Standard 4.NF.1 Cluster (M/S/A)

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Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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.

Essential Skills/Concepts Teaching Notes/Strategies Resources Understand the meaning of the Use manipulatives and drawings to Manipulatives: numerator and denominator of a given generate models of equivalent fractions. Linking Color Cubes fraction. 4th grade is responsible for Fractions (Circles, Tiles, and Bars) denominators of 2, 3, 4, 5, 6, 8, 10, 12, Use number lines to show equivalent and 100. fractions (see website for examples) Example of “Equivalent Fractions” models available at: Use fraction manipulatives, models, Play matching games of equivalent http://illuminations.nctm.org/ drawings, and number lines to show fractions. activitydetail.aspx?id=80 equivalency. Use a multiplication table to find multiplication table Know how to recognize equivalent equivalent fractions fractions and explain why they are Math journals equivalent, even when the numerators Math journal/partner talk as to why and denominators are different. fractions are equivalent. Anchor chart of property of equivalent fractions and how to generate them (see Understand the property of equivalent Use models to write equations using the left column ). fractions and how to generate them property of equivalent fractions (see (n x a)/(n x b) when n ≠ 0 left column ). Creating Equivalent Fractions OR Fraction Wall Game both available at a = nxa Solve contextual word problems that http://www.k- b nxb when n ≠ 0 use equivalent fractions (i.e., Maria is 5mathteachingresources.com making a cake, but she only has a ½ size Generate equivalent fractions using the measuring cup. The recipe calls for 4/8 multiplication equation(s) above. of a cup of sugar. How many ½ cups of sugar will Maria need to put in the cake?) Standard 4.NF.2 Cluster (m/s/a)

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole . Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Essential Skills/Concepts Teaching Notes/Strategies Resources How to find equivalent fractions (see PDIM on unit fractions Manipulatives: 4.NF.1) Use fraction circles, tiles, and/or bars, Fractions (Circles ,Tiles, and Bars) Understand unit fractions fraction strips, fraction number lines, or drawings to compare and order unit Fraction strips (on paper) fractions and fractions with different Compare two fractions with different numerators/denominators. Fraction number lines numerators and different denominators:  using a benchmark fraction PDIM to explain benchmark fractions  finding common denominators Use number lines with benchmark fractions to place other fractions on the number line. Math Journals Students can explain their reasoning and record results using <. =, and > Write equivalent fractions to produce a symbols. common denominator to compare fractions. Anchor chart on different methods for Have students explain reasoning in math comparing fractions (benchmark, finding journals/with partners. common denominators)

Do error analysis of comparisons and explain.

Birthday Fractions Solve contextual problems using comparison (i.e., Two different size pizzas. Each person Pattern Block Fractions at a different amount. Who ate more?) Who Ate More? Fraction Compare Fraction Cards Which is Larger? Snack Time

All available at http://www.k- 5mathteachingresources.com Standard 4.NF.3a Cluster (m/s/a)

Understand a fraction a/b with a>1 as a sum of fractions 1/ b . (decomposing a fraction).

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Essential Skills/Concepts Teaching Notes/Strategies Resources Understand what a unit fraction is. Using linking cubes to represent units Manipulatives: (1/b) and put cubes together to Understand that a whole is the same as represent adding units to make wholes. Linking Color Cubes 5/5, 8/8, 10/10, etc. Fractions (Circles and Tiles) Compare linking cubes to fraction bars Understand that unit fractions can be or circles. added to make a whole (i.e., ¼ + ¼ + ¼ + ¼ = 4/4 = 1) Use Fraction cards (see Resource Fraction Cards: Use index cards to make  column ) to teach unit fractions (i.e. fraction cards in the following way. Understand that 1/b + 1/b + 1/b = 3/b given a fraction 3/5, the unit is fifths. (i.e. ¼ + ¼ + ¼ = ¾) Write the equation 5/5 = 3/5 + ___ and 10 cards with 1/10 written on each have students model the addition Understand that unit fractions can be equations using fraction cards. Using 9 cards with 1/9 written on each subtracted from the whole the same cards, write different addition (i.e. 4/4 – ¼ = ¾) problems to show 5/5. Repeat activity Continue pattern through to model subtraction from the whole. 2 cards with ½ written on each

Using linking cubes to represent units of a whole (b/b) and remove cubes to represent subtracting units from wholes Math journals (i.e. 3/3 – 1/3 = 2/3). Add/Subtract Fractions with Like Students explain concepts in math Denominators journal/partner talk. Adding Fractions Using Pattern Blocks The Chocolate Bar Problem Sense or Nonsense (1) Sense or Nonsense (2)

All available at http://www.k- 5mathteachingresources.com Standard: 4.NF.3b Cluster (m/s/a)

Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model, Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8

Essential Skills/Concepts Teaching Notes/Strategies Resources Understand what a unit fraction is. Using linking cubes to compose and Manipulatives: decompose fractions. Each cube Understand that a whole is the same as represents a unit (1/b). Write the Linking Color Cubes 5/5, 8/8, 10/10, etc. equations the composing and Fractions (Circles and Tiles) decomposing represent ( i.e., ¼ + ¼ + ¼ + Understand that unit fractions can be ¼ = 4/4 and/or 3/3 – 1/3 = 2/3). added to make a whole (i.e., ¼ + ¼ + ¼ + ¼ = 4/4 = 1) Use Fraction cards (see Resource Fraction Cards: Use index cards to make  column ) to teach unit fractions (i.e. fraction cards in the following way. Understand that unit fractions can be given a fraction 3/5, the unit is fifths. subtracted from the whole Write the equation 5/5 = 3/5 + ___ and 10 cards with 1/10 written on each (i.e. 4/4 – ¼ = ¾) have students model the addition problem using fraction cards. Using the 9 cards with 1/9 written on each Understand how to decompose a same cards, write different addition fraction into the unit parts (see equations to show 5/5. Repeat activity Continue pattern through examples written in the standard itself). to model subtraction from the whole.) 2 cards with ½ written on each Extend to include improper fractions, such as 5/4.

Continue modeling and/or fraction cards Math journals by adding wholes (1) (i.e. 3 ¾ = 1 + 1 + 1 + ¼ + ¼ + ¼ and/or 3 ¾ - 1 = 2 3/4 )

Use Math journals to record different Decomposing fractions addition/subtraction drawings and related equations from models. Pizza Share

both available at http://www.k- 5mathteachingresources.com

Standard: 4.NF.3c Cluster (m/s/a)

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.

Essential Skills/Concepts Teaching Notes/Strategies Resources Add or subtract fractions with like Using fraction manipulatives, model how Manipulatives: denominators, including mixed numbers. to model adding/subtracting fractions with like denominators including mixed Linking Color Cubes Understand the relationship between numbers (i.e. ¼ + ¼ + ¼ = ¾; ¼ + ¼ + ¼ + ¼ Fractions (Circles and Tiles) adding and subtracting fractions by + ¼ = 5/4 = 1 ¾ ) using the properties of operations. Use fraction number lines to model adding/subtracting. Number lines should Fraction number lines not only show the fractions, but also the equivalencies (i.e. 4/4 = 1; 1 ¼ = 5/4, etc.) Math journals Write equations from modeling

Use the properties of operations to Mixed Number Word Problems (like show the relationship between addition denominators) and subtraction. Adding Mixed Numbers Subtracting Mixed Numbers Use linking cubes as a model for addition and subtraction and to show the All available at http://www.k- relationship between the two 5mathteachingresources.com operations.

Discuss with a partner/write in math journal.

Standard: 4.NF.3d Cluster (m/s/a)

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

Essential Skills/Concepts Teaching Notes/Strategies Resources Using prior learning of Sample problem: Mary ate 3/6 of a Manipulatives: adding/subtracting fractions, students pizza and Lacey at 2/6 of the same can solve word problems involving pizza. How much of the pizza did the Linking Color Cubes addition and subtraction of fractions girls eat altogether? Fractions (Circles and Tiles) using like denominators. Use the PUSD Universal Problem Solving Students can model word problems using Strategy to analyze word problems. visual models and equations. Fraction number lines Students use manipulatives, a fraction number line, or draw a diagram to show how to solve the problem. Problem Solving Strategy Anchor Chart Students write the equation to solve the problem. Math journals Students explain their solutions to a partner and write in their math journal.

Fraction Word Problems (like denominator) Addition Word Problems with Fractions Subtraction Word Problems with Fractions

All available at http://www.k- 5mathteachingresources.com

Standard: 4.NF.4a Cluster (m/s/a)

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1 /b . For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).

Essential Skills/Concepts Teaching Notes/Strategies Resources Prior Knowledge: Understand the Review basic concept of multiplication as Manipulatives: concept of multiplication (3 x 7 is 3 previously learned (see left ) Fractions (Circles and Tiles) groups of 7 objects; 3 x 7 is 7 + 7 + 7) See standards 4.NF.4a, 4b, and 4c for Fraction number lines Understand that a fraction has teaching notes/strategies to continue multiples (i.e., 1/5 has the multiples of the process of how to teach students to Math journals 1/5, 2/5, 3/5, 4/5, 5/5, 6/5, 7/5, etc.) multiply fractions by whole numbers. Models for fraction multiplication Understand that an improper fraction, Use manipulatives and drawings to available at http://www.k- such as 5/4 is 5 x (1/4) or the equation demonstrate multiples and repeated 5mathteachingresources.com 5/4 = 5 x (1/4). addition of fractions, relating it to Another way to see this using repeated students prior learning about addition is: multiplication (repeated addition, __ 5/4 = ¼ + ¼ + ¼ + ¼ + ¼ = 5 x ¼ groups of ___).

Relate drawings to equations.

Partner talk/math journals to explain repeated addition or __groups of ___ for a given problem.

Standard: 4.NF.4b Cluster (m/s/a)

Understand a multiple of a/b as a multiple of 1 /b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a)/b.)

Essential Skills/Concepts Teaching Notes/Strategies Resources Use prior learning from standard Use manipulatives and drawings to Manipulatives: 4.NF.4a to continue the process of demonstrate how to multiply fractions Fractions (Circles and Tiles) learning how to multiply fractions by by whole numbers whole numbers: Math journals ¼ x 4 = 1 or 6/5 = 6 x (1/5) Multiplying a number by a fraction Relate drawings to equations. available at http://www.k- 5mathteachingresources.com (n x (a/b) = (n x a)/b)

OR using real numbers:

3 x (4/5) = (3 x 4)/5

12 x 1/5 = 12/5

To make meaning of the product of a whole number and fraction, drawings can be used. For 3 x 2/5, these drawings could be made: Then repeated addition could be modeled using the drawing:

2/5 + 2/5 + 2/5 = 6/5 OR 3 x 2 = 6/5

Partner talk/math journals to explain drawings/equations/solutions. Standard: 4.NF.4c Cluster (m/s/a)

Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Essential Skills/Concepts Teaching Notes/Strategies Resources Students use models to demonstrate a Use PUSD Universal Problem Solving Manipulatives: word problem involving multiplication of Strategy to solve word problems. Fractions (Circles and Tiles) a fraction by a whole number. (see sample problem in standard) Model with manipulatives or drawings Number line with whole numbers how to solve a word problem involving Show answers on a whole number line multiplication of a fraction by a whole Math journals (between what two whole numbers does number. your answer lie?) Whole number x fraction word problems Write the equation for the problem. available at http://www.k- 5mathteachingresources.com Show the answer on a number line.

Partner talk/math journal to explain problem solving strategy.

Standard: 4.NF.5 Cluster (m/s/a)

Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

Essential Skills/Concepts Teaching Notes/Strategies Resources Compare decimal fractions with a Use base 10 blocks to demonstrate why Manipulatives: denominator of 10 or 100 3/10 = 30/100 and to compare. Base 10 blocks

Add decimal fractions by converting Anchor charts them to fractions with the same denominator. (Ex: express 3/10 as PDIM converting fractions with 10 as a Math journals 30/100 before they add denominator into fractions with 100 as a 30/100 + 4/100 = 34/100) denominator. 10th/100ths grid paper (see Math Survival Kit)

Sums of 1 Make charts for students to fill in Equivalent Fractions with a denominator conversions of 10ths to 100ths and vice of 100 problems versa. available at http://www.k- 5mathteachingresources.com Tenths Hundredths 2/10 50/100 9/10 70/100 90/10 900/100 Have students write in their journals why 9/10 = 90/100 and show a model to prove it. Standard: 4.NF.6 Cluster (m/s/a)

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.

Essential Skills/Concepts Teaching Notes/Strategies Resources Understand decimal notation for PDIM decimal notation and the place Manipulatives: fractions and can read/write decimals. values, reading and writing decimals. Number lines Base 10 blocks Understand that the number of digits to the right of the decimal point Anchor charts indicates the number of zeros in the Review using base 10 blocks to denominator. demonstrate why 0.3 = 0.30 Math journals

Students make the connections between 10th/100th grid Paper (see Math Survival fractions with denominators of 10 and Kit) Use a number line to represent values in 100 and place value. fraction and decimal form. Decimals in Money Add decimal fractions by converting Representing Decimals with Base 10 them to fractions with the same Blocks denominator. (Ex: express 3/10 as PDIM expressing fractions with Decimal Riddles 30/100 before they add Metric Relationships denominators of 10 and 100 as decimals 30/100 + 4/100 = 34/100) All available at http://www.k- 5mathteachingresources.com Convert fractions to decimals (with a denominator of 10 or 100) and then PDIM converting fractions with 10 as a compare. denominator into fractions with 100 as a denominator and their decimal equivalents

Make charts for students to fill in conversions of 10ths to 100ths and vice versa in fraction and decimal form

Tenths Dec Hundredths Dec 2/10 0.2 50/100 0.50 9/10 0.90 0.7 70/100 90/10 900/100 Have students write in their journals why 0.9 = 0.90 and show a model to prove it.

Standard: 4.NF.7 Cluster (m/s/a)

Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using the number line or another visual model. CA

Essential Skills/Concepts Teaching Notes/Strategies Resources Compare two decimals to hundredths by Order decimals and/or fractions on a Manipulatives: reasoning about their size. number line, first converting any Number line decimals/fractions so that all of them Base 10 blocks Compare decimals using the meaning of a have the same “whole”. decimal as a fraction (fractions with the (Ex: to compare 2/10, 40/100, and 0.3, Anchor charts same denominator and the “wholes” you would need to convert 2/10 to being the same). 20/100 and 0.3 to 30/100 Math journals Ex: Can only compare 3/10 to 4/100 if OR you change 3/10 to 3/100. convert 40/100 to 4/10 and 0.3 to 3/10) 10th/100th grid paper (see Math Survival Kit) Use base 10 blocks to convert fractions/decimals and/or to compare Comparing Decimals fractions/decimals. Decimal Sort Both available at http://www.k- Use contextual problems. 5mathteachingresources.com

Do error analysis.

Students talk with partners/write in math journals to show their reasoning when comparing decimals/fractions. This may include drawings.