Running head: PEER TEACHING MODULE

Peer Teaching Module: Chapter 17 – Developing Concepts of Decimals and Percents

Heather Christie

St. Thomas University

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Peer Teaching Module: Chapter 17 – Developing Concepts of Decimals and Percents

Chapter 17 Summary (Pg. 333-351)

This chapter opens by describing the various ways that decimals are a part of people’s

everyday lives, yet many people have misconceptions about decimals. Thus, the aim of this

chapter is to exemplify how to elicit a conceptual understanding of decimals. Such a conceptual

understanding is best established through linking ideas and understandings of fractions to

decimals, which is the main focus of this chapter. A crucial first step in developing that

conceptual understanding is to review and extend whole-number place values in terms of how

the 10-to-1 relationship extends infinitely in both directions and remains the same between any

two place values. The next major piece is discerning the role of the decimal point, which is to

separate whole units (left of the decimal) from fractional units (right of the decimal). It is also

important for students to recognize that adding a zero to the left of a whole number, and to the

right of a decimal fraction, will not change the value. Next, students must appreciate that the

decimal point locates the unit’s place.

Now students are ready to learn about the connection between fractions and decimals,

which is done best by using base-ten fractions (3/10, 76/100, etc.) and then familiar, common

decimal fractions (1/2 = 0.50; 1/4 = 0.25; 1/3 = 0.33333… (to introduce infinitely repeating

decimals)). Fractions and decimals are both systems that represent the same concepts. The use

circular blocks, square grids, or base-ten blocks are great tools to introduce decimals through.

Number lines and meter sticks are also good length models for decimal fractions, and

calculators play a significant role in developing decimal concepts as well. It is imperative that

students are taught to verbalize decimal fractions correctly, which will help them hear the

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connections between fractions and decimals. For example, seven and nine tenths rather than

seven over nine or seven point nine. Students also need to be introduced to various names and

formats (twelve and thirty-four hundredths is the same as 12 34/100, which is also the same as

10 + 2 + 3/10 + 4/100). Because decimals in the real world rarely have exact equivalent

fractions, it is extremely important to introduce approximation and estimation; first

benchmarks should be 0, ½, and 1. Comparing and ordering decimal fractions is the next step to

further developing an understanding of decimal numeration and place value concepts. But, one

also needs to be aware of, and on the look-out for the 6 common misconceptions: longer is

larger, shorter is larger, internal zero, less than zero, reciprocal thinking, and equality.

The next step is to move on to decimal computations. Again, the role of estimation here

is essential (rounding to whole numbers or base-ten fractions), thus it should always be the first

step. When adding, subtracting, or multiplying decimals it is important to start with relevant

contexts and then move to problems without contexts. For multiplication, it is important to get

students to do the computation as if the numbers were whole and then get them to place the

decimal by estimation. Division should be approached in the exact same way as multiplication.

Percent is the final aspect discussed in this chapter. Percent has a denominator of 100,

thus it is a substitute for ‘hundredths.’ Hence, it is not a new concept; it is a new notation and

term. Appropriate models to make the connection to fractions and decimals are needed (base-

ten blocks, rational number wheel, 10 by 10 grid). Again, contextual problems are very

important. In addition, common fractions (and decimals) should become familiar percents as

well (1/2, 0.5, and now 50%). Repeatedly, estimation is critical especially in real-life situations

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(tips, taxes, and discounts). One way to estimate is to substitute for a close percent that is easy

to work with. Another option is to select numbers that are compatible to the percent.

Focus Concepts: Extending the Place Value System & Connecting Fractions to Decimals, pages 333 – 339.

Reviewing the place value system and fractions helps students make the future

connections of decimals to fractions and whole numbers. Therefore, the initial focus of this

lesson is to review fractions with a specific emphasis on proper wording (ie., nine tenths rather

than 9 out of, or over, 10, which will later help students when verbalizing the decimal 59.23 -

fifty-nine and twenty-three hundredths, not fifity-nine point/decimal twenty-three). Proper

wording enables students to hear the connection between decimals and fractions as a result of

them being pronounced the same; they are solely different ways of representing the same

concept. Therefore, when students hear nine-tenths they should come to think of 0.2 AND

2/10, rather than just 2/10. Language also helps students make the connection between

decimals and whole numbers because when they say seventy-four and six-tenths, they hear the

extension decimals add to whole numbers (decimals are extensions of whole numbers, NOT a

new concept)! Studets will come to see the decimal point as an indicator of the speration

between whole units and fractional parts.

Another important aspect of this lesson is helping students see that a 10-to-1

relationship extends infinitely to the right and left of the ones (unit) place. That is, the

relationship between two adjacent pieces is the same, regardless of which two adjacent pieces

are being considered. An emphasis will also be given to the different ways a decimal can be

written/said, depending on the choice of unit being used to count the entire collection (ie.,

2.35, 2 35/100, two and thirty-five hundredths, 235/100, and two hundred thirty-five

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hundredths all mean the same thing). As is evident through the examples provided so far, base-

ten fractions will be the focus of this lesson because this lesson is an introduction, thus

commonly known, contextual fractions will allow for an easier transition for students. Access to

concrete base-ten models, to physically represent what is being taught/learned about, is also

fundamental.

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Lesson Plan: Introduction to Decimals, and the Connection to Fractions and Whole #’s

Subject: Math Grade: 4 Lesson Length: 1.5 hrs Date: Feb. 1, 2016

Stage 1: Pre-Lesson PreparationLesson Rational/Outcome:

- This lesson will provide students with various contexts to explore connections between decimals, fractions, and whole numbers. Through this, students will begin to develop a conceptual understanding of fractions and decimals as an extension of whole numbers rather than something separate. By the end of the lesson, students will be able to read decimals as fractions and represent them using various models. Students will also begin to order decimals by attempting to make largest numbers.

Objectives:- NCTM Content Standards:

Grades 3-5 Expectations (Number and Operations); all students should: Understand numbers, ways of representing numbers, relationships

among numbers, and number systemso Understand the place-value structure of the base-ten number

system and be able to represent and compare whole numbers and decimals;

o Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on numbers lines, and as divisions of whole numbers;

o Recognize and generate equivalent forms of commonly used fractions, decimals, and percents;

- New Brunswick Curricular Outcomes: GCO: Number (N): Develop Number Sense

SCO: N9: Describe and represent decimals (tenths and hundredths) concretely, pictorially, and symbolically.

SCO: N10: Relate decimals to fractions (to hundredths).

Content:- Information gathered from Elementary and Middle School Mathematics: Teaching

Developmentally by Van de Walle, Karp, Bay-Williams, McGarvey, & Folk (2015), P. 333 – 339.

Teacher Materials (all required materials are in the accompanying blue bin in labelled bags):- Fraction review sheet (appendix A); one transparent copy for teacher and a paper

copy for each student- Overhead projector- Caramilk bars (1 for each student) - review- Dry erase marker

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- Computer with Internet access- Youtube video - https://www.youtube.com/watch?v=0JB3bNfLqEM - review- Fraction/Decimal recording charts – appendix B (one transparent copy for teacher and

a paper copy for each student) – activity 1- Base-ten blocks; 1 set for each student and the teacher (set should include: 1

hundreds square, 10 tens rods, and at least 10 single cubes) – activity 1 If base-ten blocks are not available, print-outs can be used (appendix C)

- Decimal place value mat – appendix D – activity 1- USB stick with figures from the textbook – activity 1- Colored painters tape (to make the place value chart on the floor) – activity 2- Ball, or another object that can be used as a decimal point – activity 2- Printed and laminated numbers 1-9; a printable copy can be found at the link listed in

appendix E – activity 2- String (to attached to the numbers 1-9 for students to hang around their neck –

activity 2- Various numbers (2-5 digits with a decimal) written on cue cards (ie., 45.81) – activity

2- “I have… Who has…” cards – activity 3

A blank copy that can be used with various concepts is included in appendix F- Standard decks of playing cards (1 per group) with K, Q, and J’s removed – activity 4- Score Cards for decimal place value game – appendix H – activity 4

Laminated allows these to be reused by writing on them with dry erase markers

- Rules for decimal place value game – appendix G – activity 4- Cue cards for exit slips – wrap-up

Student Materials:- Pencils- Math journals- Individual white boards- Dry erase markers

Stage 2: Lesson Planning and ImplementationWarm-up ( 10 minutes ):

- Fraction review Have a copy of the fraction review sheet (Appendix A) on transparency paper

to use for this whole class review (display on overhead) Have the ‘helper of the day’ hand out a review sheet (Appendix A), and a

chocolate bar to each student (DO NOT EAT THE CHOCOLATE YET); instruct students to have a pencil out as well

Read the directions aloud and have each student cover the appropriate parts with a piece of candy in each block.

1. Tell the students to cover 6 parts; ask students to write the name of the part of the whole that is covered (6/8); ask one of the students to verbalize the part of the whole they recorded (six-eighths: correct them if they say 6 out of, or over, 8; having the correct language is

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essential for later when connecting decimals to fractions and whole numbers); as students respond, fill in the projected overhead copy so everyone can self-check their work

2. Tell the students to cover 3 parts; ask students how many parts are covered (3); now ask what fraction of the chocolate bar is not covered; next, ask one of the students to verbalize the part of the whole that is not covered (three-tenths; again correct improper use of language)

3. Have students cover 4 parts; ask students to write how many parts are NOT covered (2); now get them to write the fraction to represent the covered parts; ask one of the students to verbalize the fraction for the number of parts that are covered (four-sixths; correct improper language)

4. Ask students to cover 5 parts; get students to write the name of the part of the whole that is covered (5/10 OR 1/2); ask if there another way to write it? ask one of the students to verbalize the part of the whole they recorded (five-tenths; one-half); now ask students to try and write this as a decimal; ask one student to provide their answer and an explanation

- Play the fractions to decimals song on YouTube (https://www.youtube.com/watch?v=0JB3bNfLqEM)

Activities:- 1: Connecting fractions to decimals (20 minutes):

Have a copy of appendix B (fraction/decimal charts) on transparency paper, displayed on the overhead

Get a couple helpers to distribute a base-ten set (manipulatives or cut outs (appendix C)) AND appendix B (fraction/decimal charts) AND appendix D (place value chart) to each student

Have students attach appendix B (fraction/decimal charts) in their math journals

Tell students to first think of the flat as one whole unit and get them to place it under the unit column on their place value chart – appendix D (Explain to them that it may represent a cake for a group of students. So, if they divide it into 10 equal parts, each part will be 1/10 or 0.1)

Discuss and show (USB images projected on the screen) them that the “10-makes-1 rule” continues indefinitely in either direction

Insert Image******************************* (figure 17.1) To exemplify, ask students to find the number of rods that is equivalent to one

flat, by placing rods on top of the flat. Teacher can be doing this at the same time on the overhead

Next, have students remove the rods from the flat, place them under the tenths column on the place value chart (appendix D) and then take one rod and place it on the flat (refer to appendix B) and ask:

What part of the flat would one rod be? (one-tenth – BE CONSCIOUS of

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LANGUAGE); have them fill this in under the second column on the 1st chart on appendix B

How would we write this as a fraction? (1/10); have them also record this under the second column on appendix B

Tell students another way to represent a fractional part is as a decimal. Numbers written after a decimal represent parts of the whole, just as fractions represent parts

To write one-tenth as a decimal, get students to write (under the third column in appendix B) a zero to show there is no whole number, then the point to show the division between whole numbers and parts of whole numbers, and then the number of tenths (rods) - 1.

Continue the previous steps with other examples (7 rods, 4 rods, 9 rods, etc.) Encourage students to start on one side and to keep their rods

together (not spread all over the flat) Have students take turns coming up to fill in the chart, while the rest of

the students are still filling in their own. Practice writing and saying the fractions and decimals each time as a whole class (LANGUAGE IS ESSENTIAL).

Now add hundredths (cubes) to the picture. Demonstrate, and get students to follow along with their blocks; Change it so that the rod now equals one whole unit, which means the cubes now equal a tenth of the new whole, and a hundredth of the initial flat. Ensure that students are physically moving their manipulatives on their place value chart to get a better understanding

As done previously, get students to work through a few examples, filling in their 2nd chart on appendix B

Now have students find the number of cubes in a flat (flat is the unit again, rods are tenths, and cubes are hundredths), and what part of the flat each cube would represent (1/100; one-hundredth) – fill in on 3rd chart on appendix B

Tell students to write one-hundredth as a decimal, they need to first write a zero, then the decimal point, then a zero to represent no tenths, then a one to represent hundredths (still on 3rd chart)

As done previously, get students to work through a few examples, filling in their 3rd chart on appendix B

Have students place one rod and 10 units in the column next to the rod on the flat. Discuss how many units are equal to one rod (10) to have students discover 1/10 = 10/100, just as 0.1 = 0.10. What would be another name for three rods? (0.3, 0.30) What would be another way to say 70 units? (0.7, 0.70)

Have students place one rod and four units on the flat, starting at the left side of the flat and keeping them together. Lead discussion for ways to write this. Encourage students to think in terms of units (10 units + 4 units = 14 units = 14/100 = 0.14). Continue showing examples for practice

* As with most math concepts, it's not enough to introduce a concept to students with

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hands-on materials and then move on to the next lesson. Students have to practice the terminology and work with the concepts until they are fluent with them. That's where math games come in handy. Thus, these next three activities are used to reinforce what they just learned in the previous guided lesson. If the previous lesson takes a little longer than expected due to confusions and more time being required for clarification, the following math games could be extended into the next days lesson. Such games can also be minorly adapted and turned into center activities for further continued practice.

- 2: Human Place Value Chart (15 minutes): The goal of this activity is to provide students with awareness and practice

with different place value positions when decimals are involved. Tape off a simple place value chart on the floor (I am only doing 5 sections

(hundreds to hundredths)) and place a ball (or another object to represent the decimal) at the bottom of the chart between the unit (one whole) position and the tenths position

Hand out the individual numbers (1-9; or a sample of these numbers if it is a smaller group) to students to hang around their neck (printables are also available through the link in appendix E)

The rest of the students have individual white boards and dry erase markers to fill in as numbers are called out as well

Get a different ‘white board’ student to pick and call out a number from the various numbers written on cue cards each round

Students wearing the large numbers around their neck have to place themselves in the place value chart on the floor

‘White board’ students are to fill the numbers in on their white board place value chart

After each number is called and discussed, switch a couple students out (a couple of the ‘white board’ students switch with a couple of the ‘human numbers’); continue until everyone has had at least one turn being a ‘human number’

An addition to the game is to call on various students to tell you the value of certain digits (for example, if the number was 34.29 – ask what the 9 represents (hundredths) and so on).

- 3: ‘I have… Who has…” (15 minutes): Can be done as a whole group (if it is quite small) or with several smaller

groups (multiple sets of cards would be needed – a blank template is also provided, see appendix F) – We will be doing it as one whole group

Shuffle the cards and pass all of them out to the entire group The player with the card that says start reads their card (ensure they are using

the correct LANGUAGE) The student that has the ‘answer’ reads their card and so on until the student

with the finish card reads theirs

- 4: Decimal Place Value with Playing Cards (15 minutes):

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Students compete against one another to form the highest decimal number using playing cards. This game challenges students to think critically about the place value of digits in decimal numbers and how each digit's placement impacts the total value of the number. It's also a strategic game that keeps students thinking about the odds of drawing certain cards as they try to form the highest decimal number.

divide the class up, as evenly as possible, into groups of 2-4 students Provide each group with a large baggie (attached), which contains: rules

(appendix G) a shuffled deck of playing cards with all the J, Q, and K’s removed AND enough score cards for each player to have one (appendix H).

Conclusion/Wrap-up ( 15 minutes ): - Have students clean up all their materials and bring them back to the front table.- Have students go back to their seats and as a class review the major concepts they

were introduced to (tenths, hundredths, decimals (and the connection to both whole numbers and fractions), and language (4 and 9 tenths or 49 tenths)).

- Next, have each student fill out an exit slip. Hand them each a cue card (in attached baggie) and have them write a like (something they learned and understand), and a wonder (something they are still not completely sure about).

- Once they are done have them bring them to the front table and then they can go get washed up for lunch.

Assessment:- The initial review activity will be used as a type of formative assessment to see what

the students remember from the fractions unit. If there are particular students who are having difficulty, especially with the proper language of how to express a fraction, it is imperative that notes be made about this and such notes be included in the math assessment binder (further instruction will be needed for them). The review sheets will also be passed in, as another way for the teacher to ensure student understanding.

- Again, formative assessment (as a type of pre-assessment here) will take place during activity 1, as students will be required to hand in their fraction/decimal charts (appendix B) for the teacher to review and use to inform further instruction.

- During the 3 follow-up games, anecdotal notes need to be made (see appendix I) in terms of each students’ strengths and needs; teacher needs to be circulating during this time to take notes and provide assistance.

Differentiation:- Hands-on engagement with the entire class is essential and a huge part of this entire

lesson, which helps to keep attention focused; this lesson also includes serveral activities that the class will move through rather than staying on one activity for an extended period of time (flexible context).

- In addition to the hands-on, kinesthetic activities, visuals are being provided, along with explicit oral instructions, guidelines, and expectations (various input means help with they diversity of learners in the class)

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- Various connections to real-life and contexts familiar to students are also made throughout the lesson – this enhances learning for everyone

- The students will also be practicing and working together collaboratively (peer support); they know they always have the option of ‘phoning’ their elbow friend for help. This also fosters social inclusion and more ‘risk’ taking

- Flexible grouping is also being used throughout, but especially for activity 3 and 4 (I have… Who has… and the decimal card game)

Look to the person next to them if they are unsure of how to say what is represented on their card, for example, during I have… Who has…

Thus, time can be given before this game to provide all students with the chance to look at their cards, practice their numbers, and ask the person beside them for help if they are struggling with one

During the decimal card game, group members can provide peer assistance and support

- For exit slips, if someone is having a hard time getting their thoughts down on paper, the teacher can scribe for them

- In addition, depending on particular special needs in the classroom, teachers can make adjustments accordingly, prior to implementing this lesson

Stage 3: Post-Lesson Reflection- Preparation and Research – Was I well prepared? What could I have done differently?- Written Plan – Was I organized? What did I learn that will help me in the future?- Presentation – Were the students involved? Was I clear in my presentation? How was

the pacing?- Assessment – What did the class do? How do I know if they were successful? What

should I change for next time?

Resources:

For the Teacher:

- Literature Connections (for students as well):

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o Newspapers and magazines are filled with decimals and percents, which can be

used in the classroom to provide real-life connections.

o The Phantom Tollbooth (1961) by Norton Juster

o Piece = Part = Portion: Fraction = Decimal = Percent (2008) by Scott Gifford

o Fractions, Decimals, and Percents (2010) by David Adler

o Do you know Dewey?: Exploring the Dewey Decimal System (2012) by Brian

Cleary

- Articles:

o Cramer, K., Monson, D., Wyberg, T., Leavitt, S., & Whitney, S. B. (2009). Models

for initial decimal ideas. Teaching Children Mathematics, 16(2), 106-117.

o Suh, J. M., Johnston, C., Jamieson, S., & Mills, M. (2008). Promoting decimal

number sense and representational fluency. Mathematics Teaching in the

Middle School, 14(1), 44-50.

- Websites:

o Base Blocks – Decimals:

http://nlvm.usu.edu/en/nav/frames_asid_187_g_2_t_1.html?

open=instructions&from=topic_t_1.html

o Fraction Model – Version 3: http://illuminations.nctm.org/activitydetail.aspx?

ID=11

For the Student:

- Websites

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o IXL Math: https://www.ixl.com/math/grade-4/compare-decimal-numbers;

https://www.ixl.com/math/grade-3/what-decimal-number-is-illustrated

o Math Play decimal games: http://www.math-play.com/decimal-math-

games.html

o Math Playground: http://www.mathplayground.com/index_fractions.html

o Interactive sites for education: http://interactivesites.weebly.com/decimals.html

o Spelling City: http://www.learninggamesforkids.com/math_games.html?

utm_source=SpellingCity&utm_medium=Link&utm_term=Resource&utm_conte

nt=mathvocab&utm_campaign=#post-131

References

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Field, B. (2015). Decimal place value with playing cards. Games4Gains. Retrieved from

http://games4gains.com/blogs/teaching-ideas/41379652-decimal-place-value-with-

playing-cards

Kawas, T. (2010). More place value activities. MathWire. Retrieved from

http://mathwire.com/numbersense/morepv.html

National Council of Teachers of Mathematics. (2016). Principals and standards for school

mathematics: Grade 3-5 expectations. Retrieved from http://www.nctm.org/Standards-

and-Positions/Principles-and-Standards/Principles,-Standards,-and-Expectations/

New Brunswick Department of Education and Early Childhood Development. (2008). New

Brunswick Mathematics Curriculum: Grade 4. Retrieved from

http://www2.gnb.ca/content/dam/gnb/Departments/ed/pdf/K12/curric/Math/Math-

Grade4.pdf

Ohio Department of Education. (n. d.). Fraction and Decimal Equivalency – Grade Four.

Retrieved from

http://ims.ode.state.oh.us/ODE/IMS/Lessons/Web_Content/CMA_LP_S01_BB_L04_I01_

01.pdf

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M., McGarvey, L. M., & Folk, S. (2015).

Developing concepts of decimals and percents (333-351). Elementary and middle school

mathematics: Teaching developmentally (4th Canadian Edition). Toronto, ON: Pearson

Appendices

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Appendix AFraction Review

Name: _________________________________ Date: _________________________

1. Cover 6 parts of the chocolate bar below. What fraction is of the chocolate bar is covered? ____________

2. Cover 3 parts of the chocolate bar below. How many parts are covered? __________ What fraction of the chocolate bar is NOT covered? __________

3. Cover 4 parts of the chocolate bar below. How many parts are NOT covered? _________

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Write the fraction for the part that IS covered ______________.

4. Cover 5 parts of the chocolate bar below. How many parts of the whole are covered? ________; Is there another way to record this? Yes or No? If so, how? ___________; Try writing it as a decimal? ______________

Appendix B

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Connecting Fractions to Decimals

Name: ___________________________________ Date: _________________________

Number of rods: Fraction of flat covered: Decimal:

Number of small cubes: Fraction of rod covered: Decimal:

Number of small cubes: Fraction of flat covered: Decimal:

Appendix C

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Appendix D – Decimal Place Value Mat

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Appendix E

For digit cards go to: http://mathwire.com/numbersense/digitcards.pdf

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Appendix F

Appendix G

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Decimal Place Value Game with Playing Cards!Players: 2-4 per deck of cards

Materials: - Score Sheet (1 per player) – see attached- Deck of cards with ALL J, Q, and K’s removed (1 deck per group)- Pencil for each player

*Note – 10’s represent zeros

Procedure:- Shuffle the deck of cards and place them face down in the middle of the table- Each player takes a turn turning over a card from the top of the pile and placing it face-

up, directly in front of them so that all other players can see the card- After the first player draws their card, they determine which place value spot to put that

number in (tens, ones, tenths, or hundredths) – REMEMBER, players want to try and make the BIGGEST number.

o The player can only write the number under one place value column, and once it is recorded it cannot be changed at any point during the game

o Players should not let other players see where they wrote their number- Then the second player draws a card from the top of the pile, places it face up in front of

them, they determine which place value spot to put that number in (tens, ones, tenths, or hundredths); then the third, and so on; this process continues until all players have all four columns filled in (each player has done this 4 times)

- Players all reveal their final numbers and determine who has the highest number- The player with the highest must read their decimal aloud CORRECTLY to gain the point

(ie., eighty-five and ninety-three hundredths, NOT eighty-five point/decimal ninety-three)

o If the player does not read their number correctly, they do not receive a point and everyone else does.

- The player(s) who get a point each round, circles that particular round’s number so points can be tallied at the end

- Cards are all put back into the deck, reshuffled, and then the whole process is repeated for every round.

o Play ends when a pre-determined time is up, or when all ten rounds are complete

Appendix H

Decimal Place Value Game Score Card

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Round Tens Ones Tths Hths

1

2

3

4

5

6

7

8

9

10

*Circle any rounds that you earned 1 point.

Appendix I

Human Place Value I have… Who has… Decimal Card Game

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Class List: Strengths Needs Strengths Needs Strengths Needs1. Alexa A.

2. Haley B.

3. Emily B.

4. Adam B.

5. Amy C.

6. Shawna C.

7. Mark. L

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