Northfield Community School
Math Curriculum
Unit Title: Numbers and Operations Fractions
Unit 3
Grade Level: 5
Summary of Unit: The focus of this unit is to use equivalent fractions as a strategy to add
and subtract fractions. Apply and extend previous understandings of
multiplication and division to multiply and divide fractions.
Stage 1 – Desired Results
CCSS for Mathematical Practices:
MP1. Make sense of problems and persevere in solving them.
MP2. Reason abstractly and quantitatively.
MP3. Construct viable arguments and critique the reasoning of others.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.
CCSS for Mathematical Content:
5.NF.A.1
Add and subtract fractions with unlike denominators (including mixed
numbers) by replacing given fractions with equivalent fractions in such a
way as to produce an equivalent sum or difference of fractions with like
denominators.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions
referring to the same whole, including cases of unlike denominators, e.g.,
by using visual fraction models or equations to represent the problem.
Use benchmark fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers.
5.NF.B.3 Interpret a fraction as division of the numerator by the denominator
(a/b = a ÷ b). Solve word problems involving division of whole numbers
leading to answers in the form of fractions or mixed numbers, e.g., by
using visual fraction models or equations to represent the problem.
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a
fraction or whole number by a fraction.
5.NF.B.4a 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.
5.NF.B.4b Find the area of a rectangle with fractional side lengths by tiling it with unit
squares of the appropriate unit fraction side lengths, and show that the area
is the same as would be found by multiplying the side lengths. Multiply
fractional side lengths to find areas of rectangles, and represent fraction
products as rectangular areas.
5.NF.B.5 Interpret multiplication as scaling (resizing), by:
5.NF.B.5a Comparing the size of a product to the size of one factor on the basis of the
size of the other factor, without performing the indicated multiplication.
5.NF.B.5b Explaining why multiplying a given number by a fraction greater than 1
results in a product greater than the given number (recognizing
multiplication by whole numbers greater than 1 as a familiar case);
explaining why multiplying a given number by a fraction less than 1 results
in a product smaller than the given number; and relating the principle of
fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying
a/b by 1.
5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed
numbers, e.g., by using visual fraction models or equations to represent the
problem.
5.NF.B.7 Apply and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.
5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and
compute such quotients.
5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such
quotients.
5.NF.B.7c Solve real world problems involving division of unit fractions by non-zero
whole numbers and division of whole numbers by unit fractions, e.g., by
using visual fraction models and equations to represent the problem.
Understandings (ENDURING):
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Fractional parts are equal-sized
parts of a unit, where the unit can
be a region, a set of objects, or a
length.
A fraction can be represented as
division and vice versa.
Fractions in which the numerator is
greater than the denominator may
be expressed as mixed numbers or
as improper fractions.
Benchmark fractions are familiar
fractions that are easy to visualize,
such as halves, thirds, and fourths.
Equivalent fractions name the same
part of a whole, part of a set, or
same location on the number line.
Essential Question(s):
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How can you name part of a
whole?
How can a fraction name a point
on a number line?
How can you divide 3 objects into
equal parts?
What is a mixed number?
How can you locate fractions and
mixed numbers on a number line?
How can you name a part in more
than one way?
How can you find equivalent
fractions?
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Equivalent fractions can be found
by multiplying or dividing the
numerator and denominator by the
same non-zero number.
The greatest common factor is the
largest factor that two or more
numbers have in common and is
used to help express fractions in
simplest terms.
Mixed numbers can be expressed
as
improper fractions.
Fractions and decimals can be
represented on a number line.
When adding or subtracting
fractions with like denominators,
you are adding or subtracting
pieces or portions of the same size,
so you can add the numerators-the
number of pieces or portionswithout changing the denominator.
To add or subtract fractions or
mixed numbers with unlike
denominators, change the number
sentence to a simpler one with like
denominators.
The least common denominator is
the least common multiple of the
denominators that are added or
subtracted.
Multiplying a whole number by
fraction involves division as well as
multiplication.
When we multiply two fractions
that are both less than 1, the
product is smaller than either
fraction.
When we divide by a fraction that
is
less than 1, the quotient is greater
than the number being divided (the
dividend).
A fraction is another representation
for division.
Equivalent fractions represent the
same value.
 How can you find the greatest
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common factor?
What does it mean for a fraction
to be in simplest form?
How can you compare and order
fractions and mixed numbers?
How do you add or subtract
fractions with like denominators?
How can you find the least
common denominator of two
fractions?
How can you use the least
common denominator to add or
subtract fractions with unlike
denominators?
How can you estimate with mixed
numbers?
How is adding mixed numbers
like adding fractions and whole
numbers?
How do you subtract a mixed
number from a mixed number?
How do you subtract a mixed
number from a whole number?
How can you use mental math to
find a fraction of a whole
number?
What are some ways to estimate
the product of fractions?
How can you find the product of
mixed numbers?
How can you find the quotient of
fractions?
How can you find the quotient of
mixed numbers?
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Whole numbers can be classified
into subsets.
Divisibility rules can be used to
find equivalent fractions quickly.
Fractions and decimals are different
representations for the same
amounts and can be used
interchangeably.
Students will be able to:
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Identify and show fractional parts of regions and sets and locations on a number
line.
Use division to divide objects into equal parts where the parts are fractions of
the whole.
Express fractions greater than one as mixed numbers or improper fractions.
Convert improper fractions to mixed numbers and vice versa.
Estimate fractional parts of regions.
Identify and locate fractions and mixed numbers on a number line.
Identify and write equivalent fractions.
Determine common factors and greatest common factor of numbers.
Identify fractions that are in simplest form and find the simplest form of a
fraction by dividing the numerator and denominator by their greatest common
factor.
Determine which of two fractions is greater or less and write a comparison.
Compare and order fractions and mixed numbers.
Add and subtract fractions with like denominators.
Find a common denominator for two fractions.
Add and subtract fractions with unlike denominators.
Add and subtract mixed numbers with and without renaming.
Estimate sums and differences of mixed numbers.
Estimate sums and add mixed numbers.
Estimate differences and subtract mixed numbers.
Use models or mental math to find fractions of whole numbers.
Use compatible numbers and mental math to estimate the product of a whole
number and a fraction.
Use models and paper and pencil to multiply fractions.
Multiply and divide fractions.
Multiply and divide mixed numbers.
Use models and mental math to divide fractions.
Solve problems involving too much information by using only the information
that is needed, and decide when there is not enough information to solve a
problem.
Use the information given in the problem to make conclusions.
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Solve a problem by choosing an operation.
Review and apply key concepts, skills, and strategies learned in this and
previous units.
Stage 2 – Assessment Evidence
Assessment Evidence will be comprised of, but not limited to, the following
suggested activities:
Performance Task(s):
Title:
Lots of Chocolate
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
Amy, Elizabeth, Katie, Gretchen, and Deb love chocolate. One afternoon at a meeting,
each of them brought their own large candy bar. Each candy bar was the same size.
Throughout the meeting, all of the ladies munched on their candy bars. At the end of
the meeting, everyone was moaning and groaning about having stomach aches.
Here is a list of what the ladies consumed:
☺ Amy: two-sixths of her candy bar
☺ Elizabeth: two-thirds of her candy bar
☺ Katie: three-fourths of her candy bar
☺ Gretchen: one-half of her candy bar
☺ Deb: one-third of her candy bar
Determine which of the ladies ate the most chocolate. Who ate the least? How
much chocolate did they eat all together?
Student Task Worksheet :
http://www.rda.aps.edu/MathTaskBank/pdfs/tasks/35/t35LotsChoc.pdf
Teacher Instructions with Rubric:
http://www.rda.aps.edu/MathTaskBank/pdfs/instruct/35/i35LotsChoc.pdf
Title:
Candy Party
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
You are having a party and 36 friends have been invited. You surveyed the group and
5
found out that 49 of them like Jolly Ranchers ® and 12
of them like Hershey's Kisses ®.
You want to get the candy liked by most guests. Which candy would you buy and
why?
Student Task Worksheet :
http://www.rda.aps.edu/MathTaskBank/pdfs/tasks/35/t35candy.pdf
Teacher Instructions with Rubric:
http://www.rda.aps.edu/MathTaskBank/pdfs/instruct/35/i35candy.pdf
Title:
Dog Food
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
Kathy’s family has two dogs, Charlie and Spot. Charlie is a big dog and eats a whole
can of dog food each day. Spot is smaller and eats only 23 of a can each day. If dog
food costs $2.00 for three cans, how much will it cost Kathy’s family to feed the dogs
each month (30 days)? You may draw pictures to assist you. Show your work clearly
so another person could tell how you arrived at your answer.
Taken from A Collection of Performance Tasks – Upper Elementary School
Mathematics by Charlotte Danielson. Teacher instructions and rubric are on pages 84
through 93. Copy for students is on page 185.
Title:
Notable Fractions
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
Show a whole note, half note, quarter note and eighth note. The names of musical
notes are given by fractions to tell how long to play the note. A whole note is played
twice as long as a half note. A quarter note is played half as long as a half note. Make
pairs of different or equivalent note combinations. For example, one card could show
one half note and another card can show two quarter notes. Play concentration with
your cards. Play until one player makes three matches.
Taken from SF Mathematics Grade 5 Teachers Edition Volume 3, page 410B
Title:
Mixed Number Match Up
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
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Have each partner write 5 pairs of equivalent mixed numbers such as 3 12 and 2 2, and 4
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and 3 74 as shown on page 472B. The second mixed number shows a renaming used in
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subtraction. Have students mix up the cards and make a 5 by 4 array with their cards.
Students should take turns finding matches, keeping the matches they find. If a student
chooses two unequal mixed numbers, he or she loses that turn. Play continues until all
matches are made.
Taken from SF Mathematics Grade 5 Teachers Edition Volume 3, page 472B
Title:
Multiplying Fractions of the States
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
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How many states are there in the United States? If 25
of the states entered the Union
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before 1800, how many is that? If 10 entered before 1900, how many entered after
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1900? Can you name them? If 25
of the states have names that start with M, how many
is that? Can you name them?
Create two questions about our states using fractions. Provide an answer key.
Challenge another student to solve your problems.
Taken from SF Mathematics Grade 5 Teachers Edition Volume 3, page 506B
Title:
Follow The Bouncing Ball
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
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A ball rebounds 2 of the height from which it is dropped.
Assume the ball is dropped 128 feet from the top of a school and keeps bouncing.
How far will the ball have traveled up and down when it strikes the ground for the third
time?
How far will the ball have traveled up and down when it strikes the ground for the
fourth time?
How far will the ball have traveled up and down when it strikes the ground for the fifth
time?
Draw a diagram or create a table to organize the drops and rebounds.
Answer using complete sentences.
Title:
Farmer Goes To Market
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
1
An apple farmer went to town to sell a truck full of apples. At his first stop, he sold 3
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of his apples. On the way to the next stop, he wrecked his truck and lost 3 of the
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remaining apples. Of the apples he had left, the farmer gave away 2 to some local
children. He then had 80 apples left.
How many apples did the farmer have when he started his trip?
Write a full explanation of your answer including all calculations used to solve.
Title:
Eating Habits
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
During a recent school survey of two middle school classrooms, 13 of the students
reported that they bring their lunch to school. Another 14 reported that they buy the
lunch on the menu in the cafeteria and 16 reported that they only buy the snack from the
snack stand for lunch. The remaining 18 students reported that they don’t eat lunch.
How many students were surveyed from the two classes at the middle school?
Write a full explanation of your answer including all calculations used to solve.
Title:
All Gassed Up
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
When the mechanic at a car dealership checked the cars on the lot, he found that some
were 12 full of gas and some were 14 full of gas.
There were a total of 85 cars but only enough gas to fill 30 tanks. All of the cars have
the same size gas tank. How many cars were 12 full?
Write a full explanation of your answer including all calculations used to solve.
Title:
On and Off
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
1. An airplane departs from an airport. At its first stop, 12 the passengers leave the
plane and 10 new passengers board. There are now 124 passengers on board
the plane.
How many passengers began the trip? How many departed on the first stop?
2.
A certain number of passengers board a cruise ship in Miami. When the ship
reaches Jamaica, 12 of the passengers leave the ship and 40 new passengers
board. At the Bahamas port, 14 of the passengers leave and 52 new passengers
board. When the ship reaches Bermuda, 15 of the passengers leave and 35 new
passengers board.
There are now 163 passengers on board the ship. How many passengers began
the cruise in Miami?
Write a full explanation of your answer including all calculations used to solve.
Title:
How Many Teeners?
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
The South Teener factory workers can make 36 small Teeners or 14 large Teeners in
one hour. The North Teener factory can produce Teeners 1 12 times faster.
How many more large Teeners can the North Teener factory produce in an 8-hour day
than the South Teener factory?
Write a full explanation of your answer including all calculations used to solve.
Title:
A Treacherous Ride
Bloom’s Taxonomy: Analysis & Evaluation
Gardner’s MI: Logical/Mathematical, Visual/Spatial
A San Francisco streetcar turns curves at the bottom of hills very rapidly. One day a
careless streetcar driver turned the first of three curves so rapidly that he lost 12 of the
passengers. At the second and third curves he again lost 12 of his remaining passengers
at each curve. Despite all of this, 8 passengers managed to hang on until the end of the
ride.
How many passengers started this treacherous streetcar ride?
Write a full explanation of your answer including all calculations used to solve.
Performance Task Rubric
General Scoring Rubric (to be used for all applicable tasks):
Score of 3: Student shows a correct and/or appropriate answer and shows work
and/or an explanation that demonstrates full and complete understanding.
Score of 2: Student has minor flaws in the answer, but the work and/or
explanation is acceptable and the reasoning is appropriate.
Score of 1: Student does not have a reasonable explanation or show sufficient
work, resulting in a demonstration of only limited understanding.
Score of 0: Student shows no understanding of the problem or how to arrive at a
solution.
OTHER EVIDENCE:
Daily Warm-Up and Open-Ended Questions
Daily Homework
Observation of Classroom Activities
Quizzes
Tests
Diagnostic Assessments
District Assessments
State Assessments
Stage 3 – Learning Plan
Select a performance task from Stage 2 based on individual progress in this unit.
Day 1- Introduction to Chapter 6
Day 2- Lesson 6.1- Investigate- Addition with Unlike Denominators
Use models to add fractions with unlike denominators.
Day 3- Lesson 6.2- Investigate- Subtraction with Unlike Denominators
Use models to subtract fractions with unlike denominators.
Day 4- Lesson 6.3- Estimate Fraction Sums and Differences
Make reasonable estimates of fraction sums and differences.
Day 5- Lesson 6.4-Common Denominators and Equivalent Fractions
Find a common denominator or a least common denominator to write
equivalent
Fractions.
Day 6- Lesson 6.5- Add and Subtract Fractions
Use equivalent fractions to add and subtract fractions.
Day 7- Mid Chapter Checkpoint- Quiz on Lessons 6.1 through 6.5
Day 8- Lesson 6.6- Add and Subtract mixed numbers
Add and subtract mixed numbers with unlike denominators.
Day 9- Lesson 6.7- Subtraction with Renaming
Rename to find the differences of two mixed numbers.
Day 10- Lesson 6.8- Algebra- Patterns with Fractions
Identify, describe, and create numeric patterns with fraction.
Day 11- Lesson 6.9- Problem Solving- Practice Addition and Subtraction
Solve problems using the strategy work backward.
Day 12- Lesson 6.10- Algebra-Use Properties of Addition
Add fractions and mixed numbers with unlike denominators using the
properties.
Day 13- Review on Lessons 6.1 through 6.10
Day 14- Test on Chapter 6
Day 15- Introduction to Chapter 7
Day 16- Lesson 7.1- Find Part of a Group
Model to find the fractional part of a group.
Day 17- Lesson 7.2- Investigate- Multiply Fractions and Whole Numbers
Model the product of a fraction and a whole number.
Day 18- Lesson 7.3- Fraction and Whole Number Multiplication
Multiply fractions and whole numbers.
Day 19- Lesson 7.4- Investigate- Multiply Fractions
Multiply fractions using models.
Day 20- Lesson 7.5- Compare Fraction Factors and Products
Relate the size of the product compared to the size of one factor when
Fractions.
Day 21- Lesson 7.6- Fraction Multiplication
Multiply fractions.
Day 22- Mid Chapter Checkpoint- Quiz on 7.1 through 7.6
Day 23- Lesson 7.7-Investigate- Area and Mixed Numbers
Use a model to multiply two mixed numbers and find the area of a
rectangle.
Day 24- Lesson 7.8- Compare Mixed Number Factors and Products
Relate the size of the product to the factors when multiplying fractions
greater than one.
Day 25- Lesson 7.9- Multiply Mixed Numbers
Multiply mixed numbers.
Day 26- Lesson 7.10- Problem Solving- Find Unknown Lengths
Solve problems using the strategy guess, check, and revise.
Day 27- Review on Lesson 7.1 through 7.10
Day 28- Test on Chapter 7
Day 29- Introduction to Chapter 8
Day 30- Lesson 8.1- Investigate- Divide Fractions and Whole Numbers
Divide a whole number by a fraction and divide a fraction by a whole
number.
Day 31- Lesson 8.2- Problem Solving- Use Multiplication
Solve problems using the strategy draw a diagram.
Day 32- Lesson 8.3- Connect Fractions to Division
Interpret a fraction as division and solve whole-number division problems
that result in a fraction or mixed numbers.
Day 33- Mid Chapter Checkpoint- Quiz on lessons 8.1 through 8.3
Day 34- Lesson 8.4- Fraction and Whole- Number Division
Divide a whole number by a fraction and divide a fraction by a whole number.
Day 35- Lesson 8.5- Interpret Division with Fractions
Represent division by drawing diagrams and writing story problems and
equations.
Day 36- Review on Lesson 8.1 through 8.5
Day 37- Test on Chapter 8
Unit Resources/References Needed:
Textbook References:
Go Math! -Grade 5 Houghton Mifflin Harcourt Publishing Company
Chapters 6, 7, & 8
Materials Needed:
Printed Materials from Go Math! Grade 5
Connected Mathematics 2- Bits and Pieces II- Computing With Decimals and
Percents by Lappan, Fey, Fitzgerald, Friel, Phillips
Common Core State Standards
National Council for Teachers of Mathematics Standards
Various Materials for Performance Tasks and Instructional Activities
SMART Technology
Technology Integration:
Go Math! Grade 5 Online, http://www-k6.thinkcentral.com
www.hmheducation.com/gomath
www.smckids.com
Illuminations activities on for Grade 5.
http://illuminations.nctm.org/Activities.aspx?grade=2
SMART Software and Hardware Products
SMART Exchange
http://exchange.smarttech.com/#tab=0
IXL Math – Practice and Lessons
http://www.ixl.com/
Research, Development and Accountability, Mathematics Performance Task Bank
ww.rda.aps.edu/mathtaskbank/fi_html/pfactask.htm
AAA Math:
Interactive mathematical practice opportunities with place value of
decimals
http://www.aaamath.com/plc51b-placevalues.html
Illuminations activities on Number Sense and Operations
http://illuminations.nctm.org/WebResourceList.aspx?Ref=2&Std=0&Grd=0
NCTM Technology Principles and Standards for School Mathematics
http://standards.nctm.org/document/eexamples/index.htm#6-8
4. INDIVIDUAL ACCOMMODATIONS
Extra support:
Refer to Houghton Mifflin Harcourt Go Math! Grade 5 Mathematics Teacher’s Edition for
“Extra Support.” The section labeled “If students need lesson support…” provides resources
by chapter for those who need additional assistance. If time and supervision allows, break
students into small groups and re-teach the lesson. Have the students work together in pairs or
small groups to try to work through the problems. Use appropriate manipulatives to explain
the concepts in a different way.
Enrichment or early finishers:
Refer Houghton Mifflin Harcourt Go Math! Grade 5 Mathematics Teacher’s Edition for
“Enrichment Strategies.” The section labeled “If students are ready for enrichment…”
provides resources by chapter for enrichment and extension. If time and supervision allows,
give students the opportunity to peer tutor students who are struggling. This will allow
students to demonstrate what they have learned. Completion of IXL topics will reinforce and
introduce concepts.
Various Learning Styles:
Refer to Houghton Mifflin Harcourt Go Math! Grade 5 Mathematics Teacher’s Edition for
“Differentiated Instruction.” Teach each lesson with components that appeal to students
various learning styles. Focus on kinesthetic, auditory, visual, and tactile. If time and
resources allow, employ the use of manipulatives, math journals, modeling, group discussions,
and other appropriate types of alternative assessment.
Limited English proficiency:
Refer Houghton Mifflin Harcourt Go Math! Grade 5 Mathematics Teacher’s Edition for
“Language Support”. Ensure that vocabulary is previewed with the student. If possible,
provide pictures for the student to help reinforce the vocabulary.
5. TEACHER REFLECTION
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Were my students talking about the subject, or was I doing all of the talking and
students were just listening to me?
Were my students engaged at the beginning of the lesson?
How much time did I spend reviewing homework, and how much time did I spend on
new material?
Did the students respond to “How” and “Why” questions?
Did my students have an opportunity to discuss and/or write about the topic?
What changes would I make next time the lesson is taught?
What steps do I need to take next in this topic?