2012-13 and 2013-14 Transitional Comprehensive Curriculum
Grade 4
Mathematics
Unit 6: Fractions and Decimals
Time Frame: Approximately five weeks
Unit Description
This unit develops an understanding of fractions with denominators through twelfths and decimals through hundredths. The unit focuses on fraction and decimal equivalents and adding, subtracting, and multiplying fractions.
Student Understandings
Students develop a strong understanding of fractions with denominators through twelfths. They use this understanding to compare and order fractions and to find equivalent fractions. Students are able to read, write and relate decimals through hundredths and connect them to fractions. Students are able to decompose fractions from mixed numbers and improper fractions into fractional units to add, subtract and multiply.
Guiding Questions
1. Can students model, read, write, compare, order and represent fractions with denominators through twelfths using region and set models?
2. Can students estimate fractional amounts from pictures, models, and diagrams?
3. Can students read, write and represent decimals through hundredths?
4. Can students connect decimals with decimal fractions and find equivalent decimals for , ,?
5. Can students generate equivalent fractions?
6. Can students add, subtract fractions and mixed numbers with like denominators and add fractions with denominators of 10 and 100?
7. Can students multiply a fraction by a whole number?
8. Can studnts solve word problems involving fractions?
Unit 6 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)
Grade-Level ExpectationsGLE # / GLE Text and Benchmarks
Number and Number Relations
5. / Read, write, and relate decimals through hundredths and connect them with corresponding decimal fractions (N-1-E)
6. / Model, read, write, compare, order, and represent fractions with denominators through twelfths using region and set models (N-1-E) (A-1-E)
7. / Give decimal equivalents of halves, fourths, and tenths (N-2-E) (N-1-E)
9. / Estimate fractional amounts through twelfths, using pictures, models, and diagrams (N-2-E)
CCSS for Mathematical Content
CCSS # / CCSS Text
Number and Operations – Fractions (NF)
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.
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: =++; =+; 2=1 + 1+=++.
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.
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.
4.NF.4 / 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 as the product 5´(), recording the conclusion by the equation = 5´().
b. 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´() as 6´(), recognizing this product as . (In general, n´()=.
c. 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?
4.NF.5 / 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 asand add +=
Measurement and Data
4.MD.4 / Make a line plot to display a data set of measurements in fractions of a unit (,,). Solve problems involving addition and subtraction of fractions by using information presented in line plots.
ELA CCSS
CCSS # / CCSS Text
Writing Standards
W.4.2d / Write informative/explanatory texts to examine a topic and convey ideas and information clearly. Use precise language and domain-specific vocabulary to inform about or explain the topic.
Speaking and Listening Standards
SL.4.1 / Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 5 topics and texts, building on others’ ideas and expressing their own clearly.
b. Follow agreed-upon rules for discussions and carry out assigned roles.
c. Pose and respond to specific questions by making comments that contribute to the discussion and elaborate on the remarks of others.
d. Review the key ideas expressed and draw conclusions in light of information and knowledge gained from the discussions.
Sample Activities
Activity 1: Fraction and Decimal Vocabulary Cards (CCSS: W.4.2d)
Materials List: paper, pencil, index cards, zip-top bag or envelope for the vocabulary cards.
Have students create fraction and decimal vocabulary cards, (view literacy strategy descriptions) for the following terms: numerator, denominator, unit fraction, equivalent fractions, decimals, tenths, and hundredths. Vocabulary knowledge is one of the most essential pieces of understanding mathematics. Vocabulary cards will help students learn the content-specific terminology necessary for higher order understanding. Each vocabulary card has four parts: the definition, the characteristics, an example, and an illustration. These vocabulary cards will be used throughout the unit to review the key terms for fractions and decimals and to serve as future reference cards to deepen the understanding of fractions and decimals. Have students store the vocabulary cards in a zip-loc bag or in an envelope.
Vocabulary card example:
Definition Characteristicsthe number of parts of the number above the line
the whole that are being in a fraction
considered
numerator
Illustration Example
numerator 1
2
numerator
Activity 2: Fractions as Regions and Sets (GLEs: 6)
Materials List: Fractions as Regions and Sets BLM
Review with students the definitions of numerator and denominator. Tell students that the denominator is the number of parts in a whole and that the numerator is the number of parts of the whole that are being considered. Ask students to draw a model of . Some students may draw a square divided into 4 equal parts, with 1 part shaded. Others might draw a circle or a rectangle divided into 4 equal parts with 1 part shaded. These students have drawn a region model of the fraction . Some students might draw 4 circles with 1 shaded in. Others might draw 4 stars or flowers or other objects, with 1 of the objects shaded in. These students have drawn a set model of the fraction . Both models are very important to the understanding of fractions.
Provide the students with the Fractions as Regions and Sets BLM. Ask students to tell what they notice about the models on the BLM. Explain to the students that the parts that are being considered in these pictures are the shaded parts of the whole. Ask students to identify the numerator and the denominator for each of the examples. Discuss the similarities and differences among the four examples. Guide the discussion to have students realize that all four examples show . The fourth example could also be interpreted as . Discuss the difference between the region model and the set model of fractions. A region model shows the whole divided into congruent parts. A set model uses a group of congruent objects as a whole. One way to illustrate the set model is to use students themselves. Call 4 students to come to the front of the class. Tell the other students to name a fraction that shows the number of students wearing tennis shoes to the whole group (possibly ). Ask other questions about fractional parts of the group of 4 students, such as:
· “What fraction of the group is wearing glasses?” (possibly )
· “What fraction of the group has blue eyes?” (possibly )
· “What fraction of the group is under 12 years old?” (possibly )
Provide students with multiple opportunities to identify the fractions in regions and sets as well as have students draw regions and sets for a given fraction.
Activity 3: Exploring Fractions and Decimals (GLEs: 5, 7; CCSS: 4.NF.1)
Materials List: base-10 blocks, Hundredths Grid Paper BLM, Place Value Chart BLM, pencils
Students will use their knowledge of place value to explore how fractions and decimals are related using base-10 blocks. Provide students with the Hundredths Grid Paper BLM and base-10 blocks. Explain to the students that one flat will represent 1 whole unit. Have them place 1 unit cube on the flat or on the grid paper. Ask the number of cubes it would take to fill the entire grid. (100) Ask them how much 1 cube would represent (1 square out of 100 squares or). Show them that can be written as the decimal, 0.01.
Place 10 cubes in a row on the flat or grid paper. Ask students to name this fraction (). Tell them that this can be written as the decimals 0.10. Ask them how many columns are covered (1) and how many columns are shown on the flat or grid paper (10). Have them replace the 10 cubes with 1 rod. Ask them to name the fraction covered now (1 column out of 10 columns or ). Tell students that fraction is written as the decimal 0.1. Tell students that = = 0.10 and 0.1. Introduce the word equivalent. Have students place 3 rods on top of the hundredths grid paper or shade in 3 columns on the paper. Display a place-value chart similar to the one below and distribute the Place Value Chart BLM to students. Have them state all of the fractions and decimals that name this amount (,, 0.30, 0.3). Call out the fraction and have students record it in the place value chart. Discuss how to write the decimal in word form and how to read the decimal. Have students record 0.3 and state the decimals in words. Tell students that decimals less than 1 whole are written with a zero in front of the decimal point to show that there are no whole numbers. Tell students that 0.30 should be read as three hundredths, not as zero point three zero.
Example:
Hundreds / Tens / Ones / . / Tenths / Hundredths0 / . / 3 / 0
0 / . / 3
0 / . / 0 / 5
0 / . / 4 / 3
Continue modeling by showing that 0.05 is 5 hundredths . Discuss how there is an 0 in the tenths place, so there must be an 0 between the decimal and the hundredths digit. Record this number in the place-value chart. Model 0.43 to show that it is the same as 4 tenths and 3 hundredths, , or and . Discuss how the number can be shown using expanded form (0.4 + 0.03). Record this number in the place-value chart. Call out additional numbers for students to record on the Place Value Chart BLM.
Have students model on the grid paper. Ask how many cubes would be needed to cover of the grid paper (50). Ask students to write an equivalent fraction and decimal for . (and 0.50) Ask how many rods would be needed to cover of the grid paper (5). Ask students to write another equivalent fraction and decimal for . (and 0.5) Continue showing the fractions, and , relating them to their fractional and decimal equivalents.
Provide students with multiple examples for practice, stopping to discuss how each fraction is related to its decimal equivalent as needed. Observe how the students read, write, and model each fraction and decimal.
Extend the activity by incorporating whole numbers with the decimals. Model how to read, write, and model each of the numbers. Gradually have students work only with pencil and paper and take away the base-ten blocks so they can rely on their knowledge of place value to explain how they found each digit’s value.
Activity 4: Fractions and Decimals on Grids (GLEs: 5, 7; CCSS: 4.NF.1)
Materials List: Fractions and Decimals on Grids BLM (2 pages), crayons, math learning log, pencil
Use the Fractions and Decimals on Grids BLM to demonstrate fractions as parts of a whole. Help students understand the connection between fractions and decimals by relating them through money. Discuss the denominators that will be used to show varying amounts of money (the denominator for pennies in a dollar is 100, for dimes in a dollar is 10, for quarters in a dollar is 4, and for a 50-cent piece is 2.) Since students do not have to know denominators of 20, do not include nickels in this activity. Demonstrate for students how to color in 50¢ on the first grid of the Fraction and Grids BLM. Discuss the equivalent decimals and fractions for 50¢ (, 0.50, , 0.5, ). Have the students complete the other grids on their own, stopping for class discussions when needed. (For example, 1 quarter is 0.25 or , or of a dollar. Show that is 25 cents out of 1 dollar and since there are 4 quarters in 1 dollar, 25 cents can also be written as of a dollar.) Use other examples such as 6 dimes is .60 or , or of a dollar and 3 quarters is .75 or , or of a dollar, etc.