Fourth GradeAlicia Klaich and Deanna LeBlanc

Progression

4.NF.1 Equivalent Fractions

• 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.

Equivalent Fractions• Students subdivide the equal parts of a fraction,

resulting in a greater number of smaller parts. • Students discover that this subdividing has the

effect of multiplying the numerator and denominator by n.

1 x 4 parts and 4 x 4 parts.

Equivalent Fractions• Students learn that they can also divide the

numerator and denominator equally to generate equivalent fractions.

• Using a model, they accomplish this by equally combining smaller parts to create larger parts.

4.NF.2 Comparing Fractions

• 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 1/2. 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.

Comparing Fractions• Students create common numerators or common

denominators by renaming one fraction or both.

• When comparing fractions with the same numerator, they use prior knowledge about the relative sizes of fractional parts.

2 and 15 3

How would you rename these fractions to compare them?

Comparing Fractions

• Complete the following comparisons without using equivalent fractions. Make a note of how you did them…

What are some student misconceptions about comparing fractions?

1 and 88 9

7 and 58 6

4 and 39 4

4.NF.3 (a-b) Fractions as a Sum of Unit Fractions

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

b. 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.

Describe a common misconception for this problem:

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Fractions as a Sum of Unit Fractions

• All fractions can be seen as a sum or difference of two other fractions.

• Think in terms of adding/subtracting copies of 1/b.

• This line of thinking enables students to clearly understand why only the numerators are added or subtracted.

“Just like 3 dogs + 9 dogs is 12 dogs, or 3 candies + 9 candies is 12 candies, or 3 children + 9 children is 12 children, 3 fifths + 9 fifths is 12 fifths” (Small, 2014, p. 51).

Fractions as a Sum of Unit Fractions

• Students rename mixed numbers as improper fractions and vice versa.

• Students decompose fractions and mixed numbers in more than one way:

mrs-c-classroom.blogspot.com Before teaching the “shortcut,”

allow students plenty of time to reason with models.

4.NF.3 (c-d) Adding and Subtracting Mixed Numbersc. 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.

d. 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.

Adding and Subtracting Mixed Numbers

• Students rename mixed numbers as improper fractions, then add or subtract.

But…• They notice that sometimes it is easier to add or

subtract the whole number and fraction separately.

How would you solve each problem? Why?

Adding and Subtracting Mixed

Numbers• Students can “count up” to find the difference

between mixed numbers (see page 52 in Uncomplicating Fractions).

OR• They might prefer to regroup part of the whole

number in the greater mixed number.

Explain how to regroup the first mixed number in this problem.

4.NF.B.4 Multiplying Fractions by a Whole Number

• CCSS.Math.Content.4.NF.B.4a 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 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

• CCSS.Math.Content.4.NF.B.4b 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 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

• CCSS.Math.Content.4.NF.B.4c 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?

Interpreting the meaning of

multiplication• It is important to let students model and solve

these problems in their own way, using whatever models or drawings they choose as long as they can explain their reasoning.

• Once students have spent adequate time exploring multiplication of fractions, they will begin to notice patterns.

• Then, the standard multiplication algorithm will be simple to develop. Shift from contextual problems to straight computation.

Kristen ran on a path that was ¾ of a mile in length. She ran the path 5 times. What is the total distance that Kristen ran?

How can you solve the following problem? How many different ways can you solve it?

Interpreting the meaning of

multiplication• Adding 4/5 3 times ( 4/5 +4/5 +4/5)• 4 fifths + 4 fifths + 4 fifths = 12 fifths, or 12/5• The result of three jumps of 4/5 on a number line,

beginning at 0

• The number of fifths of a 2-D shape if 3 groups of 4 fifths are shaded.

Understand decimal notation for

fractions, and compare decimal

fractions• CCSS.Math.Content.4.NF.C.5 Express a fraction wi

th denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

• CCSS.Math.Content.4.NF.C.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.

• CCSS.Math.Content.4.NF.C.7 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 >, =, or <, and justify the conclusions, e.g., by using a visual model.

Interpreting decimals• Representing tenths and hundredths and

decimals as a sum. The 10-to-1 relationship continues indefinitely.

What is a common misconceptions students have about decimal place value?

Students must understand the equivalent relationship between tenthsand hundredths.

Representing Decimals on a Number

line

One of the best length models for decimalfractions is a meter stick. Experiencesallow students to compare decimals andthink about scale and place value.

Comparing Decimals• Reason abstractly and quantitatively. Develop

benchmarks; as with fractions: 0, ½, and 1. For example, is seventy-eight hundredths closer to 0 or ½, ½ or 1? How do you know?

• Using decimal circle models. Multiple wheels may be used to conceptualize the amount. Or, cut the tenths and hundredths and the decimal can be built.

• Why do many studentsthink .4 < .19?

Activity: The Unusual Baker

• George is a retired mathematics teacher who makes cakes. He likes to cut the cakes differently each day of the week. On the order board, George lists the fraction of the piece, and next to that, he has the cost of each piece. This week he is selling whole cakes for $1 each. Determine the fraction and decimal for each piece. How much will each piece cost if the whole cake is $1.00?

The Unusual Baker• CCSS 4.NF.A.1 Equivalent fractions• CCSS 4. NF.A.2 Compare two fractions• CCSS 4. NF.B.3a Understand addition and

subtraction of fractions• CCSS4.NF.B.3b Decompose a fraction into a sum

of fractions with the same denominator.• CCSS4. NF.C.6 Use decimal notation for fractions

with denominators 10 or 100.• CCSS4.NF.C.7 Compare two decimals to

hundredths by reasoning about their size.

Assessment/Resources• Howard County Wikispaces• https://grade4commoncoremath.wikispaces.hcpss.org/

• Learning Trajectories versus Proficency