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MTH15_SE04_CA_T12_L11.in
Another Example
Explain ItExplain It
Draw a Picture and
Write an Equation
Problem Solving
Marie and her mother hiked three
trails, the Ansel Trail, The Briar Trail, and
the River Trail. How far did they walk
in all?
Lesson
12-
11 ___
of a mile in all
___^1
x
Regina and Don are hiking a trail. They have
already hiked
___^1
of a mile. How much farther do
they have to travel to reach the
11 ___
Write an equation and subtract the fractions.
Let x 5 how much farther they have to travel.
11 ___
= the total distance of the hike
___ 1
= the distance already hiked
11 ___
___^1
5 x
x 5
10 ___
5 __
Regina and Don need to hike
5 __
of a mile farther
to reach the
11 ___
1. How could you find how much farther Regina and Don
will have to hike to reach one mile?
2. Reason If Regina and Don turn around and hike back
___ 1
of a mile, how can you find the difference between
the total length they traveled and
11 ___
of a mile?
Common
Core
4.NF.B.3a Understand a fraction a_ b with a 1 as a sum of fractions _ b^1. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Sample answer: Subtract
___ 1
from 1.
See margin.
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76 Overview and Implementation Guide
Problem Solving
Research says that conceptual understanding,
computational fluency, and problem-solving skills are each
essential and mutually reinforcing, influencing performance
on such varied tasks as estimation, word problems, and
computation (National Mathematics Advisory Panel, 2008).
enVisionMATH California Common Core was built on a
foundation of problem solving. From the Problem-Based
Interactive Learning to the Problem Solving exercises,
each lesson offers students multiple opportunities to develop
proficiency with sense-making, solution plans, and solution
explanations and justifications, key skills of effective problem
solvers and proficient math thinkers.
Each topic includes one or more Problem Solving lessons.
These lessons offer students opportunities to integrate and
apply concepts and skills learned in earlier lessons as they
draw on and strengthen their problem-solving abilities.
An important Teaching Tool is the Problem-Solving
Recording Sheet. Students in Grades 3–6 can use the
Recording Sheet to organize and record their thinking as they
work through their solutions.
Program Foundations
A new approach to solving word
problems is to use bar diagrams
as visual representations that
show how quantities in
a word problem are related.
—Dr. Randall Charles ”
The Standards for Mathematical
Practice stress the importance of strong
problem-solving and reasoning abilities for
mathematically proficient thinkers
▲ Student Edition
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Read & Understand Plan & Solve
Do you know HOW? Do you UNDERSTAND?
Let x 5 total miles hiked
Write an equation and add the fractions. x 5 7 __ 8
3 __
8
5 __
8
15 ___
8 or 1 __ 7 8 miles Marie and her mother walked 1 7 __ 8 miles in all. Draw a picture and write an equation to solve.
- Heather ran 2 __ 6 of a mile. Rick ran __^1 6 of a mile. How much farther did Heather run than Rick?
- Writing to Explain If you were asked to find how far Marie and her mother walked on the Briar and River Trails alone, would the model be different?
- Write a Problem Write a problem that you can solve by drawing a picture and writing an equation. Model Draw a picture and write an equation to solve.
- Barb connected a wire extension that is ___^7 12 foot long to another wire that is 11 ___ 12 foot long. How long is the wire with the extension?
- The smallest female spider measures about ___^5 10 millimeter (mm) in length. The smallest male spider measures about ___^4 10 mm in length. How much longer is the female spider than the male spider?
What do
I know? What am I asked to find?
Marie and her mother
hiked 3 trails.
Ansel Trail 5 7 __
mi
Briar Trail 5 3 __
mi
River Trail 5 5 __
mi
How far did Marie
and her mother walk in all?
x feet
___^7 12 ___^11 12 ___^5 10 mm long ___^4 10 x
Applying Math Practices
Guided Practice*
Independent Practice
- (^) What am I asked to find?
- (^) What else can I try?
- (^) How are quantities related?
- How can I explain my work?
- How can I use math to model the
problem?
- Can I use tools to help?
- (^) Is my work precise?
- (^) Why does this work?
- (^) How can I generalize?
x miles in all
7 __ 8 3 __ 8 5 __ 8 MATHEMATICAL
PRACTICES
MATHEMATICAL
PRACTICES
See margin.
See margin. See margin. See margin. See margin.
*For another example, see Set K on page 321. Lesson 12-11 315
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Problem-Solving Recording Sheet
Program Guide 77 Bar Diagrams Research has found that bar diagrams help students understand the relationships between quantities in a problem, and this helps students choose a correct operation to solve the problem (Diezmann and English, 2001). enVisionMATH California Common Core provides:
- Visual models for ways to make a number in
Grade K to build a foundation for addition and subtraction. 8 is 3 and 5. 8 is 4 and 4.
- Visual models for addition and subtraction
situations in Grades 1–2 to help children see relationships between quantities. In the model, children place objects, then later draw dots, and then later write numbers. Add to There are 3 birds. 2 more fly in. How many in all? There are 3 birds. More fly in. Then there are 5 in all. How many flew in? There are some birds. 2 more fly in. Then there are 5 in all. How many were there to begin with? (^5 ) Take from There are 7 birds. 3 birds fly away. How many are left? There are 7 birds. Some fly away. Then 4 birds are left. How many flew away? There are some birds. 3 birds fly away. Then 4 birds are left. How many were there to begin with? 7 3 7 4 3 4 The models help children think about adding to and taking from as Putting together or taking apart. Children add when the whole is unknown and subtract when a part is unknown. Comparison problems and joining-equal groups problems are also introduced.
▲ Problem-Solving
Recording Sheet
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Program Guide 79
Joining
Equal Groups
Unknown Product
Kim has 4 photo albums.
Each album has 85 pictures.
How many photos are in her
4 albums?
Group Size Unknown
Pam put the same number of
apples in each of 4 bags. She
ended up with 52 apples in
bags. How many apples did
she put in each bag?
Number of Groups
Unknown
Fred bought some books.
Each book cost $16. He spent
$80 on books. How many
books did he buy?
Separating
Equal Groups
Unknown Dividend
Kim had some cards. She put
them into piles of 35 and was
able to make 4 piles. How
many cards did she start with?
Group Size Unknown
Bryan got 45 pigeons. He put
them in 5 pens with the same
number of pigeons in each pen.
How many pigeons are in
each pen?
Number of Groups
Unknown
A total of 108 children signed
up for soccer. The coach put
them into 18-person teams.
How many teams were made?
Comparison
Larger Amount Unknown
Alex has 17 toy cars. Keisha
has 3 times as many. How
many cars does Keisha have?
Smaller Amount Unknown
Barney has 24 old coins.
This is 3 times as many coins
as Steve has. How many
old coins does Steve have?
Multiplier Unknown
Ann’s teacher is 39 years old.
Ann is 13 years old. How many
times as old as Ann is Ann's
teacher?
- Focused instruction on bar diagrams is in problem-solving lessons,
in lessons on meanings of operations, and in lessons on mental math.
- A variety of problems as shown here are found throughout the
lessons to help students develop the quantitative reasoning called for in
the Common Core State Standards for Mathematics.
Product
Group
size
Number
of groups
Dividend
Group
size
Number
of groups
Larger
amount
Smaller
amount
Multiplier
3 times
as many
3 times
as many
?? times
as many
Bar Diagrams: Multiplication-Division
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3 ∙ 3 ∙ 6
You can change
the order of the
addends. You will see
that the sum is the
same.
You can write
two number
sentences.
4 plus 2 equals 6.
4 and 2 is 6.
2 plus 4 equals 6.
2 and 4 is 6.
1 ____ 2
1 ____ 4
You can model repeated
addition with an array.
Arrays have equal rows. Arrays have equal
amounts in each row.
Write the
addition sentence.
____^3 ____^3 5 ____^6
There
are 6!
80 Overview and Implementation Guide
Algebra Across the Grades
Program Foundations
Kindergarten Visual Learning Bridge
Grade 1 Visual Learning Bridge
Grade 2 Visual Learning Bridge
Research has found that rule-based
instructional approaches that do not give
students opportunities to create meaning for
the rules or to learn when to use them can
lead to weak conceptual understanding,
unsystematic errors, reliance on visual clues,
and poor strategic decisions (Mathematics
Learning Study Committee, 2001).
In 2006, then-President Bush charged the
National Mathematics Advisory Panel
with advising him and the Secretary of
Education on the best way to advance the
teaching and learning of mathematics. In
its 2008 final report, the Panel noted: “To
prepare students for Algebra, the curriculum
must simultaneously develop conceptual
understanding, computational fluency, and
problem-solving skills. These three aspects
of learning are mutually reinforcing and
should not be seen as competing for class
time” (p. 19).
The National Mathematics Advisory Panel
recommended that the K–8 mathematics
curriculum focus on the Critical Foundation
of Algebra, specifically:
(1) proficiency with whole numbers,
(2) proficiency with fractions, and
(3) particular aspects of geometry and
measurement.
Students in Kindergarten through Grade 6
can develop algebraic thinking by using
patterns to make generalizations
about numbers, and by using
mathematical symbols to describe
relationships.
enVisionMATH California Common Core
provides:
- lessons involving patterns and
generalizations with numbers.
- multiple representations of relationships.
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