This lesson is intended to help you assess how well students are

able to identify equivalent decimal fractions.

This lesson is adapted from the Kentucky Department of Education

Mathematics Specialists and revised by PA-MDC Writing Committee.

PA – MDC GRADE 4

FRACTION AND DECIMAL

CONNECTIONS

Concept

Development

Formative

Assessment

Lesson

With Sincere thanks and appreciation for the effort and work of the Members of the PA- MDC Writing

Committee and for the unwavering support from their home districts: Camp Hill School District,

Cumberland Valley School District, Lower Dauphin School District and Shippensburg School District.

Joan Gillis, State Lead MDC

Dan Richards, Co- Lead MDC

PA –MDC Writing Committee

Jason Baker Heather Borrell Susan Davis Carrie Tafoya

Richard Biggs Carrie Budman Miranda Shipp

Review Committee

Katherine Remillard, Saint Francis University Renee Yates, MDC Specialist, Kentucky

Carol Buckley, Messiah University Josh Hoyt, Berks IU

Dr. Karla Carlucci, NEIU Kate Lange, CCIU

Lori Rogers, CAIU

If you have any questions please contact Joan Gillis at jgillis@caiu.org

1

FRACTION AND DECIMAL CONNECTIONS

Mathematical Goals: This lesson unit is intended to help you assess how well students are able

to identify equivalent decimal fractions. Students will:

Recognize and generate equivalent fractions.

Use equivalent fractions to add and subtract fractions with like denominators. Use

decimal notation for fractions with denominators 10 and 100.

Use words to indicate the value of the decimal.

Use decimal fractions and locating them on the number line.

Use area models to represent equivalent fractions and decimals

Common Core State Standards: This lesson involves mathematical content in the standards

from across the grade, with emphasis on:

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual

fraction models, with attention to how the number and size of the parts differ even though the

two fractions themselves are the same size. Use this principle to recognize and generate

equivalent fractions.

4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. c. Add and subtract mixed

numbers with like denominators, e.g., by replacing each mixed number with an equivalent

fraction, and/or by using properties of operations and the relationship between addition and

subtraction.

4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite

0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

This lesson involves a range of Standards for Mathematical Practice with emphasis on:

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

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

MP.7 – Look for and make use of structure

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

PA Core Standards:

PA CC.2.1.4.C.1: Extend the understanding of fractions to show equivalence and ordering

PA CC.2.1.4.C.3: Connect decimal notation to fractions, and compare decimal fractions

This Formative Assessment Lesson is designed to be part of an instructional unit. This task should be

implemented approximately two-thirds of the way through the instructional unit. The results of this

task should then be used to inform the instruction that will take place for the remainder of your unit.

2

Introduction: This lesson is structured in the following way:

A day or so before the lesson, students work individually on an assessment task that is

designed to reveal their current understanding and difficulties. When complete, you

review their work and formulate questions for students to answer, to help them improve

their solutions.

During the lesson, students work in pairs to match the fraction and addition problems

with fraction and decimal equivalencies, the correct number line that represents the

fraction/decimal, and an area model representation.

In a whole-class discussion, students will share and justify their answers.

You may choose to have students revisit the original assessment task, and try to improve

their own responses.

Time Needed – Estimated 80 – 105 minutes

Timings are approximate: exact timings will depend on the needs of your students

Pre-assessment: no more than 15 minutes

Whole Class Introduction: 10 – 15 minutes

Collaborative Activity: 30 – 60 minutes (time is dependent upon amount of cards used)

Whole Class Discussion 10 – 15 minutes

Post Assessment: 15 minutes

Materials required

Each student will need 2 copies of the assessment to use a pre-assessment and a revisit.

Each student will need a mini-whiteboard, an erasable marker, and an eraser (white board

could be sheet of white paper in a sheet protector)

Each pair of students, during the collaborative lesson, will need a packet of Card Set A –

G. (Start with Card Sets A and B. After students can demonstrate their reasoning for the

matches, give them the next ‘layer’ of cards. You may want to make copies of the card

sets on different color card stock to assist with organization.The card sets should be cut-

up before the lesson.

Glue/tape and chart paper

Before the lesson

Pre-Assessment Task: Equivalent Fractions (no more than 15 minutes)

Have the students do this task in class a day or more before the Formative Assessment Lesson.

This will give you an opportunity to assess the work and to find out the kinds of difficulties

students have with it. Then you will be able to target your help more effectively in the follow-up

lesson.

Framing the Task:

Give each student a copy of Pre-Assessment. Introduce the task briefly help the class to

understand the problem and its context.

3

Spend fifteen minutes on your own, answering these questions. Don’t worry if you can’t

figure it out.

There will be a lesson on this material [tomorrow] that

will help you improve your work. Your goal is to be able

to answer these questions with confidence by the end of

that lesson.

It is important that students answer the question without assistance,

as far as possible. If students are struggling to get started, ask them

questions that help them understand what is required, but do not do

the task for them and be conscientious to not lead or provide the

thinking for your students.

Assessing Students’ Responses

Collect students’ responses to the task. Make some notes on what their work reveals about their

current levels of understanding. The purpose of this is to forewarn you of the issues that will arise

during the lesson, so that you may prepare carefully.

We suggest that you do not score students’ work. The research shows that this is

counterproductive, as it encourages students to compare scores, and distracts their attention from

how they may improve their mathematics.

Instead, help students to make further progress by asking questions that focus attention on aspects

of their work. Some suggestions for these are given on the next page. These have been drawn

from common difficulties anticipated.

We suggest that you write your own lists of questions, based on your students’ work, using the

ideas below. You may choose to write questions on each student’s work.

If you do not have time to do this, select a few questions that will be of help to the majority of

students. These can be written on the board when you return the work to the students at the start

of the lesson.

4

Common Issues Suggested questions and prompts:

No fractional understanding

Correctly identifies fraction but does

not know equivalent fractions.

(Question 1)

Define a fraction for me

What are the parts of a fraction?

What does the numerator tell you? Denominator?

Draw a box and a line 1/2 through it. Draw a second

box the same size and divide it into fours and shade two

of the squares, what do you notice about 1/2 and 2/4?

Students incorrectly identify fractional

(or decimal) representations on the

number line (Question 2)

How can you tell the number of equal divisions there

are between 0 and 1 on the number line?

Can you find ½ on the number line? (anchor fraction)

Students apply an algorithm without

having understanding of what it

means to add fractions (conceptually).

Students treat each part of the fraction

(both numerator and denominator) is

treated as a different single-digit

whole number (Question 3)

Draw a model of one-tenth plus one-tenth?

If I have 10 candy pieces all together and 𝟑/𝟏𝟎 are red

and the rest are yellow. Does the total amount change?

Can you express this as a number sentence?

(3

10+

7

10=

10

10)

Please use the blanks to add your class misconceptions and directed questions.

5

Suggested lesson outline

Whole-class Introduction (10 – 15 minutes)

Give each student a mini-whiteboard, an erasable marker, and an eraser (white board could be

sheet of white paper in a sheet protector).

Explain to the class that in this lesson they will be working with fraction decimals in different

forms (fraction, decimal, word and picture form) and you will also be asked to locate them on a

number line. Before we begin this activity we will do a quick review. Ask students to write on

their mini-whiteboards the answers to questions relating to the upcoming activity, such as the

ones below. Each time, ask students to explain their method.

What is the difference between a decimal fraction and a fraction? Give me an example of

each. (Decimal fractions have a denominator of 10, 100, 1000, etc. and a fraction is a

numerical quantity that is not a whole number) Why do you think they are identified

differently?

Draw a number line to compare 2/5 and 3/10 – ask several students to explain their

comparison. (Did anyone change 2/5 to 4/10 to make the comparison or place on number

line easier to do?)

Write a decimal which is equivalent to 7/10 – ask a student to explain how they did this.

Write a fraction which is equivalent to 3/4 – ask a few students to explain how they know

their fraction is equivalent.( Share the steps you took to make an equivalent fraction and

what that means)

Collaborative Lesson activity (30 to 60 minutes) – lesson activity time is dependent upon the

amount of cards used and the ability of the class.

Organize the class into groups of two students. Deliberately pair the students in homogenous

groups of two, determined from the pre-assessment, so that students are grouped by similar

misconceptions.

Explain to students how they should work together, making sure that each student can articulate

why the card is placed where it is, even if that student did not place the card.

Today we are going to explore your knowledge of fraction decimals, and how you

can express them in various ways. You and your partner are going to match sets

of cards to the original Set A cards. I want you to add or subtract the fractions on

Set A and find its solution on Card Set B. Once you find the solution for Set A

line- up Card B and begin work on Card Set C, which is naming an equivalent

fraction to card set B.

6

Give each pair Card Set A: Addition/Subtraction, Card Set B: Solutions and Card Set C:

Fraction Equivalence.

NOTE: Depending on how students performed on the pre-test, you may want to hold Card Set C

until students have started matching and can articulate how they started matching card Set A to

Card Set B.

Your tasks during the small group work are to make a note of student approaches to the task, to

support student problem solving and to monitor progress. Note any difficulties that emerge for

more than one group; these can be discussed later in the lesson.

Once your students are satisfied with their card placement, you may instruct them to glue/tape

their cards down on chart paper.

If you notice several pairs struggling over certain equivalent fractions, you could have one

person from one group remain at their desktop and defend their choices, and have the other

student ask another group what they chose and why, prior to gluing their final response on chart

paper.

If you anticipate this lesson taking more than one class period you may also decide to use the

graphic organizer (page 9) to keep track of their card match-ups

Notice when students are ready for Card Set D: Decimals and Card Set E

You will continue to take note of student’s approaches to the task, to support student’s problem

solving and to monitor progress. Use this time to determine which pairs will present in the

Whole Class discussion.

Card Set F: Number Lines bring the lesson together.

Card Set G: Area Models Students should make connections between the area models, number

lines, and fractional and decimal representations once the sort is complete.

Whole Class Discussion (10 – 15 minutes)

Allow time for a brief gallery walk. Pairs of students are given post-it notes to leave

questions/comments for groups to respond when called upon.

Choose two or three pairs to present their posters. Your job is to facilitate the conversation of

your class.

Some suggestions:

How did you approach this task? (organization skills)

Were any cards, in particular, easier or more difficult for your group?

Would it have been easier if you had Card Set G first, why or why not?

7

What advice would you give to others who were going to do this Classroom

Challenge?

If I asked you to connect these card sets to a real life situation what

subject/situation could you base it on? (money, baking , running and swimming

meets, measuring, weight –metric would be great with decimal fractions , why?

( base ten)

What have you learned from this activity?

Post Assessment (10 – 15 minutes)

Return the pre-assessment along with a second blank copy. Post the common prompts from the

misconceptions chart and allow for individual reflection on original responses. Give no more

than 15 minutes for students to improve upon their response on the blank copy.

8

Fractions: Relating Fractions to Decimal Fractions

Card Match-Up Solutions Page

(Good idea to create graphic organizer in case you run out of class time)

A1

B1

C 3

D5

E1

F4

G7

A2

B6

C2

D4

E2

F1

G2

A3

B5

C5

D7

E9

F3

G5

A4

B4

C6

D8

E4

F6

G4

A5

B7

C1

D1

E3

F2

G1

A6

B 10

C8

D2

E6

F5

G 10

A7

B3

C9

D3

E7

F8

G9

A8

B2

C4

D6

E8

F7

G6

A9

B9

C 7

D9

E5

F 10

G3

A 10

B8

C 10

D 10

E 10

F9

G8

9

Fraction and Decimal Connections

Group Recording Sheet for ___________________________________________

A 1

A 2

A 3

A 4

A 5

A 6

A 7

A 8

A9

A 10

10

Name:_______________________________________

Fill in the missing parts on the number line

Write two equivalent fractions for the shaded portion of the whole.

Explain how they are equivalent in the box, below.

Pre-Assessment/Post-Assessment

11

CARD SET A

A1

𝟐

𝟏𝟎 +

𝟑

𝟏𝟎

A2

A3

A4

A5

A6

A7

A8

A9

A10

𝟏

𝟓 +

𝟏

𝟓

𝟗

𝟓 -

𝟐

𝟓

𝟖

𝟏𝟎 +

𝟐

𝟏𝟎

𝟏

𝟏𝟎 +

𝟏

𝟏𝟎

𝟕𝟐

𝟏𝟎𝟎 -

𝟒𝟐

𝟏𝟎𝟎

𝟏𝟕

𝟏𝟎𝟎 +

𝟓𝟑

𝟏𝟎𝟎

𝟏𝟎

𝟓 -

𝟔

𝟓

𝟏𝟒𝟖

𝟏𝟎𝟎 -

𝟑𝟖

𝟏𝟎𝟎

𝟏𝟎

𝟏𝟎 -

𝟒

𝟏𝟎

12

CARD SET B

B1

B2

B3

B4

B5

B6

B7

B8

B9

B10

𝟒

𝟓

𝟕

𝟓

𝟏𝟏𝟎

𝟏𝟎𝟎

𝟐

𝟓

𝟑𝟎

𝟏𝟎𝟎

𝟔

𝟏𝟎

𝟐

𝟏𝟎

𝟕𝟎

𝟏𝟎𝟎

𝟏𝟎

𝟏𝟎

𝟓

𝟏𝟎

13

CARD SET C

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

𝟏

𝟐

𝟏𝟏

𝟏𝟎

𝟓

𝟓

𝟒

𝟏𝟎

𝟖

𝟏𝟎

𝟕

𝟏𝟎

𝟑

𝟏𝟎

𝟑

𝟓

𝟏𝟒

𝟏𝟎

𝟏

𝟓

14

CARD SET D

D1

0.2

D2

0.6

D3

0.8

D4

1.0

D5

0.5

D6

1.1

D7

0.4

D8

1.4

D9

0.3

D10

0.7

15

CARD SET E

E1

five-tenths

E2

one

E3

two-tenths

E4

one and four-tenths

E5

three-tenths

E6

six-tenths

E7

eight-tenths

E8

one and one-tenths

E9

four-tenths

E10

seven-tenths

16

F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

CARD SET F

17

CARD SET G

G1

G2

G3

G4

G5

G6

G7

G8

G9

G10

18

Tear-Off Sheet with Suggestions for Helping Students Access Information

Barrier to Learning Suggested Strategy

Student lacks understanding of

math language

Review domain specific vocabulary; create a picture

dictionary with student.

Student lacks basic perimeter/area

knowledge

Review prior lessons in perimeter and area; use

manipulatives to explore perimeter/area concepts.

Students have problems in

understanding math word

problems (reading comprehension)

Use key words and "picture stories" to help students

identify the appropriate operation.

Build vocabulary through repeated classroom use and

picture dictionary.

Work on reading and understanding problems through

modeling in small groups and peer-to- peer situations.

Student struggles with multi-step

problems

Break the problem into smaller tasks, an understandable

sequence.

Student struggles with writing

explanations and math reasoning

Continued use of Math Journaling and Share Time in

which classmates critique each other can help strengthen

this. Explain to the student that written explanation

takes the place of verbal communication and the reader

needs to understand how you solve the problem.

Student struggles with creating a

rectangle with a given perimeter.

Identify the perimeter of objects by measuring with a

ruler. Develop skill through use of geoboard. Have

student count each line drawn creating a rectangle.

Student struggles with identifying

the area of a rectangle.

Provide multiple methods of find area: counting

squares, repeated addition, composite units, multiplying

the length and width.