
Name:_____________________________ Date:___________________ Parent Initials_______

In mathematics, we often compare quantities, representations, strategies, and solutions. We do this
to gain a deeper understanding of the problems we are working with. In this lesson, you will focus
on comparing different arrangements of pennies or dots and the quantities they represent. As you
think about comparison and equality with your team, ask the following questions to help focus your
discussion.




Which is greater?
Are they the same?
Which representation best describes the quantity?
1-51. Cody and Jett each have a handful of pennies. Cody has arranged his
pennies into 3 sets of 16, and has 9 leftover pennies. Jett has 6 sets of 9 pennies,
and 4 leftover pennies. Each student thinks he has the most pennies.
a. Work on your own to draw a diagram and write an expression with
numbers that represent the way Cody could have arranged his pennies (3 sets of 16 with 9
leftover pennies). Do this in the space below. Is there a different way you could have
represented this same number of pennies with a diagram and number expression?
Then do the same for Jett’s pennies (6 sets of 9 pennies with 4 leftover pennies).
b. Compare your results with your team.

Copy the different numerical expressions for each student from your team to your
paper. Here:

How many different ways did your team find to represent the number of pennies
with diagrams and numerical expressions?

With your team, decide which arrangements best represent the groups of pennies
held by Cody and Jett.
c. Which student has more pennies?
How did you figure this out?
d. Jett decided to rearrange all of his pennies into groups of 10, even though one group will not
be complete. How many groups can he make?
 How can he represent his new grouping with a number expression?

1-52. Use the numerical expressions from Lesson 1.2.1 (problem 1-44) to calculate how many
pennies each team had. Which team received the greatest number of pennies? Which team
received the least? Where was your team in comparison to the other teams?
. Did any teams have the same number of pennies? If so, write a number sentence to express the
value of each team’s numerical expressions. Your number sentence should show that the values are
the same. What symbol is used to show that two values are the same?
a. Compare your team’s value to at least two other teams. Express each of your comparisons
with both a number and a word sentence.
For example: Team A may have written “5 sets of 25 pennies, or 5(25),” while Team B may
have written “5 sets of 17 pennies and 2, or 5(17) + 2.” To compare, you would use the less
than (<) or greater than (>) symbol to write a number sentence like: 5(25) > 5(17) + 2. The
corresponding word sentence might be, “5 sets of 25 pennies is greater than 5 sets of 17
pennies with two more.”

1-53. Hilda knows about the “less than” (<) and “greater than” (>) symbols that are used to write
comparison statements, but she can never remember which is which.

“I have a way to remember that makes perfect sense,” Artie said. “Just write down the two things
you are comparing, side by side with space for the symbol between them. Then in the space
between the two quantities, put two dots (one above the other) next to the biggest value and one dot
by the smallest. Now just connect the dots.”

Then he wrote the following diagrams to show his method for part (b) of
problem 1-52:

What do you think of Artie’s method? Will it always work? Try Artie’s method with several
expressions and values before deciding.

1-54. Allen and Dwayne would like help comparing two piles of
pennies. The pennies are arranged and represented in the diagrams at left.
. Which pile has more pennies? How do you know?
a. With your team, write expressions that represent each pile. Then write a comparison of
these expressions showing if one is greater than the other or if they are the same.

1-55. Doreen and Delilah were comparing pennies from two teams. They wrote the comparison
statement 6(16) + 3 > 4(9) + 4(9) + 2. What arrangements could this represent? Can you find more
than one possibility? Work with your team to draw diagrams of the arrangements of pennies that
Doreen and Delilah could have been comparing.

1-56. The figure below is reprinted from problem 1-15. Explore using 1-56 Student eTool (CPM).
Using different colors within the pattern may help your team find various patterns.
.
Working alone or with a partner, write as many numerical expressions as you can to describe
the number and organization of dots in this figure. How many different ways can you see the
pattern?
a. Now compare your numerical expressions with those from the rest of your team. Are some
easier to match to the diagram than others? As a team, choose two numerical expressions
that represent the dots in the figure in very different ways. Be sure that everyone has these
two expressions written on their own papers.
b. Find the value of both expressions. How do they compare?


Comparisons

Mathematical symbols are used to compare quantities. The most commonly used symbols are the
two inequality signs (< and >) and the equal sign (=). You can see how these symbols are used
below.
greater than: >
7>5
less than: <
3<5
equal to: =
1+2=3
greater than or equal to: ≥
4 ≥4
8≤9
less than or equal to: ≤

1-57. Which is greater: three sets of (5 − 2) or two sets of (2 + 3)? Draw diagrams to support your
conclusion. Homework Help ✎

1-58. The diagrams at left represent piles of pennies. Homework
Help ✎
a. Which pile has more pennies? Explain your reasoning.
b. Write two different number expressions to represent the number of pennies in each pile.
c. Write a number comparison (using > , < , or = ) to show if the number of pennies in one
pile is greater than the other or if they are the same.

1-59. Connie is helping her mother gather together the loose change (coins) in their house. They
have collected a large jar full of quarters and dimes. Connie wants to arrange the coins on the table
to make it easy to know how much money they have. Homework Help ✎
. On your paper, draw a diagram to show Connie how she can arrange her quarters and dimes to
make her money easy to count.
a. Draw another arrangement that would also be easy to count.
b. Which of your arrangements do you think is easiest to count? Explain why.

1-60. Use your knowledge of place value to place the correct inequality sign (< or >) between
each pair of numbers. Homework Help ✎
. 16.5 ___ 16.52
a. 4.110 ___ 4.10
b. 5.963 ___ 5.9
c. Write the numbers given in part (b) in words.

1-61. Aria and 19 of her friends plan to go to a baseball game. They all want to sit together. Aria
wants to order the seats in the shape of a rectangle, but she cannot decide on the best
arrangement. She starts by considering one row of 20 seats.

Draw a diagram showing Aria’s idea for a seat arrangement. Then draw all of the other possible
rectangular arrangements for 20 seats. Label each arrangement with its number of rows and the
number of seats in each row. Are all arrangements practical? Explain. Homework Help ✎