2012-13 and 2013-2014 Transitional Comprehensive Curriculum
Grade 5
Mathematics
Unit 4: Number Theory and Equivalent Fractions
Time Frame: Approximately four weeks
Unit Description
The focus of this unit is equivalent fractions, comparison of fractions, and the number theory properties that provide the basis for such equivalencies and comparisons. Activities presented in this unit will give students opportunities to represent fractions in various ways that will help them gain fluency with manipulating proper and improper fractions and mixed numbers. This unit also gives them activities that allow them to use their understanding and skills and apply them to real-world, problem-solving situations.
Student Understandings
Students develop an understanding of different representations of fractions such as parts of wholes, parts of collections, locations on number lines, as ratios, and as divisions of whole numbers. Students recognize and generate equivalent forms of commonly used fractions, mixed numbers, and decimals. Students use models, benchmarks, and equivalent forms to judge the size of fractions.
Guiding Questions
1. Can students identify fractions using region models, set models, and linear models?
2. Can students identify or develop equivalent fractions related to a given fraction?
3. Can students write ratios as fractions?
4. Can students convert between decimals and fractions or mixed numbers?
5. Can students compare fractions?
Unit 4 Grade Level Expectations (GLEs) and Common Core State Standards (CCSS)
Grade-Level ExpectationsGLE # / GLE Text and Benchmarks
Number and Number Relations
2. / Recognize, explain, and compute equivalent fractions for common fractions (N-1-M) (N-3-M)
4. / Compare positive fractions using number sense, symbols (i.e., <, =, >), and number lines (N-2-M)
6. / Select and discuss the correct operation for a given problem involving positive fractions using appropriate language such as sum, difference, numerator, and denominator (N-4-M) (N-5-M)
CCSS for Mathematical Content
CCSS# / CCSS Text
Operations and Algebraic Thinking
Number and Operations in Base Ten
5.NBT.3 / Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3´100+ 4´10+7´1+3´(1/10)+9´(1/100)+2´(1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Number and Operations- Fractions
5.NF.3 / Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
ELA CCSS
CCSS# / CCSS Text
Writing Standards
W.5.2a / Write informative/explanatory texts to examine a topic and convey ideas and information clearly.
a. Introduce a topic clearly, provide a general observation and focus, and group-related information logically; include formatting (e.g., headings), illustrations, and multimedia when useful to aiding comprehension.
Speaking and Listening Standards
SL.5.1c / 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.
c. Pose and respond to specific questions by making comments that contribute to the discussion and elaborate on the remarks of others.
SL.5.4 / Report on a topic or text or present an opinion, sequencing ideas logically and using appropriate facts and relevant, descriptive details to support main ideas or themes; speak clearly at an understandable pace.
Language Standards
L.5.6 / Acquire and use accurately grade-appropriate general academic and domain-specific words and phrases, including those that signal contrast, addition, and other logical relationships.
Sample Activities
Activity 1: All About One-Half (GLEs: 2, 4, 6)
Materials List: paper, pencils
Have students use brainstorming (view literacy strategy descriptions) to determine what they know about the fraction. Some examples are the following: = , 1 is the numerator, 2 is the denominator, < 1, > , + = , ; in simplest terms, = 0.50, of $10 is less than of $1000, is 1 ÷ 2, etc. This activity provides insight about students’ knowledge of fractions. Have each student draw a picture of . Using their drawings, discuss the different models for fractions such as area or regions, sets, linear models, or as a division problem.
Activity 2: What about Fractions? (GLEs: 2, 6; CCSS: L.5.6)
Materials List: What about Fractions? BLM, pencils
Before beginning the fraction activities, have students complete a vocabulary self-awareness (view literacy strategy descriptions) chart. Provide students with the What about Fractions? BLM. Do not give students definitions or examples at this point.
Word/Phrase / + / Ö / – / Example / Definitionnumerator
denominator
mixed number
improper fraction
equivalent
fraction
simplest form
Ask students to rate their understanding of each word with either a “+” (understands well), a “Ö” (some understanding), or a “–” (don’t know). During, and after completing fraction activities, students should return to the chart and fill in examples and definitions in their own words. Some words may have a “–”, a “Ö”, and a “+” by the end of the activities. The goal is to have plus signs for all words at the end of the activities.
Activity 3: Fractions of Shapes (GLEs: 2, 4, 6)
Materials List: Circle BLM, Square BLM, pencils, fraction tiles or circles (optional), The Internet (optional)
Using a display of the Circle BLM, demonstrate dividing the circle in half. Label each half with using colored overlays. To give the activity a more real-like feel, call the circle a pizza. Discuss the terms numerator and denominator. The denominator shows the number of pieces that the pizza has been cut into and the numerator shows the number of pieces that you get to eat. Draw a line perpendicular to the first line through the center of the circle and ask students what the pizza is divided into now (fourths). Continue drawing lines to show eighths and sixteenths. Shade parts of the circle and ask what fraction would describe it. An example would be , which could also be called or .
Give the students the Square BLM (maybe call it a pan of brownies) and have them divide it the same way (halves, fourths, eighths, and sixteenths). Have students make up various problems and use them in number sentences. Having colored fraction tiles and circles for the students will make this activity easier. Next, have students write comparison statements using their fractional pieces. For example, they could write. Informally introduce addition and subtraction, by saying . You want students to see that equals , not . Continue informally with addition and subtraction questions. Each time, write the equation on the board.
The website www.nctm.org has a lesson on the region model of fractions. Go to the lesson, http://illuminations.nctm.org/LessonDetail.aspx?ID=L343, Fun with Pattern Blocks Investigating Fractions with Pattern Blocks
Activity 4: Creating Equivalent Fractions (GLE: 2)
Materials List: Equivalent Fractions BLM, paper, pencils, The Internet (optional), fraction circles (optional)
Use student sets of fraction circles, if available. If not, provide each student with the Equivalent Fractions BLM. Have students cut out the three circles. Have students create a model of by having them fold one circle along a diameter to make two congruent parts. Using another circle, have students fold it into four congruent parts. Now, have students repeat folding with a third circle to create eighths. Once the folding is complete, have students cut the fractional parts and label them. Using the fractional parts, have students create number sentences by showing that, . Having students working in pairs, challenge them to create as many equivalencies as they can with their models. The activity can be repeated with thirds, sixths, and twelfths. Students can think of a clock to estimate and. They can draw lines from the center to 12:00, to 4:00, and to 8:00 to make thirds. Another way to fold the circles to get sixths is to fold the circle in half and then make two folds to make equal sections. Make sure that the edges meet to ensure equal sections. This divides the circle into six equal parts.
As an extension, students can visit http://www.learningplanet.com/sam/ff/index.asp to practice matching equivalent fractions in a game format. The game is called Fraction Frenzy.
Activity 5: Sets of Fractions (GLEs: 2, 6)
Materials List: paper, pencils, The Internet (optional)
Have students draw 3 small circles on their papers and shade 2 of the circles. Ask what fraction represents the number of shaded circles. () Ask what the numerator, 2, represents and what the denominator, 3, represents. Underneath the first set, have students draw a second set exactly like the first set. Ask questions, such as: How many circles in all? How many circles are shaded? What fraction represents the number of shaded circles? () Discuss the fact that these fractions are equivalent.
Show that 2 sets of means . Continue modeling with other fractions, emphasizing equivalent fractions. The website www.nctm.org has a lesson on the set model of fractions. Go to the lesson, Fun with Fractions: Another Look at Set Model Using Attribute Pieces, http://illuminations.nctm.org/LessonDetail.aspx?ID=L339.
Activity 6: Fractions as Division: Writing Ratios (GLEs: 2, 6; CCSS: 5.NF.3)
Materials List: paper, pencils, paper cups, 2-color counters
Have students count different items in the classroom, such as the number of boys, girls, teachers, desks, chairs, doors, windows, and clocks. On the board, make a table of their counts. Discuss that ratios are comparisons of two quantities, and that they can be used to compare a part to another part, a part to the whole, or the whole to a part. Ratios can be written in 3 forms. Examples are 15 to 14, 15 : 14, and . Have students choose 2 items to compare, such as boys and girls in the classroom, and have them write as many ratios as they can about the two quantities. Each ratio should be written in all 3 forms.
Provide pairs of students with 20 two-color (e.g., red/yellow) counters and a paper cup. Have the first student shake the counters in the cup and then empty the counters onto a desk. Ask the other student to determine the ratio of red counters to yellow counters. Once the ratio has been determined, the first student should give another equivalent ratio. For example, if the 20 counters are emptied from the cup and produce 14 red counters and 6 yellow counters, then the ratio is . An equivalent ratio would be 7 red counters to 3 yellow counters. This can be shown by creating 2 stacks of red counters with 7 in each stack versus 2 stacks of yellow counters with 3 in each stack. Once the ratio is in simplest terms, repeat the shake. When adequate time is given, have students write about a situation in their own lives in which they could use a ratio to describe the situation. Have them share the experience with the class.
Activity 7: Fractions as Division: Ratios in Patterns (GLEs: 2, 6; CCSS: 5.NF.3)
Materials List: pattern blocks, paper, pencils
Display a repeating pattern such as this:
core
The pattern should show the core and 3 repetitions. Create a table such as this:
Number of blocks / In Core / After 1st Repetition / After 2nd Repetition / After 3rd RepetitionNumber of Triangles / 1 / 2 / 3 / 4
Number of Squares / 4 / 8 / 12 / 16
Ratio of Triangles
to Squares
Ask students if they see any relationships between the number of triangles and squares. They should say something such as, for every triangle there are 4 squares. Ask questions such as these: If you continued the pattern completing each repetition until you had 5 triangles, how many squares would you have? (20) Write a ratio between the number of triangles and squares. If you continued the pattern until you had 28 squares, how many triangles would you have used? (7) Write this as a ratio of triangles to squares. Explain that all of these ratios show the same comparison, and are called equivalent ratios. Using pattern blocks, have students make their own patterns and a table for their patterns. Lead students to see that they can simply use the original ratio and multiply the numerator and denominator by the same number to get an equivalent ratio.
Activity 8: Fractions as Division: Ratios in Recipes (GLE: 2; CCSS: 5.NF.3)
Materials List: Sample Recipes BLM, paper, pencils
Display the sample recipe for lemonade from the Sample Recipes BLM. Have students write ratios for different ingredients, such as 6 cups of water to 2 cans of juice is a 6:2 ratio. Ask them what the equivalent ratio would be if they doubled the recipe or tripled it. (12:4, 18:6) Distribute the Sample Recipes BLM to students. Have students make a table showing equivalent ratios for different pairs of ingredients from any of the recipes. If there are two ingredients that are multiples of 2, ask students what would be the ratio if they divided the recipe in half. If other recipes are desired, write ratios from the ingredients that are whole numbers.