Mathematics: Geometry
3.G.1
Cluster Heading: Reason with shapes and their attributes.
Content Standard: Understand that shapes in different categories may share attributes, and that the shared attributes can define a larger
category. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not
belong to any of these subcategories.
Practice Standards: MP 5 Use appropriate tools strategically, MP8 Look for and express regularity in repeated reasoning.
Problem/Task Suggestions
Quadrilateral Questions
Create a single design on a geoboard using only 1 rubber band.
Say “I have a category or characteristic in mind. Guess the category based on the two
groups of geoboards I display”. (The teacher will use 8-10 of the shapes created by the
students and will separate these geoboards into 2 groups -- quadrilaterals and nonquadrilaterals. If a student or group thinks they know the category the teacher has in
mind, rather than stating what the category is, they would place another geoboard
into the existing groups to check their theory. This will allow more students to keep
engaged in thinking about the problem. After it is clear most students have predicted
the category, spend time in whole group processing the vocabulary of quadrilateral
and sub-categories of quadrilaterals. Allow students to create their own definition of
quadrilateral and provide counterexamples of shapes that fit their definition to show
the need to refine their definition. For instance, draw an open shape with 4 sides or
one that has curved sides. Now that the students have seen the process modeled, they
will work in their groups to create geoboard categories for others to guess)
If you wish to do these activities in a written format, say “The geoboards on the left
have the characteristic, the geoboards on the right do not. Describe this
characteristic in writing. In your description, include a drawing of one example that
has the characteristic and one example that does not.”
Differentiation
Support
• Give the student a word bank of possible characteristics or categories.
Extension
•
Formative Assessment Suggestions
Observation of Students
Is the student able to
• Generalize a property by looking at specific designs?
• Verbalize an appropriate definition for a geometric property? MP6
• Create an appropriate drawing(s) for the property?
• Represent a generalization from specific drawings? MP8
• Organize general shapes by attributes?
Questions to Guide Student Thinking
• In non-math terms, describe the shapes on the left and on the right.
• What characteristics have we talked about in class? Describe to me.
• Could you put a geoboard in both the right and the left categories?
Misconceptions
Students may
• Not represent a common characteristic
• Not know the properties or attributes of shapes well enough to
distinguish among them.
• List a characteristic that works for some, but not all.
Vocabulary
• Categories: quadrilateral, squares, rectangles, parallelogram,
triangles, etc.
• Characteristics: containing a right angle, parallel lines or a line of
symmetry
Create examples and non-examples on geo-board dot paper and ask another
group member to guess the characteristic.
Adapted from: Math Solutions. Burns, Marilyn. Mathematics: For Middle School. New Rochelle, N.Y.: Cuisenaire Co. of America, 1989.
Mathematics: Measurement and Data
3.MD.4
Cluster Heading: Represent and interpret data.
Content Standard: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data
by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
Practice Standards: MP1 Make sense of problems and persevere in solving them, MP3 Construct viable arguments and critique the reasoning
of others, MP5 Use appropriate tools strategically.
Problem/Task Suggestion
The Long and the Short
Give each student a baggie and instruct him or her to fill the baggie with ten items
that are less than ten inches in length. They may find items in the room or cut
items to fit the criteria (strips of paper, string, etc). Items in the bag should include
items of the same length when possible.
Students will exchange bags with a partner and measure each item to the quarter
inch. Students will create a line plot with the measurement data. When finished
they will work with their partner to check both sets of measurements and line
plots. Students should talk about what they noticed about their line plots and how
they would correct any errors. Students will write individual summaries of the
math they learned below their own line plots and include a reflection of how they
could improve their work.
Ask the group how to recognize a good line plot, and a good math summary.
Create a list of class suggestions on the board and give them time to make changes
to their papers.
Differentiation
Supports
• Have sample line plots posted or available that students can use as a
reference.
• Have rulers available that have only inch markings and not centimeters.
• Ask students what the shortest and longest length they have in their collection
and how that will determine what increment marks they will need on their
line plot.
• Provide a ready-made line plot with hash marks in place but only a couple
labeled.
Extension
• Create two line plots with the same items, one line plot in centimeters and
one in inches.
Created by: Illinois State Board of Education Math Content Specialists
Formative Assessment Suggestions
Observations
• Does the student understand the problem, including measurement
restrictions and requirements?
• Is the student using his/her ruler to accurately measure items?
• Is the student communicating appropriately about the math related to
the task with a partner?
Evaluating each student’s written work
• Does the line plot have a title?
• Are increments marked on the line plot, and increments consistent in
length?
• Are math summary statements correct and extensive?
• Are statements of how to improve work constructive?
Misconceptions
• Measure from the edge of the rule instead of the 0 mark.
• Measure some items in inches and others in centimeters.
• Think there has to be a vertical scale as well as a horizontal scale on line
plots.
• Not understand once they have established an increment length on their
line plot that they have to consistently use the same increment distance
Vocabulary
• Line plot, increment
Mathematics: Measurement and Data
3.MD.8
Cluster Heading: Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area
measures.
Content Standard: Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the
side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area
and different perimeters.
Practice Standards: MP2 Reason abstractly and quantitatively, MP 1 Make sense of problems and persevere in solving them.
Problem/Task Suggestions
Puppy Problem
Max is Ms. Weber’s new puppy. She decided to build him a pen so that he can
be outside while she is at school. She has 50 feet of fencing in her storage
shed to use. What size should the pen be to maximize the area for the puppy?
Differentiation
Supports
• Have students use manipulatives, such as color tiles, unifix cubes,
ribbon, yarn, geoboards or graph paper.
• Let students solve a similar problem, e.g., with 24 ft. of fencing.
• Ask students to solve within whole number feet.
Extension
• Have students write a letter to the principal, or to a local pet shop,
generalizing the solution strategies, using three sides of fencing and
four sides of fencing.
• Have students research and report on how fencing is sold.
• Read Measuring Penny by Loreen Leedy
Solution
The pen with the largest area is 12.5 ft. x 12.5 ft.
The largest pen with whole number sides is 12 ft. x 13 ft.
Answers may vary for students who decide to use the side of the house as one
side of the fence or non-rectangular solutions.
Formative Assessment Suggestions
Ask students to write journal responses to the following:
• What are some of the ways she can set up the pen to use all of the
fencing?
• What would be the dimensions of a rectangular pen with the most space
available for Max to play? Defend your answer.
• Write a letter to Ms. Weber explaining the choices she has and which pen
you recommend for her to make. Be sure to show how you made your
decision and include a mathematical representation to support your
solution.
Questions to Guide Student Thinking
• What numbers have you tried so far?
• How will you represent your solutions?
• How will you make your recommendations?
• Are there any other possibilities?
• How will you show your work?
Misconceptions
• Students may include a drawing, but forget to include actual
measurements.
• Students may include formulas, but forget to include a mathematical
representation.
• Students might assume sides of the pen have to be in whole number
lengths.
• Students may not label the measurements clearly.
Source for Student Page: http://www.rda.aps.edu/mathtaskbank/pdfs/tasks/3-5/t35PuppyProb.pdf
Source for Teacher Pages: http://www.rda.aps.edu/mathtaskbank/pdfs/instruct/3-5/i35PuppyProb.pdf
Mathematics: Operations and Algebraic Thinking
3.OA.3
Cluster Heading: Represent and solve problems involving multiplication and division.
Content Standard: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and
measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Practice Standards: MP7 Look for and make use of structure, MP1 Make sense of problems and persevere in solving them.
Problem/Task Suggestions
Terrific Tiles
Mrs. McKegney wants to tile her kitchen floor.
The square tiles are 1 ft. x 1 ft. The kitchen floor is a 10 ft. x 9 ft. area. How many
tiles does she need?
Tiles are sold in bundles of 10. Each bundle costs $3.00. How many bundles are
needed? How much will she need to pay for the tiles?
Pretend you are her contractor and write a letter to Mrs. McKegney clearly
explaining your solution to her tile problem and how you went about solving the
problem. Be sure to include a visual representation of your solution and
observations that you made.
Differentiation
Support
• Have students use a room with smaller dimensions.
• Let students solve a problem based on measurements of a room in a
dollhouse
• Have students use color tiles or graph paper.
Extensions
• Have students create a rubric for grading this problem.
• Have students create their own dimensions for various rooms and then
solve.
• Ask students to measure rooms in their houses and write a problem
involving square tiles.
• Use smaller tiles and challenge students to show how they would
determine the number of tiles in a square foot before they solve a larger
problem.
Formative Assessment Suggestions
Student Journal Prompt
Write a letter to explain to your parents what you learned today. Draw out a
sample solution and explain the strategies you used. Show how you
organized your work to keep track of each question.
Questions to Guide Student Thinking
• How could you organize your work for this problem?
• How could you represent the room and the tiles?
• What do the numbers in the problem and in your solution represent?
• What different types of computation will you be using? Explain your
thinking.
Misconceptions
• Some students may manipulate the numbers presented, but not
comprehend the situation.
• Some students may neglect to find the number of bundles needed
and the total cost.
• Some students may have trouble writing a letter to Mrs. McKegney
and communicating a solution clearly.
• Some students may not include a visual representation or their
observations.
Solution: 90 tiles are needed; 9 bundles are needed.
She will need 9 bundles x $3.00 per bundle for the tiles, or $27.00.
Adapted from: http://www.rda.aps.edu/mathtaskbank/pdfs/tasks/3-5/t35tiles.pdf
Source for Similar Problems and Teacher Pages: http://www.rda.aps.edu/mathtaskbank/pdfs/instruct/3-5/i35tiles.pdf
Mathematics: Measurement and Data
4.MD.3
Cluster Heading: Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Content Standard: Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
Practice Standard: MP5 Use appropriate tools strategically.
Problem/Task Suggestions
Four Square Court Challenge
A physical education teacher wants to set up a four square tournament in a
large room. We will need to calculate how many four square courts can be
taped onto the floor for the tournament. We want to use as many courts as
possible. The dimensions of a four square court is 8’ x 8’ or 64 square feet. In
pairs, measure the room and figure out how many four square courts would
fit into the large room/area. Show, on chart paper, how you came up with
your solution and be able to justify the solution.
Differentiation
Supports
1. Give students a specified size for the Court that includes a specific
area for observers, such as 10’ x 10’.
2. Have the class decide on a reasonable size room and share the
dimensions prior to the start of the task.
Extensions
1. Prior to the start of this task, have students research the history of the
Four Square Court, as well as recommended space considerations,
and write a summary for the class.
2. Have one of the gym teachers come to class to select the most
appropriate design and explanation.
3. If this was a school event, what other space considerations might be
necessary?
Solutions will vary, depending on how much space is allowed for observers.
Sample solution with back-to-back Four Square Courts: If the gym was 100 ft
x 60 ft, or 6000 square feet, there would be room for 12 across and 7 down,
84 Four Square Courts. With a size of 6000 square feet, and 10’ x 10’ allocated
for each Court, there would be 6000/100 or 60 Four Square Courts available.
Adapted from: http://illuminations.nctm.org/LessonDetail.aspx?id=L860
Formative Assessment Suggestions
Observation of Students
Collect data on the following
• Can students justify their tournament design orally?
• Do students recognize that there is more than one correct answer?
Examining student written work
Do students
• Reflect on at least two things they observed about their group’s design?
• Relate how they would modify their design based on others’ designs?
Questions
To guide incorrect or incomplete responses
• How many courts do you think would fit in the large room/area?
• How could you determine the desired grid size on your graph paper?
• How much room is needed around each game court for players to wait
their turns?
To extend correct responses
• Compare your design with another group’s design. Select which you
think is the best. Take this one and compare it to another group and
select your favorite again. Defend your decision orally and in writing.
Write a letter to the physical education teacher to explain why a design
is most appropriate for the upcoming tournament.
Misconceptions
Students may
• Overlap squares or place squares adjacent to each other without any
space in-between.
• Confuse perimeter and area.
Mathematics: Measurement and Data
4.MD.6
Cluster Heading: Geometric measurement: understand concepts of angle and measure angles
Content Standard: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
Practice Standards: MP1 Make sense of the problem, MP5 Use appropriate tools strategically
Problem/Task Suggestion
Formative Assessment Suggestions
Protractor Game
The Protractor Game is played with partners. The first partner gives the
second partner a number between 0 and 180. The second partner then
attempts to draw an angle with the given number of degrees using only a
straight edge. Once the angle has been drawn, both students use a protractor
to measure it and the score for the student who drew the angle is the
difference between the actual measurement and the given
measurement. Play five rounds with each person getting a chance to give the
number of degrees five times and draw an angle 5 times. Add up the scores
and the lowest score wins.
Create a score sheet with the following headings and a total for the
differences at the bottom:
Given Number
Actual Measure
Difference
Assessment
• Put drawings of angles in multiple orientations with several possible
measurements next to each angle as choices for answers. Have students
write which measurement they think is the correct measurement and
why.
• Put numbers representing degrees in four locations on the page and ask
students to sketch an angle with that number of degrees and explain
how they made their sketch.
Differentiation
Support
• Have the students use toothpicks or a virtual geoboard to model angles
that are 90 degrees, greater than 90 degrees and less than 90 degrees.
• Have the students draw a dotted line of a 90 degree angle to use as a
reference for drawing any angle. Ask them to consider where 45 degrees
would be and use this as another reference.
• Talk about the two different scales on a protractor. Using 90 degrees as a
reference, decide if their answer should be more or less than 90 degrees.
i.e. Compare 20 and 160 degrees.
• Have the students only give angles in increments of 5 degrees.
Extension
• Have students create drawings with different angles within it. Exchange
drawings and have students find an acute, obtuse, and 90 angle within the
drawing.
Solutions: Answers will be related to student angle choices.
Task created by: Illinois State Board of Education Content Area Specialists
Questions to Guide Student Thinking
• Can you use two straws to demonstrate an angle greater than 90° and an
angle less than 90°?
• Can you draw a very large picture of a very small angle?
• Can you draw a tiny picture of a large angle?
• Does the length of the legs of the angle have an effect on angle size?
• How do you decide, when using a protractor, which number to use?
Misconceptions
• Students may think that if you change the orientation of the angle it
affects the measurement of the angle.
• Students will often draw a picture of a triangle when asked to draw an
angle.
• Students may think that lengthening the ray of an angle will make the
angle bigger.
• Students will call the rays of an angle lines and not rays.
Vocabulary
Vertex, ray, degree, acute, obtuse
Mathematics: Numbers and Operations in Base Ten
4.NBT.4
Cluster Heading: Use place value understanding and properties of operations to perform multi-digit arithmetic.
Content Standard: Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Practice Standard: MP2 Reason abstractly and quantitatively.
Problem/Task Suggestions
Ages Away
After reading a book.
Based on the copyright date, find out if you or the book you read is older and
how much newer or older. Show your work and explain in words.
Differentiation
Support
• Find the copyright date for the student.
Extensions
• Using the Internet or the classroom library, find a book that is exactly
ten years older than you.
• Have the students use several books to compute the age difference
and put them in order from closest to their age to farthest from their
age.
Solutions:
• Students will correctly find the copyright date and subtract such as 20031984 to realize that the book is 19 years older than them.
• Students will correctly find the copyright date and create a number
sentence such as 1984 +
= 2003 and realize that the blank must be
19.
Created by: Illinois State Board of Education Content Area Specialists
Formative Assessment Suggestions
Observation of Students:
• Are students familiar with books enough to know where to locate the
copyright date?
• Are students able to set up either an addition or subtraction problem to
solve?
• Are students fluently adding and subtracting within the problem?
• Are students able to write a sentence that correctly states if they or the
book are older and how much older?
Questions to Guide Student Thinking
• Where in a book is the copyright located?
• Is the larger or smaller date older?
Misconceptions
Students may:
• Be unable to find the copyright date.
• Try to subtract the number that is larger from the number that is smaller.
• Believe that the larger number is older by relating age to the birth year.
• State their sentence backwards, e.g., if the book is 19 years older, they
may say that they are 19 years older.
• Be confused at the concept that there could be a greater difference than
their age is.
• Be unable to subtract 4-digit numbers.
Vocabulary Considerations
• Copyright, thousands, hundreds, tens, ones
Mathematics: Number and Operations - Fractions
4.NF.2
Cluster Heading: Extend understanding of fraction equivalence and ordering.
Content Standard: Compare two fractions with different numerators and different denominators.
Practice Standard: MP3 Construct viable arguments and critique the reasoning of others.
Problem/Task Suggestions
Roll a Fraction
Challenge students to build the greatest fraction. Arrange students in pairs.
Distribute a traditional six-sided die to each pair of students. Ask students to set
up a piece of paper as shown:
•
Formative Assessment Suggestions
Assessment
• Check to be sure students are correctly finding the greater fraction.
• Have students explain their reasoning.
• Collect and read student strategies to decide which students
recognized that larger numbers should go in the numerator to create
a greater fraction.
Questions to Guide Student Thinking
• Can you show me the fraction with fraction strips?
• What does the numerator tell you?
• What does the denominator tell you?
• How do you know which fraction is greater?
• What is the best roll to get? Why?
Direct students to take turns rolling a die and placing the number
into one of the boxes. They should continue until both players’
boxes are filled in and they each have a fraction.
• Ask students to determine the greater fraction by explaining and/or
using drawings or number lines.
Misconceptions
• Ask students to place a greater than (>), less than (<), or equal
(=) sign between the fractions
Students may
• Have students play the game multiple times, and develop some
• Think that the bigger the denominator will result in the greater fraction
strategies for building the greatest fraction. Have students write
• Have trouble finding the fractions on a number line
their strategies, and then repeat the game.
• Have trouble with fractions like 3/1 or 5/3, which result in numbers
Differentiation
greater than 1
Supports
• Misunderstand fractions with the same numerator and denominator,
• Supply fraction manipulatives, number line with fractions or grid paper and
resulting in the number 1
encourage students to build each fraction before comparing
• Vary dice per student need (use sticky dots 1, 2, 4, 4, 8, 8)
Vocabulary Considerations
Extensions
Numerator, Denominator, Equivalent Fractions, Greater Than, Less Than,
• Create dice with any of the following numbers, such as 1, 2, 3, 4, 5, 6, 8, 10,
Equal, Improper Fraction
12, 100
• Let students decide what numbers to place on the die
• Have a recording sheet with five empty boxes to indicate 2 fractions and a
reject box. The students add the two fractions to determine which player had
the largest sum.
Adapted from The Comparing Game, Marilyn Burns Teaching Arithmetic: Lessons for Extending Fractions, Grade 5, Math Solutions Publications, 2003,
pp. 56–63, 145–152.
Mathematics: Numbers and Operations in Base Ten
5.NBT.4a
Cluster Heading: Understand the place value system.
Content Standard: Read, write, and compare decimals to thousandths. b. Compare two decimals to thousandths based on meaning of the
digits in each place, using >, =. And < symbols to record the results of comparisons.
Practice Standard(s): MP1 Make sense of the problem and persevere in solving it.
Problem/Task Suggestions
Comparing is in the Cards
Place the following cards into the blanks to make the greatest number and the
smallest number. Write these two numbers on a piece of paper with a <, >, or
= to make a true statement. (Cards are 8, 3, 5, 1, and 7)
.
Differentiation
Supports
• Ask the student to first create the greatest possible two-digit whole
number. Then, use the remaining three to find the greatest possible
decimal number.
• Give the students an additional problem that only compares numbers
to the tenths.
Extensions
• What would be the second largest number with the same digits?
• Write a number that would come between the two numbers.
Solutions:
• 87.531 > 13.578
• 13.578 < 87.531
Formative Assessment Suggestions
Observation of Students:
• Do students realize that the digit farthest to the left affects the value
of the number the most, and the number farthest to the right affects
the value the least?
• Are students able to correctly use <, >, and/or =?
Questions to Guide Student Thinking
• What would the greatest number be with the same blanks and any
cards?
• Compare your answer to other students and decide if there is the
possibility for a larger or smaller number with the same digits. Be
prepared to justify your answer. MP3
• What are the place values of each of the blanks and what do they
mean?
Misconceptions
Students may:
• Put the numbers in reverse order for each.
• Put the cards in whatever order they pick them up.
• Believe that the digit after the decimal place is the most important,
e.g., thinking 13.875 would be the greatest).
• Misunderstand the numbers after the decimal place. (i.e. 87.135
would be the highest.
• The student may put in his/her own numbers, e.g., creating a high
number of 99.999.
Vocabulary
• Tens, ones, tenths, hundredths, thousandths, decimal, decimal point
Adapted from: Tokyo Shoseki Mathematics for Elementary School 5A p. 6
Mathematics: Numbers and Operations – Fractions
Cluster Heading: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Content Standard: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Interpret
5.NF.4a
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.
Practice Standard: MP1 Make sense of problems and persevere in solving them.
Problem/Task Suggestions
Formative Assessment Suggestions
Sixes are Wild
Complete the math equation below by using four of the cards provided:
×
=6
(The teacher needs to create cards 1, 2, 3, 4, 5, 6, 7, 8, 9)
Differentiation
Supports
 Could you make an equation with the following 4 cards? (Teacher
gives student(s) the four cards: 1, 2, 3, and 9.)
 Can you make one improper fraction that equals a whole number?
Extensions
 Is there another possibility? How many can you find?
 What different fractions would work if you were allowed to use a card
twice?
Solutions
o
9
3
×2=6
4
9
3
×4=6
o
3
1
×2=6
o
4
3
× =6
o
4
1
×2=6
8
4
3
1
×4=6
9
2
8
3
× =6
3
8
1
×4=6
Observation of Students
 Is the student able to use mental math to do simple computation or do
they need to use another tool? MP5
 Does the student recognize that there is more than one correct answer?
 Does the student recognize that he/she cannot use the card more than
once in an equation and that he/she does not have to use every card in
each equation?
 Does the student include a correct equation?
 Does the student include multiple correct equations?
 Does the student recognize the pattern that the product of the
numerators divided by the product of the denominators equals 6? MP7
Questions to Guide Student Thinking
 Multiply the fractions you created and check to see if they equal 6.
 Describe features that are necessary to guarantee that you have created
a true equation.
 What operations are involved in this problem?
 Is there a problem like this that you know how to solve? (Allow the
student to write a problem that does not have to equal six)
8
9
4
Misconceptions
Students may
 Use the same card more than once.
 Create fractions that do not have a product of 6.
3
Vocabulary
 Numerator, denominator, equation
Adapted from: Sugiyama, Yoshishige. Tokyo Shoseki's Mathematics for Elementary School Grade 6a. Madison, NJ: Global Education Resources, LLC, 2006.
Print.