problem solving • find the whole group using unit fractions · f c r focus: common core state...
TRANSCRIPT
493A Chapter 8
About the MathProfessional Development
LESSON AT A GLANCE
Interactive Student Edition
Personal Math Trainer
Math on the Spot
iTools: Fractions
HMH Mega Math
Problem Solving • Find the Whole Group Using Unit Fractions
LESSON 8.9
Why Teach ThisThere are many different strategies that can be used to solve problems. The different strategies should be modeled and practiced so that they become familiar to students. Then students can apply their knowledge of problem-solving strategies to determine which strategy is best to solve a particular problem.
Visualizing a problem with objects or pictures can make a solution to a problem more evident. Drawing a diagram is similar to making a picture.
In this lesson, students will solve problems that involve a fraction of a group and the number of objects in that fractional part. They will draw the number of fractional parts in a whole group. Then they will use what they know about the number of objects in one fractional part to draw the items in all the parts. The total number of objects they draw is the number of objects in the whole group.
Professional Development Videos
Learning ObjectiveSolve fraction problems by using the strategy draw a diagram.
Language ObjectiveStudents write in their Math Journal an example of the strategy draw a diagram and explain how it is used to solve fraction problems.
MaterialsMathBoard
F C R Focus:Common Core State Standards3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/ b
as the quantity formed by a parts of size 1/b.
MATHEMATICAL PRACTICESMP1 Make sense of problems and persevere in solving them. MP4 Model with mathematics.MP5 Use appropriate tools strategically. MP6 Attend to precision.
F C R Coherence:Standards Across the GradesBefore2.G.A.3
Grade 33.NF.A.1
After4.NF.A.1
F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency.......On Your OwnLevel 3: Applications..................................Think Smarter and Go Deeper
F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 441J.
FOCUS COHERENCE RIGOR
ENGAGE1
Lesson 8.9 493B
Daily RoutinesCommon Core
Daily RoutinesCommon Core
How can you use the strategy draw a diagram to solve fraction problems?
Fluency BuilderMaterials index cards numbered 0–9
Mental Math Give each pair of students a set of cards numbered 0–9. Mix the cards and place them facedown. Students take turns choosing 2 cards, naming the sum, the other related addition fact, and the two related subtraction facts.
Problem of the Day 8.9Jordan has 32 trophies. He wants to put his trophies on a bookcase that has 4 shelves. How many trophies should he put on each shelf to have an equal number of trophies on each shelf?
______
Vocabulary• Interactive Student Edition• Multimedia Glossary e
with the Interactive Student Edition
Essential QuestionHow can you use the strategy draw a diagram to solve fraction problems?
Making ConnectionsInvite students to share what they know about fractions and drawing diagrams. Display a drawing of twelve fish. Circle four of the fish. Then ask:
• What fraction of the fish is circled? 1 __ 4
• Describe how you could draw a picture that shows one half of the fish are striped? Separate the fish into two equal groups. Draw stripes on one of the groups.
Learning ActivityWhat is the problem the students are trying to solve? Connect the story to the problem.
• One third of the airplanes Pedro saw were large. He saw 8 large airplanes. If the rest are small, what information is missing? the total number of airplanes Pedro saw
• Use circles to represent the large airplanes Pedro saw. How many circles should you draw to show 1 __ 3 of the whole group? 8
Literacy and Mathematics• Have students write a problem involving finding the total number
of items given a unit fraction and the number of items in one part.
8 trophies
Common Core Fluency Standard 3.NBT.A.2
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EXPLORE2
Name
1. Max has 3 beta fish in his fish tank. One half of his fish are beta fish. How many fish does Max have in his tank?
2. Two boys are standing in line. One sixth of the students in line are boys. How many students are standing in line?
Problem Solving • Find the Whole Group Using Unit Fractions
There are 3 apple juice boxes in the cooler. One fourth of the juice boxes in the cooler are apple juice. How many juice boxes are in the cooler?
Read the Problem Solve the Problem
What do I need to find?
I need to find are in the cooler.
Describe how to draw a diagram to solve.
The denominator in 1 _ 4
tells you that there are parts in the whole group. Draw 4 circles to show parts.
Since 3 juice boxes are 1 _ 4
of the group, draw counters in the first circle.
Since there are counters in the first circle, draw counters in each of the remaining circles. Then count all of the counters.
So, there are juice boxes in the cooler.
What information do I need to use?
There are apple juice boxes.
of the juice boxes are apple juice.
How will I use the information?
I will use the information in the problem to draw a diagram.
Lesson 8.9Reteach
12
33
3
4
4how many juice boxes
12 students6 fish
One fourth
3
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8-21 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company
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Name
What Part of the Group?
Draw a quick picture to solve.
1. Lisa’s dog has 4 squeaky toys. Two thirds of the dog’s toys are squeaky toys. How many dog toys does Lisa’s dog have in all?
2. Sam has 9 yellow pencils in his desk. Three fourths of his pencils are yellow. How many pencils does Sam have in his desk?
3. Julia has 8 red barrettes. Two fourths of her barrettes are red. How many barrettes does Julia have?
4. Antonio has 15 pennies in his pocket. Five eighths of his coins are pennies. How many coins does Antonio have in his pocket?
5. Stretch Your Thinking One half of the birds at a pet store are yellow. Tara buys one of the yellow birds. Then one third of the birds at the store are yellow. How many yellow birds were at the pet store before Tara bought one? Explain how you know.
Lesson 8.9Enrich
Check students’ work.
2; Possible explanation: when you buy 1
yellow bird, the fraction of yellow birds
changes from 1 _ 2 to 1 _ 3 . So, there were 4 birds
to start with, 2 of which were yellow.
24 coins16 barrettes
12 pencils6 dog toys
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8-22 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company
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Unlock the ProblemUnlock the Problem
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Read the Problem
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Solve the Problem
Cameron has 4 clown fish in his fish tank. One third of the fish in the tank are clown fish. How many fish does Cameron have in his tank?
Use the graphic organizer to help you solve the problem.
What do I need to find?
I need to find ___ are in Cameron’s fish tank.
What information do I need to use?
Cameron has _ clown fish.
__ of the fish in the tank are clown fish.
How will I use the information?
I will use the information in the problem
to draw a __.
Describe how to draw a diagram to solve.
The denominator in 1 _ 3 tells you
that there are _ equal parts in the whole group. Draw 3 circles to
show _ equal parts.
Since 4 fish are 1 _ 3 of the whole group,
draw _ counters in the first circle.
Since there are _ counters
in the first circle, draw _ counters in each of the remaining circles. Then find the total number of counters.
Chapter 8 493
Problem Solving • Find the Whole Group Using Unit FractionsEssential Question How can you use the strategy draw a diagram to solve fraction problems?
So, Cameron has _ fish in his tank.
PROBLEM SOLVINGLesson 8.9
Number and Operations—Fractions—3.NF.A.1
MATHEMATICAL PRACTICESMP1, MP2, MP8
how many � sh
4
3
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12
One third
diagram
Possible drawing:
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DifferentiatedInstruction1
2
3
493 Chapter 8
Enrich 8.9Reteach 8.9
Unlock the Problem MATHEMATICAL PRACTICES
To introduce the lesson, have students watch the Real World Video, Feeding Big Cats at the Zoo. Have a class discussion about how zookeepers care for the animals at the zoo, and how it is similar to having pets, such as fish in an aquarium.MP1 Make sense of problems and persevere in solving them. Guide students to read each question in the graphic organizer and answer it before they solve the problem.
• Why do you draw 3 circles? Because the 4 fish are one third of the fish in the tank. Draw 3 circles to show the 3 equal parts of all the fish.
• Why do you draw 4 counters in each of the 3 circles? Four fish are one third of the fish in the tank, so there are 4 fish in each third.
• How could you check your answer with addition or multiplication? Possible answers: I can check with addition by adding 4 three times— 4 + 4 + 4 = 12. I can check with multiplication by multiplying 3 × 4 to get the answer 12.
MP4 Model with mathematics.•Cameron said that 1 __ 6 of his fish are angel
fish. How can you use the answer of 12 fish and a diagram to determine the number of angel fish Cameron has in his tank? Since 1 __
6
of the fish are angel fish, the denominator tells you to divide the 12 counters into 6 even groups. Each group will have 2 counters. So, Cameron has 2 angel fish.
ELL Strategy: Illustrate Understanding
Students can use drawings to solve fraction problems.
•Select a problem from the lesson.
•Explain how to use the denominator to know how many circles to draw and model it for the group.
•Model and explain how to read the problem to see how many counters to draw inside the circle. Have students draw them.
•Then, model how to use the drawing to figure out how many items there are in total and have students draw them.
LESSON 8.93.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
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Meeting Individual NeedsDifferentiated
Instruction
COMMON ERRORS
Try Another Problem
MathTalk MATHEMATICAL PRACTICES 1
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494
Read the Problem Solve the Problem
A pet store has 2 gray rabbits. One eighth of the rabbits at the pet store are gray. How many rabbits does the pet store have?
What do I need to find?
What information do I need to use?
How will I use the information?
1. MATHEMATICALPRACTICE 8 Draw Conclusions How do you know that your
answer is reasonable?
2. How did your diagram help you solve the problem? ____
Make Sense of Problems Suppose 1 _ 2 of the rabbits are gray. How can you find the number of rabbits at the pet store?
I need to fi nd how many rabbits are at the pet
store.
Since the denominator in 1 _ 8 is 8, there are
8 equal parts in the whole group. Draw
8 circles to show 8 equal parts.
Since 2 rabbits are 1 _ 8 of the whole group,
draw 2 counters in the fi rst circle.
Since there are 2 counters in the fi rst circle,
draw 2 counters in each of the remaining
circles. Then fi nd the total number of
counters.
Possible explanation: since half of the rabbits
are gray, there are two groups with 2 rabbits
in them. So, I can multiply the 2 rabbits by
2 groups to fi nd 4 rabbits in the whole group.
So, the pet store has 16 rabbits.
The pet store has 2 gray rabbits. One eighth
of all the rabbits are gray.
Possible answer: since the whole group is divided into 8 equal parts, I can multiply
8 groups of 2 to fi nd there are 16 rabbits in all. 16 = 16, so my answer is reasonable.
me see how many counters I needed to draw to fi nd the total number of rabbits.
Possible answer: it helped
I will use the information in the problem to
draw a diagram.
Advanced Learners
Lesson 8.9 494
Error Students might not know how many groups to draw.
Example I drew 2 groups with 2 counters each. So, the pet store has 4 rabbits.Springboard to Learning Have students circle the word that describes the denominator in the fraction. Then have students draw that many groups. The denominator is 8, so there should be 8 groups of 2 counters each, or 16 counters in all.
Try Another ProblemHave students read the problem about the rabbits. Then have students answer the questions in the graphic organizer and solve the problem. Invite students to share their diagrams and explanations.
• How many equal groups did you show with your counters? Explain. Possible answer: I made 8 equal groups because one eighth of the rabbits are gray.
• How many counters did you draw in all? Explain. 16; because 2 rabbits are one eighth of the rabbits, I showed 2 rabbits in each of the 8 groups, and 8 × 2 = 16.
MP8 Look for and express regularity in repeated reasoning.• How many of the rabbits are not gray?
How do you know? 14; I found that there are 16 rabbits in all. If 2 are gray, then 16 − 2, or 14, rabbits are not gray.
MathTalk Use Math Talk to focus on
students’ understanding of the relationship between the denominator and the number of equal groups in a whole.
• If you know the fraction of the rabbits that are gray, what other information do you need to know to find the total number of rabbits? the number of rabbits that are gray
You may suggest that students place completed Try Another Problem graphic organizers in their portfolios.
• Present students with the following problem:
If you know that 1 __ 8 of a number is 3, how can you find the number?
• Have students solve the problem. 24
• Then have students write a rule they can use to find a whole given a unit fraction of it. Have students share their rules. Possible answer: Multiply the denominator of the unit fraction by the number part given.
Visual / KinestheticIndividual / Partners
EXPLAIN3
Quick Check
If
Rt I 1
2
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Quick Check
If
Rt I 1
2
3
Then
Share and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and Show MATHBOARDMATHBOARDMATHBOARDMATHBOARDMATHMATHMATHMATHBOARDBOARDBOARDBOARD
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√ Circle the question.
√ Underline important facts.
√ Put the problem in your own words.
√ Choose a strategy you know.
1. Lily has 3 dog toys that are red. One fourth of all her dog toys are red. How many dog toys does Lily have?
First, draw _ circles to show _ equal parts.
Next, draw _ toys in _ circle since
_ circle represents the number of red toys.
Last, draw _ toys in each of the remaining circles. Find the total number of toys.
So, Lily has _ dog toys.
2. SMARTER What if Lily has 4 toys that are red? How many dog toys would she have?
3. The pet store sells bags of pet food. There are 4 bags of cat food. One sixth of the bags of food are cat food. How many bags of pet food does the pet store have?
4. Rachel owns 2 parakeets. One fourth of all her birds are parakeets. How many birds does Rachel own?
Chapter 8 • Lesson 9 495
Unlock the Problem
4
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12
16 dog toys
24 bags
8 birds
1
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495 Chapter 8
MP2 Reason abstractly and quantitatively. To extend their thinking, have students tell about the unidentified parts in each problem. For example, if one fourth of the dog toys are red, what can you tell about the remaining toys? Possible answer: the remaining 9 toys are three fourths of the group and are a color different than red.
Share and Show MATHBOARDMATHBOARD
The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.
SMARTER
Exercise 2 gives students an opportunity to analyze how a change in the number of objects in a part affects the whole number of objects in the group.Use the checked exercises for Quick Check.
a student misses the checked exercises
Differentiate Instruction with • Reteach 8.9
• Personal Math Trainer 3.NF.A.1
• RtI Tier 1 Activity (online)
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ELABORATE4
EVALUATE5 Formative Assessment
WRITE Math Show Your Work
MATHEMATICAL PRACTICES COMMUNICA E CONSTRUCT ARGUMENTS
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On Your OwnOn Your Own
_ hats
8. SMARTER Two hats are 1 _ 3 of the group. How many hats are in the whole group?
5. SMARTER Before lunchtime, Abigail and Teresa each read some pages from different books. Abigail read 5, or one fifth, of the pages in her book. Teresa read 6, or one sixth, of the pages in her book. Whose book had more pages? How many more pages?
6. MATHEMATICALPRACTICE 2 Represent a Problem Six
friends share 5 meat pies. Each friend first eats half of a meat pie. How much more meat pie does each friend need to eat to finish all the meat pies and share them equally? Draw a quick picture to solve.
7. DEEPER Braden bought 4 packs of dog treats. He gave 4 treats to his neighbor’s dog. Now Braden has 24 treats left for his dog. How many dog treats were in each pack? Explain how you know.
Teresa’s book; 11 more pages
Has been eaten
Need to eat
6
Possible drawing for 6:
Possible answer: 1 third, or 2 sixths
7 dog treats; possible explanation: I began
with the 24 treats left for Braden’s dog.
I added the number of treats he gave
away: 24 + 4 = 28. Then I divided that
number by 4 packs: 28 ÷ 4 = 7.
1 2 3 4 5 6
1 2
3
4 5
6Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.
Differentiated Centers Kit
DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD
Lesson 8.9 496
Essential QuestionUsing the Language ObjectiveReflect Have students write an example in their Math Journal to answer the Essential Question.How can you use the strategy draw a diagram to solve fraction problems? Possible answer: I can draw a diagram to represent all the equal groups in a whole. Then I can draw the number of objects in each group. Finally, I can add or multiply to find how many objects are in the whole.
Math Journal WRITE MathWrite a problem about a group of objects in your classroom. Tell how many are in one equal part of the group. Solve your problem. Draw a diagram to help you.
On Your Own MATHEMATICAL PRACTICES
If students complete the checked exercises correctly, they may continue with the On Your Own section.
SMARTER
Students complete blue Activity Card 11 by finding
fractional parts of a group of pattern blocks.t
Students read the book and model fractional parts.
LiteratureThe Whole Picture
ActivitiesFraction Action
MP2 Reason abstractly and quantitatively. Exercise 6 requires students to compare the number of wholes and the number of shares in order to find equal shares.
SMARTER
This item assesses a student’s ability to find a whole group given a unit fraction. Students should understand that the model represents three equal parts of the whole group, with two hats making up one of the parts. Students who give two as the answer likely confused the part of the group with the whole.
Math on the Spot Video TutorUse this video to help students model and solve this type of Think Smarter problem.
Cross-Curricular
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Chapter 8 497
Problem Solving • Find the Whole Group Using Unit Fractions
Draw a quick picture to solve.
1. Katrina has 2 blue ribbons for her hair. One fourth of all her ribbons are blue. How many ribbons does Katrina have in all?
_____
2. One eighth of Tony’s books are mystery books. He has 3 mystery books. How many books does Tony have in all?
_____
3. Brianna has 4 pink bracelets. One third of all her bracelets are pink. How many bracelets does Brianna have?
_____
4. Ramal filled 3 pages in a stamp album. This is one sixth of the pages in the album. How many pages are there in Ramal’s stamp album?
_____
5. Jeff helped repair one half of the bicycles in a bike shop last week. If Jeff worked on 5 bicycles, how many bicycles did the shop repair last week?
_____
6. WRITE Math Write a problem about a group of objects in your classroom. Tell how many are in one equal part of the group. Solve your problem. Draw a diagram to help you.
8 ribbons
Lesson 8.9Practice and Homework
COMMON CORE STANDARD—3.NF.A.1 Develop understanding of fractions as numbers.
24 books
12 bracelets
18 pages
Check students’ work.
10 bikes
Check students’ work.
Practice and HomeworkUse the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers.
497 Chapter 8
SCIENCE SOCIAL STUDIES
• Take a class walk through a local park. Have students classify the animals they see according to birds, reptiles, and amphibians. Remind students that animals like alligators and turtles are reptiles, and animals like frogs and salamanders are amphibians.
• Suppose 1 __ 6 of the animals they saw were reptiles. If they saw 2 reptiles, what was the total number of animals they saw? 12 animals
• Discuss how people use money to purchase goods and services. Goods are items you can see and touch, like food, clothing, or a book. Services are provided by other people like the doctor, the person who cuts your hair, or the person who fixes the washing machine.
• Ask students how they buy items they want.
• Lindsay saved $5 to buy a game. She has 1 __ 4 the amount of money she needs. How much money does the game cost? $20
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Personal Math Trainer
FOR MORE PRACTICE GO TO THE
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498
Lesson Check (3.NF.A.1)
1. A zoo has 2 male lions. One sixth of the lions are male lions. How many lions are there at the zoo?
2. Max has 5 red model cars. One third of his model cars are red. How many model cars does Max have?
Spiral Review (3.OA.A.3, 3.NBT.A.1, 3.NBT.A.2, 3.NF.A.1)
5. Ken has 6 different colors of marbles. He has 9 marbles of each color. How many marbles does Ken have in all?
6. Eight friends share two pizzas equally. How much of a pizza does each friend get?
3. There are 382 trees in the local park. What is the number of trees rounded to the nearest hundred?
4. The Jones family is driving 458 miles on their vacation. So far, they have driven 267 miles. How many miles do they have left to drive?
458 − 267
__
12 lions
400 trees
54 marbles
15 model cars
191 miles
2 eighths
Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section.
Lesson 8.9 498