy strategy descriptions) as the story is read. Give the students red beans (red ants)
to discover arrays for other numbers. Have them record their arrays and write the
corresponding fact families in their math log. Drawing the arrays and writing the
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corresponding fact families in their math log will help them better understand the concept
of multiplication and the relationship of multiplication to division. Have them discuss
numbers that were attempted but were not able to be made into arrays. Help them to
discover that these numbers cannot be divided evenly. These numbers will always have a
remainder.
Activity 4: Multiplication Strategies (GLEs: 4, 10)
Materials List: chart paper, markers
Students will brainstorm,(view literacy strategy descriptions) to construct strategies that
can be used to help with multiplication. Example: Multiplying by two’s is the same as
doubling a number. Use the minute hand on the clock to help you multiply by five.
Display the multiplication strategy chart in the room as an easy reference when students
are having difficulty. Students should be able to come up with ideas similar to these:
Multiplication Strategies
x 0---Any number multiplied by 0 is 0. Example: 435 x 0 = 0
x 1---Any number multiplied by 1 is that number. Example: 64 x 1 = 64
x 2---Any number multiplied by two just double that number.
Example: 7 x 2 = (7+7=14)
or Skip count by 2’s that many times.
Example: 4 x 2 = (2, 4, 6, 8)
x 3 Any number multiplied by 3 just multiply that number by 2 and add that number to the
answer. Example: 3x7= (2x7=14 so 14+7=21)
x4---To multiply by four, you double the double
Example: 4 x 6 = (You think 2 x 6 = 12 so4 x 6 = 12 + 12 which is 24 because 4 x 6 is the same
as 2 x 6 + 2 x 6)
x5---To multiply by five, use the minute hand on the clock.
Example: 8 x 5 = 40 (The minute hand is on the 8 so it is 40 minutes past the hour.) or Skip
count by 5’s that many times.
Example: 8x5= (5, 10, 15, 20, 25, 30, 35, 40)
x6--- just multiply any number by 5 and add that number to the answer.
Example: 6x7= (Think 5x7=35 so 35+7=42)
x7---You only have to learn 7x7=49 and 7x8=56 All other multiplication facts for 7 you already
know. (Because of the commutative property)
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54
36x8---You only have to learn 8x8=64 All other multiplication facts for 8 you already
know. (Because of the commutative property) Use this poem to help you---8x8 Open the
door because I know it’s 64.
x9---The digits add up to 9 Example: 6 x 9 = (Think 1 less than 6 is 5. Write 5 down. Now think
“what number can I add to 5 to get 9”? (4) Write 4 behind the 5. Answer 6 x 9 =5
4(Because 5 + 4 = 9)
or
Hand jive---Place your hands on a flat surface. Look at the multiplication sentence, example 4x9.
Start with your pinky and count the number that is being multiplied by nine (4). Fold your fourth
finger under. Count the number of fingers in front of your fourth finger (3); put that number in
the tens place. Count the number of fingers behind your fourth finger (6); that number is in the
ones place so 4x9=3 6
or
Think Backwards---Example: 6x9 = (Think 6 x 10 = 60 so 60 – 6 = 54)
x10---Add a 0 behind the number you are multiplying. Ex.: 6 x 10 = 6 with a 0 behind it. (60)
Think of 10 as a dime and count the number of dimes you are multiplying 10 by. Example: 4 x
10 = 4 dimes
Activity 5: Collect and Compute (GLEs: 4, 10, 14, 19)
Materials List: chart paper, markers
Have students collect aluminum cans for recycling. Set a goal for the allotted collection
time period. Students record the number of cans collected each day and multiply by the
amount paid for each (e.g., 7 cans × 7¢). Have students repeat the calculation for each
day of the week. Create a classroom chart. Have students calculate how many more cans
are needed to reach their goal.
Activity 6: Multiplication Wheels (GLEs: 4, 19)
Materials List: construction paper, scissors, dry erase markers, glue
Have students draw and cut out two circles: a large circle and a medium size circle, use a
different color of construction paper for each circle. Draw a small inner circle in the
middle of the medium circle. Write the numbers 0 through 12 along the edge of the
medium circle (as if on a clock). Glue the medium circle on top of the large circle.
Laminate the circles. (You could laminate the sheets of construction paper ahead of time.
That way, once the circles are cut, the students can immediately begin using them.) When
ready to use, write a number (factor) in the small center circle with an
erasable marker. Students complete their wheels by writing the product
54
36
of the two factors on the outer segment of the wheel. Example: Student
9
6
puts a 6 in the middle circle. He thinks 6x9, and writes 54 in the outer
6
circle.
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Activity 7: Multiplication Stories (GLEs: 4, 10, 14, 19)
Materials List: Each Orange Has Eight Slices, paper, pencils, crayons (for illustrating
books)
Read the story, Each Orange Has Eight Slices by Paul Giganti, Jr., or a similar book.
Discuss the different multiplication stories from the book along with other real-life
multiplication stories. Have students work in groups of four to construct their own
multiplication story chain (view literacy strategy descriptions) using the book Each
Orange Has Eight Slices as a model. The first person will write the first page, and then
pass it on to the next person to add a page. They will continue taking turns writing a page
until the book is complete. You may want to have each group choose a topic to write
about before beginning their story chain to ensure a variety of stories. Have each group
share their book with the class and then place them in a center for students to read again
and again. The real-world application of multiplication shown in the story chains will
help to give relevance to multiplication.
Activity 8: Domino Games (GLE: 4)
Materials List: dominoes or make dominoes on card stock (a template can be found at
this site: http://www.first-school.ws/theme/printables/dominoes-math.htm),
manipulatives (beans, cubes, chips), calculator, pencil, paper
Use dominoes in a variety of games to practice multiplication facts. The two numbers on
the domino tiles are factors. Students draw a tile and state the product. The person
having the largest product gets a point. An incorrect answer causes a person to lose their
turn. If the other person can state the correct answer, he steals that point and then takes
his regular turn. Make sure that manipulatives such as cubes, beans, chips are available
for students to use to determine the products, if needed. A calculator should also be made
available to check answers when there is a disagreement.
Activity 9: Fence It In! (GLEs: 4, 14, 19)
Materials List: Fence It In! BLM, pencils
Students, working in groups, will complete a project using the Fence It In! BLM. They
will utilize division to complete this backyard activity learning the importance of division
in our everyday lives.
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Activity 10: Catalog Shopping (GLEs: 13, 14, 19)
Materials List: catalog, book club order, or grocery/department store circulars, Catalog
Shopping BLM, pencils
Students work in small groups to place an imaginary order from a catalog, book club
order, or grocery/department circular. They will use the Catalog Shopping BLM to order
ten items, spending no more than $25 and no less than $20. Students should use
estimation to determine if their “order” meets the prescribed criteria before completing
the actual order. This activity has an even greater effect if the students are placing real
orders, such as items for the school store or a classroom book order.
Activity 11: What’s Best? (GLEs: 13, 14)
Materials List: index cards with items and the price of the items glued on the cards, box,
calculator
Prepare index cards with items and the items’ prices glued on them ahead of time and
place the cards in a box. One student will then select a card. They will use that card to
play professor know-it-all (view literacy strategy descriptions). Another student will be
chosen to be the “professor.” Professor know-it-all will either be asked (1) to give the
exact amount or the estimated cost for buying several items at the given price; or (2) to
give the exact amount or the estimated unit price when given the total cost of several of
the same items. The “professor” will then
o state if the actual answer or the estimated answer would be best (easiest to figure
out)
o state the actual or estimated answer
o explain how they figured out the answer.
Students may use calculators to check the Professor Know-It-All’s answers for accuracy.
Example: Student 1 draws this card and asks “The Professor” to give the actual or
estimated cost of buying 6 suckers.
Sucker---36¢
The “professor” states, “It’s best to give an estimated answer. The answer is $2.40. I got
my answer by rounding 36¢ to 40¢ and multiplying by 6.
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Activity 12: Multiplication/Division Equation Games (GLEs: 4, 10, 19)
Materials List: vary according to game chosen
Adapt popular games to solve multiplication/division equations.
● Concentration can display cards showing equations ( 3 m  15 , 32  4  y ) that
correspond with cards showing solutions (5, 8). Students must find the
corresponding missing part of the equation rather than the identical pair.
● Bingo cards can also have numbers that students may cover if they solve equations
read by the caller. Students can create these games themselves in small groups
making sure there is only one possible answer for each equation. Laminate and
place in a center for remediation or just for fun.
● Example of Bingo card made by
 121
 40
students:
 21
 65
Cards read by caller: n x 11= 10 x =
7 x n = 3 x n = 12 x n = n x 4 =
 6
 64
6xn= nx8= 9xn=
● Buzz---This game is used to review a specific fact family. It can be played in a
small group or the entire class. The leader chooses a number between 2 and 9. The
leader says 1, the next player says the 2, and so on. When they reach a multiple of
the number chosen, the player says "buzz" instead of the number. If a player forgets
to say buzz or says it at the wrong time, he or she is out. Play continues until the
group reaches the last multiple of the number times 9. Example: The leader chooses
“3.” The first person begins, “1” the next person says, “2” the next person says,
“Buzz” (because 3 is a multiple of 3). The game continues with each person taking
a turn until someone misses saying “Buzz.” (1, 2, Buzz, 4, 5, Buzz, 7, 8, Buzz, etc.)

56

16

72
● Around the World---Large group flash cards are great for "Around the World."
Students sit in a circle and choose a starting person. This student stands behind the
next student in the circle. The teacher holds up a flash card. The first student to say
the answer stands behind the next person in the circle. If a sitting student says the
answer first, the standing student sits down in the winner’s chair. This process
continues until at least one student makes it completely around the circle.
Students need a great deal of practice to become proficient with multiplication and
division. Try these Internet sites for more game ideas:
http://www.multiplication.com/classroom_games.htm (Many of these games can be
adapted for division.) and/or http://www.edhelper.com/division.htm
Activity 13: Multiple Patterns (GLEs: 4, 42)
Materials List: hundred boards, crayons, Multiple Patterns BLM
The students, using a hundred board, will circle the multiples of 2 in red. Next on the
same hundred board, they will draw a blue square around the multiples of 3. Last they
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will draw a green triangle around the multiples of 4. They will make a list of all the
common multiples of 2 and 3; 2 and 4; 3 and 4; and 2, 3, and 4. Have them look for the
patterns that are formed when you multiply even numbers times even numbers, even
numbers times odd numbers, and odd numbers times odd numbers. At a later date, the
same thing can be done to find multiples of other numbers.
Sample Assessments
General Assessments
 Maintain portfolios containing student work.

Record anecdotal notes on students as they complete tasks.

Give prompts such as the ones that follow, and students will record their thoughts
in their personal math journals.
 Ask students to demonstrate comprehension of multiplication and division
concepts in real-world problems such as equal division of materials
(candy, pencils, money, etc.) among participants.
 Include topics such as budgeted shopping trips, dissemination of monies
earned through a fundraiser, or calculating how many cars are needed for a
class field trip.
Activity-Specific Assessments
 Activity 3: The students will demonstrate the relationship between
multiplication and division by writing the fact families from 1–12.
Example:
3 x 8=24
8 x 3=24
24 ÷ 8=3
24 ÷ 3=8

Activity 7: The students will create multiplication picture books for
younger students (If two wagons each have three puppies in them, how
many puppies are there in all?) or real-world word problem books for their
peers (The student bought four pencils at 25¢ each and three folders at 50¢
each at the school store. He gave the cashier $5.00. How much change
did he get back?).

Activities 9, 10: Rubrics for group activities
Score 3---Project was successfully completed. There was evidence
that all members were actively involved.
Score 2---Project was successfully completed, but all members
were not actively involved.
Score 1---Project was attempted but not successfully completed.
All members were not actively involved.
Score 0---Project was not attempted.
Grade 4 MathematicsUnit 3Understanding Multiplication and Division
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Grade 4
Mathematics
Unit 4: The Multiplication Algorithm
Time Frame: Approximately four weeks
Unit Description
Students learn the multiplication algorithm for 2-digit by 1-digit numbers, 3-digit by 1digit numbers, and 2-digit by 2-digit numbers. This builds on the understanding of the
distributive property from the previous grade as well as work with facts and the concept
of multiplication. Work should proceed from partial products and work with objects and
models representing partial products. Patterns that work with multiplying by 10 also
need to be included.
Student Understandings
Students can solve multiplication problems involving 2-digit by 1-digit numbers, 3-digit
by 1-digit numbers, and 2-digit by 2-digit numbers based on their work with the concept
of multiplication and models representing how place value and the multiplication
operation are combined with the distributive property.
Guiding Questions
1. Can students model multiplication of 2-digit by 1-digit and 2-digit by 2-digit
numbers?
2. Can students use the multiplication algorithm to solve problems involving 2digit by 1-digit numbers, 3-digit by 1-digit numbers, and 2-digit by 2-digit
numbers?
3. Can students rewrite products as the sum of two products to illustrate the
distributive property of multiplication over addition?
4. Can students write a number sentences containing a variable for
multiplication?
Grade 4 MathematicsUnit 4The Multiplication Algorithm
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Unit 4 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
4.
Know all basic facts for multiplication and division through 12 x 12 and 144 ÷
12, and recognize factors of composite numbers less than 50 (N-1-E) (N-6-E)
(N-7-E)
10.
Solve multiplication and division number sentences including interpreting
remainders (N-4-E) (A-3-E)
11.
Multiply 3-digit by 1-digit numbers, 2-digit by 2-digit numbers, and divide 3digit numbers by 1-digit numbers, with and without remainders (N-6-E) (N-7E)
13.
Determine when and how to estimate and when and how to use mental math,
calculators, or paper/pencil strategies to solve multiplication and division
problems (N-8-E)
14.
Solve real-life problems, including those in which some information is not
given (N-9-E)
Algebra
15.
Write number sentences or formulas containing a variable to represent real-life
problems (A-1-E)
16.
Write a related story problem for a given algebraic sentence (A-1-E)
17.
Use manipulatives to represent the distributive property of multiplication over
addition to explain multiplying numbers (A-1-E) (A-2-E)
Patterns, Relations, and Functions
42.
Find and describe patterns resulting from operations involving even and odd
numbers (such as even + even = even) (P-1-E)
Sample Activities
Activity 1: Multiplication Vocabulary Bingo (GLE: 4)
Materials List: paper, pencil, index cards, (optional)zip lock bag or envelope for the
vocabulary cards
Have students create multiplication vocabulary card, (view literacy strategy descriptions)
for the following terms: factor, product, multiple, array, multiplication, distributive
property, property of zero, identity property, composite number, prime number, digit,
even, odd. These vocabulary cards will be used to play Multiplication Vocabulary Bingo
as well as to serve as future reference cards to deepen the understanding of
multiplication.
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Vocabulary card example:
An arrangement
of objects in
rows and columns.
A rectangular
picture made
To show a multiplication
problem.
array
6 columns
( “array”ngement)
arrangement that
shows multiplication.
3 rows
xxxxxx
xxxxxx
xxxxxx
This is an array for 3 x 6 = 18
o Have the students fold a sheet of paper into eighths. Then have them choose any
eight multiplication vocabulary cards and have them write one vocabulary word
on each space on their paper to create their own bingo card. State definitions or
examples of a vocabulary word. The student will cross out the corresponding
vocabulary word on their card. The first person to cross out their entire bingo
card wins.
Activity 2: Multiplication Target Game (GLEs: 4, 10, 11)
Materials List: Multiplication Target Game BLM, pencil, calculators,
Two students play this game using the Multiplication Target Game BLM. Select a 3digit target number. Players take turns choosing two factors they think will result in a
product close to the target number. Calculators are used to check their guesses, and the
result is recorded as either “too high” or “too low.” The winner is the student who finds
two correct factors or the one who guesses closest to the target number. This game can
be adapted to use constant factors. (Play this game several times.) Closely monitor groups
to ensure that the students are writing their guesses first and then using the calculator.
Target Product: 192
Turn Factors
1
2
3
12 × 12
12 × 18
×
Product
144
216
Difference
Too High
Difference
Too Low
48
24
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Activity 3: Understanding Multiplication I (GLEs: 4, 11, 17, 42)
Materials List: base 10 blocks or grid paper, pencil, paper
Provide students with base 10 blocks or grid paper on which they will create rectangular
arrays of dots that illustrate multiplication problems. Have students break the arrays
along place value lines.
For example 3 × 12 would look like the following:
* * * * * * * * * * |* *
* * * * * * * * * * |* *
* * * * * * * * * * |* *
Notice that the array is such that to the left of the vertical lines are the tens and to the
right are the ones. There are three rows since the multiplier is 3. Now, help students to
see that 3 × 12 = 3 × 10 + 3 × 2 = 30 + 6 = 36. Note that the arrays could be broken in
other ways. For example, instead of the array 3 × (10 + 2), 3 × (7 + 5) could have been
used. A discussion could ensue about why it is easier to break the arrays along place
value lines. Repeat this activity with other 2-digit by 1-digit multiplication problems.
o Students along with the teacher will create a word grid (view literacy strategy
descriptions), to help them better understand multiplication. Once the word
grid is formed, the students will suggest 2-digit by 1-digit multiplication
problems to be used in the grid. (Students can continue to build the
multiplication problems using the Base 10 blocks or draw the corresponding
array on grid paper, if needed.)
Example:
Multiplication
problem
Both
factors are
even
Both
factors are
odd
5 x 26
8 x 12
One factor
is even and
one factor
is odd
X
X
15 x 6
X
Another way to break
this multiplication
problem
The
product is
even.
(5 x 20) + (5 x 6)
X
(8 x 10) + (8 x 2)
X
(10 x 6) + (5 x 6)
X
The
product is
odd
Activity 4: Any Way You Slice It! (GLE: 17)
Materials List: grid paper, pencil
Students will use square color tiles or centimeter grid paper to illustrate various ways to
compute a multiplication fact. Assign each pair of students a product such as 4 × 5.
Students will create a rectangular array with the tiles or on the grid paper and then
determine all of the different ways to make two rectangles by drawing a line through the
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original rectangle. For 4  5 , students could create two rectangles one of which is 4 × 2
and the other 4 × 3. Thus, 4 × 5 = 4 × 2 + 4 × 3. Students should write the individual
products in the arrays. Repeat this activity with several products.
Example:
4x2
4x3
8
20
12
4x5
20 squares
8 squares
+
12 squares = 20 squares
Activity 5: Multiplying Large Number Patterns (GLEs: 4, 10, 11, 13, 14, 15, 42)
Materials List: Anno’s Mysterious Multiplying Jar, pencils
Read the story, Anno’s Mysterious Multiplying Jar by Mitsumasa Anno, and have the
students listen to it - stop after reading the question: "But how many jars were in all the
boxes together?"
a. Have the students estimate the answer to the question and write it down.
b. Re-read the story, but this time have the students take notes using split-page
notetaking (view literacy strategy descriptions) and calculators to compute
what will happen in each step of the story as things multiply. They should list
one island, two countries, three mountains, four walled kingdoms, etc. By
asking how many total mountains (1 x 2 x 3 = 6), and how many total walled
kingdoms, etc. (1 x 2 x 3 x 4 = 24), they should recognize the pattern.
c. They can record their findings and discuss the patterns that develop.
Example:
1 mountain, 2 walled
1x2=2
kingdoms
1 mountain, 2 walled
kingdoms, 3 __________
1x2x3=6
1 mountain, 2 walled
kingdoms, 3 ___________,
4 _____________
1 x 2 x 3 x 4 = 24
Teacher note: What is created through the multiplication pattern in this book is called
factorials. The students do not need to learn this term, but for the high achievers, an
explanation of factorials follows:
The factorial of a number is the product of all the whole numbers, except
zero, that are less than or equal to that number. For example, to find the
factorial of 7, you would multiply together all the whole numbers, except
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zero, that are less than or equal to 7. Like this: 7 x 6 x 5 x 4 x 3 x 2 x 1 =
5,040 The factorial of a number is shown by putting an exclamation point
after that number. So, 7! is a way of writing “the factorial of 7” (or “7
factorial”). Here are some factorials: (1! = 1 = 1) (2! = 2 x 1 = 2) (3! =
3 x 2 x 1 = 6) (4! = 4 x 3 x 2 x 1 = 24)
(5! = 5 x 4 x 3 x 2 x 1 = 120) (6! = 6 x 5 x 4 x 3 x 2 x 1 = 720) (7! = 7 x
6 x 5 x 4 x 3 x 2 x 1 = 5,040) (8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 =
40,320)
Activity 6: Understanding Multiplication II (GLEs: 11, 17)
Materials List: graph paper or base 10 blocks, pencil, paper
Extend Activity 3 to 3-digit by 1-digit and 2-digit by 2-digit multiplication problems.
Instead of dot arrays, students should draw rectangles or use base 10 blocks, as shown
below, to show the problems. Make sure the rectangles are broken along place value
lines for both numbers. Repeat this activity several times with various multiplication
problems. Notice the use of the distributive property: 11 × 52 = (10 + 1) × (50 + 2) =
10×50 + 10×2 + 1×50 + 1×2 = 572. (This will take two to three days of practice.)
For example, 11 × 52 would be represented as:
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Activity 7: Multiplication using Expanded Notation (GLEs: 11, 13, 17)
Materials List: Multiplication Using Expanded Notation BLM, pencil, calculator
Have students work with a partner to complete Multiplication Using Expanded Notation
BLM. Both decide on a good estimate for the problem. Then one person restates one of
the two-digit factors in expanded notation form and multiplies. The other person uses
the calculator to check the answer. Answers are compared. Together, they decide which
method would be the best way to solve the problem. When the Multiplication Using
Expanded Notation BLM is completed, the students will discuss the various methods
used and explain situations when each method might be better utilized.
Example:
Multiplication
Problem
65
x 23
Estimated
Answer
70
x20
1400
Restated Using
Expanded Form
65
x 20
1300
+
Calculator
Check
65
x 3
195 =1495
65
x 23
1495
Activity 8: Turn the Table (GLEs: 10, 11, 13, 16)
Materials List: index cards with scenarios, paper, pencil, calculators
Student groups will create a math story chain (view literacy strategy descriptions) that
can be solved by the given multiplication problems. Put students into cooperative groups
of three students. Give each group a selection of multiplication statements that include 2digit by 1-digit, 3-digit by 1-digit, and 2-digit by 2-digit problems along with a group
scenario. (They may choose to use the scenario and multiplication statements given or
they may create their own following the above guidelines.) Use some problems that lend
themselves to mental math techniques, some to the use of a calculator, and some to paper
and pencil. Once the stories are created, groups will exchange their work with other
groups and solve the story problems. Solutions should contain the multiplication
sentence and the computation technique used. That is, students will indicate whether they
used mental math, calculators, or paper and pencil to do the computations. A discussion
about the computational method could be held.
Example:
Group 1 is given the scenario that they are putting together a fourth grade party, and they
are to incorporate these multiplication statements 48 x 2 = X, 16 x 12 = X, 103 x 50 = X.
Student 1 begins by saying that 2 schools will take part in the party. Each school has a
total of 48 students. How many fourth graders will participate? (He uses mental math for
this part of the story because some students may be absent or more students may enroll.
He thinks 50 x 2 is 100 and says, “There will be about 100 students at the party.)
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Student 2 says there are 16 tables. He wonders if there will be enough room for everyone
to sit down and have some seats leftover for guests. Twelve students can sit at each table.
There are 16 tables. (He uses a calculator to decide---16 x 12 = 192 seats. He says,
“There is more than enough room for everyone to sit down.”) Student 3 says to help pay
for the party, each student and teachers would collect cans to be recycled and the money
received would go towards the party. (She writes, there are 103 people who will
participate, if each of them collects 50 cans, that would be 103 x 50 = they will have
5,150 cans.
Activity 9: Higher or Lower? (GLEs: 11, 13)
Materials List: index cards with multiplication problems, calculators, paper, pencil
Prepare several multiplication problems using a 2 digit by 1 digit, 2 digit by 2 digit, or a 3
digit by 1 digit on index cards. Write an estimated answer on the card. Then choose one
student to come up to be “The Professor” to play professor know it all (view literacy
strategy descriptions). “The Professor” will draw a card. He will write the multiplication
problem on the board. Then he will read the answer on the card. He will then say if the
given answer is higher or lower than the exact answer. The class will use paper/pencil,
mental math, or the calculator to check for the correctness of his answer. Include several
cards where the exact answer can be found using mental math strategies. Include several
exercises where the computation strategy would require using a calculator. Have
students compare the strategies for exact computations. Engage students in a discussion
about when the appropriate calculation strategy may be a calculator, mental math, or
paper/pencil. (This game should be played more than once.)
For example, “The Professor” pulls the following card:
345
x 23
He writes the problem on the board and says, “The answer is 600 on the
card. I think that answer is lower than the exact answer because 300 x 20
would be 600. 345 is more than 300 and 23 is more than 20, so the exact
answer is higher.
Answer: 600
Activity 10: Your Purchase, Please! (GLEs: 4, 14, 15)
Materials List: toy store ads, pencil, paper
Give students toy store ads. Have them round the cost of each toy to the nearest whole
dollar amount. Guide the students through the process of making toy purchases when
more than one of a given toy is bought. Have students write number sentences using a
missing factor that illustrate purchases at the toy store.
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Example: 3 m  12 could illustrate the situation of buying three of the same items for a
total cost of $12. In the example, 3 x m = 12, the students need to understand that they
have to actually find a toy’s price (m), multiply it by 3 and get the answer of $12. Have
students continue to write algebraic equations that illustrate the various scenarios given
that involve the use of multiplication due to purchasing several of the same items.
Activity 11: Roll to Win (GLEs: 4, 10, 11)
Materials List: number cubes, paper, pencil, calculator
This game is played with a partner. Partner 1 rolls a number cube (dice) and records his
number. He then rolls again and records that number. The numbers are then multiplied.
Partner 2 uses the calculator to check Partner 1’s answer. The partners then change roles,
and the activity is repeated. The person with the largest product wins that round.
Practice multiplication for 2-digit by 1-digit or 2-digit by 2-digit by rolling two number
cube to generate two digit factors and one cube to generate one digit factors.
Activity 12: Beat the Calculator (GLEs: 10, 11, 13)
Materials List: overhead calculator, overhead pens
The class is divided into two groups. A player from each group comes up to the front of
the room. One student will be assigned to use the overhead projector calculator. The
other student will have to use mental math (rounding to the nearest ten or nearest
hundred) to compute the problem.
The game begins when the teacher shows the players an index card with a multiplication
problem on it. (Start with 1-digit by 1-digit problems and progress to 2-digit by 1-digit,
then 2-digit by 2-digit problems.) The person with the calculator must type the actual
numbers in the computer and get the exact answer. The person using mental math only
needs to write down the estimated answer.
The first person that writes the correct answer (exact answer for the calculator side or
estimated answer for the mental math side) wins a point for his team.
Rotate which team uses the calculator and which team uses mental math. This game helps
students understand that mental math is a quick way of getting an answer. It can even
beat a calculator. Have students brainstorm to think of situations when an estimated
answer may be all that is needed.
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Sample Assessments
General Assessments
 Maintain portfolios containing samples of student’s work.

Record anecdotal notes on students as they complete tasks.

Give prompts such as the ones that follow, and have students record their thoughts
in their personal math journals.
 Explain how you can use 10 x 6 to solve 12 x 6.
 Write a story problem that shows it would be easier to multiply than to add.
 Write a story to illustrate the property of multiplication we used in class today.
(This can be repeated for Distributive Property, Commutative Property,
Associative Property, Zero Property, and Identity Property.)
Activity-Specific Assessments

Activity 3: Assign each pair of students a large 1-digit by 1-digit
multiplication fact to solve. The students will show how smaller arrays can be
used to make it easier to find the product of a more difficult multiplication
fact. Display the solution arrays to be used as a reference by other students.
Example: 8 x 7 = could be displayed as
5 x 7=35
+

3 x 7=21
8 x 7=56
Activity 9: The students will be given several 2-digit by 2-digit multiplication
problems. They will determine the estimated answer by using mental math,
then state if the actual answer is higher or lower. Next, they will work the
problem using paper and pencil. They will then exchange papers and use
calculators to check for accuracy.
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
Activity 10: The students will take a Virtual Shopping Trip. Have students
choose a group of people they would like to buy things for, e.g., family, ball
team, classmates. Determine what that factor is and find advertisements for
ten items that include unit prices. Ask them to write a story problem with
correct multiplication sentences for each of the ten items to determine cost for
the group for each item, and determine the total cost of the virtual shopping
trip.
Example:
My birthday will be in a few days. I have invited all 20 children in my
class to come. My mom says I will have to help get things ready, for my
big celebration. I want to give each person a bag of party favors but my
mom says I can’t spend too much money. I will have to find some
inexpensive things to fill my bags. I found some smiley face pencils that
are 10 cents a piece. If I buy 20 of them that would be 20 x 10 = $2.00. I
also found some small yo-yos that cost 4 for $1.50. I would need to buy 5
packs at $1.50 a pack, or 5 x $1.50 = $7.50 (Child may continue with
added ideas.)
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Grade 4
Mathematics
Unit 5: Dividing by 1-Digit Divisors
Time Frame: Approximately four weeks
Unit Description
This unit examines division algorithms for 2-digit and 3-digit numbers divided by 1-digit
numbers, with and without remainders developed through the use of repeated subtraction,
the creating of fair shares, working with manipulatives, and drawing models to represent
divisive settings.
Student Understandings
Students understand the concept of division and are able to divide 2- and 3-digit numbers
by 1-digit numbers, with and without remainders, using a variety of methods..
Guiding Questions
1. Can students model division of 2-digit and 3-digit numbers by a 1-digit
number with and without a remainder?
2. Can students use the division algorithm to solve problems that can be
represented by division of 2-digit and 3-digit numbers by a 1-digit number,
with and without a remainder?
Unit 4 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
4.
Know all basic facts for multiplication and division through 12 x 12 and 144 ÷ 12,
and recognize factors of composite numbers less than 50 (N-1-E) (N-6-E) (N-7-E)
5.
Read, write, and relate decimals through hundredths and connect them with
corresponding decimal fractions (N-1-E)
6.
Model, read, write, compare, order, and represent fractions with denominators
through twelfths using region and set models (N-1-E) (A-1-E)
7.
Give decimal equivalents of halves, fourths, and tenths (N-2-E) (N-1-E)
9.
Estimate fractional amounts through twelfths, using pictures, models, and
diagrams (N-2-E)
10.
Solve multiplication and division number sentences including interpreting
remainders (N-4-E) (A-3-E)
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11.
13.
15.
16.
18.
19.
Multiply 3-digit by 1-digit numbers, 2-digit by 2-digit numbers, and divide 3-digit
numbers by 1-digit numbers, with and without remainders (N-6-E) (N-7-E)
Determine when and how to estimate and when and how to use mental math,
calculators, or paper/pencil strategies to solve multiplication and division problems
(N-8-E)
Write number sentences or formulas containing a variable to represent real-life
problems (A-1-E)
Write a related story problem for a given algebraic sentence (A-1-E)
Identify and create true/false and open/closed number sentences (A-2-E)
Solve one-step equations with whole number solutions (A-2-E) (N-4-E)
Sample Activities
Activity 1: Understanding Division (GLEs: 10, 11)
Materials List: paper, pencil
Have students maintain a vocabulary self-awareness chart (view literacy strategy
descriptions) for division. Provide this list of words to the students and have them
complete a self-assessment of their knowledge of the words using a chart like the one
below. Do not give students definitions or examples at this stage. Ask students to rate
their understanding of each word with either a “+” (understands well), a “√” (limited
understanding) or a “-“(don’t know). Over the course of this unit, students should be told
to return to the chart and add new information to it. The goal is to replace all the check
marks and minus signs with a plus sign. Because students continually revisit their
vocabulary charts to revise their entries, they have multiple opportunities to practice and
extend their understanding of key division terms.
Word
+ √ _
Example
Definition
division
divisor
quotient
dividend
division
algorithm
division
notations
division as
sharing
division as
fractions
division as
repeated
subtraction
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Ask students to describe situations that would involve a division problem such as 48 ÷ 6.
One example might be, “When forming six teams from 48 students, how many members
would each team have?” Have the students decide how they might find the answer.
Some students may use repeated subtraction or some may form sets of 6 from 48 dots on
a paper. Some students may use an algebraic equation to solve the problem, 6 x N = 48.
Discuss all the methods used. Once students have an understanding of the division
process, help them understand the traditional division algorithm. Explain to the students
that the algorithm is simply a way of keeping track of the numbers as you go through the
process of dividing.
Activity 2: Division Notations (GLEs: 4, 10, 11, 13, 15, 19)
Materials List: paper, pencil
To develop the students’ understanding of the ways division can be expressed, introduce
the variety of notations for division (12÷2, 12/2, 2 x n =12, as well as the notation for
long division.) Using simple division problems, have students represent each division
problem in a variety of ways and estimate the answers. For students having difficulties
with estimating the answers, have them write the division problem as an algebraic
equation and then have them apply their knowledge that division is the opposite of
multiplication to help solve the problem. Allow practice writing fact families if needed.
Student thinks of
Example: 41 ÷ 6 = N
nearest fact family
6 x N ≈ 41
6 x 7 = 42
7 x 6 = 42
42 ÷ 6= 7
42 ÷ 7 = 6
Activity 3: Division Dollars: Division as Sharing (GLEs: 5, 6, 7, 10, 11)
Materials List: paper money, plastic coins, paper, pencil, learning log
Tell students they will begin keeping a record in their math learning log (view literacy
strategy descriptions) of different ways division can be used. This will create a quick
reference that they will be able to use to help clarify problems. It will also serve as a
concrete way to see division as more than an algorithm. Have them label this section,
Division as Sharing. Give each group of two students a zip lock bag with paper money
and plastic coins in it. Next give them this problem to complete. You and a friend have
been washing cars. Together you have made $34. How can you share the $34 between
the two of you? You only have $10 bills and $1 bills.
Solution: They can share one $10 bill with each of the two people and trade the
remaining one $10 bill in for ten $1 bills. These ten $1 bills, plus the original
four $1 bills, are shared, giving seven $1 bills to each person; hence, 34  2 = 17.
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Should the original problem have involved a remainder, the remainder represents dollars
left over. Additional problems can be made that involve paper money for $100, $10, and
$1 bills. After students can easily perform these fair shares, they are ready to move to the
algorithm, using the dollar bill place value understandings to develop the standard
algorithm. Have them discover the many different ways of dividing a dollar. Plastic
coins can be added to discover the decimal amounts each person would receive in a given
situation, as well as being able to see the fractional part of the dollar each person could
receive. Have them record both the decimal amount for each situation and the equivalent
fraction.
Activity 4: Division as Fractions (GLEs: 6, 10, 11)
Materials List: Jump, Kangaroo, Jump, paper, pencil, crayons or markers, learning log
Read the story, Jump, Kangaroo, Jump by Stuart Murphy. Have students act out the
story. Use 12 children to represent the campers on Field Day. Have them divide into
equal member groups as the event denotes (If the book is unavailable, have the students
name various Field Day events and have them form these different groups: 2 groups-1/2
of the campers in each group; then 3 groups-1/3 of the campers in each group; then 4
groups-1/4 of the campers in each group; 6 groups-1/6 of the campers in each group; 12
groups-1/12 of the campers in each group). As in the book, explain the corresponding
fraction for each situation. Have students notice what happens to the groups as the
denominator increases. Help them make the connection between the size of the group
and the size of the denominator. Have them discover the order of the fractions from the
story.
o Have the students break into groups of four and write their own “Field Day” story
using a story chain (view literacy strategy descriptions). The group would choose
a number of participants for their field day. The first person will write about a
field day event and tell how many groups the students will have to be divided
into. The second person would add another event and how many groups the
students would be divided into for that event. Then the third and fourth person
will add their events. If time permits, have them illustrate their stories.
Example:
First person writes: We are having a Field Day on April 17. My favorite event is
the potato relay race. A potato is placed in a spoon and you must carry it to a line,
then turn around and carry it back to your next team member. The team that
finishes first wins. There are 24 students in our class. We will have to break into
4 groups. How many people will be in each group? (solution---24÷4=6 There
will be six people in each event.) Second person writes: My favorite event is the
20 yard dash. Eight of us compete at one time. If there are 24 students in our
class and there are eight people in each race, how many races will be held?
(solution---24÷8 =3 There will be 3 races run.) Third person writes: My favorite
event is the wheelbarrow. We will have to be put in groups of two for this event.
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One person is the wheelbarrow with their hands on the ground. Their partner will
hold their feet up high and steer them to the finish line. If there are 24 students
and they are put into groups of 2, how many groups will we have? (solution--24÷2=12 There will be twelve groups for the wheelbarrow relay.) Fourth person
writes: I like the water balloon toss. We will make circles with 6 people in each
circle. The circles start off close together, and then we move further away from
one another as we toss the water balloon. The winning group is the group that
doesn’t pop the balloon. If there are 24 students and 6 people are in each circle,
how many circles will we have? (solution---24÷6=4 There will be 4 circles for
this event.)
Have them record their story chain in their math learning log (view literacy strategy
descriptions) and label this section Division as Fractions. At a later time, have them
share their stories with the class. The class will work the problems that go with each
story and check for accuracy.
Activity 5: A Remainder of One (GLEs: 10, 11)
Materials List: A Remainder of One, manipulatives (black beans or red beans make great
“ants”), dry erase marker, whiteboard
Supply 25 manipulatives for each student and allow students to recreate the
corresponding groups of ants from the book, A Remainder of One, by Elinor Penczes, as
it is read. They can predict what will happen and give an estimated answer for the
solution before using the manipulatives to prove the results. As an extension of this book
or if the book is unavailable, the students can play the game “Mingle.” To play
“Mingle,” call a given number of children to the front of the room. Have them walk
around until you call a number. When the number is called, they must stop walking and
quickly put themselves in groups of that designated number. Another student then writes
the corresponding division sentence on the board. Continue the game with the same
group of children but call different numbers to form other groups. Example: Thirteen
children are mingling and the teacher calls 3. The students stop and make groups of 3.
After all groups are formed, it is noted that 1 is left over (remainder). Another student
writes the corresponding division sentence 13÷3= 4 R 1 on the board.
Activity 6: Meaningful Remainders (GLEs: 10, 13, 19)
Materials List: Divide and Ride, paper, pencil
Read, discuss, and have students act out Divide and Ride by Stuart Murphy to illustrate
division with remainders in a real-life situation. Next, provide students with several
division word problems that include remainders; however, make sure these remainders
have meaning. For example: A class trip is planned where transportation must be
arranged by using parent volunteers and their vehicles. If only 4 students can ride in a car
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and there are 30 students in the class, how many cars will they need? Discuss this
situation with the students and help them see that 7 cars will not be enough and so the
remainder of 2 students requires that 8 cars be used. Have students estimate the answer or
use mental math before using paper and pencil. Discuss which is needed---an estimate or
an exact answer for each problem.
Activity 7: Division as Repeated Subtraction (GLEs: 10, 11)
Materials List: paper, pencil, newspapers or magazines, calculators
Present a division problem as a repeated subtraction situation such as “Sally has 56 dolls
and would like to give her friends 4 dolls each. How many friends will get 4 dolls?”
This can be solved by repeatedly subtracting 4 from 56 until the remainder is either less
than 4 or zero. Have students use the calculator to work the repeated subtraction and then
record each time they subtract by using tally marks. In this case, you can subtract 4 from
56 fourteen times. So, Sally can give 14 of her friends 4 dolls each. Have students work
in groups of four and use pictures from the newspaper or magazines to create a story
chain (view literacy strategy descriptions) that lends itself to repeated subtraction.
For example: One group may cut out a bag of candy. Then together they write a story by
each contributing one sentence. The first one writes---It was Christmas and the class was
going to have a party. The second writes---John brought a bag of candy to share with his
class. The third writes---His teacher said he could give four pieces to each student. The
fourth writes---There were 23 students and 96 pieces of candy in the bag. Was there
enough for everyone? Have them record their story chain in their math learning log (view
literacy strategy descriptions) and label this section Division as Repeated Subtraction.
Activity 8: Candy Division (GLEs: 15, 16, 19)
Materials List: candy (real or cutouts), paper, pencil
Use candy bars that are already evenly divided (for example, Hershey bar, Tootsie Roll,
etc.) to reinforce division. Assign a certain number of bars to various groups. Students
create word problems based on the scenario. Two bars divided into four pieces and
shared equally among a group of three, would result in 8 ÷ 3 = 2 r. 2 ( 8 ÷ 3 = N).
Divide a bag of small circle candies such as M&Ms or Skittles among a group of students
and the problem might be 50 ÷ 5 = 10 (50 ÷ 5 = N). A classroom pizza party would be
another opportunity to practice dividing with food. Five pizzas sliced into eighths
divided among 20 students, would result in 40 slices ÷ 20 students = 2 slices for each
student (40 ÷ 20 = N). When using real food items as manipulatives, ensure that students
have washed their hands, or make sure that students do not eat the “manipulatives.” Have
some extras (that all the students haven’t touched) to be given as treats, if desired, after
the activity is completed.
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Activity 9: Division as Part of a Whole (GLEs: 6, 9, 10)
Materials List: Division as Part of a Whole BLM, brown crayons, scissors, pencil
Have your students color the Division as Part of a Whole BLM brown then cut the12
small brown squares out to use to recreate fractions from the story Hershey’s Milk
Chocolate Fractions Book by Jerry Pallotta. The large rectangle “chocolate bar” is used
to represent the whole bar, so comparisons can more easily be made. As you read the
book, have students divide the 12 small brown rectangles as they relate to the story.
Have them compare the division of the whole candy to the corresponding fraction in the
book. Discuss the number of pieces of the candy that make up its corresponding fraction,
noting that there are fewer pieces of candy to correspond to a given fraction as the
denominator increases. Have students discuss the equivalent fractions stated in the book
as another way of describing the pieces they have. They can make up similar stories
using other “wholes” that can be divided. (a bag of cookies, a set of cards, a box of
crackers, etc.) Have them complete the bottom of Division as Part of a Whole BLM.
Have them draw a whole chocolate bar with varying fractional amounts displayed in their
math learning log (view literacy strategy descriptions) and label this section Division as
Part of a Whole.
Example:
6/12 = 1/2
4/12 =1/3
8/12 = 2/3
Activity 10: Divide and Recycle (GLEs: 15, 16, 19)
Materials List: paper, pencil
Integrate science and social studies with math by computing some recycling problems.
Example:
o If paper products are packaged in 25 lb. bags, how many bags are needed for 100
lbs. of paper?
o If the recycling center gave us 7¢ per pound for white paper, and they gave us
$14.49, how many pounds of paper did we take to the center?
o The workers at the recycling center recycle 80 loads of materials in a 5-day
workweek. On average, how many loads do they recycle in one day? If they
want to recycle 20 loads per day, how many days will it take? Students should be
led to write the associated number sentences and relate their operational thought
to see how problems, such as 80  5  x or 80  y  20 , were solved.
To make this assignment authentic, contact your local recycling center and get real data
to use.
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Activity 11: Professor Know-It-All (GLEs: 4, 10, 13, 19)
Materials List: number cards, dry erase marker, whiteboard
Play a modified version of professor know-it-all (view literacy strategy descriptions) Put
students in groups of four. Have the group choose one person in their group to be the
“professor.” Each group will be given a number card. The other members of the group
will give clues to their “professor.” The Professor has three chances to guess the number.
If he cannot guess the number by that time, a professor from another group may try. If
they are correct, that team steals the point and then continues with their group’s turn. For
example: If the quotient, The Professor (group member 1) had to guess, was 45, Group
member 2 could say, “It’s a multiple of 5. Group member 3 could say, “It is greater than
40 but less than 50.” Group member 4 could say, “Nine is a one of the factors.”
Professor know-it-all would write his answer on the board.
Activity 12: True or False (GLE: 4, 18)
Materials List: paper, pencil, teacher-made cards with division sentences, extra index
cards
In preparation for the True or False game, have the students create a Divisibility Rules
word grid (view literacy strategy descriptions). This activity will help students review
the important concepts of the divisibility rules. It will also serve to reinforce prime and
composite numbers. Have students generate three or four digit numbers to use for the
word grid. Through student participation, fill in the word grid by placing “+” in the
space of any divisibility rule that applies for that number and determine if that number is
prime or composite.
Divisibility Rules Word Grid
Number
Divisible
Divisible
by 2
by 3
325
630
503
+
+
Divisible
by 5
+
+
Divisible
by 10
Prime
Number
Composite
Number
+
+
+
+
o Next have the students work with a partner. Give one card from the set of cards
with division sentences you created to each group. Students, working with a
partner, read the sentence, then rewrite the sentence using math symbols when
possible, state if the number sentence is true or false, and explain their reasoning.
These activities should involve estimations where students have to determine if
the estimation is appropriate and give a rationale for why or why not.
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Examples of cards could include:
 57 divided by 4 is less than 57 The group writes 57 ÷ 4 < 57 The number
sentence is true because when you divide 57 into 4 groups no group can have
all 57 in it.
 The remainder when 545 divided by 5 results in 5. The group writes 545 ÷ 5 =
N with a remainder of 5 is not true because if you have 5 left you can divide
the number one more time. (Or it is not true because 545 is a multiple of 5 so
545 can be evenly divided by 5. There would be no remainder.)
 9,578 is divisible by 2.
 2,467 is divisible by 2.
 578 divided by 8 is greater than 586 divided by 8.
 9 divided by 0 is 9.
When students have completed these cards, they could make up a similar set of cards to
exchange with another group and repeat the process. When the sets are completed, they
would be returned to the original group to check.
Activity 13: Unwrapping Equations (GLEs: 10, 16)
Materials List: paper, pencil
Given an equation of the form 30 candies ÷ x = 10 candies for each student, students need
to ask if 30 is shared among some number of students, and each receives 10, how many
students could there have been? Alternately, one might ask if a given number of students
each receives 10 candies in a sharing and there was a total of 30 to begin with, how many
students are there? Getting students to think through the motions of sharing, both
forward and backward, provides a basis for later work in algebraic solving of these and
related sentences in symbolic form.
Sample Assessments
General Assessments:

Maintain portfolios containing samples of student work that illustrates the various
approaches to solving division problems.

Record anecdotal notes on students as they complete tasks.

Give prompts such as the ones that follow, and have students record their thoughts
in their personal math journals.
 Your class is planning a party. You want each person to have at least three
cookies. There are 48 cookies in a bag. There are 24 children in your class.
Will one bag be enough? Why or why not?
 Explain why 12 R2 is not reasonable for 242 ÷ 2.
 Your class is making gingerbread boys to sell. How many gingerbread boys
will you be able to make with 112 raisins for eyes?
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Activity-Specific Assessments

Activities 5, 6: Create a thematic bulletin board display that illustrates real-life
division with remainder scenarios. The book, Divide and Ride by Stuart
Murphy can serve as a model for ideas. Have them write ways to solve
problems when there is a remainder.

Activity 7: Give the students real-life division problems. The students will
work the division problems in more than one way. (Repeated subtraction,
picture, using multiplication to work backwards, etc.)

Activity 10: Plan and adopt a project, such as recycling or collecting for the
Food Bank. Each student will be responsible for writing a division problem
that applies to the project. Example:
 How many people will be assigned to each committee?
 How many crates will be needed to haul 50 lbs. of food?
 What was the average amount collected each day
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Grade 4
Mathematics
Unit 6: Geometry and Measurement
Time Frame: Approximately six weeks
Unit Description
In this unit, students extend their ability to measure precisely and to apply their
knowledge of space to finding perimeters, areas, and volumes. They draw, identify and
classify angles, perform specified transformations, identify and describe properties of two
and three dimensional shapes, and find points on a coordinate graph in the first quadrant.
Student Understandings
Students identify, describe the properties of, and draw circles, polygons, angle
measurements, and coordinate graphs. They make and test predictions regarding
transformations. Students select and use the appropriate customary and metric
measurement tools and convert measures within the same systems.
Guiding Questions
1. Can students recognize, select, and apply appropriate measurement concepts
and tools to measurement settings?
2. Can students use geometric knowledge of shapes to subdivide and translate
pieces of those shapes in finding and calculating perimeters and areas?
3. Can students find the perimeters and areas of rectangular objects?
4. Can students measure and classify angles according to their measures, and use
these measures in rotating objects in the plane?
5. Can students plot and interpret points and paths in the first quadrant of the
coordinate plane?
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Unit 6 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
14.
Solve real-life problems, including those in which some information is not given
(N-9-E)
Algebra
15.
Write number sentences or formulas containing a variable to represent real-life
problems (A-1-E)
16.
Write a related story problem for a given algebraic sentence (A-1-E)
18.
Identify and create true/false and open/closed number sentences (A-2-E)
19.
Solve one-step equations with whole number solutions (A-2-E) (N-4-E)
Measurement
20.
Measure length to the nearest quarter-inch and mm (M-2-E) (M-1-E)
21.
Describe the concept of volume, and measure volume using cubic in. and cubic cm.
and capacity using fl. oz. And ml (M-2-E) (M-3-E)
22.
Select and use the appropriate standard units of measure, abbreviations, and tools to
measure length and perimeter (i.e., in., cm, ft., yd., mile, m, km), area (i.e., square
inch, square foot, square centimeter), capacity (i.e., fl. oz., cup, pt., qt., gal., l, ml),
weight/mass (i.e., oz., lb., g, kg, ton), and volume (i.e., cubic cm, cubic in.) (M-2E), (M-1-E)
24.
Recognize the attributes to be measured in a real-life situation (M-2-E) (M-5-E)
25.
Use estimates and measurements to calculate perimeter and area of rectangular
objects (including squares) in U.S. (including square feet) and metric units (M-3-E)
26.
Estimate the area of an irregular shape drawn on a unit grid (M-3-E)
27.
Use unit conversions within the same system to solve real-life problems (e.g., 60
sec. = 1 min., 12 objects = 1 dozen, 12 in. = 1 ft., 100 cm = 1 m, 1 pt. = 2 cups) (M4-E) (N-2-E) (M-5-E)
Geometry
28.
Identify the top, bottom, or side view of a given 3-dimensional object (G-1-E) (G3-E)
29.
Identify, describe the properties of, and draw circles and polygons (triangle,
quadrilateral, parallelogram, trapezoid, rectangle, square, rhombus, pentagon,
hexagon, octagon, and decagon) (G-2-E)
30.
Make and test predictions regarding transformations (i.e., slides, flips, and turns) of
plane geometric shapes (G-3-E)
31.
Identify, manipulate, and predict the results of rotations of 90, 180, 270, and 360
degrees on a given figure (G-3-E)
32.
Draw, identify, and classify angles that are acute, right, and obtuse (G-5-E) (G-1-E)
33.
Specify locations of points in the first quadrant of coordinate systems and describe
paths on maps (G-6-E)
Data Analysis, Probability, and Discrete Math
38.
Solve problems involving simple deductive reasoning (D-3-E)
40.
Determine the total number of possible outcomes for a given experiment using lists,
tables, and tree diagrams (e.g., spinning a spinner, tossing 2 coins) (D-4-E) (D-5-E)
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Sample Activities
Activity 1: Test Those Measurements (GLEs: 14, 15, 16, 20, 22, 24, 25, 27)
Materials List: chart paper and markers, objects that have a traditional measurement,
customary and metric rulers, catalogs, paper, pencil
Have the class brainstorm (view literacy strategy descriptions) traditional measurements that
are accepted standards: 8 ½ x 11-inch paper, 3 x 5 index cards, 3 ½ -inch floppy disk, etc.
Ask students to check those measurements and record the results. Using rulers marked in ¼
inch, students should record their measurements to the nearest ¼ inch. Repeat the
measurement activities using a ruler marked in mm. Explore what other traditional
measurements are considered standard: letter-size envelope or folder and legal-size envelope
or folder, 8 x 10 photograph, 5x7 photograph, etc.
As an extension to this activity, students create story chains (view literacy strategy
descriptions) using some of the customary measurements listed in the brainstorming
activity, or they may come up with another object that is made in a traditional measurement
that they find in a catalog. Model the exercise, and then students create their own problems.
Example:
Student 1-Angie was planning a party, and she was going to decorate the tables with
tablecloths. Student 2-When she went to the store she didn’t know what size tablecloth to
get. Student 3-She went back home and measured her table. Student 4-The table was 4 ft. by
5 ft. The tablecloth had to be at least 4 ft. x 5 ft. to work. Students together (after looking up
the dimensions of tablecloths)-The tablecloth that she bought was 50 inches by 70 inches.
Would it fit? Students write: 4 ft. = X in., 5 ft. = X in. 50 in. x 70 in. ≥ N (Size of the table)
Answer: Yes because 4 ft. = 48 in. and 5 ft. = 60 in. and 50 in. x 70 in. > 40 in. x 60 in.
Activity 2: Scale drawings (GLEs: 20, 22, 25, 27)
Materials List: customary and metric rulers, drawing paper, learning log notebook, pencil
Students will need metric and customary rulers and drawing paper. They will be drawing 3
quadrilaterals each time they are given numbers for dimensions---one using inches, one using
centimeters, and one using millimeters.
Example: The students are given the numbers 8 by 6. They will draw a quadrilateral that is 8
in. x 6 in. Then they will draw another quadrilateral that is 8 cm x 6 cm. Last they will draw a
quadrilateral that is 8 mm x 6 mm.
The correct ruler will be chosen, the quadrilaterals drawn and dimensions labeled. This will
create scale drawings. Have students predict the area and perimeter of each shape. Then
have them check their predictions by actually measuring and recording the perimeter and the
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area of each shape. Be sure to include lengths to the nearest quarter inch. Give some
dimensions that would require unit conversions. Example: A quadrilateral that measures 9
inches by 8 inches would have a perimeter of 34 inches or 2 feet 10 inches. It would have an
area of 72 square inches.
Have students begin a measurement table in their math learning log (view literacy strategy
descriptions). This table will be used often throughout this unit as a quick reference and
study guide. It should include benchmarks for a better understanding of measurement sizes,
abbreviations, and a list of linear measurements by size from smallest to largest.
Example:
Linear Measurement
Metric System
1 millimeter = about the thickness of a dime.
1 centimeter = about the thickness of a crayon
1 meter = about the width of a front door
1 kilometer = about the length of 10 football
fields
U. S. Customary System
1 inch = about the length of a small paper clip
1 foot = about the length of a license plate
1 yard = about the width of a front door
1 mile = about the length of 15 football fields
Metric Units
10 millimeters = 1 centimeter
Customary Units
1 foot = 12 inches
100 centimeters = 1 meter
1 yard = 3 feet
1,000 meters = 1 kilometer
1 yard = 36 inches
1 mile = 5,280 feet
1 mile = 1,760 yards
Abbreviations of linear measurements according to size:
mm (millimeter)
cm (centimeter)
in. (inches)
ft. (foot)
m (meter)
km (kilometer
mi. (mile)
yd. (yard)
Perimeter is the distance around a shape. Perimeter of a rectangle or square is
P = length + width + length + width or (P = 2 length + 2 width) or P = 2L + 2W
Area is the number of square (sq.) units inside a given shape. Area is measured in square
units. Area of a rectangle or square is Area = length x width. For all other shapes, count the
squares that are inside the shape.
Activity 3: Finding Areas of Irregular Shapes (GLEs: 20, 22,25, 26)
Materials List: geoboard (optional), grid paper, paper, pencil
Using a geoboard (or grid paper), students will create irregular polygons. Students will
estimate the area by counting whole square units and combining partial units used to make
approximate whole units to complete their count. Have them measure and record the
dimensions of their irregular polygons to the nearest ¼ inch or millimeter on grid paper.
Repeat this activity several times.
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Activity 4: Baking Areas (GLEs: 14, 22, 24, 26)
Materials List: grid paper, several cookie cutters, crayons, pencil, Baking Areas BLM
Have students use Baking Areas BLM and cookie cutters to draw an outline of a variety of
cookies (irregular polygons). Have them count the whole square units and combine the
partial units to determine the area of the cookie to frost. Have them record the estimated
area. Have them check their work by allowing them to use crayons to frost their cookies as
they count the estimated area again. Have them compare the cookie shapes that have similar
areas but different dimensions. Discuss why knowing the area of a cookie would be
important when baking and decorating cookies, especially for a bakery.
Activity 5: Area vs. Perimeter (GLEs: 19, 25, 40)
Materials List: square tiles, grid paper, pencil
Provide square tiles and grid paper for students. Give them an area measurement and have
them create all the possible quadrilaterals with that area out of the square tiles. Have them
record their findings on the grid paper. After they have found all the possible quadrilaterals
with that area, have them record the dimensions and the perimeter of each shape. Have them
discover if the perimeters are the same when the areas are the same. Have them repeat this
activity but this time all the quadrilaterals must have the same perimeter. Have them
discover if the areas of the quadrilaterals remain the same when the perimeter is the same.
Example: If the area given is 12 square units, the possible dimensions for a quadrilateral
with that area are: 12x1, 1x12, 2x6, 6x2, 3x4, 4x3.
The area is 12 sq. units. The perimeter is 26 units.
12 by 1
The area is 12 sq. units. The perimeter is 16 units.
2 by 6
The area is 12 sq. units. The perimeter is 14 units.
3 by 4
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Activity 6: The Shape of Pizza (GLEs: 14, 22, 24, 25, 26, 40)
Materials List: grid paper, paper, pencil, ruler
Have students work in groups to design the best shaped pizza for their “Pizza Parlor.” The
pizza must have an area of 36 square inches. They are to draw as many different shaped
pizzas as possible with this area recording the perimeter and area of each pizza on grid paper.
They must decide on which shape pizza would appeal most to the customer.
Next, have them create a menu for their store. Using an organized list, a table or a tree
diagram, they will determine all possible pizza combinations that could be ordered. Limit
them to thin, thick and stuffed crust along with only two or three choices of toppings.
Activity 7: Box Puzzles (GLEs: 14, 22, 24, 25, 28)
Materials List: variety of empty rectangular boxes, markers, grid paper, pencil, ruler
Each student will construct a part of a “building” using an empty rectangular box
(rectangular prisms). Before building, students measure the length and width of each face of
their box and label them clearly with a marker. Students, working in groups, place their
boxes together, creating a “building,” and draw an outline of their building from a side view,
a front view, and a top view on grid paper. Students then draw a scale model of their
creations. Students could use a scale of 1 ft. = 1 in. Students should compute the area and
perimeter of their structures from two perspectives—front, top.
Activity 8: Plotting Quadrilaterals (GLEs: 15, 22, 25, 33)
Materials List: grid paper, ruler, pencil
Combine exploring perimeter and area with coordinate graphing. Provide students with
Quadrant I of a coordinate graph. Following a lesson on correct labeling and order (first x,
then y, or “over and up”), provide the coordinate points of a square. Students plot and join
the points, using a straight edge (ruler). Discover the number of linear units in the perimeter
and square units in the area. Allow students to explore the concept of area by examining (or
coloring) rows, columns. Introduce the formulas for perimeter and area. Repeat the
procedure with other quadrilaterals. Explore similar shapes by increasing or decreasing the
lengths and widths by one unit.
Example: Mark these co-ordinates on the grid.
(1, 1)
(1, 4)
(4, 1)
(4, 4)
Now connect the dots. What quadrilateral have you drawn?
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Activity 9: Making a Map (GLEs: 14, 33,)
Materials List: grid paper, pencil
Each student will design a simple map on grid paper. They will create a legend using
different symbols to mark schools, parks, churches, post office, their house, etc. Give
coordinates and tell the students to place a specific symbol at that point. For example: Place
a school on the coordinates (3, 5.) Continue to give such directions until all symbols have
been placed. After the map is completed, ask questions such as: How far is it from the school
to the post office? What is the shortest distance from the park to your house? etc.
As a follow-up activity, students can use this same idea and create a map of the things in
their classroom, school, bedroom, or another room in their house.
Activity 10: Trading Spaces (GLEs: 22, 24, 25)
Materials List: floor plan (optional), grid paper, rulers (foot, yard, meter), measuring tape,
pencil
If possible, bring in a real floor plan to discuss with the class. Students will then use grid
paper to create a floor plan of their existing bedroom. They will need to create a scale, select
appropriate measurement tools and find the area and perimeter of their room. They will need
to measure all pieces of furniture that occupy floor space. Students then create a second floor
plan “Trading Spaces” of their dream room with new pieces of furniture.
Note: Measurements should be rounded to whole units for finding area and perimeter.
Activity 11: Customary and Metric Measurements (GLEs: 24, 27)
Materials List: Internet or encyclopedias, Customary and Metric Measurements BLM, paper,
crayons, pencil, chart paper, learning log notebook
Have students create a vocabulary self-awareness (view literacy strategy descriptions) chart
using Metric Measurements BLM, about measurement. Do not give students definitions at
this time. Ask students to rate their understanding of each word with either a “+”
(understands well), a “√” (limited understanding), or a “-“ (don’t know). Over the course of
this unit, students will return to this chart and add new information to it. The goal is to
replace all √ and – with +. Because they will be returning often to this chart, they will have
multiple opportunities to practice and extend their knowledge.
Example:
Measurement Vocabulary Self Awareness Chart
Word
+ √ - Example
capacity
√
fluid ounce
cup
+
Cup of hot chocolate
Definition
How much something can hold.
8 ounces
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Students devise measurement questions like the following, research the answers, and provide
customary and metric measurements (linear, weight, capacity or volume measurements).
Example: How tall is the Eiffel Tower? How long does an anaconda grow? How much
does a baby rhinoceros weigh? Display the results of the activity with pictures and
measurements in a classroom measurement chart, from shortest to longest, or lightest to
heaviest.
Have students add to their Measurement Table in their math learning logs (view literacy
strategy descriptions) information about volume, weight, and capacity.
Example:
Volume is the amount of space inside a three-dimensional object. Volume of a rectangular
prism or a cube is the length times the width times the height. (V=l x w x h) Volume is
measured in cubic (cu.) units.
Weight Measurement
Customary Units
Metric Units
1 pound = 16 ounces
1 kilogram = 1000grams
1 ton = 2,000pounds
1 ounce is about the weight of 30 cm cubes
1 pound is the weight of 1 bag of coffee
1 gram is about the weight of 1 cm cube
1 kilogram weighs about 2 ¼ bags of coffee
Abbreviations of weight measurements according to size:
g (gram)
oz. (ounce)
lb. (pound) kg (kilogram)
Capacity is the amount a container can hold.
Capacity Measures
Customary Units
Metric Units
2 cups = 1 pint
1 milliliter = about one drop from an
eye dropper
2 pints = 1 quart
1 liter = about the size of 1 quart
4 cups = 1 quart
1 gallon = 2 half gallons
1 gallon = 4 quarts
1 gallon = 8 pints
1 gallon = 16 cups
Abbreviations of capacity measurements according to size:
ml (milliliter) c (cup)
pt. (pint)
qt. (quart)
l (liter)
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Activity 12: Transformations (GLE: 30)
Materials List: acetate sheets, di-cut letters or shapes, index cards, paper, pencil
Introduce transformations by having students make vocabulary cards (view literacy strategy
descriptions) that will help them connect prior knowledge of transformations from their third
grade terms (slide, turn and flip) to the new mathematical terms (translation, rotation, and
reflection). The vocabulary cards will also aid students in the identification of
transformations in future activities as well as a way of reviewing terms for assessments.
Example: Vocabulary Card
A reflection moves a figure
by “flipping” it over a line.
Mirror image
Flip
Reflection
Next, have students use sheets of acetate, or di-cut letters to explore the effects of flips,
slides, and turns. If using acetate sheets, provide simple drawings of geometric shapes.
Students trace the shapes onto the acetate and manipulate it to attain the desired
transformation. Rotate figures one-quarter turn, one-half turn, clockwise, and counterclockwise. Flip and slide shapes in different directions. Given an assortment of simple
shapes and a straight line, have students draw the flip image on the opposite side of the line.
This activity could be done on the computer using Microsoft Word. Click on auto shapes and
choose a shape. Then click on draw, choose flips and rotations and click on the desired
command.
Activity 13: Rotations (GLEs: 30, 31)
Materials List: construction paper or di-cut shapes, paper, pencil, scissors
Have students draw a variety of recognizable images: capital letter B, a scalene triangle, an
irregular pentagon that resembles the outline of a house, a heart, a circle, a square, etc. On a
sheet of paper, have students identify a pivot point, and name it “P.” Students predict what
the shape will look like after a given rotation takes place:
 90º
 180º
 270º
 360º
Trace the original shape on construction paper and cut it out or use di-cut letters or shapes to
test the predictions by manipulating the shape to demonstrate the rotations. Discuss the
observations students make.
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Activity 14: Casting Shadows (GLEs: 28, 31)
Materials List: 3-dimensional shapes, flashlight, drawing paper, pencil
Students will work in groups of three using three dimensional shapes, a flashlight, and a large
sheet of drawing paper. One person will hold the shape, the other will hold the flashlight,
and the third person will trace the shadow that is cast by the shape on the large sheet of
paper. The shadow will be labeled as the top, side, or bottom view of the three dimensional
shape. Extend this activity by having students observe and record what happens to the
shadow when the same top, side, or bottom view is rotated 90º, 180º, 270º, and 360º.
Activity 15: Tessellations (GLE: 30)
Materials List: heavy paper (old folders), scissors, paper, tape, colors, pencil
Use the computer to show students some Escher tessellations. A gallery of his tessellations
can easily be found by typing Escher tessellations in the search engine bar. Discuss with
students when and how they have seen tessellations used (ties, clothes, floor tiles, wall paper,
etc.). Students will explore the concepts of transformations by creating Escher-like
tessellations. First, have students cut out a 1-inch square out of heavy paper (old folders
work well for this). Next, have them draw an irregular shape on one side of the square, from
corner to corner, cut out the shape, slide it to the opposite side of the square and tape it
securely. They can create a piece of art by tracing the design, and sliding it to trace it again
(translation). Repeat several times. With a second 1-inch square, cut another irregular shape
from one side, rotating it on a corner and taping it to the adjacent side. Trace this piece,
rotating it each time on a corner before tracing it again (turn). Finally, with a third 1-inch
square, have students draw and cut out another irregular shape. This time, slide it to the
opposite side, and then flip it before taping it to the edge (reflection). Once again, students
repeatedly trace their tessellation piece, placing each new piece tightly against the previous
tracing, leaving no gaps. Students can creatively color and display their creations.
Activity 16: Exploring Angles (GLE: 32)
Materials List: straws, index cards, triangle cut outs (scalene, isosceles, equilateral), pencil
Introduce angles by having students use two bendable straws. Put the small end of the first
straw inside the long end of the second straw. Begin by having them make a right angle.
Explain that a right angle is 90°. Next, have them make an acute angle. Explain that an
acute angle is less than 90°. Last, have them make an obtuse angle with the straws. Explain
that an obtuse angle is over 90°. Practice calling out different degrees and having the
students use the straws to show you an estimated example of that angle. Have them state if
the angle is acute, right, or obtuse. Next, have them make angle vocabulary cards (view
literacy strategy descriptions). The vocabulary cards will help students in the identification
of angles in future activities as well as a way of reviewing terms for assessments.
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Example: Vocabulary Card
An acute angle is
less than 90°.
.
It is “a cute, little” angle.
Acute Angle
acute angle
Explore the angles in a triangle. Use equilateral, scalene, and isosceles triangles. Students
will use a “square corner” tool (like an index card) to classify the angles in triangles.
Students will decide if each angle of a triangle is right, acute, or obtuse by placing their tool
vertex to vertex with each of the angles.
Activity 17: What’s My Angle? (GLEs: 29, 32)
Materials List: Greedy Triangle, toothpicks (10 per child), marshmallows (10 per child),
paper, pencil, napkins
Give the students ten toothpicks and ten small marshmallows on a napkin. Discuss line
segments and vertices. Explain that they will be making polygons today using their
toothpicks as line segments and their marshmallows as vertices. Read the story, Greedy
Triangle by Marilyn Burns. Have students make each polygon as you read about it. Have
them create a chart with the name of the polygon, the number of sides, the number of
vertices, and the number of angles. Discuss the types of angles (acute, right, or obtuse) that
are formed in each shape.
Example:
Name of polygon
Number of sides
Number of vertices
Number of angles
triangle
quadrilateral
pentagon
3
4
5
3
4
5
3
4
5
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Activity 18: Graphic Organizer of Polygons (GLEs: 29, 32)
Materials List: large sheet of paper per student, pencil
Have students create a graphic organizer (view literacy strategy descriptions) to clarify their
understanding of polygons. Discuss each part of the graphic organizer as the students record
their own graphic organizer on a large sheet of paper.
Example of a graphic organizer:
Polygons
Triangle
3 sides
Quadrilateral
4 sides
Scalene
Isosceles
Equilateral
Rectangle
Square
Rhombus
Trapezoid
Pentagon Hexagon Heptagon Octagon Nonagon Decagon
5 sides
6 sides
7 sides
8 sides
9 sides
10 sides
Discussion Examples: A scalene triangle has no equal sides, an isosceles triangle has two
equal sides, and an equilateral triangle has three equal sides. Quadrilaterals have four sides
and four angles. Discuss that a rectangle has two pair of parallel lines and four right angles.
Discuss a square is a rectangle. It has two pairs of parallel lines and four right angles. A
rhombus has two pairs of parallel lines, but it does not have right angles. A trapezoid has
only one pair of parallel lines. A rectangle, square, and rhombus are all parallelograms. A
trapezoid is not a parallelogram because it does not have two pairs of parallel lines.
Activity 19: Classifying Polygons (GLEs: 29, 32)
Materials List: various size circles and polygons, paper, pencil
Provide each pair or small group of students with a set of various circles and polygons. Have
the students classify the polygons into sets with similar features. Have students discuss the
similarities and differences of the polygons. Students should be noticing properties such as
that the opposite sides of a rectangle are parallel, all sides of a square are equal, all angles in
a rectangle and a square are right angles, etc. Have each student choose two polygons and
write a description of their similarities and differences. As each group of students reports
their conclusions, record the names and properties of the polygons on the board. At the end
of their study of circles and polygons, students should be able to identify, draw, and describe
the properties of the figures.
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Activity 20: What’s My Capacity? (GLEs: 21, 27)
Materials List: variety of containers, paper, pencil
Provide students with several irregularly shaped empty containers that hold a specified
number of fluid ounces or milliliters. Provide students with standard measures of capacity
(liquid cold medicine cup marked off in ounces, eye dropper marked off in milliliters, cup,
pint, quart, liter, and gallon). Working in pairs, students will discover the capacity of each of
the containers and label each one appropriately. Have students create a table showing unit
conversions for capacity.
(2 cups = 1 pint, etc.)
Activity 21: What’s My Volume? (GLEs: 15, 18, 20, 21)
Materials List: variety of empty boxes, centimeter or inch cubes, paper, pencil
Have students bring in several different shoe boxes, cereal boxes, etc. Working in pairs,
students will discover the volume of these containers by stacking centimeter cubes or inch
cubes inside the boxes. After student pairs have calculated the volume by using the cubes,
then have them measure the length, width, and height of the boxes to the nearest quarter inch
or millimeter. Using the measurements obtained, students will write a number sentence that
describes the volume of the boxes.
Activity 22: Can It Be True? (GLEs: 18, 19, 21, 25)
Materials List: paper, pencil
Students working in cooperative groups will create number sentences that relate to finding
perimeter, area, or volume of a set of objects. The number sentences can be either true or
false. The challenge is for other student groups to determine if the number sentences are
correct.
Example:
Group 1--- A box is 12 inches long and 8 inches wide. To find its perimeter we would add
12 inches and 8 inches. A group member writes on the board 12 + 8 = P.
Another Group responds---That is a false number sentence. To find the perimeter of that box
you would add 12 in. + 8 in. + 12 in. + 8 in. A group member writes 12 + 8 +12 + 8 = P.
Another Group responds---Group 1’s number sentence is false. To find the perimeter of that
box you could multiply 12 by 2 and 8 by 2, then add that together. (2 Lengths + 2 Widths)
A group member writes 2(12) + 2(8) = P .
Both teams would receive a point. The team with the most points after all number sentences
are read is the winner.
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Activity 23: Simple Logic (GLE: 38)
Materials List: paper, pencil, simple games, Internet
Use riddles, puzzles (Chinese Brain Teasers), and simple games (tic-tac-toe, connect 4,
Chinese checkers, chess, etc.) to help students develop deductive reasoning skills. One
example is the following game with two players. Using the grid which follows as the playing
board, each student takes a turn coloring in one square or two squares with a common side.
Whoever colors in the last square wins.
Each player will take a turn to color in
or
or
using a grid like this one.
Deductive reasoning activities can be found at this Internet site:
http://www.internet4classrooms.com/skills_4th_original.htm. Click on Brain Teasers
Sample Assessments
General Assessments



Maintain portfolios containing samples of student’s graphic organizers, tables, grid
drawings of dream room, etc.
Record anecdotal notes on students as they complete tasks.
Give prompts such as the ones that follow, and have students record their thoughts in
their personal math journals.
 Do polygons that have the same area also have the same perimeter? Why or why
not.
 Make a riddle for your friend to solve that describes a polygon.
 Describe a time when someone could use tessellation.
 What quadrilateral is not a parallelogram? Explain.
 Mary has a gallon of lemonade. Does she have enough for each of the 23 children
in her room to have a cup? Explain your answer.
 Terrell and Tyrisha ran a race. Terrell ran 2 miles in ¼ of an hour. Tyrisha ran 2
miles in 18 minutes. Who ran the fastest? Explain your answer.
Activity-Specific Assessments

Activity 1: Create a book of measurement problems about their school with
questions such as:
 How long and wide are our halls?
 What is the length and width of the doors?
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 What are the dimensions of the floor tiles?

Activity 2: Estimate, using grid paper, the area of other non-rectangular polygons.
Extension: Draw a circle on the grid paper, tracing a CD or other pattern.
Estimate the area of the circle.

Activity 8: Use a tangram pattern to cut out their own tangrams. Have them color
one side of each piece. Leave the other side white. This will make it easy to see
the rotations that occur. Integrate art, social studies, and math by creating tangram
picture puzzles as the ancient Chinese did. Have the students describe the
rotations that were needed for creating their picture.

Activities 11, 12, 13: Make a polygon mobile. They will display their knowledge
of two-dimensional and three-dimensional shapes by constructing a mobile. The
teacher will provide a checklist of required shapes to be included. The students
will be encouraged to use real-world objects such as film canisters for cylinders,
ping-pong balls for spheres, etc., but that may also be allowed the use folded
paper shapes. They will label all shapes represented (including faces) and types
of angles—acute, obtuse, and right angles. This activity may be accomplished
individually or in cooperative pairs. Assess student products with rubrics that
clearly explain the scoring criteria. Display completed mobiles in the classroom
as reference models for future lessons.
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Grade 4
Mathematics
Unit 7: Fun with Fractions and Chance
Time Frame: Approximately three weeks
Unit Description
This unit develops an understanding of fractions with denominators through twelfths and
decimals through hundredths. The major application of fractions as it relates to its equivalent
decimal and its corresponding percent in this unit is in the interpretation of probability.
Student Understandings
Students develop a strong understanding of fractions with denominators through twelfths and
decimals through hundredths demonstrated through use in probability and interpretation of
data. They are able to represent fractions and decimals in sets of objects, rectangular arrays,
measurements, on the number line, and in written and verbal forms.
Guiding Questions
1. Can students represent fractions through twelfths with all forms of models,
including written and verbal?
2. Can students estimate fractional amounts from sets, pictures, and arrays?
3. Can students represent and interpret decimals through hundredths?
4. Can students connect decimal fractions with fractional representations, including
decimals for 12 , 14 , 101 ?
5. Can students discuss the probability of an event along the 0 (never), 12 (equally
likely), and 1 (certain interval)?
Unit 7 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
5.
Read, write, and relate decimals through hundredths and connect them with
corresponding decimal fractions (N-1-E)
6.
Model, read, write, compare, order, and represent fractions with denominators
through twelfths using region and set models (N-1-E) (A-1-E)
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GLE #
7.
8.
GLE Text and Benchmarks
Give decimal equivalents of halves, fourths, and tenths (N-2-E) (N-1-E)
Use common equivalent reference points for percents (i.e., ¼, ½, ¾, and 1 whole)
(N-2-E)
9.
Estimate fractional amounts through twelfths, using pictures, models, and
diagrams (N-2-E)
Data Analysis, Probability, and Discrete Math
35.
Find and interpret the meaning of mean, mode, and median of a small set of
numbers (using concrete objects) when the answer is a whole number (D-1-E)
36.
Analyze, describe, interpret, and construct various types of charts and graphs using
appropriate titles, axis labels, scales, and legends (D-2-E) (D-1-E)
37.
Determine which type of graph best represents a given set of data (D-1-E), (D-2-E)
41.
Apply appropriate probabilistic reasoning in real-life contexts using games and
other activities (e.g., examining fair and unfair situations) (D-5-E) (D-6-E)
Sample Activities
Activity 1: Fractions on Grids (GLEs: 5, 6, 7, 9 )
Materials List: Fractions on Grids BLM, crayons, math learning log, pencil
Use Fractions on Grids BLM to demonstrate fractions as part of a whole. Help students begin
to understand the connection between fractions, decimals, and percents by relating them
through money. Discuss the denominators that will be used to show varying amounts of
money (the denominator for pennies in a dollar is 100ths, dimes in a dollar would be 10ths,
quarters in a dollar could be 4ths, etc.) Demonstrate for students how to color in 50¢ on the
first grid of the Fraction and Grids BLM. Take them through each step as you discuss the
equivalent decimals, fractions, and percents for 50¢. Have the students complete the other
grids on their own, stopping for class discussions when needed. (1 quarter as .25 or 25/100,
or 14 or 25% of a dollar; 6 dimes as .60 or 60/100, or 106 , or 60% of a dollar; 3 quarters as .75
or75/100, or 34 or75% of a dollar, etc.) Have the students create an Equivalence Table in their
math learning log (view literacy strategy descriptions). This table will be used as a reference
for future activities as well as a study guide for assessments. Have students add other money
amounts and their corresponding fractions and any other equivalences as the unit continues.
Example:
Money amount
25 pennies = $0.25
1 penny = $0.01
2 quarters = $0.50
1 dime = $0.10
Equivalent decimal
.25
.01
.50 = .5
.10 = .1
Equivalent fraction
25/100 = 1/4
1/100
50/100 = 5/10 = 1/2
10/100 = 1/10
Grade 4 MathematicsUnit 7Fun with Fractions and Chance
Equivalent percent
25%
1%
50%
10%
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Activity 2: Ordering Decimals (GLEs: 5, 6)
Materials List: zip-lock bags, plastic coins, index cards with decimals written on them, pencil
Give each student the following money amounts and have them put them in order by value,
starting with the smallest amount (10¢, 25¢, 75¢, 5¢, 50¢, 1¢, $1.00). Help them connect
these amounts to the corresponding equivalent decimal by putting the decimal amounts under
each money amount. Discuss how decimals are ordered. Explain the place value of decimals
to the hundredths place. Next, explain that a decimal is another way of writing a fraction.
Have the students state the equivalent fraction for given decimal amounts. Record their
responses on the board. (.01 = 1/100, .05 = 5/100 or 1/20, .10 = 10/100 or 1/10, .25 = 25/100
or 1/4, .50 = 50/100 or 1/2, .75 = 75/100 or 3/4) Working in pairs, give students a zip-lock
bag with three to four cards in it with a decimal amount on each card and its equivalent
fraction. Have the students take out the cards and put them in order. Allow them to use their
plastic coins if needed. Check for accuracy. Have students exchange bags and repeat the
activity.
Example:
1¢
.01
1/10
5¢
.05
.5
5/10
10¢,
.10
.1
1/10
25¢,
.25
1/4
50¢,
.50
.5
1/20
75¢,
.75
3/4
$1.00
1.
Activity 3: Fraction Strips (GLEs: 6, 9)
Materials List: construction paper cut into 12 inch strips in a variety of colors, scissors, math
learning log, pencil
Using 12” strips of construction paper, have students create individual fraction strips. Begin
with one color, fold in half, and mark each segment with the corresponding fraction. With
another color, fold in half, and then in half again. Mark each segment 14 . Continue with
eighths, thirds, sixths, twelfths, etc. Cut each fraction strip at the creases. Use the fraction
pieces to make comparisons. Comparing the 12 strip with the 14 strip, help students
understand that it takes two 14 ’s to make a 12 . Having them compare fraction pieces with
other fraction pieces helps students develop fraction “sense” by exploring the size
relationship between two fractions, for example, 13 and 14 . Have them record their findings in
their math learning log (view literacy strategy descriptions), as well as any observations they
make about the equivalent fractions. Demonstrate on the board how to construct the
following table.
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Fraction
Equivalent
fraction
Equivalent
fraction
Equivalent
fraction
Equivalent
fraction
Equivalent
fraction
1/2
1/3
2/4
3/6
2/6
4/8
5/10
6/12
4/12
Observation:
The denominators of equivalent fractions are multiples of the first denominator.
Activity 4: Fractions, Decimals, and Percents in Print (GLEs: 5, 8)
Materials List: newspaper, magazines, construction paper, glue, scissors, paper, pencil
Have students brainstorm (view literacy strategy descriptions) times they have seen a
fraction, decimal, or percent used in real life. Next have students, working in groups, use
newspapers and magazines to cut out as many examples of decimals, fractions, and percents
as they can. Have them create a story chain (view literacy strategy descriptions). from one of
the examples they cut out. Have students mount stories on construction paper and create a
bulletin board display of their real-life uses of decimals, fractions, or percents. Allow
students to do a gallery walk to look at the bulletin boards and check the story chain
problems for accuracy and logic.
Example of story chain:
Student 1---Todd wanted a new fishing pole.
Student 2---The fishing pole he wanted was $40.
Student 3---He had saved $20. He thought it would be a long time before he had
enough money to buy the fishing pole.
Student 4---He was so excited when he saw an ad in the paper that said, “All Fishing
Poles ½ Off!”
Activity 5: Two-Color Counter Fractions (GLE: 6)
Materials List: 12 two-color counters for each student, Two-Color Counter Fractions BLM,
pencil
Explain that a fraction can also be part of a set. Using12 two-color counters have students
create a chart using Two Color Counter Fractions BLM to demonstrate fractions of a set. Red
and yellow counters are used as an example. If other colors are used, change the colors on
the BLM.
0
Place all counters on the desk with the red side facing up. Complete the chart: 12; 12
12 ; 12 .
11 1
Turn one counter over and repeat the process. Thus, you have 12; 12
; 12 . Notice that the
number in the “Total in Set” column is always 12 since 12 counters make up the set.
Continue with all possibilities. Have students compare the two fractions (fraction red and
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fraction yellow) by just comparing the numerators since the denominators are the same.
Repeat this activity often using a different number of two-color counters to explore other
fractions with other denominators.
Activity 6: Fraction/Decimal/Percent Dominoes (GLEs: 6, 7, 9)
Materials List: Fraction/Decimal/Percent Dominoes BLM, card stock
Make a set of fraction/decimal/percent domino cards by running the
Fraction/Decimal/Percent BLM on card stock. Run enough copies for students to work in
pairs. Place all dominoes face down. Have each player choose three dominoes. Taking turns,
have players place dominoes against other dominoes with equal value. If a player cannot
make a play, he or she draws a domino from the pile until a play is possible. Play continues
until one player has no more dominoes or until no more plays can be made.
Activity 7: Professor Know-It-All (GLEs: 5, 7, 9)
Materials list: index cards, pencils
Students will work in groups of threes. They will choose a fraction through twelfths and
record the fraction and its corresponding decimal and percent on an index card to be used to
play professor know-it-all (view literacy strategy descriptions). One group will come to the
front of the room and take on the role of “Professor.” Another group will read their fraction,
decimal or percent. The “Professor” group will have to state the two missing parts. Rotate
groups of know-it-alls after 5 minutes to include all groups.
Example: Group 1 has written
¾
.75
75%
Group 1 says, “75%.”
Group 2 “The Professor” responds, ¾ and .75
Activity 8: It’s Not Fair! (GLEs: 37, 41)
Materials List: It’ Not Fair! BLM, paper, pencil, probability cards, colors, paper clips
Have students use the It’ Not Fair! BLM to create a vocabulary self-awareness chart, (view
literacy strategy descriptions), about probability. Do not give students definitions at this
time. Ask students to rate their understanding of each word with either a “+” (understands
well), a “√” (limited understanding), or a “-” (don’t know). Over the course of this unit, have
students return to this chart and add new information to it. The goal is to replace all √ and –
with +. Because they will be returning often to this chart, they will have multiple
opportunities to practice and extend their knowledge of the key terms.
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Example:
Measurement Vocabulary Self-Awareness Chart
Word
+ √ - Example
Definition
chance
+
There is a chance of rain. Something that could happen.
probability
After completing and discussing the vocabulary self-awareness chart, draw a number line on
the board or on a chart and label it with 0 and 1 only. Then guide the students to create a
number line similar to the one below. Make probability cards with these probability words,
percents, and fractions (Impossible, Less likely, Possible, More Likely, Certain, Never,
Sometimes, Equally Likely, Always, Absolutely Not, Absolutely, ½, ¼, ¾, 100%, 25%,
75%, 50%, 0%) on them and give them to students in random order. Have them place their
card on the number line.
Example:
0
1/4
1/2
3/4
0%
25%
50%
75%
Impossible
Never
Absolutely Not
Less likely
Possible
More Likely
Equally Likely
Sometimes
1
100%
Certain
Always
Absolutely
Teacher note:
Chance is the possibility that something may happen. Probability is a number from 0 to 1
that tells the chance that an event will happen. The closer a probability is to 1; the more
likely the event is to happen. You may state the probability of something happening in
words, or as a percent, or as a fraction.
To practice probability, adapt simple board games such as Sorry®, Monopoly® Chinese
Checkers®, etc., where the chance of winning can be controlled by the configuration of a
spinner. For example, play the game as usual but use a spinner that is 23 red and 13 blue to
determine whose turn it is to play. Tell students each time the spinner lands on his/her color
to move one space forward and then he/she gets another turn. If the same color is spun on the
next turn, keep taking another turn. Before playing the games, have students discuss what
they think the outcome will be. Ask student pairs to construct spinners that would give the
player no chance to win, an even chance of winning, or a certain chance of winning. Have
them predict the outcomes of using a specific spinner. To make spinners: 1. Place the point of
a pencil through a large paper clip. 2. Place the point of the pencil on the center of the
spinner. 3. Adjust the paper clip so that the end of the paper clip is on the center of the
spinner. 4. “Flick” the paper clip to spin it. Have them write if their prediction was correct.
Have students test their spinners by spinning them 30 times and recording the results by
using a table or graph.
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Activity 9: How Lucky Are You? (GLEs: 8, 41)
Materials List: poster board cut into fourths, paper clips, inch cubes, index cards, markers,
crayons, scissors
Have students work in groups of four to design their own board games. They must decide
how they will make it a game of chance. They will create the rules of the game and how the
winner is determined. They may design fair or unfair spinners, dice, or cards. Have them
field test their new games by playing them several times. After the game has been played
several times, have them record probability statements about the game, using fractions and
percents. Have them include statements about the effect their spinner, dice or cards had on
their game and if the game was fair or unfair.
Example: Jasmine won ½ of the games, or 50% of the time.
Carlos lost ¾ of the games, or 75% of the time.
Kimberly lost every game, or 100% of the time.
Leon won ¼ of the games, or 25% of the time.
The spinner was not fair. Kimberly’s space was the smallest.
Activity 10: Mouth-watering Math (GLEs: 5, 6, 7, 8, 9, 35, 36, 41)
Materials List: color coated candy, crayons, paper, pencil
Show students a spoonful of colored coated candy such as mini M&Ms®. Have them write
down their estimate of the amount of candy in the spoon. Then have the class create a line
plot of their estimates on the board. Discuss mean, mode, and median of the estimated data.
Count candies to find the actual number of candies in the spoon. Make a new line plot with
the actual data. Find the mean, mode, and median for the new data. Compare the estimated
and actual data.
Next, give each pair of students a small bag of the color coated candy and ask them to
estimate the amount of candy in the bag, using the spoonful of candy as their benchmark.
After recording their estimates, have them count the candy, record the actual amount of
candy in the bag, and find the difference between their estimates and the actual amount.
Have students classify the candy by color, record the fractional part of each color, and list the
fractions in order from the least to the greatest.
Have students construct a pie (circle) graph using 25 of the candies. Have them put the
candies together in the circle by color. Make a legend showing that for this graph
1 candy = 4% candies since a pie graph represents 100%.
Then have each pair of students record the pie graph on a sheet of paper using the
corresponding crayon that matches the candy. Have them draw the appropriate lines on the
pie graph where one color begins. Have them write in each fractional part of the graph along
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with the corresponding decimal and percent amounts. Have them check that their graph
contains all necessary parts-title, labels, scale, legend. Discuss why this type of graph is good
to use when you are showing parts of a whole. Discuss other times a pie graph may be used.
Have them write probability statements about each color of candy based on their graph.
Activity 11: Food Probability (GLEs: 6, 8, 41)
Materials List: paper, pencil, calculator, bag, Cloudy with a Chance of Meatballs or
Probably Pistachio
Read the story Cloudy with a Chance of Meatballs by Judi Barrett or Probably Pistachio by
Stuart J. Murphy and discuss the probability events in the story. After reading the story, have
students work in groups to create a monthly school menu. Ask each student to write down
their three favorite foods on individual strips of paper. The captain will collect the strips and
reveal to the group the results. Have each group record the probability of drawing each strip
from a bag as a fraction and as a percent. (A calculator can be used to help determine the
percents.) Then ask them to take turns pulling a strip and recording that food on a blank
monthly menu form. (Make sure the strip is returned to the bag each time so that the
probability will not change.) When they complete their monthly menu, have them check their
predicted probability against what actually happened.
Activity 12: Weather Probability (GLEs: 8, 41)
Materials List: Internet, paper, pencil
Have students work in groups using the Internet to find the rain probability of five different
cities around the country on a given day. Have them record their gathered data on a graph or
table. Next, have students record the chance of rain as a percent and create an ordered list of
the cities with the smallest probability of rain to the largest probability of rain. Based on
what they know about reference points (e.g., 14  25%, 12  50%, 34  75%,1  100% ) have
students find the city or cities which have about a 1 in 4 chance of rain, a 1 in 2 chance of
rain, etc. Have them give oral reports to the class and display their findings on the bulletin
board.
Activity 13: Test Taking Probability (GLEs: 8, 41)
Materials List: Test Taking Probability BLM, pencil, calculator
Give students the Test Taking Probability BLM. Tell the students to circle the correct
answers. No questions will be provided. Use this opportunity to discuss how their chance of
being correct increases when they read the questions. That is why students are asked during
a test to “read the questions carefully.” They must determine what chance they have of being
correct. (1 out of 4, or 25%) After they have circled the answers, call out your own choices
Grade 4 MathematicsUnit 7Fun with Fractions and Chance
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as the “correct” answers. Have students determine the percent correct. Have them discuss
how close they were to the predicted probability of correct answers.
Next, tell the students you will let them take the test again. This time tell them that two
choices will be eliminated. Explain the value of eliminating choices that can’t be correct.
Discuss how the probability of getting the right answer will change. Determine the new
chance for a correct answer (1 out of 2, or 50%). Have them retake the test knowing this time
that the answers b and c have been eliminated as the correct answers. Again call out your
choice of the new correct “answers” and let each student determine their percent correct.
Have them compare their two scores and discuss the use of elimination to increase their
probability of being correct in any given situation.
Sample Assessments
General Assessments



Maintain portfolios containing student reports and samples of student work that
included representations of fractions and decimals using regions or set models.
Show the student cards with a picture, a model, or a diagram on them. The student
will estimate the fraction through twelfths that is represented.
Give prompts such as the ones that follow and the student will record his/her
thoughts in a personal math journal.
o Give real life examples of events that use fractions, decimals, and percents.
o What is your favorite game to play? Describe the probability of your winning.
Is there anything you can do to increase your probability to win?
o How can your understanding of probability and weather help you plan an
event?
Activity-Specific Assessments

Activity 4: The students will work in cooperative groups to create a fraction and
decimal bulletin board using pictures and photographs from magazines,
newspapers, etc. Each picture will be partitioned with a line, a strip of paper, or
yarn. The class will identify each segment using a fraction and a decimal.

Activity 6: Provide shaded grids. The student will describe the shaded area using
a fraction and a decimal.

Activity 10: The student will demonstrate his ability to construct pie (circle)
graphs by choosing from a variety of manipulatives. He will arrange the
manipulatives by color to make a pie (circle) graph and will record the graph on
paper displaying the fraction, decimal, and percent of each color section. Possible
manipulatives to be used are square tiles, colored bears, two colored chips,
colored cubes, colored linking chains.
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Grade 4
Mathematics
Unit 8: Algebraic ThinkingPatterns, Counting Techniques, and Probability
Time Frame: Approximately four weeks
Unit Description
In this unit, students look at patterns, evaluate and generate rules for other patterns. This unit
involves estimation and the pulling together of numbers—especially multiplication and
division—in application settings. Algebraic thinking is applied to solve real-life problems,
including those in which some information is not given. The unit also introduces systematic
counting with tree diagrams, lists, tables, and elementary experiments and simulations.
Student Understandings
Students will understand what operation is needed and which representation is appropriate
for a given setting as demonstrated throughout various real-life activities. Students will use
the development of systematic counting with lists, tables, tree diagrams, finding averages,
and quantification of chance.
Guiding Questions
1. Can students apply multiplication and division facts and algorithms studied earlier
in year?
2. Can students systematically count the possible outcomes to an experiment with
lists, tables, and trees?
3. Can students interpret probabilities in simple real-life situations?
4. Can students determine appropriate methods of determining answers to
computational questions?
5. Can students apply algebraic thinking to create equations to solve real-life
problems in which some information is not given?
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Unit 8 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Number and Number Relations
4.
Know all basic facts for multiplication and division through 12 x 12 and 144 ÷ 12,
and recognize factors of composite numbers less than 50 (N-1-E) (N-6-E) (N-7-E)
10.
Solve multiplication and division number sentences including interpreting
remainders (N-4-E) (A-3-E)
13.
Determine when and how to estimate and when and how to use mental math,
calculators, or paper/pencil strategies to solve multiplication and division problems
(N-8-E)
14.
Solve real-life problems, including those in which some information is not given
(N-9-E)
Algebra
15.
Write number sentences or formulas containing a variable to represent real-life
problems (A-1-E)
18.
Identify and create true/false and open/closed number sentences (A-2-E)
19.
Solve one-step equations with whole number solutions (A-2-E) (N-4-E)
Geometry
33.
Specify locations of points in the first quadrant of coordinate systems and describe
paths on maps (G-6-E)
Data Analysis, Probability, and Discrete Math
35.
Find and interpret the meaning of mean, mode, and median of a small set of
numbers (using concrete objects) when the answer is a whole number (D-1-E)
38.
Solve problems involving simple deductive reasoning (D-3-E)
39.
Use lists, tables, and tree diagrams to generate and record all possible
combinations for 2 sets of 3 or fewer objects (e.g., combinations of pants and
shirts, days and games) and for given experiments (D-3-E) (D-4-E)
40.
Determine the total number of possible outcomes for a given experiment using
lists, tables, and tree diagrams (e.g., spinning a spinner, tossing 2 coins) (D-4-E)
(D-5-E)
41.
Apply appropriate probabilistic reasoning in real-life contexts using games and
other activities (e.g., examining fair and unfair situations) (D-5-E) (D-6-E)
Patterns, Relations, and Functions
43.
Identify missing elements in a number pattern (P-1-E)
44.
Represent the relationship in an input-output situation using a simple equation,
graph, table, or word description (P-2-E)
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Sample Activities
Activity 1: One for the Ages! (GLEs: 10, 14, 39)
Materials List: paper, pencil
Students examine ages using algebraic expressions. For example, if you are 9 years old, how
old were you n years ago? Using the idea of inputs and outputs, replace the variable with a
variety of numbers and generate several answers. Have students work in groups of three to
construct a “One for the Ages” story chain, (view literacy strategy descriptions).
Example:
The first student would write: My father is three times my age.
The second student would write an equation to solve the problem using a variable-Father’s
age (N) = 3 x 9 (my age) then the second student would add to the story: How much older
will I be in the year 2020?
The third student would write an equation using a variable to solve the problem. Then the
third student would add to the story chain: If the sum of the ages of three children in our
family equals 32 and none are younger than 9, what are the possible combinations?
The first student would set up a list, table, or tree diagram to determine the possibilities and
together the group would solve.
Activity 2: Unknowns in Real Life GLEs: 14, 39)
Materials List: math learning log, book Safari Park, tickets
Read the story Safari Park by Stuart Murphy. In this story, five cousins are each given 20
tickets to spend at the grand opening of Safari Park. The cousins must decide what
combination of rides, activities, or treats they will use with their tickets. As you read the
story, have the students act out the different combinations that the cousins choose using their
own hand-made paper tickets to solve each number sentence for the unknown. Model and
discuss how to set up number sentences or number patterns to find the unknown. Then have
students record their findings by writing the algebraic number sentence that goes along with
the events of the story in their math learning log (view literacy strategy descriptions).
After hearing the story, have students extend this activity by finding and recording in their
math learning log, different possible combinations for spending their 20 tickets based on the
number of tickets it took for each ride. Note: If the book is unavailable, this activity can be
modified. Allow the students to make up their own park with rides, activities and treats and
designate a ticket amount for each of them. Then have them follow the above activity using
the 20 tickets.
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Activity 3: Guess Your Number (GLEs: 4, 10, 19)
Materials List: two sets of number cards (0-12)
This game is played with a group of three children. They will need a set of number cards to
play the game. One person is the leader. The leader gives one card to each of the other two
students. Without looking at their card, they place their card on their forehead so that the
number of their card shows only to their partner. The leader looks at the two numbers and
says, “The product of these two factors is ___.” From this clue, the two card holders must
guess their own number by solving for the missing number. (They would know the product
and their partner’s number, so their number is the number missing from the equation.
Example: Partner’s number = 7, Product given = 21, N= my number, thus 7 x N = 21. My
number is 3. Or the problem could have been solved by thinking 21 ÷ 7 = N; N = 3 My
number is 3) The first person to guess their number correctly wins that round and becomes
the leader for the next round. Have the winning student explain their thinking to the class.
Activity 4: Explore Weight on the Moon (GLEs: 13, 44)
Materials List: paper, pencil
Tell students that the moon weight equals Earth weight ÷ 6. Given this information, students
will use simple deductive reasoning to determine what they and other common objects would
weigh on the moon. Students should represent the relationship as an “input-output” situation,
in a chart, or in an equation. Encourage students to use an appropriate calculation technique
when computing what they and several other objects would weigh on the moon. For
example, they could use mental math to compute “moon weight” when the “earth weight” is
a multiple of 6. Science could be integrated by researching the effects of gravity and
atmospheric pressure at sea level.
Example:
Input-output table
In (Earth’s
wt.)
66 lbs
Out (Moon’s
wt.)
11 lbs
27 lbs.
9 lbs.
Chart
Object
Me
Dog
Earth
66 lbs.
27 lbs.
Moon
11 lbs.
9 lbs.
Equations
66÷6 = 11 lbs.
27÷6 = 9 lbs.
If I weigh 66 lbs. on earth, I would
weigh 11 lbs. on the moon.
If my dog weighed 27 lbs. on the
earth, he would weigh 9 lbs.
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Activity 5: Fun with Factors (GLE: 4)
Materials List: paper, pencil
Students can use factor trees to list the factors of a given number. Students list all possible
factors of a number in pairs. Make sure they include one and the number as a factor pair (8:
1 and 8, 2 and 4; 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6). Notice that all numbers at the tips
of the branches are prime numbers. Multiply the prime numbers. What is the product?
Introduce prime factorization. Have students work with a partner to complete factor trees for
other given numbers.
Example: 24: 1, 2, 3, 4, 6, 8, 12, 24
24
⁄\
2 12
⁄ \
2 6
⁄ \
2 3
2  2  2  3=24
Activity 6: Ice Cream Combinations (GLE: 39)
Materials List: The Sundae Scoop book, construction paper, glue, scissors, paper, pencil
Read and discuss the book, The Sundae Scoop by Stuart Murphy. Using split-page
notetaking (view literacy strategy descriptions) have the students construct the same tree
diagrams that are in the book as the story is being read. Have a student explain how to read
the diagram each time. After the story is read, have the students show another way for
determining the combinations of ice cream (a table, chart, draw a picture, etc.). Working with
a partner, each pair of students could use construction paper to make one of the sundae
combinations to be displayed on a bulletin board along with the corresponding table, tree
diagram, list, or picture. Note: If the book is unavailable, this activity can be modified. The
class can generate their own story about ice cream sundae combos. With the split-page
notes, demonstrate for students how they can review the information by covering one column
while using the information in the other to recall the covered information. Students can also
be given time to quiz each other over the content of their notes in preparation for tests and
other class activity.
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Example: of split-page notes for The Sundae Scoop
The children decided to have 2 kinds of
Sundae Combos
ice cream and 2 kinds of sauces.
Vanilla
Hot Fudge
Caramel
Chocolate
Hot Fudge
Caramel
Activity 7: Lunch Choices (GLEs: 39, 40)
Materials List: Lunch Choices BLM, pencil
Use the Lunch Choices BLM to have students explore the possible choices of lunch
combinations. Students create a table, a tree diagram or make a list to explore all possible
lunch combinations. Students should look for patterns that help them stay organized as well
as patterns that relate to the solution. Once students have found all possible lunch choices,
have them answer the probability questions about the probability of someone’s making a
particular choice. For example, “What is the probability that a student chosen at random in
the cafeteria will be eating a hamburger and milk?” Students should express these
probabilities as fractions or percents. As an extension of this activity, a third category such
as desserts could be added. Variations on this activity include choices of three colors of
pants, shirts, and shoes, or three potted plants with all the possible arrangements on a
windowsill.
Activity 8: Growing Combinations (GLEs: 39, 40)
Materials List: connecting cubes, crayons, paper, pencil
Give each pair of students, 3 different color connecting cubes to represent the uniform shirts
for their soccer team (red, blue, yellow). Give them two other colors of connecting cubes
(brown, black) to represent the soccer uniform pants. Have them use the cubes to create all
possible color combinations of uniforms. Then have each group record with crayons all the
possible combinations they found.
Example:
red
brown
red
blue
black
yellow
black
brown
blue
yellow
brown
black
Next, using SQPL (view literacy strategy descriptions) to determine the depth of student
understanding about combinations, write this statement on the board: If you have 1 more
additional color cube for shirts and 1 more additional color cubes for pants, you will be able
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to generate 8 possible color combinations. Students should pair up and based on the
statement generate 1-2 questions they would like answered. Record the questions on the
board and have the students make an effort to answer any of them they think they can.
Example of student generated questions:
Could there be more than eight combinations?
Could you mix the new shirt and new pants with the other shirts and pants?
Is there a pattern to figure this out?
If I multiply the number of shirts or pants by two, could I get the answer?
Next, give each pair of students a fourth color connecting cube. Have them make 8 color
combinations as the statement on the board suggests. Ask if they all have the same 8
combinations. Record the combinations on the board that have been made thus far. Based on
this new information, have the students predict the possible combinations. Have each group
create and record the possible combinations using a table, a tree diagram, a chart or a picture.
Have a whole class discussion of observations that were made. Have students answer any
unanswered questions that were generated using SQPL.
Activity 9: Plot a Picture (GLE: 33)
Materials List: Plot a Picture BLM, pencil
Students work in pairs. Using the Plot a Picture BLM, have each student draws a simple
picture on the first quadrant of the coordinate grid. Identify as many coordinate points as
possible, in the order in which they are connected. List the points on the X/Y table.
Exchange the X/Y tables only. Partners attempt to recreate the other’s drawing on a new
piece of graph paper.
Activity 10: Where in the City Are You? (GLE: 33)
Materials List: Internet, grid paper, pencil
Working with a partner, have students search for their school on the computer using a map
service. Have them print a copy of the map. Using this information, have the students create
their own neighborhood map of the school and its surroundings on a grid. Next, have them
mark as many places as they can on their grid, trying to keep the map as accurate as possible.
Have them create a key/legend for their map. Then have each group develop 3 to 4 questions
for other groups to answer. Include questions that can be answered by finding points on the
grid and questions that involve paths on a map. Have groups exchange maps and answer
each other’s questions.
Example of student questions:
Name a street that is parallel to our school street.
What building is closest to coordinates 4, 5?
Name two streets that form a right angle?
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Activity 11: What Does This Mean? (GLE: 35)
Materials List: connecting cubes, paper, pencil
Students are presented with five stacks of connecting cubes, two stacks that are 3 cubes high,
one 6 cubes high, and two 4 cubes high. Ask students to determine the mode height, the
median height, and the mean height for this set of data. Students will determine the mode by
grouping the stacks according to height. The stacks with the same number of cubes in it that
appears most often is the mode. Students will determine the median by putting the stacks in
order from shortest to longest and then selecting the stack in the middle. (Keeping the
number of stacks odd facilitates getting a whole number median.) Students will determine
the mean by “leveling” the stacks. In this case, they could make all the stacks have 4 cubes.
Allow the students to make up their own problems, using the cubes. Make sure students state
the mean, mode, and median for each set of cubes they use.
Activity 12: A Pattern Machine! (GLEs: 43, 44)
Materials List: Pattern Machine BLM, calculator, pencil
Students can create their own pattern machine by using the Pattern Machine BLM. Have
students work in pairs. Each member of the pair will secretly generate a “rule” that governs
their pattern machine. For example, a student might use “3 more than twice the number” as
their rule. Each student will supply the first, second, third, and fifth term in the pattern. The
object is for the other student to deduce the missing term and to state the rule that was used.
Students can use a calculator to generate more complex numbers or to supply larger terms.
Example:
Student 1
In
4
(N)
Out 11
(Y)
5
6
8
13
15
19
Student 2 will put 7 in and put 17 out for the fourth term.
He will state the rule: N x 2 + 3 = Y
Activity 13: “What’s My Rule?” (GLEs: 43, 44)
Materials List: calculator, paper, pencil
Students work with a partner to play “What’s My Rule?” One person uses the calculator and
puts in any number and then puts in a rule (i.e. The person could put in 46 then put in +6, or 3 and then presses =). Both the input and output number are recorded. That person repeats
this procedure for two more numbers without changing the rule. Each time a number must
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be recorded. It may either be the input or output number. The calculator is then handed to
the partner who must figure out the rule and complete the missing numbers in the table.
Once the rule and missing numbers are discovered, the partners switch roles and play again.
Example:
Input
Solution:
Output
Input
Output
56
53
56
53
28
___
28
25
___
11
14
11
Rule: Subtract 3
Activity 14: Probability Experiments (GLEs: 40, 41)
Materials List: circles cut for spinners, paper clips, paper, pencil, Probability Experiments
BLM
Use the Probability Experiments BLM to make a spinner. 1. Place the point of a pencil
through a large paper clip. 2. Place the point of the pencil on the center of the spinner. 3.
Adjust the paper clip so that the end of the paper clip is on the center of the spinner. 4.
“Flick” the paper clip to spin it. Have the students, working with a partner; divide the first
circle into four equal sections with a different color in each section. Have students decide on
all possibilities of the outcome of a spin. Students should realize that each color has an equal
chance of being “spun.” Next, have students create a spinner that contains three colors, but
such that one color is twice as likely than either of the other two colors (e.g., a spinner with
half red, one fourth green and one fourth yellow).
Challenge: Have students create spinners that produce specified outcomes (a probability of 1
(always) for a color, a probability of 2 3 for a color, a probability of 0 (never) for a color etc.)
Additional challenge: Have students create two different spinners. Have them record the
possible outcomes if both spinners are spun at the same time.
Activity 15: How Many Coins? (GLEs: 14, 15, 18, 19)
Materials List: small box, play money, paper, pencil
Have students work in groups of three to play this game. Provide each group with a small
empty box and play money. One person is the cashier and will choose an amount of coins to
put in the box. The cashier will then give the other two people clues, such as I have 53 cents.
I have 1 quarter and three pennies. The rest of the coins are nickels. How many nickels do I
have? The other two people must write an algebraic number sentence using the clues.
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Example: (1 quarter + x nickels + 3 pennies = 53¢). The first person to solve the problem
takes over as the cashier and the game continues.
Activity 16: Mystery Numbers (GLEs: 4, 18, 19, 38)
Materials List: math learning log, calculator, pencil
Students work with a partner to play this game. The mystery number riddles and their
solutions will be recorded in their math learning log, (view literacy strategy descriptions). .
They may use a calculator to help create an equation that has an unknown. The person
creating the equation with the unknown will give clues to their partner about the mystery
number. The partner will use the clues to write an equation that contains the unknown to
discover the mystery number.
Example:
The first person says, I’m thinking of a number that when it is multiplied by 4 it equals 64 ÷
2. The second person writes, 4 x n = 64 ÷ 2. The second person then thinks, 4 x n = 32. (He
knows this from simplifying 64 ÷ 2 = 32. Now he is able to discover the mystery number by
thinking 4 x n = 32. He then solves the equation to discover the mystery number is 8. The
partners then change roles, and the game continues.
Activity 17: Math Logic Puzzles (GLE: 38)
Materials List: paper, pencil
Create simple logic problems for students to solve. Show them how to use a grid to help
solve the logic problems. Offer them many opportunities to practice this skill providing them
with more complex problems as they become more skilled.
Example:
John has more money than Tim.
Tim has more money than Cindy.
Cindy has the least money.
Felicia has more money than Cindy but less than John.
50¢
45¢
42¢
John
Tim
Felicia
Cindy
40¢
√
X
X
X
X
√
X
X
X
X
√
X
X
X
X
√
More logic problems can be found at http://www.edhelper.com/Math_Logic_Puzzles.htm.
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Sample Assessments
General Assessments

Maintain an inclusive student portfolio of math skills acquired throughout the
year adding student work that reflects algebraic thinking along with a student
reflection of their individual progress in mathematical understanding attained this
year.

A portfolio of group project reports, for all group activities, noting the activity
completed, the observations, and the discoveries made will be on file.

Prompts such as the ones that follow will be given, and students will record their
thoughts in their in their math learning log, (view literacy strategy descriptions).
 The volleyball team has just received their uniforms. They have a blue shirt
and a white shirt and they have black shorts and blue shorts. How many
different combinations of their uniforms can they wear?
 If you receive a dollar for each year you are old, how much money will you
have saved by the time you are ten if you do not spend any of it? (Hint: Make
an input/output table.)
Activity-Specific Assessments

Activity 5:
A spinner will be used and spun twice to create a two digit number to factor.
A factor tree will be constructed to find all the factors.

Activities 6, 7, 8:
Have students construct possible pizza combinations when given one crust
choice and choice of any two toppings. For instance the three toppings,
(pepperoni, ground beef, or sausage) and three crust types, (thin, pan, or thick)
could be used.

Activities 12, 13:
Input/output tables will be constructed by the students that will demonstrate a
specific rule given by the teacher.
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