Number, Operation, and Quantitative Reasoning
Underlying Processes and Mathematical Tools
25 days: 45 minutes per day
5.2 Use fractions in problem-solving situations.
5.3 Add, subtract, multiply, and divide to solve meaningful problems.
5.6 Describe relationships mathematically.
5.14 Apply Grade 5 mathematics to solve problems connected to everyday experiences and activities in and outside of school.
5.15 Communicate about Grade 5 mathematics using informal language.
5.16 Use logical reasoning.
TEKS / TAKS Obj. / Instructional Scope / Possible Resources GT=Game Time; CC=Cross Curricular /
Region 4 Instruction / Region 4 Assessment / KISD Suggested Resources /
5.2A
Generate a fraction equivalent to a given fraction such as 1/2 or 3/6 or 4/12 and 1/3.
5.16A
Make generalizations from patterns or sets of examples and nonexamples
5.16B
Justify why an answer is reasonable and explain the solution process.
5.5A
Describe the relationship between sets of data in graphic organizers such as lists, tables, charts, and diagrams. / 1
6
6
2 / Equivalent Fractions· Generate equivalent fractions.
Use a concrete or pictorial model to generate an equivalent fraction.
Example:
Prompt the students to name a fraction that is equivalent to .
Prompt the students to use concrete models such as fraction circles to find a fraction equivalent to .
Possible Answer: =
Example:
Ask the students, “What is one fraction that is equivalent to ?”
Prompt the students to use concrete models such as fraction strips to represent a fraction equivalent to .
Answer: and are equivalent fractions.
· Use the identity property for multiplication and division to generate equivalent fractions.
Identity Property for Multiplication – The identity property states that any number times 1 equals that number.
Example:
Note: The number 1 can be expressed as any fraction that has the same numerator and denominator. For example:
Identity Property for Division – The identity property for division states that any number divided by 1 equals that number.
Example:
Prompt the students to determine that an equivalent fraction can be generated by multiplying or dividing the fraction by any fractional representation of 1.
· Make generalizations from sets of examples and nonexamples.
Example:
Prompt the students to look at the two sets of fractions below.
Ask the students, “Into which set would you place the fraction ? Explain your thinking.”
Possible Answer: “The fraction would go in Set B because all of the fractions in Set B are equivalent to .
I can prove this answer by building models of the fractions or by using the identity property for multiplication and division to determine whether or not the fractions are equivalent.”
Example:
The fractions , , and are each equivalent to . What conclusion can be made about the relationship between the numerator and denominator in each fraction that is equivalent to ?
Possible Strategy: Create a table
Numerator / Denominator
2 / 6
3 / 9
5 / 15
Possible Answer: The denominator is
3 times the numerator. / TAKS Mathematics Preparation Grade 5, “Fractions Lesson.”; pg 34
Rethinking Elementary Mathematics Grades 3-5, “Fraction Frame Game.”
TAKS Mathematics Preparation Grade 5: Patterns, Relationships, Algebraic Thinking Lesson, pg 123 / TAKS Mathematics Preparation Grade 5, “Objective 1: Number, Operation, and Quantitative Reasoning Selected Response Questions.” www.mathbenchmarks.org / TAKS Informational Booklet: http://www.tea.state.tx.us/student.assessment/taks/booklets/math/g5e.pdf
TEKS Clarifying Activities:
http://www.utdanacenter.org/mathtoolkit/instruction/activities/5.php
Macmillan Texas Mathematics:
9-3; GT9; 9-4; 9-5
Name That Portion, Investigation 1: Sessions 1-7
http://illuminations.nctm.org/
http://www.lab.brown.edu/investigations/
http://standards.NCTM.org/document/eexamples/chap5/5.1/indes.htm
Macmillan Texas Mathematics:
9-8
Groundworks, Algebraic Thinking: Function Factory pg 80-86
5.2C
Compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators.
5.14D
Use tools such as real objects, manipulatives, and technology to solve problems.
5.3E
Model situations using addition and/or subtraction involving fractions with like denominators using concrete objects, pictures, words, and numbers. / 1
6
1 / Comparing Two Fractional Quantities
· Compare two fractional quantities in problem-solving situations.
Prompt the students to use an instructional strategy such as using concrete models to compare two fractions.
Example:
Candy has 2 lasagna recipes. One recipe calls for cup of parmesan cheese, and another recipe calls for
cup of parmesan cheese. Which recipe calls for the greater amount of parmesan cheese?
Prompt the students to use concrete models such as fraction bars to represent each fraction and then compare the fractions.
Answer: < ; > , so the recipe requiring cup cheese requires the greater amount of cheese.
Prompt the students to use an instructional strategy such as finding common denominators in order to compare two fractional quantities.
Example: Which fraction is greater:
or ?
Prompt the students use concrete models to rename each fraction with a common denominator.
Answer: Since < , then < .
Since > , then > .
Addition and Subtraction of Fractions
· Use concrete and pictorial models to develop the concept of addition and subtraction of fractions with like denominators. / TAKS Mathematics Preparation Grade 5, “Fractions Lesson.”
TAKS Mathematics Preparation Grade 5, “Addition and Subtraction of Fractions.” Pg 52-58
/ TAKS Mathematics Preparation Grade 5, “Fractions Lesson - Evaluate.”
TAKS Mathematics Preparation Grade 5, “Objective 1: Number, Operation, and Quantitative Reasoning Selected Response Questions.” www.mathbenchmarks.org
TAKS Mathematics Preparation Grade 5, “Addition and Subtraction of Fractions - Evaluate.”
TAKS Mathematics Preparation Grade 5, “Objective 1: Number, Operation, and Quantitative Reasoning Selected Response Questions.” www.mathbenchmarks.org / Macmillan Texas Mathematics:
8-5; CC8; Explore 9-9; 9-9
Name That Portion, Investigation 1: Sessions 1-7
http://www.lab.brown.edu/investigations/
http://standards.NCTM.org/document/eexamples/chap5/5.1/indes.htm
Macmillan Texas Mathematics:
Explore 9-9; 9-9
Macmillan Texas Mathematics:
Explore 10-1; 10-1; Explore 10-2; 10-2; 10-4; 10-5; 10-6; GT10; 10-8; CC10
Name That Portion, Investigation 2: Sessions 1-9
http://www.lab.brown.edu/investigations/
http://standards.NCTM.org/document/eexamples/chap5/5.1/indes.htm
5.6A
Select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations.
5.14D
Use tools such as real objects, manipulatives, and technology to solve problems.
5.6A
Select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations.
5.6A
Select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations.
5.14B
Solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness.
5.6A
Select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations.
5.6A
Select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations. / 2
6
2
2
6
2
2 / Example:
Rudy’s kitten is ill. The doctor recommends that Rudy give his kitty tube of antibiotics on the first day and tube on the second day. What fraction of the tube of medicine is Rudy giving his kitten?
Prompt the students to determine that two quantities of medicine are being combined (addition). If m represents the fraction of the tube of medicine that Rudy gave his kitty, then the problem situation could be represented with the expression:
.
Prompt the students to use concrete models such as fraction circles to represent the problem situation.
Answer: or tube
Example:
Mindy is making one batch of peanut butter bars and one batch of peanut butter cookies for her class party. The batch of peanut butter bars requires 1 cups of peanut butter, and the batch of peanut butter cookies requires cup of peanut butter. How many cups of peanut butter will Mindy need for both recipes?
Prompt the students to determine that we are looking for the total of two quantities (addition). If p represents the total amount of peanut butter needed, then the problem situation could be represented with the expression:
Prompt the students to use concrete materials such as area models to represent the problem situation.
Answer: or cups
Example:
Martie needs meter of ribbon to decorate a project. If she already has meter of ribbon, how much more ribbon will Martie need?
Prompt the students to determine that the problem can be solved by subtracting the amount of ribbon that Martie already has from the total amount of ribbon needed. If r represents the amount of ribbon that Martie needs to buy, then the problem situation can be represented by the expression:
.
Prompt the students to use concrete models such as fraction circles to represent the problem situation.
Answer: or meter
Example:
Courtney uses pints of beef broth to marinate a roast. If she has already poured pint of beef broth onto the roast, how many more pints of beef broth will Courtney need to marinate the roast?
Understanding the Problem:
· Ask the students to restate the problem using their own words.
· Ask the students, “What are we being asked to find out?”
Possible Answer: “We are trying to figure out how many more pints of beef broth Courtney will need to marinate the roast.”
Making a Plan:
· Ask the students, “Are we combining or separating quantities?”
· Ask the students, “Which operation(s) will we use to find the solution?”
Possible Answer: Since we know the whole amount of beef broth needed and we know that one part has already been added, we need to subtract to determine the part still needed. If b represents the amount of beef broth needed, the problem situation can be represented by the expression:
.
Prompt the students to use concrete materials such as fraction bars to represent the problem situation.
Carrying out the Plan:
Ask the students, “How are you going to solve the problem?”
Prompt the students to remove from .
Ask the students, “Why is this not possible with this model?”
Possible Answer: “There is only one piece, and we need two pieces.”
Ask the students, “What might we do with the whole to create more pieces?”
Possible Answer: “Rename the 1 whole to .”
Prompt the students to remove from .
Answer: pint
Evaluating for Reasonableness:
· Ask the students, “How do you know that your answer makes sense?”
· Add and subtract fractions in problem solving situations.
Example:
Donald is mixing a solution in the science lab. He adds cups of solution A and cups of solution B together in a container. How many cups of solution are in the container?
Ask the students, “What expression could be used to represent the situation in the problem?”
Possible answer: “If s represents the number of cups of solution in the container, then
.”
Prompt the students to use concrete materials such as fraction bars to represent the problem situation.
Ask the students, “How many cups do you have? How do you know?”
Possible Answer: “We have cups.”
Ask the students, “What is of a cup equivalent to?”
Possible Answer: “One whole”
Prompt the students to rename as 1 whole and add 1whole to 3 wholes.
Answer: 4 cups
Example:
Mickie has to drink cups of water prior to going to the doctor. If she has already drunk cups, how much more water does Mickie have left to drink?
Prompt the students to use concrete materials such as fraction bars to represent the problem situation.
Prompt the students to remove from .
Ask the students, “Why is this not possible with this model?”
Possible Answer: “There is only one piece, and we need three pieces.”
Ask the students, “What might we do with one of the wholes to create more pieces?”
Possible Answer: “Rename one whole as and add the to .”
Prompt the students to remove from .
Answer:
Answer: or cups / Macmillan Texas Mathematics:
10-3; 10-7
5.2B
Generate a mixed number equivalent to a given improper fraction or generate an improper fraction equivalent to a given mixed number. / 1 / Mixed Numbers and Improper Fractions
· Use pictorial models to change improper fractions to mixed numbers and mixed numbers to improper fractions.
Example:
What mixed number is equivalent to ?
Prompt the students to use concrete models to solve the problem.
Answer:
Example:
What improper fraction is equivalent to ?
Answer: / Macmillan Texas Mathematics:
8-1; Explore 8-2; 8-2; GT 8; 8-4
Name That Portion, Investigation 1: Sessions 1-7
Number, Operation, and Quantitative Reasoning
Underlying Processes and Mathematical Tools
*Curriculum-Based Assessment 3
*For each student expectation (SE) that incorporates the use of a concrete model/object and/or a mathematical tool, students should be encouraged to use a concrete model/object and/or a mathematical tool to model items on the curriculum-based assessment (CBA).
Name That Portion, Investigation 1: Sessions 1-7
NOTE: RELATING FRACTIONS AND DECIMALS ARE NEW SKILLS FOR FIFTH GRADE. ADEQUATE TIME MUST BE SPENT IN ORDER TO BUILD THE CONCEPTS SO THAT STUDENTS CAN APPLY THEIR KNOWLEDGE IN A PROBLEM SOLVING CONTEXT. STUDENTS MUST BE ABLE TO PUT FRACTIONS IN THEIR SIMPLEST FORMS.
Although percents are not a 5th grade TEK, this lesson is important because it uses decimal grids to teach students the relationship between fractions and percents. It also helps students understand that decimals, percents, and fractions are all fractional parts of a whole. Great real-life connection.
ACTIVITY 1. Name That Portion, Investigation 1: Session 1 "Connecting Fractions, Decimals, and Percents"
Students discuss common uses of fractions, decimals, and percents and their meanings. Then, in pairs, they solve a set of word problems and discover how much they already know about fractions, decimals and percents. Their work focuses on:
* interpreting common uses of fractions, decimals, and percents
* expressing the same quantity as a fraction, decimal, and percent
Materials:
Chart paper
Student sheets 1&2
Calculator
NOTE: Common usage of fractions, decimals and percents. The teacher note at the end of this lesson stresses the importance of being comfortable with all three. This knowledge leads to a richer understanding of each and better enables students to choose which one is appropriate for computing or communicating a particular idea. The entire teacher note has wonderful information. A good reinforcer would be to put student grades on their papers both as fractions and percents. (Example: 5/6 and 83%.)
ACTIVITY 2. Name That Portion, Investigation 1: Session 2 "Percent Grid Patterns"
As students explore further the connection between fractions and percents, they use fractions to describe a portion of a group. They then represent 1/2 of a group as 50 percent and discuss the meaning of percent. They color 10-by-10 grids in different patterns to represent percents and use these colored grids to find equivalent fractions. Their work focuses on:
* using fractions to describe how many in a group share a particular
characteristic
* partitioning a whole
* making patterns and identifying percents on 10-by-10 grids
* identifying equivalent fractions and percents
Materials:
Equivalents chart
Grids transparency
Student sheets 3-4
Grids
Colored pencils, crayons, or markers,
Overhead projector and pens
Calculator
NOTE: Although percents are not a 5th grade TEK, this lesson is important because it uses decimal grids to teach students the relationship between fractions and percents. It also helps students understand that decimals, percents, and fractions are all fractional parts of a whole. Great real-life connection.
ACTIVITY 3. Name That Portion, Investigation 1: Sessions 3-4 "Fraction and Percent Grids"
Student color more 10-by-10 grids, this time to represent common fractions (thirds, fourths, fifths, sixths, and eighths); they then find the equivalent percents. Their work focuses on:
* representing common fractions on 10-by-10 grids
* identifying equivalent fractions and percents
Materials:
Grids
Grids transparency
Students sheets 5-7
Colored pencils, crayons, or markers
Overhead projector and pens
Calculator
NOTE: Students will extend their knowledge of equivalent fractions and relate that knowledge to decimals and percents.
Vertical alignment note: A similar activity was done in fourth grade using fractions on a geoboard. The lesson was called "Crazy Cakes."
ACTIVITY 4. Name That Portion, Investigation 1: Session 5-6 "Percent Equivalent Strips"
Students mark a paper strip of percents to show equivalent fractions. They use their strip as a reference for the In-Between Game, in which they develop their sense of the relative size of fractions as they lay out fraction cards in order. Their work focuses on:
* finding equivalent fractions and percents
* ordering fractions and percents around landmark numbers