Fifth Grade Application Problems - Module 4
Multiplication and Division of Fractions and Decimal Fractions
Topic A: Line Plots of Fraction Measurements
Lesson 1: Measure and compare pencil lengths to the nearest 1/2, 1/4, and 1/8 of an inch and analyze the data through line plots.
The following line plot shows the growth of 10 bean plants on their second week after sprouting.
(See page 4.A.5 in the fluency activity packet.)
a. What was the measurement of the shortest plant?
b. How many plants measure inches?
c. What is the measure of the tallest plant?
d. What is the difference between the longest and shortest measurement?
Note: This Application Problem provides an opportunity for a quick, formative assessment of student ability to read a customary ruler and a simple line plot. As today’s lesson is time-intensive, the analysis of this plot data is necessarily simple.
Topic B: Fractions as Division
Lesson 2: Interpret a fraction as division.
(You will need page 4.B.3 in the fluency activity packet for this activity.)
The line plot shows the number of miles run by Noland in his PE class last month, rounded to the nearest quarter mile.
a. If Noland ran once a day, how many days did he run?
b. How many miles did Noland run altogether last month?
c. Look at the circled data point. The actual distance Noland ran that day was at least ____mile and less than ____mile.
Note: This Application Problem reinforces the work of yesterday’s lesson. Part (c) provides an extension for early finishers.
Lesson 3: Interpret a fraction as division.
Hudson is choosing a seat in art class. He scans the room and sees a 4-person table with 1 bucket of art supplies, a 6‐person table with 2 buckets of supplies, and a 5‐person table with 2 buckets of supplies. Which table should Hudson choose if he wants the largest share of art supplies? Support your answer with pictures.
Note: Students must first use division to see which fractional portion of art supplies is available at each table. Then students compare the fractions and find which represents the largest value.
Lesson 4: Use tape diagrams to model fractions as division.
Four grade-levels need equal time for indoor recess, and the gym is available for three hours.
a. How many hours of recess will each grade level receive? Draw a picture to support your answer.
b. How many minutes?
c. If the gym could accommodate two grade-levels at once, how many hours of recess would each grade-level get?
Note: Students practice division with fractional quotients, which leads into the day’s lesson. Note that the whole remains constant in (c) while the divisor is cut in half. Lead students to analyze the effect of this halving on the quotient as related to the doubling of the whole from previous problems.
Lesson 5: Solve word problems involving the division of whole numbers with answers in the form of fractions or whole numbers.
NO APPLICATION PROBLEM
Topic C: Multiplication of a Whole Number by a Fraction
Lesson 6: Relate fractions as division to fraction of a set.
Olivia is half the age of her brother, Adam. Olivia’s sister, Ava, is twice as old as Adam. Adam is 4 years old. How old is each sibling? Use tape diagrams to show your thinking.
Note: This Application Problem is intended to activate students’ prior knowledge of half of in a simple context as a precursor to today’s more formalized introduction to fraction of a set.
Lesson 7: Multiply any whole number by a fraction using tape diagrams.
Mr. Peterson bought a case (24 boxes) of fruit juice. One-third of the drinks were grape and two-thirds were cranberry. How many boxes of each flavor did Mr. Peterson buy? Show your work using a tape diagram or an array.
Note: This Application Problem requires students to use skills explored in G5–M4–Lesson 6. Students are finding fractions of a set and showing their thinking with models.
Lesson 8: Relate fraction of a set to the repeated addition interpretation of fraction multiplication.
Sasha organizes the art gallery in her town’s community center. This month she has 24 new pieces to add to the gallery.
Of the new pieces, 1/6 of them are photographs and 2/3 of them are paintings. How many more paintings are there than photos?
Note: This Application Problem requires students to find two fractions of the same set—a recall of the concepts from G5–M4–Lessons 6–7 in preparation for today’s lesson.
Lesson 9: Find a fraction of a measurement, and solve word problems.
There are 42 people at a museum. Two-thirds of them are children. How many children are at the museum?
Extension: If 13 of the children are girls, how many more boys than girls are at the museum?
Note: Today’s Application Problem is a multi-step problem. Students must find a fraction of a set and then use that information to answer the question. The numbers are large enough to encourage simplifying strategies as taught in G5– M4–Lesson 8 without being overly burdensome for students who prefer to multiply and then simplify or still prefer to draw their solution using a tape diagram.
Topic D: Fraction Expressions and Word Problems
Lesson 10: Compare and evaluate expressions with parentheses.
Bridget has $240. She spent 3/5 of her money and saved the rest. How much more money did she spend than save?
Note: This Application Problem provides a quick review of fraction of a set, which students have been working on in Topic C, and provides a bridge to the return to this work in G5–M4–Lesson 11. multi-step problem.
Lesson 11: Solve and create fraction word problems involving addition, subtraction, and multiplication.
NO APPLICATION PROBLEM GIVEN
Lesson 12: Solve and create fraction word problems involving addition, subtraction, and multiplication.
Application Problem: Please see page 4.D.33 in the fluency activity packet enclosed for the problem.
Topic E: Multiplication of a Fraction by a Fraction
Lesson 13: Multiply unit fractions by unit fractions.
NO APPLICATION PROBLEM GIVEN
Lesson 14: Multiply unit fractions by non-unit fractions.
Solve by drawing an area model and writing a multiplication sentence.
Beth had 1/4 box of candy. She ate 1/2 of the candy. What fraction of the whole box does she have left?
Extension: If Beth decides to refill the box, what fraction of the box would need to be refilled?
Note: This Application Problem activates prior knowledge of the multiplication of unit fractions by unit fractions in preparation for today’s lesson.
Lesson 15: Multiply non-unit fractions by non-unit fractions.
Kendra spent 1/3 of her allowance on a book and 2/5 on a snack. If she had four dollars remaining after purchasing a book and snack, what was the total amount of her allowance?
Note: This problem reaches back to addition and subtraction of fractions as well as fraction of a set. Keeping these skills fresh is an important goal of Application Problems.
Lesson 16: Solve word problems using tape diagrams and fraction-by- fraction multiplication.
NO APPLICATION PROBLEM GIVEN
Lesson 17: Relate decimal and fraction multiplication.
Ms. Casey grades 4 tests during her lunch. She grades 1/3 of the remainder after school. If she still has 16 tests to grade after school, how many tests are there?
Note: Today’s Application Problem recalls the previous lesson’s work with tape diagrams. This is a challenging problem in that the value of a part is given and then the value of 2 thirds of the remainder. Possibly remind students to draw without concern initially for proportionality. They have erasers for a reason and can rework the model if they so choose.
Lesson 18: Relate decimal and fraction multiplication.
An adult female gorilla is 1.4 meters tall when standing upright. Her daughter is 3 tenths as tall. How much more will the young female gorilla need to grow before she is as tall as her mother?
Note: This Application Problem reinforces that multiplying a decimal number by tenths can be interpreted in fraction or decimal form (as practiced in G5─M4─Lesson 17). Students who solve this problem by converting to smaller units (centimeters or millimeters) should be encouraged to compare their process to solving the problem using 1.4 meters.
Lesson 19: Convert measures involving whole numbers, and solve multi- step word problems.
Angle A of a triangle is the size of angle C. Angle B is the size of angle C. If angle C measures 80 degrees, what are the measures of angle A and angle B?
Note: Because today’s fluency activity asks students to recall the content of yesterday’s lesson, this problem asks students to recall previous learning to find fraction of a set. The presence of a third angle increases complexity.
Lesson 20: Convert mixed unit measurements, and solve multi-step word problems.
A recipe calls for 3/4 lb of cream cheese. A small tub of cream cheese at the grocery store weighs 12 oz. Is this enough cream cheese for the recipe?
Note: This problem builds on previous lessons involving unit conversions and multiplication of a fraction and a whole number. In addition to the method shown, students may also simply realize
that 3/4 is equal to 12/16 .
Topic F: Multiplication with Fractions and Decimals as Scaling and Word Problems
Lesson 21: Explain the size of the product, and relate fraction and decimal equivalence to multiplying a fraction by 1.
Carol had 3/4 yard of ribbon. She wanted to use it to decorate two
picture frames. If she uses half the ribbon on each frame, how many feet of ribbon will she use for one frame? Use a tape diagram to show your thinking.
Note: This Application Problem draws on fraction multiplication concepts taught in earlier lessons in this module.
Lesson 22: Compare the size of the product to the size of the factors.
In order to test her math skills, Isabella’s father told her he would give her 6/8 of a
dollar if she could tell him how much money that is and what that amount is in decimal form. What should Isabella tell her father? Show your calculations.
Note: This Application Problem reviews G5–M4–Lesson 21’s Concept Development. Among other strategies, students might convert the eighths to fourths, and then multiply by 25/25, or they may remember the decimal equivalent of 1 eighth and multiply by 6.
Lesson 23: Compare the size of the product to the size of the factors.
Jasmine took 2/3 as much time to take a math test as Paula. If Paula took 2 hours to take the test, how long did it take Jasmine to take the test? Express your answer in minutes.
Note: Scaling as well as conversion is required for today’s Application Problem. This both reviews G5–M4– Topic E and prepares students to continue a study of scaling with decimals in today’s lesson.
Lesson 24: Solve word problems using fraction and decimal multiplication.
NO APPLICATION PROBLEM GIVEN.
Topic G: Division of Fractions and Decimal Fractions
Lesson 25: Divide a whole number by a unit fraction.
The label on a 0.118-liter bottle of cough syrup recommends a dose of 10 milliliters for children aged 6 to 10 years. How many 10-mL doses are in the bottle?
Note: This problem requires students to access their knowledge of converting among different size measurement units—a look back to Modules 1 and 2. Students may disagree on whether the final answer should be a whole number or a decimal. There are only 11 complete 10-mL doses in the bottle, but many students will divide 118 by 10, and give 11.8 doses as their final answer. This invites interpretation of the remainder since both answers are correct.
Lesson 26: Divide a unit fraction by a whole number.
A race begins with 2 1/2 miles through town, continues through the park
for 2 1/3 miles, and finishes at the track after the last 1/6 mile. A volunteer
is stationed every quarter mile and at the finish line to pass out cups of water and cheer on the runners. How many volunteers are needed?
Note: This multi-step problem requires students to first add three fractions, then divide the sum by a fraction, which reinforces yesterday’s division of a whole number by a unit fraction. (How many 1/4 miles are in 5 miles?) It also reviews adding fractions with different denominators (G5– Module 3).
Lesson 27: Solve problems involving fraction division.
NO APPLICATION PROBLEM GIVEN.
Lesson 28: Write equations and word problems corresponding to tape and number line diagrams.
NO APPLICATION PROBLEM GIVEN.
Lesson 29: Connect division by a unit fraction to division by 1 tenth and 1 hundredth.
Fernando bought a jacket for $185 and sold it for 1 1/2 times what he paid. Marisol spent 1/5 as much as Fernando on the same jacket, but sold it for 1/2 as much as Fernando sold it for.
How much money did Marisol make? Explain your thinking using a diagram.
Note: This problem is a multi-step problem requiring a high level of organization. Scaling language and fraction multiplication from G5–M4–Topic G coupled with fraction of a set and subtraction warrant the extra time given to today’s Application Problem.
Lesson 30: Divide decimal dividends by non‐unit decimal divisors.
Alexa claims that 16 ÷ 4, 32/8 , and 8 halves are all equivalent expressions. Is Alexa correct? Explain how you know.
Note: This problem reminds students that when you multiply (or divide) both the divisor and the dividend by the same factor, the quotient stays the same or, alternatively, we can think of it as the fraction has the same value. This concept is critical to the Concept Development in this lesson.
Lesson 31: Divide decimal dividends by non‐unit decimal divisors.
A café makes ten 8-ounce fruit smoothies. Each smoothie is made with 4 ounces of soy milk and 1.3 ounces of banana flavoring. The rest is blueberry juice. How much of each ingredient will be necessary to make the smoothies?
Note: This two-step problem requires decimal subtraction and multiplication, reviewing concepts from G5–Module 1. Some students will be comfortable performing these calculations mentally while others may need to sketch a quick visual model. Developing versatility with decimals by reviewing strategies for multiplying decimals serves as a quick warm-up for today’s lesson.
Topic H: Interpretation of Numerical Expressions
Lesson 32: Interpret and evaluate numerical expressions including the language of scaling and fraction division.
Four baby socks can be made from 1/3 skein of yarn. How many baby socks can be made from a whole skein? Draw a number line to show your thinking.
Note: This problem is a partitive fraction division problem intended to give students more experience with this interpretation of division.
Lesson 33: Create story contexts for numerical expressions and tape diagrams, and solve word problems.
NO APPLICATION PROBLEM GIVEN.