Download K-5 Tasks - Possible Solutions and Notes

K-5 Tasks - Possible Solutions
and Notes
Quarterly Meeting # 2
November 2013
Alabama Department of Education Task: Counting Balls
Kindergarten Cedric and Caleb like to play with different kinds of sports balls. Cedric has 5 tennis balls and Caleb has 5 golf balls. Cedric says that he has more balls because the tennis balls are bigger. Do you think Cedric is right? Explain your answer with words and pictures. (See below for pictorial representation of the size of the balls) Teacher Notes: Use of actual tennis balls and golf balls or cut outs of the attached figures as manipulatives is recommended. Conservation of number is essential to conceptual understanding of counting. College- and Career-Ready State Standards for Mathematical Content
Count to tell the number of objects.
4.Understand the relationship between numbers and quantities; connect counting to cardinality.[K-CC4]
a.When counting objects, say the number names in the standard order, pairing each objectwith one and only one
number name and each number name with one and only one object.[K-CC4a]
b.Understand that the last number name said tells the number of objects counted. Thenumber of objects is the same
regardless of their arrangement or the order in which theywere counted. [K-CC4b]
c.Understand that each successive number name refers to a quantity that is one larger.[K-CC4c]
5.Count to answer “how many?” questions about as many as 20 things arranged in a line, arectangular array, or a
circle, or as many as 10 things in a scattered configuration; given a numberfrom 1-20, count out that many objects.
[K-CC5]
Standards for Mathematical Practice 1. Make sense of problems and
persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in
repeated reasoning.
Essential Understandings • Counting includes one‐to‐one correspondence, regardless of the kind of objects in the set and the order in which they are counted.
• When counting objects in a group/set, the last number stated names the total number of objects in that group/set.
Explore Phase Possible Solution Paths Assessing and Advancing Questions Student will count each set of balls and label or name them with the correct number. Student will then state that because there is the same number of tennis balls that there is of golf balls Cedric and Caleb have the same or equal groups so Cedric’s claim is incorrect. Assessing Question: How did you decide that Cedric was not right? What strategy did you use to determine whether Cedric had more? Advancing Question: What would you change in the problem to make Cedric’s claim correct? Does the size of the objects make a difference in this problem? Is there another way to show /explain your answer? Assessing Question: How did you know to line the balls up? Student may line the balls up using one‐to‐one correspondence to Did that make it easier? Why or why not? show that the groups/sets of tennis balls and golf balls are the Advancing Question: same. Cedric’s claim is incorrect. What would you change in the problem to make Cedric’s claim correct? Does the size of the objects make a difference? Is there another way to show /explain your answer? Assessing Question: Tell me how you decided to draw the lines to match a tennis ball and a Student may use the pictorial representation to connect one tennis golf ball. ball to one golf ball until all are paired and relate that the boys Tell me how matching a tennis ball with a golf ball helped you answer the question. have the same number.
Advancing Question: If each boy had some of the tennis balls and some of the golf balls could you use the same strategy? Explain your answer. Possible Student Misconceptions Assessing Questions: Students may think that there are more tennis balls because they Why do you think Cerdic said she had more? are larger than the golf balls. How could you find out? When placed in a line, the line of tennis balls is longer than the line Advancing Question: of golf balls so it may appear there are more. Is there a way to arrange the balls that would help you? Students may count the balls in random order and count some of If you move the balls around would that change your answer? them more than once. Entry/Extensions If students can’t get started…. If students finish early…. Assessing and Advancing Questions Assessing Question: Why do you think Cedric said he had more balls? What can you do to see how many balls each girl has? Advancing Question: Is there a way to arrange the balls that would help you see if Cedric is correct? Have the students show how each boy could have an equal number of balls but a combination of both tennis balls and golf balls. Write number sentences to demonstrate their answers. If you gave each boy two more balls would they still have an equal number of balls? Write number sentences to demonstrate their answers. Discuss/Analyze Whole Group Questions • What ways did you use to count the balls?
• How were the counting strategies that we shared similar and different?
• How did you know if you had already counted a ball? How did you keep from counting it more than one time?
• Does it matter where you start when you are counting objects? Explain your answer.
• When you say the number of the last ball in a group, what does that number tell you?
• If you rearrange the balls in a group and count them again, will you get the same number? Explain your answer.
• If you are counting things in a group, does it matter if the things are different sizes? Explain your answer.
Tennis Balls and Golf Balls Cedric’s Tennis Balls
Caleb’s Golf Balls Alabama Department of Education
1st Grade
Task: The Baseball Cards Collection
John and Isaac are collecting baseball cards. They each have a collection. Use pictures and number sentences (equations) to describe the total
number of cards in each boy’s collection.
•
John has 8 baseball cards. He receives 4 more baseball cards for his birthday. His brother gives him 3 baseball cards. How many
baseball cards does John have in his collection?
•
Isaac has 7 baseball cards. His sister gives him 5 more cards. Isaac finds 4 more baseball cards. How many baseball cards does Isaac
have in his baseball card collection?
•
John says he has more baseball cards in his collection than Isaac has in his collection. Do you agree with John? Explain why or why not.
Use the symbol >, <, or = in your explanation.
Teacher Notes:
Students may choose to solve this problem using direct modeling, counting on, or with reasoning strategies. Whole class discussions should
highlight how numbers can be decomposed and recombined to make groups of tens. 10 is a benchmark number that can make computation
easier. This discussion will also connect using the benchmark number of 10 to aid addition to comparing two numbers based on meanings of
tens and ones. When comparing the totals of 15 and 16, students should be able to identify that each 15 is 1 ten and 5 ones and 16 is one ten
and six ones. Students should have access to manipulatives, such as cubes, counters, etc., to use as needed.
Standards for Mathematical Practice
College- and Career-Ready State Standards for Mathematical Content
2. Solve word problems that call for addition of three whole
numbers whose sum is less than or equal to 20, e.g., by using
objects, drawings, and equations with a symbol for the unknown
number to represent the problem. [1-OA2]
11. Compare two two-digit numbers based on meanings of the
tens and ones digits, recording the results of comparisons with the
symbols >, =, and <. [1-NBT3]
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Essential Understandings
•
The addition of whole numbers is based on sequential counting.
•
Addition equations can be used to describe situations that involve combining quantities.
Explore Phase
Possible Solution Paths
Assessing and Advancing Questions
Direct Modeling
Student counts a set of 8 cubes (or other manipulative or drawing),
then a set of 4 cubes and a set of 3 cubes. Student then counts all
cubes to determine a total of 15 for John. Student then continues
to model with cubes to determine that Isaac has 16.
John’s Collection
Isaac’s Collection
Assessing Questions
• What is the problem asking you to find?
• What do the cubes (or other manipulatives) represent?
• What type of number sentence could describe this situation and
how do you know?
• How do you know who has the most?
Advancing Questions
• How could you represent this problem on a number line?
• How could I rearrange the cubes in groups of tens and how would
that help me know the total?
• What if John receives one more baseball card? Who will have
more and how do you know?
Student can determine using a one-to-one correspondence that
Isaac has one more than John and states that he does not agree
that John has more because 15 < 16.
Counting On
Student begins by starting at 8, counting on 4, counting on 3 more
to determine that John has 15 cards. Student then begins at 7,
counts on 5 more, and then counts on 4 more to determine that
Isaac has 16 cards.
Student may choose to represent this on a number line:
Assessing Questions
• What is the problem asking you to find?
• Describe to me how a number line can be used to help you find
the total.
• What type of number sentence could describe this situation and
how do you know?
• How do you know who has the most?
Advancing Questions
• John started with 8 cards and Isaac started with 7 cards. How is it
John’s Collection
•
possible that Isaac ends with more cards?
How many more cards would each boy need to collect 20 cards
each?
Isaac’s Collection
Student determines that Isaac has more than John because 16 is
larger than 15. Student may also determine this by noting that 16
is one unit past 15 on a number line. Student may also note that 15
is one ten and 5 ones and 16 is one ten and 6 ones, noting that 16
has one more one than 15. Student states that he does not agree
that John has more because 15 < 16.
Reasoning Strategies
Assessing Questions
• What is the problem asking you to find?
Student recognizes that cards are being joined together to make
• Why did you choose to write an addition number sentence?
one card collection and recognizes this as an addition situation.
• Describe how breaking the numbers apart helped you find the
Student uses the following equation to represent the number of
total?
cards in John’s collection:
Advancing Questions
• John started with 8 cards and Isaac started with 7 cards. How is it
8+4+3=
possible that Isaac ends with more cards?
• How many more cards would each boy need to collect 20 cards
Student could decompose the numbers in various ways to make the
each?
computation easier. Below are some examples:
• What are other ways that the number can be broken apart to help
you find the total?
John’s Collection
Looking for Groups of Ten
8+4+3
8+2+2+3
10 + 2 + 3
10 + 5 = 15
Making Doubles + 1
8+4+3
8+7
1+7+7
1 + 14 = 15
Isaac’s Collection
Looking for Groups of Ten
7+5+4
7+ 3 + 2 + 4
10 + 2 + 6
10 + 6 = 16
Looking for Groups of Ten
7+5+4
2+5+5+4
10 + 2 + 4
10 + 6 = 16
Student may note that 15 is one ten and 5 ones and 16 is one ten
and 6 ones, noting that 16 is one more one than 15. Student states
that he does not agree that John has more because 15 < 16.
Possible Student Misconceptions
Student incorrectly counts the number of cards.
Student incorrectly decomposes or incorrectly recombines
numbers.
Entry/Extensions
If students can’t get started….
If students finish early….
Does your answer seem reasonable?
John started with more cards than Isaac. Who received the most
additional cards?
Count the cards again for me, please? Let’s see if we get the same count.
Does your answer seem reasonable?
Assessing and Advancing Questions
Write key questions that can assess and advance student thinking in this
case
How many more cards will each boy need to have 25 cards in his
collection?
If each boy gave 6 cards to a friend, how many cards would each boy
have?
Discuss/Analyze
Whole Group Questions
How did you find the number of cards in each boy’s collection?
Why did you choose to write an addition number sentence (equation)?
How is it possible that students choose different ways to find the answer but they all found the same answer?
How can we determine which boy has the most cards in his collection?
Alabama Department of Education Task: Mural Paper Task
2nd Grade Mrs. Johnston measured some butcher paper for her students to paint a mural on. The paper wasn’t long enough so, she measured 20 more feet of butcher paper. Now she has 65 feet of butcher paper for the mural. How many feet of paper did she measure before? Write an equation that represents this problem. Use a symbol for the unknown number. Solve the problem using words, numbers or pictures to explain your reasoning. Teacher Notes: • Students’ understanding of addition enhances when they have opportunities to think about and model it in various ways.
•
Although it is easy to show students how we picture a situation, we learn a great deal about how they understand the quantities and
operations involved in the situation when they create their own representations of problems (Quintera, 1986).
•
The inverse relationship between addition and subtraction provides the mathematical basis for the fact families, such as the following:
10 + 5 = 15; 5 + 10 = 15; 15 – 5 = 10; 15 – 10 = 5.
Standards for Mathematical Practice College- and Career-Ready State Standards for Mathematical Content Relate addition and subtraction to length.
1. Make sense of problems and persevere in solving
18. Use addition and subtraction within 100 to solve word problems involving lengths that
them. are given in the same units, e.g., by using drawings (such as drawings of rulers) and
2. Reason abstractly and quantitatively.
equations with a symbol for the unknown number to represent the problem. [2-MD5]
3. Construct viable arguments and critique the reasoning
of others. Represent and solve problems involving addition and subtraction.
4. Model with mathematics.
1. Use addition and subtraction within 100 to solve one- and two-step word problems
5. Use appropriate tools strategically.
involving situations of adding to, taking from, putting together, taking apart, and
6. Attend to precision.
comparing with unknowns in all positions, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem. (See Appendix A, Table 1.) [2-OA1] 7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Essential Understandings • Missing numbers in a math sentence/equation or word problem can be found using addition and subtraction.
• Understand how addition and subtraction relate to one another.
• Subtraction is the inverse operation of addition and is used for different reasons:
¾ to remove one amount from another;
¾ to compare one amount to another; and
¾ to find the missing quantity when the whole quantity and part of the quantity are known.
Explore Phase Possible Solution Paths Add To, Start Unknown x + 20 = 65 20 + x = 65 Counting on; start at 20 and count up (number line, hundreds chart) 10 20 20 30 30 40 40
50 45
60
65
65 – 20 = X Making Tens; 6 tens and 5 ones and take away 2 tens 10 + 10 + 10 + 10 + 5 = 45 Assessing and Advancing Questions Assessing Questions: Tell me what you were thinking. How do you know that? Why did you choose…? What number should we start with if we want to count up? What number should we count up to? Advancing Questions: Can you write the problem as a subtraction problem? Can you explain how to start with 65 and count backwards to find the answer? What would we count backwards to? 70
Assessing Questions: Tell me what you were thinking. How do you know that? Why did you choose…? Advancing Questions: Can you write the problem as an addition problem? How much paper would Mrs. Johnston have started with if she had ended with 80 feet? Part/Part/Whole; x + 20 = 65 Assessing Questions: What is the whole in this problem? What is one part of the whole? How can you find the other part of the whole? 65‐20 = 45 Advancing Questions: Can you write the problem as a subtraction problem? Possible Student Misconceptions The student adds 20+65=85. Entry/Extensions If students can’t get started…. If students finish early…. Questions: What does the 85 stand for in the problem? Should the length you end up with be larger than the length Mrs. Johnston ended with? If 65 feet is the length you end with, is the beginning length shorter or longer than 65? If the beginning length is shorter than 65 feet, what can you do with the 20 feet and the 65 feet to get a length shorter than 65 feet? Assessing and Advancing Questions What is the question asking us to do? Who can give me an idea of how to start our model? What is the goal of our problem? Can we organize the details to help us draw a picture? Tell me what you found. What problems did you have during your work? What if Mrs. Johnston had 73 feet of paper instead of 65 feet? How will that change the problem? What if Mrs. Johnston started with 34 feet of paper, how much more would she need to have 65 feet total? Discuss/Analyze Whole Group Questions Key Understandings: • When given a total, one part can be subtracted from the total to find the unknown part.
• When given the total and one part, one can begin with the known part and count up to the total to find the missing part.
• Addition and subtraction are inversely related and can be used in problem solving.
Questions: • Can someone explain how you determined the unknown?
• How many ways can we model this problem?
• Why do all of our different types of models work?
Alabama Department of Education
Task: Birthday Party Task
3rd Grade
Ella’s mother is baking 4 pans of brownies for a birthday party. Each pan can be divided into 16 squares of brownies. Ella wants to share
them equally with her friends at the party. There are 8 children altogether. How many squares of brownies will each child get? Draw a
picture and write an equation that shows how you solved the problem.
Teacher Notes:
Models used in solution paths show different interpretations of multiplication and division. It is possible for a student to use any combination
of models shown below.
College- and Career-Ready State Standards for Mathematical Content
Standards for Mathematical Practice
Represent and solve problems involving multiplication and division.
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in
5 groups of 7 objects each. [3-OA1] Example: Describe a context in which a total number of
objects can be expressed as 5 × 7.
2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number
of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number
of shares when 56 objects are partitioned into equal shares of 8 objects each. [3-OA2]
Example: Describe a context in which a number of shares or a number of groups can be
expressed as 56 ÷ 8.
3. 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. (See Appendix A, Table 2.)
[3-OA3]
Understand properties of multiplication and the relationship between multiplication and
division.
5. Apply properties of operations as strategies to multiply and divide. (Students need not use
formal terms for these properties.) [3-OA5]
Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of
multiplication)
3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40
+ 16 = 56. (Distributive property)
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics.
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Essential Understandings
• Multiplication can be used to find the total number of objects when there are a specific number of groups with the same number of
objects.
• When multiplying two factors, either factor can be partitioned or both. Example: 4 x 16 = 4 x (10 + 6) or (2 + 2) x 16
• Division can be used to find how many equal groups (measurement - repeated subtraction) or how many are in each group (partitive sharing)
• Multiplication and division have an inverse relationship and can be used to find division or multiplication facts
Explore Phase
Possible Solution Paths
Equal Groups or Arrays:
Assessing and Advancing Questions
Students may find the total number of brownies by thinking about
4 groups of 16 brownies and using the equation, 4 x 16 = 64 and
then find the number of brownies each child will get by thinking
about 64 brownies divided into groups of 8, 64 ÷ 8 = 8. Each child
will get 8 brownies when shared equally.
Assessing Questions:
- Why did you decide to group the brownies by 16 and 8?
- What does each number represent in your equation? (Ask students to
relate numbers back to the model.)
- How does an array show equal groups?
4 groups of 16 or 4 x 16 = 64
64 is divided into groups of 8 or 64 ÷ 8 = 8
Advancing Questions:
- Why did you decide to write these equations?
- What is the relationship between multiplication and division?
- What do you notice about the product and the dividend? Why are they
the same?
- What would happen if you turned your array sideways?
- How can you make a connection between the two models (equal sized
groups and arrays)?
Equal Sharing for Each Pan
Students may think about sharing each pan of brownies with 8
children, 16 ÷ 8 = 2 so each child will receive 2 brownies from one
pan. Then the student may realize that there are 4 pans of
brownies so the student decides to multiply by 4 pans to find the
total number of brownies each child will receive at the party,
2 x 4 = 8.
16 ÷ 8 = 2
16 ÷ 8 = 2
16 ÷ 8 = 2
16 ÷ 8 = 2
2 brownie squares x 4 pans = 8 brownie squares
Assessing Questions:
- Why did you decide to group the brownies by 2 for each pan?
- What does each number represent in your equation? (Ask students to
relate numbers back to the model.)
Advancing Questions:
- Why did you decide to divide first and then multiply?
- Does this always work on all problems? Explain
- What is the relationship between multiplication and division?
- How is this model different or the same as the previous model?
Each child will receive 8 brownie squares.
Repeated Addition and Subtraction
Students may use repeated addition to find the total number of
brownie squares: 16 + 16 + 16 + 16 = 64 or 4 + 4 + 4 + 4 + 4 + 4 + 4
+ 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 64
Students may use repeated subtraction to find the number of
brownie squares each child will get when shared equally with 8
children.
Assessing Questions:
- Why did you choose to add and subtract?
- How many times did you add and subtract? Why?
- Can you explain how addition and subtraction equations relate to the
model? Explain
Advancing Questions:
- What is the relationship between addition and multiplication?
- How can you write a multiplication equation for the addition sentence?
- What is the relationship between subtraction and division?
- How can you write a division equation for the subtraction problem?
64 – 8 = 56
56 – 8 = 48
48 – 8 = 40
40 – 8 = 32
32 – 8 = 24
24 – 8 = 16
16 – 8 = 8
8–8=0
8 brownie squares for each child at the party
Area Model
Students may use the distributive property to solve 4 x 16 by
thinking about 4 (10 + 6) = 40 + 24 = 64
Assessing Questions:
- How is breaking the rectangle apart helpful in solving the problem?
- Why did you multiply to find the total number of brownie squares?
- How does your equation relate to the model?
Advancing Questions:
- Is it possible to partition (break apart) either factor or both? Explain
your thinking.
- How can you make a connection between the two models (area model
and base-ten model)?
Then solve 64 ÷ 8 = 8 by using the area model 8 x 8 = 64 or partition
64 base-ten blocks into groups of 8.
Students may use multiplication and division equations to solve the
problem.
4 x 16 = 64
64 ÷ 8 = 8
Each child will receive 8 brownie squares at the birthday party.
Possible Student Misconceptions
Students may not realize there are 4 pans of brownies and each
pan has 16 brownie squares
Students may not realize this is a two-step problem and only find
the total number of brownie squares
Assessing Questions:
- Why did you decide to multiply and then divide?
- What does each number represent in your equation?
Advancing Questions:
- How can you solve the problem another way?
- How can you draw a model to match your equations?
Assessing Questions:
- What is the question you are trying to answer?
- How can you find out how many brownies are in 4 pans?
Advancing Questions:
- Does your answer make sense?
- What do you need to do after you find the total number of brownie
squares?
Entry/Extensions
If students can’t get started….
If students finish early….
Assessing and Advancing Questions
Assessing Questions:
- How can you state the problem in your own words?
- What are you trying to find or do?
- What information do you need to solve the problem?
- What model could you draw to help you solve the problem?
Advancing Questions:
- How many brownie squares are in 1 pan? 2 pans?
Assessing Questions:
- Does your solution make sense when you look at the original problem?
- Is it reasonable? Explain your thinking.
Advancing Questions:
- Is there another way of finding the solution?
Discuss/Analyze
Whole Group Questions
Select and sequence refers to when a teacher anticipates possible student strategies ahead of time and then selects and determines the order
in which the math ideas/strategies that students will share during the whole group discussion. The purpose of this is to determine which ideas
will be most likely to leverage and advance student thinking about the core math idea(s) of the lesson.
During a whole group discussion, students are sharing their strategies that have been pre-selected and sequenced by the teacher. Strategies to
consider sharing: Equal Groups or Arrays, Repeated Addition & Subtraction, Area Model, Multiplication and Division Equations
-How are these strategies similar and different? (Use Accountable Talk to ask students to compare strategies.)
-One student used the partition model to solve the problem. How is breaking the rectangle apart helpful in solving the problem? Is it possible
to partition (break apart) either factor or both? Explain your thinking.
- What does each number represent in your equation? (Relate numbers back to the model.)
- What do you notice about the product and the dividend? Why are they the same in this problem?
- When do you use multiplication to solve a problem? When do you use division to solve a problem?
- What is the relationship between multiplication and division?
Alabama Department of Education Task: Treat Bag Task
4th grade Ashley is making treat bags for her birthday party. She is going to put 2/3 cups of peanut M&Ms in each bag. She has invited 9 friends to her party. 1. How many cups of peanut M&Ms does she need for her friends’ treat bags? Write an equation and use a visual model to explain your
reasoning.
2. Ashley decided to also include ½ of a cup of plain M&Ms in each bag. How many cups of plain M&Ms does she need? Write an equation
and use a visual model to explain your reasoning.
3. Each treat bag will hold one cup of treats. Will Ashley be able to fit all of the M&Ms in each treat bag? Justify your answer.
Teacher Notes: Students may use a reasoning strategy to decide if the M&Ms will fit in each treat bag. They are not required to add the two fractions with unlike denominators to show that the sum is greater than one. College- and Career-Ready State Standards for Mathematical Content Standards for Mathematical Practice
Extend understanding of fraction equivalence and ordering.
13. Compare two fractions with different numerators and different denominators,
e.g., by creating common denominators or numerators or by comparing to a
benchmark fraction such as . Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the results of comparisons with
symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
[4-NF2]
Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
14.
d. 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. [4-NF3d]
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Essential Understandings • Comparison to known benchmark quantities can help determine the relative size of a fractional piece because the benchmark quantity can
be seen as greater than, less than, or the same as the piece.
• A rational number is an operator when it changes or transforms another number or quantity to magnify or shrink it.
• The interpretations of the operations on rational numbers are essentially the same as those on whole numbers, but some interpretations
require adaptation, and the algorithms are different.
• A scalar definition of multiplication is useful in representing and solving problems beyond whole number multiplication and division.
Explore Phase Possible Solution Paths •
•
•
•
Assessing and Advancing Questions Assessing Questions • Explain how you got the total amount.
Part 1 and Part 2 • Explain your model.
Students may model the solution with labeled fraction bars,
• Explain how accumulating the fractional parts lead to your answer.
boxes, or circles by partitioning them into thirds and halves.
• Explain how you multiplied your whole number by the fraction.
They may color in the correct portion of each picture or
• Explain how your number line works? How many jumps did you
accumulate the fractional pieces.
show? Why?
Students may use the multiplication equation 9 X 2/3 = 18/3 or
• How did you get the mixed umber?
6 cups of peanut M&Ms. 9 X ½ = 9/2 or 4 ½ cups of plain
• She has 6 what? She has 6 cups of what?
M&Ms.
Advancing Questions Students may use the repeated addition equation 2/3 + 2/3 +
• How can you write your repeated addition equation differently?
2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 18/3 or 6 cups of
• Can you use a different operation to express the equation?
peanut M&Ms. ½ + ½ + ½ + ½ + ½ + ½ + ½ + ½ + ½ =9/2 or 4 ½
• How can you write the improper fraction differently?
cups of plain M&Ms.
• How can you model your equation?
Students may use a number line to show repeated addition or
• How can you change the improper fraction into a whole or mixed
multiplication of the fractional part by 9.
number?
Assessing Questions • Explain your model.
Part 3 • How do you know that 2/3 is greater than ½?
• Students may justify their answer by drawing a model that
• Explain how you know the total will be greater than 1 cup.
shows that 2/3 + ½ > 1.
Advancing Questions • Students may state that 2/3 is more than ½. If ½ + ½ = 1, then
• How can you write an inequality to explain your answer?
2/3 + ½ > 1.
• How can you find equivalent fractions? What property allows you
• Students may convert fractions to common denominators and
to do this?
add to find the total is 7/6 or 1 1/6 cups of M&Ms.
• How can you use equivalent fractions to write an equation?
Possible Student Misconceptions Part 1 and 2 • Students may misinterpret the units and labels. They may say
6 peanut M&Ms instead of 6 cups of peanut M&Ms.
• Students may include themselves in the number of bags.
Sharing the M&Ms in 10 treat bags instead of 9 bags.
Assessing Questions • Show me what 6 M&Ms looks like. Does that seem possible?
• How do you know how many treat bags she needs to make?
• How many friends does Ashley have? So, how many treat bags
does she need to make?
Part 1 and 2 continued Part 3 • Students may add the total cups of peanut M&Ms to the total
cups of plain M&Ms and compare this to the 1 cup that each
bag will hold.
• Students may come up with the answer 1 1/6 cups of M&Ms
but not state that it is greater then 1 cup or that the M&Ms
will not fit in the treat bags.
Entry/Extensions If students can’t get started…. Advancing Questions • How can you use a model to show the total amount of treat bags
Ashley wants to make?
• How can you use a model to show what portion of M&Ms Ashley
wants in each bag?
Assessing Questions • What is the question being asked?
• Why are you adding 6 + 4 ½? Where did those numbers come
from? Is that how much is going into each bag?
• What portion of M&Ms will go in each bag?
• How can you find the total that she wants to put in each bag?
• How did you find the total in each bag?
• Did you finish answering the question?
Advancing Questions • What is the total portion of M&Ms that Ashley wants to put in
each bag?
Assessing and Advancing Questions Assessing Questions • What is the question being asked?
• What do you already know?
• How many treat bags does Ashley need to make?
• What part of a cup of M&Ms will go in each bag?
• How can you find the total?
Advancing Questions • What is your next step?
•
If students finish early…. •
•
What portion of the M&Ms does Ashley want to put in each bag?
How do you know?
How can you write an inequality to explain your answer in Part 3?
What amount of each type of M&Ms can Ashley use that will fit in
the treat bag?
Discuss/Analyze Whole Group Questions 1. Modeling the combining of fractions.
• How did you model the combining of each portion of M&Ms in the treat bags?
• Did anyone show it a different way?
2.
•
•
•
•
Multiplication of a whole number by a fraction.
How did you turn the whole number into a fraction?
How did you multiply the whole number by the fraction?
How could you model 9/1 ?
How is this similar to multiplying two whole numbers?
3. Addition of fractions with like denominators.
• If you chose to use repeated addition, explain why the denominator stays the same and the numerator increases?
• How can you write the repeated addition equation as a multiplication equation?
4. Changing the improper fraction into a whole or mixed number.
• How can you decompose the improper fraction and recombine the unit fractions to make a whole or mixed number?
• Did anyone use a different strategy?
5.
•
•
•
•
Comparing the combining of two fractions to the benchmark 1.
Did you have to add ½ + 2/3 ? Or is there another way to know if they are greater than 1?
How did you know that the portion of M&Ms was too much to fit into the treat bag?
Which is bigger 2/3 or 1/2?
What is ½ + ½?
Alabama Department of Education: 5th Grade Task: Apple Orchard Task 1. Andrea, Anthony, and Amy were in the orchard picking apples. Andrea picked of a bushel, Anthony picked of a bushel, and Amy picked
1
bushels. How many bushels of apples did they pick? Draw a model to represent your thinking. 2. Their Grandma needs 3 bushels to make apple jelly. Did they pick enough apples? If so, how much extra do they have? If not, how much more is needed? Draw a model to represent your thinking. Teacher Comments: The intent of this task is to be an introduction into adding fractions with unlike denominators. Students are to explore their ideas from what they know about adding fractions BEFORE they are taught how to work with unlike denominators. CoůůĞŐĞͲĂŶĚĂƌĞĞƌͲZĞĂĚLJ State Standards for Mathematical Content Standards for Mathematical Practice Use equivalent fractions as a strategy to add and subtract fractions.
11. 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. [5NF1]
12. 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-NF2]
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct a viable argument and critique the reasoning of others. 4. Model with mathematics.
5. Use appropriate tools strategically. 6. Attend to precision.
7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Essential Understandings • The interpretations of the operations on rational numbers are essentially the same as those on whole numbers, but some interpretations require
adaptations and the algorithms are different.
• Estimation and mental math are more complex with rational numbers than with whole numbers.
Explore Phase Possible Solution Paths Assessing and Advancing Questions Assessing Questions: • Can you explain your equation?
1. Students may change the denominators to be “like” and add the
• How did you get ? fractions together:
1
2 bushels of apples • Why did you add?
Advancing Questions: • (If not labeled) What does the 2 represent? •
Can you draw a picture to demonstrate your thinking?
•
1. Students may add pictorially:
Assessing Questions: • Can you explain your picture? • I don’t see 2 on your picture. Can you explain how that is your
answer?
• Why are your rectangles unevenly divided? Advancing Questions: • Can you write an equation to represent your picture? How? • (If not labeled) What does the 2 represent?
•
•
2 bushels of apples 1.
Students may add on a number line
Thus, they picked 2 bushels of apples. •
They did not pick enough apples. Students may subtract to justify:
3
1
4
2
5
8
3
2
8
2
5
8
2
10
8
How many eights are in a half? In one and a half?
Can you show me how to draw the picture using only eighths?
Assessing Questions: • Can you explain what the arrows represent? • Can you explain the number line? • Can you explain the marks on your number line?
Advancing Questions: • Can you write an equation to represent your picture? How? • (If not labeled) What does the 2 represent?
•
2.
Can you add without common denominators? Why or why not?
2
5
8
5
8
Thus, they need of a bushel more apples in order to have enough. 2. **(see misconception below) They did not pick enough apples.
Students may subtract “chunks” instead of the entire amount to justify
their thinking.
Can you show me how to fill in all of the one eights on your
number line?
How many eights are in one half? In one and a half?
Assessing Questions: • Why did you subtract? • Why did you find common denominators? • Can you explain how you got 2 ? Advancing Questions: • (If not labeled) What does the represent?
•
•
Can you draw a picture to demonstrate your thinking? Can you subtract without common denominators? Why or why not?
• Is your answer reasonable?
Assessing Questions: • Why did you subtract?
• Can you explain why you subtracted three times? 3
1
4
3
4
2
1
2
1
2
1
2
1
2
• Why did you choose to subtract in this order?
Advancing Questions: • Would your answer have changed if you subtracted the before 1 3 5
1
8 8
Thus, they need of a bushel more apples in order to have enough. 2. They did not pick enough apples. Students may use a pictorial representation to justify their thinking. Start with 3 ¼ : Take away 2 (purple): the ?
•
(If not labeled) What does the represent?
•
•
•
It is possible to solve the problem with only 1 equation. How? Can you draw a picture to demonstrate your thinking? Is your answer reasonable?
Assessing Questions: • Can you explain your picture? • Why did you take away 2 ?
• Can you explain the “white” boxes?
Advancing Questions: • Can you write an equation to represent your picture? How? • (If not labeled) What does the represent?
•
That leaves unaccounted for (yellow). Thus, they need to pick of a bushel more to have enough apples. 2. Students may subtract on a number line: •
Assessing Questions: • Can you explain your number line? • Why did you start on the right hand side of the number line? • Can you explain the “tick marks” on your number line? Advancing Questions: • Can you write an equation to represent your number line? How? • (If not labeled) What does the represent?
•
Thus, they need to pick of a bushel more to have enough apples. 2. Students may use addition to find the amount needed to get from 2
to 3 :
The may calculate this pictorially, on a number line, or with an Where are the apples that Amy picked represented in your
picture? Andrea? Anthony?
Is your answer reasonable?
•
Where are the apples that Amy picked represented in your
picture? Andrea? Anthony?
Is your answer reasonable?
Assessing Questions: • Can you explain your number line (equation)? • Why did you use addition? • Can you explain what each arrow represents (on the number line)? Advancing Questions: • Can this problem be worked another way?
For example, looking at this number line, how much more would I need • Can you write an equation to represent the number line? to get to 3 ¼ ? • Can you draw a picture to represent your equation? • (If not labeled) What does the represent? equation. 3 ¼ •
Is your answer reasonable?
I would need more. Thus, they need to pick of a bushel more to have enough apples. OR as an equation: 2 + _____ = 3 Possible Student Misconceptions Students may add both numerator and denominator. This is especially typical when working with the equation. Students may lack precision in their number lines or in partitioning the rectangles leading to incorrect answers. **Students may string these together as one long connected equation. This does not preserve the property of equality and is a VERY common mistake when students work with an operation repeatedly. 3
1
4
3
4
2
1
2
1
1
2
1
3
8
5
8
Assessing Questions: • Can you explain your work? • Why did you add both the numerators and denominators? Advancing Questions: • What happens to fractions as you change their denominator? • How does your answer compare to each of the numbers of bushels picked by Andrea, Amy, and Anthony? • Is your answer reasonable?
Assessing Questions: • Can you explain the markings on your drawing? • Can you show me ¼ on your drawing? ? Advancing Questions: • Are your intervals (partitions) precise? Why or Why not?
Assessing Questions: • Can you explain your work?
• Does 3
? Advancing Questions: • Can you find a more accurate way to show your work? Show me.
Entry/Extensions If students can’t get started… Assessing and Advancing Questions Assessing Questions: • What do the numbers in the problem represent?
• What is the problem asking you to find? Advancing Questions: • Can you draw a model to represent the apples?
A bushel of apples weighs approximately 50 pounds. How many pounds of apples has Andrea picked? Anthony? Amy? If students finish early… How many mores pounds need to be picked so that Grandma has enough apples to make apple jelly? Discuss/Analyze Whole Group Questions • How are the pictorial representations connected to the equations? • Why when working with the equation was it so important to have common denominators? • Are the common denominators shown in the other representations? • How are addition and subtraction of fractions related?
• Can we estimate to check the reasonableness our answers? How? Is this important to check? Why or why not?