4.12 Creating and Solving Fraction Word Problems (Part 2)
COMMON CORE STATE STANDARDSWrite and interpret numerical expressions.
5.OA.A.1 - Operations & Algebraic Thinking
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.A.2 - Operations & Algebraic Thinking
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.4 - Number and Operations - Fractions
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
+ 1/2 = 3/7, by observing that 3/7 < 1/2.
5.NF.B.6 - Number and Operations - Fractions
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
BIG IDEA
Students will solve and create fraction word problems involving addition, subtraction, and multiplication.
Standards of Mathematical Practice
□ Make sense of problems and persevere in solving them
ü Reason abstractly and quantitatively
□ Construct viable arguments and critique the reasoning of others
ü Model with mathematics
ü Use appropriate tools strategically
□ Attend to precision
□ Look for and make use of structure
□ Look for and express regularity in repeated reasoning / Informal Assessments:
□ Math journal
□ Cruising clipboard
□ Foldable
□ Checklist
ü Exit ticket
ü Response Boards
ü Problem Set
ü Class Discussion
PREPARING FOR THE ACTIVITY / MATERIALS
□ A general instructional note on today’s problems: Today’s problems are more complex than those found in Block 11. All are multi-step. Students should be strongly encouraged to draw before attempting to solve. As in Block 11, multiple approaches to solving all the problems are possible. Students should be given time to share and compare thinking during the reflection.
□ Measurement Conversion Reference Sheet is available in block 8. / · Response Boards
· Measurement Conversion Reference Sheet
· Problem Set 4.12
· Exit Ticket 4.12
· Additional Practice 4.12
VOCABULARY
AUTOMATICITY / TEACHER NOTES
Convert Measures
1. Distribute response boards.
2. Write 1 ft = __ in How many inches are in 1 foot? (12 inches.)
3. Write 1 ft = 12 in. Below it, write
2 ft = __ in. How many inches are in 2 feet? (24 inches.)
4. Write 2 ft = 24 in. Below it, write 4 ft = __ in. How many inches are in 3 feet? (48 inches.)
5. Write the multiplication equation you used to solve it. Students write 4 ft × 12 = 48 in.
6. Continue with the following possible sequence:
a. 1 pint = 2 cups, 2 pints = 4 cups, 7 pints = 14 cups, 1 yard = 3 ft, 2 yards = 6 ft, 6 yards = 18 ft,
b. 1 gal = 4 qt, 2 gal = 8 qt, 9 gal = 36 qt.
7. Write 2 c = __ pt. How many pints does 2 cups make? (1 pint.)
8. Write 2 c = 1 pt. Below it, write 4 c = __ pt. How many pints does 4 cups make? (2 pints.)
9. Write 4 c = 2 pt. Below it, write 10 c = __ pt. How many pints does 10 cups make? (5 pints.)
10. Write the division equation you used to solve it. Students write 10 c ÷ 2 = 5 pt.
11. Continue with the following possible sequence: 12 in = 1 ft, 36 in = 3 ft, 3 ft = 1 yd, 12 ft = 4 yd, 4 qt = 1 gal, and 28 qt = 7 gal
Multiply a Fraction and a Whole Number
1. Write 9 ÷ 3 =__ Say the division sentence.
(9 ÷ 3 = 3.)
2. Write 13 × 9 =__ Say the multiplication sentence.
(13 × 9 = 3.)
3. Write 23 × 9 =__ On your boards, write the multiplication sentence. (Write 23 × 9 = 6.)
4. Write 9 × 23=__ On your boards, write the multiplication sentence. (Write 9 × 23= 6.)
5. Continue with the following possible sequence:
12 ÷ 6, 16 × 12, 56 × 12, 12 × 56, 18 × 24, 24 × 18,
24 × 38 , 23 × 12, and 12 × 34.
Write the Expression to Match the Diagram
1. Project a tape diagram partitioned into 3 equal units with 15 as the whole and 2 units shaded. What is the value of the whole? (15.)
2. On your boards, write an expression to match the diagram using a fraction. (Write 23 × 15 or 15 × 23)
2. To solve we can write 15 divided by 3 to find the value of one unit, times 2. Write 153 × 2 as you say the words. Find the value of the expression. Students write 153 × 2 = 10.
3. Continue this process for the following possible suggestion:
a. 35 × 45, 34 × 32, 56 × 54, and 78 × 64. / Select appropriate activities depending on the time allotted for automaticity.
Convert Measures: This fluency reviews Blocks 9–11 and prepares students for Block 12 content.
UDL – Multiple Means of Engagement: Continue to allow students to use the conversion reference sheet if they are confused, but encourage them to answer questions without looking at it.
Multiply a Fraction and a Whole Number: This fluency reviews Blocks 9–11.
Write the Expression to Match the Diagram: This fluency reviews Blocks 10─11.
SETTING THE STAGE / TEACHER NOTES
Application Problem
1. Present the following table for students to complete. Review as needed.
23 yds / ______feet
4 pounds / ______ounces
8 tons / ______pounds
34 gallon / ______quarts
512 year / ______months
45 hour / ______minutes
Connection to Big Idea
Today, we will continue to solve and create fraction word problems involving addition, subtraction, and multiplication. What are some of the strategies you used to solve the problems from yesterday? (Accept student responses.) Again, you will be sharing your methods and solutions with others, so you should be prepared to support your answers. / Note: The chart requires students to work within many customary systems reviewing the work of Block 9. Students may need a conversion chart to scaffold this problem.
EXPLORE THE CONCEPT / TEACHER NOTES
While Part A is relatively straightforward, there are still varied approaches for solving. Students may find the difference between the number of games played and lost to find the number of games won (24) expressing this difference as a fraction (2432). Alternately they may conclude that the losing games are 14 of the total and deduce that winning games must constitute 34. Watch for students distracted by the fractions 58 and 14 written in the stem and somehow try to involve them in the solution to Part (a). Complexity increases as students must employ the fraction of a set strategy twice, carefully matching each fraction with the appropriate number of games and finally combining the number of games that Katy played to find the total.
The increase in complexity for this problem comes as students are asked to find the number of pink flowers in the garden. This portion of the flowers refers to 1 fifth of the remaining flowers (i.e., 1 fifth of those that are not red or purple), not 1 fifth of the total. Some students may realize (as in Method 4) that 1 fifth of the remainder is simply equal to 1 unit or 16 flowers. Multiple methods of drawing and solving are possible. Some of the possibilities are pictured below.
Method 3
The way in which Lillian’s time is expressed makes for a bit of complexity in this problem. Students must recognize that she only took 16 hour to complete the assignment. Many students may quickly recognize that Lillian worked faster as 16 35. However, students must go further to find exactly how many minutes faster. The bonus requires them to give the fraction of an hour. Simplification of this fraction should not be required but may be discussed.
Working backwards from expression to story may be challenging for some students. Since the expression given contains parentheses, the story created must first involve the addition or combining of 3 and 5.
For students in need of assistance, drawing a tape diagram first may be of help. Then, asking the simple prompt, “What would a baker add together or combine?” may be enough to get the students started.
Evaluating 14× (3 + 5) should pose no significant challenge to students. Note that the story of the chef (see the 2 examples below) interprets the expression as repeated addition of a fourth where the story of the baker interprets the expression as a fraction of a set.
Again, students are asked to both create and then solve a story problem, this time using a given tape diagram. The challenge here is that this tape diagram implies a two-step word problem. The whole, 36, is first partitioned into thirds, and then one of those thirds is divided in half. The story students create should reflect this two-part drawing. Students should be encouraged to share aloud and discuss their stories and thought process for solving.
For this problem, students need to find the whole. An interesting aspect of this problem is that fractional amounts of different wholes can be the same amount. In this case, two-fifths of 10 is the same as one-third of 12. This should be discussed with students.
/ Note: Follow the steps from Block 11 suggestions on delivering instruction for the word problems for today’s Problem Set, which are the work for the rest of block.
UDL – Multiple Means of Engagement: The complexity of the language involved in today’s problems may pose significant challenges to English language learners or those students with learning differences that affect language processing. Consider pairing these students with those in the class who are adept at drawing clear models. These visuals and the peer interaction they generate can be invaluable bridges to making sense of the written word.
REFLECTION / TEACHER NOTES
1. Guide students in a conversation to debrief the Problem Set and process the lesson. You may choose to use any combination of the questions below to lead the discussion.
· How are the problems alike? How are they different?
· How many strategies can you use to solve the problems?
· How was your solution the same and different from those that were demonstrated?
· Did you see other solutions that surprised you or made you see the problem differently?
· How many different story problems can you create for Problem 5 and Problem 6?
2. Allow students to complete Exit Ticket 4.12 independently. / Look for misconceptions or misunderstandings that can be addressed in the reflection.
Source: http://www.engageny.org/resource/grade-5-mathematics-module-4
Grade 5 Unit 4: Block 12
Name: ______Date: ______
Problem Set 4.12 – page 1
1. A baseball team played 32 games, and lost 8. Katy was the catcher in 58 of the winning games and 14 of the losing games.
a. What fraction of the games did the team win?
b. In how many games did Katy play catcher?
2. In Mrs. Elliott’s garden, 18 of the flowers are red, 14 of them are purple, and 15 of the remaining flowers are pink. If there are 128 flowers, how many flowers are pink?
Problem Set 4.12 – page 2
3. Lillian and Darlene plan to get their homework finished within one hour. Darlene completes her math homework in 35 hour. Lillian completes her math homework with 56 hour remaining. Who completes her homework faster and by how many minutes?
Bonus: Give the answer as a fraction of an hour.
4. Create and solve a story problem about a baker and some flour whose solution is given by the expression 14×(3+5).
Problem Set 4.12 – page 3
5. Create and solve a story problem about a baker and 36 kilograms of an ingredient that is modeled by the following tape diagram. Include at least one fraction in your story.
6. Of the students in Mr. Smith’s fifth grade class, 13 were absent on Monday. Of the students in Mrs. Jacobs’ class, 25 were absent on Monday. If there were 4 students absent in each class on Monday, how many students are in each class?
Name: ______Date: ______
Exit Ticket 4.12
In a classroom, 16 of the students are wearing blue shirts and 23 are wearing white shirts. There are 36 students in the class. How many students are wearing a shirt other than blue or white?
Name: ______Date: ______
Exit Ticket 4.12
In a classroom, 16 of the students are wearing blue shirts and 23 are wearing white shirts. There are 36 students in the class. How many students are wearing a shirt other than blue or white?
Name: ______Date: ______
Additional Practice 4.12 - page 1
1. Terrence finished a word search in 34 the time it took Frank. Charlotte finished the word search in 23 the time it took Terrence. Frank finished the word search in 32 minutes. How long did it take Charlotte to finish the word search?