Fractions and Decimals

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Curriculum Resource Guide - Mathematics Content: Fractions and Decimals

The Curriculum Resource Guides describe how to teach the content to students with the most significant cognitive disabilities.
Used in conjunction with the Content Modules that describe the content (what is being taught) in general education they provide teachers with the necessary background knowledge to prepare students for the NCSC alternate assessment.
The Curriculum Resource Guides offer examples of how the content is taught in general education, ideas for real life use, examples of universal design for learning, and ways to promote college and career readiness.

Questions about:

• Ratio and Proportions
• Ratio and Proportions Teaching Activities
• Ratio and Proportions and Common Core Connectors
• Real World Ratio and Proportion Activities
• Promoting College and Career Readiness through Ratio and Proportions
• Making Ratio and Proportions Accessible to ALL Students

What are "fractions" and how are they taught in general education settings?

The essential knowledge in this content area

The concept of a fraction refers to equal parts of a whole region or set. Students are expected to represent fractions as equal parts of the whole. Initially this should be taught by working with "fair shares" of an item (e.g., There are 4 of us and we want to divide our apple into 4 equal parts so that we each have a fair share). Next students should only work with fractions using region models (square, etc.), and linear models are preferred. Then the symbolic representation can be taught.

$\frac{3}{8}= \frac{Numerator}{Denominator}= \frac{how\ many\ of\ the\ equal\ parts\ are\ counted}{total\ number\ of\ equal\ parts\ in\ the\ whole}$

Students should know that fractions parts must be equal in size.

Students need to learn to compare fractions:

• When the fraction is divided into more parts the size of the fractional parts gets smaller.

• Place fractions on a number line:

Divide spaces between each whole number into equal parts (i.e., 5 segments between 0 and 1)

Students identify that the distance between 0 and the first 3 segments is 3/5.

• Can be determined by using visual models.

Students should be provided many opportunities to practice making models of equivalent fractions to ensure understanding. One idea is to use fraction strips or bars to demonstrate equivalencies.

Add or subtract fractions with like denominators

• Make sure students understand that if the denominators are the same, they just add or subtract the numerators (or numbers on top) and the denominator stays the same.

Add or subtract fractions with unlike denominators

• First, teach students using fraction strips to help them visualize the concept of adding fractions with unlike denominators.

• Then, teach students to find common denominators and then add or subtract the fractions. This can be done by multiplying the denominators or by determining the least common denominator (LCD). The LCD can be found by determining the multiples of each denominator and finding the smallest number (>1) they have in common.

Write fraction as decimal

• Students learn to divide the numerator by the denominator to change a fraction to a decimal.

Multiply and divide fractions

• Students can learn these concepts first by using models. Students need to understand that $4 * \frac{2}{3}$ is asking for 4 sets of $\frac{2}{3}$. First students can draw or make several wholes divided into thirds. Then students can circle 4 sets of $\frac{2}{3}$.

In this model 2 wholes were circled and $\frac{2}{3}$ of another whole were circled. Thus:

$4* \frac{2}{3} = 2 + \frac{2}{3} = 2 \frac{2}{3}$

This could also be represented on a number line with each whole divided into thirds.

• Once students understand the underlying concept of multiplying fractions, they can begin using symbols and strategies for solving multiplication problems symbolically. For multiplication students learn to multiply the numerators, then denominators, and then find the simplest form of the fraction.

$\frac{1}{3}*\frac{3}{5}=\frac{3}{15}\div\frac{3}{3}=\frac{1}{5}$

• Students need to understand the division of fractions. For example, the problem $2\div \frac{2}{3}$ is asking, "How many $\frac{2}{3}$ are in 2?" Below is a visual model to support students' understanding. First, 2 wholes are drawn. Then, the 2 wholes are divided into thirds.

There are 3 groups of 2/3. Therefore $2\div\frac{2}{3}=3$.

• Once students understand the underlying concept of dividing fractions, they can begin using symbols and strategies for solving division problems symbolically. For division students learn to multiply by the reciprocal of the divisor and then find the simplest form of the fraction.
  • The reciprocal of a fraction is the inverse (i.e., the reciprocal of $\frac{2}{3}\ is\ \frac{3}{2}$)

$\frac{1}{4} \div \frac{3}{6} = \frac{1}{4} * \frac{6}{3} = \frac{6}{12} = \frac{1}{2}$

Common misunderstandings in this content area

Students understand whole numbers and apply this understanding to fractions. Therefore, when they see 1/6 and 1/5 they may assume that 1/6 is larger because 6 is greater than 5. Students have difficulty understanding the relationship between the numerator and the denominator and instead may comprehend them as two separate whole numbers. This can result in students adding across fractions (i.e., numerator plus numerator over denominator plus denominator). Students also might have difficulty understanding that all 1/2s are not equal. For example 1/2 of a large pizza is larger than 1/2 of a medium pizza. This relates to understanding the 1/2 always relates to the specific "whole" that is being referred to by the fraction.

Prior Knowledge/skills needed (can be taught concurrently)

• Number sense with whole numbers
• Addition, subtraction, multiplication and division
• Understanding greater than/less than
• Sequencing numbers

What are "decimals" and how are they taught in general education settings?

The essential knowledge in this content area

Students need to understand that the numerals that come after the "dot" in a number represent an amount less than one. It will be helpful to use a place value chart to teach decimal labels.

Hundreds	Tens	Ones	Decimal	Tenths	Hundredths
			●	4	5

In this example the decimal is .45. It could be represented by a fraction as 45/100 or 4/10 +5/100. Students would name this fraction/decimal as forty-five hundredths.

Similar to a fraction, a decimal represents a part of a whole.8 Students should have extensive practice with models to ensure comprehension of the concept.

Students can then learn to order decimals first by using representations, and then numerically. It is important for students to be able to convert decimals to fractions. This will be easier for them to comprehend having used models to represent the concepts.

What are some of the types of activities general educators will use to teach this skill?

Activities from General Education Resources

(CCR & Standards for Mathematical Practice Reference Table)

Fractions

• 8 Using ten counters, lay them out so that some are red and some are yellow. Tell what fraction of the counters are red and what fraction are yellow.
• 2 Have students make a quilt by gluing 16 precut squares of three different colors onto their paper. Now have them describe the number of colors used in fractions (i.e. "Use a fraction to tell me how much of the quilt is purple.").¹⁰
• 41 To explore equivalent fractions have students fold a circle in half, draw a line on the fold and color one part of the circle. Then take another circle that is the same size and fold it in half; then fold in half again. Draw lines on the folds and this time color two parts of the circle. Then ask the students to describe how the colored portions are the same and how they are different. Last have them write fractions to represent the colored portions.¹⁰
• 64 Students make a healthy fruit snack by following directions that include fractions of a cup in the ingredients list.¹⁰
• 24 Students make a bracelet using two different color beads then describe the colors used to make the bracelet in fractions.
• 4 Give students counters and ask them to demonstrate a fraction and one that is equivalent to it (e.g., 3/4 and 6/8)
• 52 Using a number line to 20, have students place counters above all the multiples of 2. Then have them place counters of a different color above all the multiples of 3. Ask them to identify the smallest number that 2 and 3 have in common.¹²
• 22 Students use grid paper to demonstrate addition of fractions with common denominators. First they trace the perimeter of the total number of squares (denominator), then they use two different colors to represent the two fractions summed, last they count the total number of squares colored and write the answer.¹²
• 44 Use pennies, dimes, and dollars to demonstrate one-hundredths, tenths, and 1 whole. Write as decimals and fractions.¹⁰
• 42 Use graph paper to model multiplying whole numbers by decimals. Have students color in the decimal and then repeat the number of times it is multiplied. Then count to find the answer.
• 23 Write several familiar fractions on note cards and several decimals that are close to the fractions but not exactly the same. Have students play a matching game matching the decimals and fractions.
• 52 Create a decimal number line. Then have students place a list of several decimals in numerical order on the number line.

Links Across Content Areas

• 64 Using measurements to conduct a science experiment.
• 43 Reading music to play an instrument.
• 5 Measuring distance between destinations in social studies.

What Connectors to the Common Core Standards Are Addressed in Teaching "Fractions and Decimals"?

Link Core Content Connectors for the Common Core State Standards This guide is not intended to comprehensively cover all of the CCCs related to this topic. For further information on other CCC's related to fractions, see the Instructional Families, Content Modules, and Element Cards/Graduated Understandings.

Grade Differentiation	Core Content Connectors	Common Core State Standards
3rd grade	3.SE.1g1 Use =, <, or > to compare 2 fractions with the same numerator or denominator.	3.NFA.A.3d
	3.NO.1l1 Identify the number of highlighted parts (numerator) of a given representation (rectangles and circles).	3.NF.A.1
	3.NO.1l2 Identify the total number of parts (denominator) of a given representation (rectangles and circles).	3.NFA.A.1
	3.NO.1l3 Identify the fraction that matches the representation (rectangles and circles; halves, fourths, thirds, and eighths).	3.NF.A.1
	3.NO.1l4 Identify that a part of a rectangle can be represented as a fraction that has a value between 0 and 1.	3.NF.A.2a, b
	3.PRF.1f 1 Determine the equivalence between number of minutes and the fraction of the hour (e.g., 30 minutes = ½ hour) .	3.MD.A.1
	3.GM.1i1 Partition rectangles into equal parts with equal area.	3.G.A.2
4th grade	4.SE.1g2 Use =, <, or > to compare 2 fractions (fractions with a denominator or 10 or less).	4.NF.A.2
	4.SE.1g3 Use =, <, or > to compare 2 decimals (decimals in multiples of .10)	4.NF.C.7
	4.SE.1h1 Express whole numbers as fractions.	3.NF.A.3c
	4.SE.1h2 Identify the equivalent decimal for a fraction.	4.NF.C.6
	4.NO.1l6 Locate fractions on a number line.	3.NF.A.2
	4.NO.1l7 Order fractions on a number line.	3.NF.A.2
	4.NO.1m1 Determine equivalent fractions.	3.NF.A.3, 4.NF.A.1
	4.NO.1n1 Select a model of a given fraction (halves, thirds, fourths, sixths, eighths).	3.NF.A.1
	4.NO.1n2 Compare up to 2 given fractions that have different denominators.	4.NF.A.2
	4.NO.1o1 Match a fraction with a denominator of 10 or 100 as a decimal (5/10 = .5).	4.NF.C.6
	4.NO.1o2 Find the equivalent decimal for a given fraction.	4.NF.C.5
	4.NO.1p1 Read, write or select decimals to the tenths place.	4.NF.C.6
	4.NO.1p2 Read, write or select decimals to the hundredths place.	4.NF.C.6
	4.NO.1q1 Compare two decimals to the tenths place with a value of less than 1.	4.NF.C.7
	4.NO.1q2 Compare two decimals to the hundredths place with a value of less than 1.	4.NF.C.7
	4.NO.2g1 Using a representation, decompose a fraction into multiple copies of a unit fraction (e.g., ¾ = ¼ + ¼ + ¼ ).	3.NF.A.1, 4.NF.B.3a, b
	4.NO.2h1 Add and subtract fractions with like denominators of (2,3,4, or 8).	4.NF.B.3a, b
	4.NO.2h2 Add and subtract fractions with like denominators (2,3,4, or 8) using representations .	4.NF.B.3a, b
	4.NO.2h3 Solve word problems involving addition and subtraction of fractions with like denominators (2, 3, 4, or 8).	3.NF.A.3d
5th grade	5.NO.1b1 Read, write, or select a decimal to the hundredths place	5.NBT.A.3a
	5.NO.1b2 Read, write or select a decimal to the thousandths place.	5.NBT.A.3a
	5.NO.1b3 Compare two decimals to the thousandths place with a value of less than 1.	5.NBT.A.3a
	5.NO.1b4 Round decimals to the next whole number.	5.NBT.A.4
	5.NO.1b5 Round decimals to the tenths place.	5.NBT.A.4
	5.NO.1b6 Round decimals to the hundredths place.	5.NBT.A.4
	5.NO.1c1 Rewrite a fraction as a decimal.	4.NF.C.6
	5.NO.1c2 Rewrite a decimal as a fraction.	4.NF.C.6
	5.NO.2b1 Add and subtract fractions with unlike denominators by replacing fractions with equivalent fractions (identical denominators).	5.NF.A.1
	5.NO.2b2 Add or subtract fractions with unlike denominators.	5.NF.A.1
	5.NO.2b3 Multiply or divide fractions.	5.NF.B.4
	5.NO.2c1 Solve 1 step problems using decimals.	5.NBT.B.7
	5.NO.2c2 Solve word problems involving the addition, subtraction, multiplication or division of fractions.	5.NF.A.2
6th grade	6.NO.1f4 Find a missing value (representations, whole numbers, common fractions, decimals to hundredths place, percent) for a given ratio.	6.RP.A.3a
	6.NO.2c3 Solve one step, addition, subtraction, multiplcation, or division problems with fractions or decimals.	6.NS.A.1
	6.NO.2c4 Solve word problems involving the addition, subtraction, multiplication or division of fractions.	5.NF.B.7c
7th grade	7.NO.1h1 Identify an equivalent fraction, decimal and percent when given one of the three numbers.	6.RP.A.3d
8th grade	8.NO.2i3 Solve one step addition, subtraction, multiplication, division problems with fractions, decimals, and positive/negative numbers.	7.NS.A.1d
	8.NO.2i4 Solve two step addition, subtraction, multiplication, and division problems with fractions, decimals, or positive/negative numbers.	7.NS.A.1d
9-12	N/A

Performance Examples and Essential Understandings for Priority CCCs
Grade 3
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Geometry: 3.GM.1i1 Partition rectangles into equal parts with equal area. (3.G.A.2)	Selects the square that is divided into 4 equal parts Choice Selection: Answers: How do you know that these parts have the same area? (The response provided by the test example is that that the square has the same number of parts in it. Again, this is not technically correct because one could divide the square into four parts of different sizes.)	Concrete Understandings: Understand concept of equal parts Partition with concrete objects Find the rectangle that is the same or match two congruent rectangles	Representation: Partition rectangles into two, three or four equal shares Understand concepts and vocabulary: equal, partition, area, rectangle, halves, thirds, half of, a third of
Numbers: 3.NO.1l3 Identify the fraction that matches the representation (rectangles and circles; halves, fourths, thirds, and eighths). (3.NF.A.1)	Student selects fraction to represent the picture. "Choose the fraction that tells how much of the square is colored in." $\frac{1}{3}\ \ \frac{2}{2}\ \ \frac{3}{4}$	Concrete Understandings: Identify part and whole when item is divided Count the number of the parts selected (3 of the 4 parts; have fraction present but not required to read ¾)	Representation: Understand how parts to whole can be expressed as fractions using numbers Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts Ability to recognize that fraction bars of equal lengths can be divided into different numbers of equal parts/units Understand a fraction a/b as the quantity formed by a parts of size 1/b Ability to recognize that the more equal parts, the smaller the part Understand concepts, symbols, and vocabulary: numerator, denominator, _/_
Symbols: 3.SE.1g1 Use =, <, or > to compare 2 fractions with the same numerator or denominator. (3.NF.A.3d)	Student selects correct symbol to compare two fractions. You are going to compare fractions using these symbols. This fraction is 2/5. This fraction is 4/5. Choose the symbol that shows your answer. Provide ("=", ">", and "<"). Is 2/5 greater than, less than, or equal to 4/5. $\frac{2}{5}\ \ \frac{4}{5}$	Concrete Understandings: Understand the concept of a fraction (a fraction is less than a whole) Understand the concept of comparison (greater than, less than, equal) Identify a concrete representation of a fractional part of a whole as greater than, less than, equal to another (divide a rectangle into fourths and compare ¼ to ¾) (e.g., Can do by showing with parts of the whole (so relates to fractions; e.g., break into 4 equal parts…which is more the 3 parts or the 1 part?)	Representation: Understand the symbols of <, >, and = Label pictorial representations of fractions
Grade 4
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Symbols 4.SE.1g2 Use =, <, or > to compare 2 fractions (fractions with a denominator of 10 or less). (4.NF.A.2)	Student selects symbol to compare two numbers. Choose the symbol that shows whether 1/3 is less than, equal to or greater than 2/6. $\frac{1}{3}\ \ \frac{2}{6}\ <\ >\ =$ Choose the symbol that shows whether 5/8 is less than equal to or greater than 2/4. $\frac{5}{8}\ \ \frac{2}{4}\ <\ >\ =$	Concrete Understandings: Understand the concept of a fraction (a fraction is less than a whole) Understand the concept of comparison (greater than, less than, equal) Identify concrete representation of a fractional part of a whole as greater than, less than, equal to another (divide a rectangle into fourths and compare ¼ to ¾) (e.g., Can do by showing with parts of the whole (so relates to fractions; e.g., break into 4 equal parts…which is more the 3 parts or the 1 part?)	Representation: Understand the symbols of <, >, and = Label pictorial representations of fractions
Numbers: 4.NO.1m1: Determine equivalent fractions. (3.NF.3, 4.NF.A.1)	Given a fraction, student selects equivalent fraction. This fraction is two-thirds. Show me the fraction that is equal to two-thirds. $\frac{2}{3}=\$____$\ \frac{1}{6}\ \ \frac{2}{6}\ \ \frac{4}{6}$	Concrete Understandings: Equivalency (what is and what is not equivalent; this may begin with numbers/sets of objects: e.g., 3=3) Concept of fraction and decimal (part to whole) Ability to identify two equivalent fractions, both represented either symbolically (i.e., in numbers) (e.g., ½ = 2/4) or as a model (i.e., picture) (e.g., pizza cut in halves and in fourths with ½ and 2/4 shaded in). If using symbolic representations, limit denominators to 2, 4, and 8 and numerators to 1-7. Use a real world context (e.g., pizza)	Representation: Understand and identify congruent units Understand how to determine equivalent fractions Represent fractions using shaded grids Understand concepts, symbols and vocabulary: fraction and equivalent fractions
Numbers: 4.NO.1n1: Select a model of a given fraction (halves, thirds, fourths, sixths, eighths). (3.NF.A.1)	Student selects picture to represent fraction. "Show me the rectangle that shows 5/6."	Concrete Understandings: Differentiate between parts and a whole Understand the structure of a fraction Identify half of a divided item	Representation: Create a model of a fraction (recognition of fractions compared to whole numbers) Match a model to a fraction Understand the components of a fraction (numerator and denominator) Understand concept, symbols and vocabulary: fraction, numerator and denominator
Grade 5
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Numbers: 5.NO.1b1: Read, write, and select a decimal to the hundredths place (5.NBT.A.3a)	"I want you to find a number that has a digit in the thousandths place. A digit is a single symbol, 0-9. Which of these numbers has a digit in the thousandths place? 45.72 4572 4.572	Concrete Understandings: Recognize part/whole using materials divided into tenths Count tenths to determine how many (e.g., 4 tenths; 0.4 have the decimal present but not required to read)	Representation: Count to 100 Understand place value to the hundredths Understand how to write a decimal point to the right of the units Understand concepts, symbols and vocabulary: decimal, decimal point, tenths place, hundredths place,
Numbers: 5.NO.1c1: Rewrite a fraction as a decimal. (4.NF.C.6)	Student rewrites a fraction as a decimal. Rewrite the fraction 35/100 as a decimal. $\frac{35}{100}=\$ ____	Concrete Understandings: Identify some subset of item divided into ten (show me 3 of these 10ths; have 3/10 and .3 present but not required to read)	Representation: Understand place value (tenths and hundreds) Understand decimal point Understand concepts, symbols and vocabulary: decimal point, tenths, and hundredths
Numbers: 5.NO.2c2: Solve word problems involving the addition, subtraction, multiplication, or division of fractions. (5.NF.A.2)	Given a word problem (teacher may read aloud), student shows work and solves the equation. Jesse and her mother baked 24 cookies. They ate ¼ of the cookies. How many cookies did they eat? Show your work as you solve the problem.	Concrete Understandings: Identify what to do with the parts when given the key word (using the fractional parts)	Representation: Identify key information in a word problem to represent the total and fraction Represent fraction problems using: Picture Models Representation cards Number sentences Mathematical word problems Graphic representation Understand concepts, symbols and vocabulary: $+,\ -,\ *,\ \div$
Grade 6
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Numbers: 6.NO.2c3: Solve one step addition, subtraction, multiplication, or division problems with fractions or decimals. (6.NS.A.1)	Present each problem one at a time and read it to the student. After reading it say "Solve this problem." Wait for the student to solve the problem, then remove the problem, present the next one, read it, and say "Solve this problem." Continue until the student has completed all four problems. $5.6 + 8.42 =$ _____________ $6.34 - 3.5 =$_____________ $10.34 * 0.4 =$ ______________ $12.34 \div 0.3 =$ ______________	Concrete Understandings: Concept of $+,\ -,\ *,\ \div$ Concept of fraction and decimal Use concrete object to represent the removal (subtraction) or addition of one half from/to a whole object.	Representation: Recognize picture representation of symbols Relates fractions and decimals to pictorial representations Understand symbols for $+,\ -,\ *,\ \div$ Understand concepts and vocabulary: for fraction and decimal (a/b, .a)
Grade 7: NONE
Grade 8
CCC	Performance Example	Essential Understandings: Concrete Understandings and Representations
Numbers: 8.NO.2i4: Solve two step addition, subtraction, multiplication, and division problems with fractions, decimals, or positive/negative numbers. (7.NS.A.1d)	Say "Now you are going to multiply and divide positive and negative numbers." Show the following problems (one at a time). For each problem presented say: "Solve this problem. Make sure to include the positive or negative sign in your answer." Then read the problem to the student. When the first problem is finished, then do the same for the second, third, and fourth problems. $(+42) * (-4) * (-11) =$ _________ $(-144) \div (+72) \div (+8) =$_________ $(-36) * (-2) \div (-6) =$ _________ $(-144) \div (+12) * (-3) =$_________	Concrete Understandings: Place an object on a number line to add and subtract.	Representation: Locate information within a word problem Read/make meaning of a word problem Understand concept, symbols and vocabulary: addition, subtraction, multiplication, division, positive/negative numbers

What are Some Additional Activities That Can Promote Use of this Academic Concept in Real World Contexts?

• 54 Measuring out fractions of ingredients for cooking.
• 62 Using a ruler to measure out fractions of a foot and inch to cut wood to make a birdhouse.
• 44 Students can learn to sort items using the Dewey decimal system in the library.¹⁰
• 24 Working as a cashier and counting money to make change.
• 43 Calculating batting averages for sports team.
• 24 Balancing a checkbook
• 64 Measuring out medications
• 13 Calculating a cost of an item in sales promotion (1/2 off sale)
• 64 Measuring someone's height in a doctor's office.
• 54 Cutting hair (customer wants 2 and ½ inches cut)

How Can I Further Promote College and Career Readiness when Teaching "Fractions and Decimals"?

Ideas for Promoting Career/ College Ready Outcomes

Communicative competence Students will increase their vocabulary to include concepts related to "fractions and decimals." In addition, they will be learning concepts such as: "half", "whole", "more", "less", "almost," "greater than," "less than," and "equal to." Fluency in reading, writing, and math Students will have an opportunity to increase their numeracy and sight word fluency while participating in problem solving related to "fractions and decimals," such as number recognition, counting, one-to-one correspondence, comparing quantities, and reading concepts that include the use and understanding of descriptors related to size. Age appropriate social skills Students will engage in peer groups to solve problems related to "fractions and decimals" that will provide practice on increasing reciprocal communication and age appropriate social interactions. For example, students might work together with their peers to measure out ingredients for a science experiment or cooking activity. Independent work behaviors By solving real life problems related to "fractions and decimals" students will improve work behaviors that could lead to employment, such as assisting in the library, measuring in construction, and preparing food in a kitchen. When providing opportunities for real life problems leave some materials out and prompt/teach the students to determine who they should ask and what they should ask for to be able to solve the problem. Skills in accessing support systems At times, students will need to ask for assistance to complete activities related to "fractions and decimals" which will give them practice in accessing supports. Students will gain practice asking for tools such as talking calculators, a digital tape measure, or measuring cups. They can ask a peer to complete the physical movements of the tasks they are not able to do themselves. Be sure to teach students to ask versus having items or supports automatically given to them.

How Do I Make Instruction on "Fractions and Decimals" Accessible to ALL the Students I Teach?

Teach Prerequisites and Basic Numeracy Skills Concurrently: Remember that students can continue to learn basic numeracy skills in the context of this grade level content.

Basic numeracy skills that can be worked on as a part of a lesson relating to fractions and decimals:

• number identification
• more and less
• one to one correspondence
• symbol identification $(+, -, =)$

Incorporate UDL: Universal Design of Learning When Teaching Fractions and Decimals

	Visual Impairment or Deaf/Blind	Physical impairment: Little/no hand use	Lacks basic numeracy concepts	Motivational/attention issues
Representation	Use a talking calculator when solving equations or converting fractions to decimals and vice versa; use objects to represent fractions and decimals (e.g., Cuisenaire rods®); use raised lines to represent portions of the whole object. Create fraction models using cardboard (rectangular and circular) with textured surfaces to indicate parts. Use items that are velcroed together to represent the whole and have the student separate the whole into parts. Another option would be to use snap cubes.	Count the parts of fractions or decimals using a step by step process which progresses through numbers; student scans an array of possible options and uses a switch to select the number to identify the numerator; use computer representation of fractions that can be manipulated with switch; place fraction representations on a slant board or eye gaze board; create a grid on a large surface on the floor that the student can walk over or ride over in wheelchair.	Use fraction and decimal manipulatives that can be separated and placed on a number line. Use real world objects that have been partitioned to represent fractions (e.g., graham cracker, candy bar). Students can use one to one correspondence to match equal number of parts on representation of fraction or decimals. Color code equations and corresponding parts of calculator to support students correctly entering numerals and equations.	Find fractions of motivating objects (e.g., pizza, coloring markers in a box, pieces of a Lego set). Incorporate technology including computer representations, videos, animations, and talking calculators. Use token economy system that embeds fractions (e.g., "You earned ¼ of your Lego piece, you have ¾ left and then you get Lego time.)
Expression	Student states answer or scans raised numbers to select correct answer; use voice output devices for student to select the correct answer; teach tangible symbols that mean fraction and decimal.	Student scans and selects number that represents numerator or denominator; uses a switch to indicate correct answers; use an eye gaze board to select answer; use a blink response to count parts or select answer; phrase questions so that they require a "yes/no" response, these can easily be answered using an eye gaze, head turn, two switches, etc.; count parts of fractions out loud having student move in some voluntary way (e.g., nod head, tap hand, tap foot) to count along.	Student selects numerals and fractions versus writing them; selection of correct answer is done after a model; student points to each part of a fraction or decimal while teacher or peer counts aloud; student answers "yes/no" questions regarding fractions or decimals after parts have been counted aloud (e.g., 1,2,3,4. There are 4 colored parts. Is this the number we write as the numerator?); matches the parts of a fraction to the correct number (matches 4 to 4).	Have students express fractions using high interest manipulatives (e.g., Legos, food items such as chocolate bar or graham crackers, stickers of favorite characters, or jewelry beads). Provide students with response options that keep them engaged (e.g., options provided on AAC device, tablet, response cards, or interactive whiteboard.)
Engagement	Teach students to use their hands to scan the raised or textured parts of each whole item; use textures or concrete objects to represent fractions; start with simple, clearly defined fractions; use items that are familiar and reinforcing to students.	Use bright colors to call attention to numerators; use a computer with AT where the student can click to answer; use figures that are large enough to accommodate the movements that the student is able to make; pair student with another student without a physical impairment and have them work together to create fraction and decimal representations.	Student uses talking calculator and AAC device or other response board, limit fractions and decimals to numerals less than 10, use bright colors to represent fractions and numerals, use familiar objects to represent fractions.	Use visuals and concrete representations and technology. Provide frequent opportunities for responding. Use token economy system that embeds fractions ("you earned ¼ of your Lego piece, you have ¾ left and then you get Lego time.").

CCR & Standards for Mathematical Practice Table

Promoting Career and College Readiness	Standards for Mathematical Practice
1Communicative Competence	1Make sense of problems and persevere in solving them.
2Fluency in reading, writing & math	2Reason abstractly and quantitatively.
3Age appropriate social skills	3Construct viable arguments and critique the reasoning of others.
4Independent work behaviors	4Model with mathematics
5Skills in accessing support systems	5Use appropriate tools strategically.
	6Attend to precision.
	7 Look for and make use of structure.
	8 Look for and express regularity in repeated reasoning