Unit 1 Organizer:
|Grade 3 Mathematics Frameworks |
|Unit 1 |
|Addition and Subtraction of Whole Numbers |
Unit 1
ADDITION AND SUBTRACTION OF WHOLE NUMBERS
(4 weeks)
TABLE OF CONTENTS
Overview……………………………………………………………………………...... 3
Key Standards & Related Standards…………………………………………………… 4
Enduring Understandings………………………………………………………………. 6
Essential Questions…………………………………………………………………….. 8
Concepts & Skills to Maintain ………………………………………………………… 9
Selected Terms and Symbols.….. ……………………………………………………... 10
Classroom Routines…………………………………………………………………….. 11
Strategies for Teaching and Learning………………………………………………….. 11
Evidence of Learning………………………………………………………………….. 11
Tasks ……………………………………………………………………..……………. 12
• Building Base Ten Numbers………………………………………………….. 13
• I Spy a Number……………………………………………………………….. 17
• Shake, Rattle, and Roll……………………….……………………………….. 22
• Happy to Eat Healthy………………….……………………………………… 27
• Mental Mathematics………….……………………………………………...... 37
• The Power of Properties…………………………………………………......... 42
• Take 100………………………………………………………………………. 48
• Perfect 500! ……………………….………………………………………….. 53
• Car Wash at Carter Elementary ...….…………………………………………. 60
• I Have a Story, You Have a Story ...………………………………………….. 65
Culminating Task
• What’s the Story Here? .....…………………………………………………… 70
OVERVIEW
In this unit students will:
• continue to develop their understanding of and facility with addition and subtraction
• recognize and use place value to manipulate numbers
• continue to develop their understanding of and facility with money
Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the five process standards; problem solving, reasoning, connections, communication, and representation, should be addressed continually as well. This first unit should establish these routines, allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.
To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks provide much needed content information as well as excellent learning activities. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.
STANDARDS ADDRESSED IN THIS UNIT
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.
KEY STANDARDS
M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.
a. Identify place values from tenths through ten thousands.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
d. Model addition and subtraction by counting back change using the fewest number of coins.
M3A1. Students will use mathematical expressions to represent relationships between quantities and interpret given expressions.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the unknown in a number sentence.
RELATED STANDARDS
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ENDURING UNDERSTANDINGS
• Addition and subtraction are inverse operations; one undoes the other.
• We can verify the results of our computation by using the inverse operation.
• Place value is crucial when operating with numbers.
• Estimation helps us see whether or not our answers are reasonable.
• Adding zero to a number or subtracting zero from a number does not change the original amount.
• Addition means the joining of two or more sets that may or may not be the same size. There are several types of addition problems, see the chart below.
• The counting up strategy can be used to make change.
|ADDITION |
|Problem Type |Action or Situation |Number Sentence |
|Join, Result Unknown |Pete puts 25 mice in a cage, and then puts in 12 more. How many are in the |25 + 12 = ( |
| |cage? (Action) | |
|Join, Start Unknown |Pete has some mice in a cage. He puts 12 more in. Now he has 37. How many | |
| |were in the cage to begin with? (Action) |( + 12 = 37 |
|Join, Change Unknown |Pete has 25 mice in a cage. He puts some more in. Now he has 37. How many did|25 + ( = 37 |
| |he add to the cage? (Action) | |
|Compare, Compare Quantity Unknown |Pete has 25 mice. Max has 12 more mice than Pete. How many mice does Max |25 + 12 = ( |
| |have? (Situation) | |
|Part-Part-Whole, Whole Unknown |Pete has 12 white mice and 13 brown mice. How many mice does he have |12 + 13 = ( |
| |altogether? (Situation) | |
|Part-Part-Whole, Part Unknown |Pete has 25 mice. 12 are white and the rest are brown. How many mice are |12 + (= 25 |
| |brown? (Situation) |(+ 12 = 25 |
ENDURING UNDERSTANDINGS (Continued)
• Subtraction has more than one meaning. It not only means the typical “take away” operation, but also can denote finding the difference between sets. Different subtraction situations are described in the chart below.
|SUBTRACTION |
|Problem Type |Action or Situation |Number Sentence |
|Separate, Result Unknown |Pete has 37 mice in a cage. 12 escape. How many are left? (Action) |37 – 12 = ( |
|Separate, Start Unknown |Pete has some mice in a cage. 12 escape. 25 are left. How many did he have at| |
| |the start? (Action) |( – 12 = 25 |
|Separate, Change Unknown |Pete has 37 mice in a cage. Some escape. 25 are left. How many escaped? | |
| |(Action) |37 - ( = 25 |
|Compare, Difference Unknown |Pete has 37 mice. Max has 25 mice. How many more mice does Max have than |37 – 25 = ( |
| |Pete? (Situation) | |
|Compare, Referent Unknown |Pete has 37 mice. He has 12 more mice than Max. How many mice does Max have? |37 – 12 = ( |
| |(Situation) | |
ESSENTIAL QUESTIONS
• How do the value of digits change when their position in a number changes?
• How can we tell which numbers are larger or smaller than others?
• How can we figure out a number we don’t know using clues about the place value of its digits?
• What can we learn about the value of a number by examining its digits?
• What is an effective way to estimate numbers?
• How can estimation strategies help us build our addition skills?
• When will estimating be helpful to us?
• How can I use addition and subtraction to help me solve real world problems?
• What estimation and mental math strategies can I use to help me solve real world problems?
• How can I verify the results of an addition or subtraction word problem?
• What is mental math?
• How does mental math help us calculate more quickly and develop an internal sense of numbers?
• What mental math strategies are available to us?
• How can we select among the most useful mental math strategies for the task we are trying to solve?
• What are the properties that relate to addition and subtraction?
• How can we verify the results of an addition problem?
• How does knowing the commutative property help us add numbers easily and quickly?
• How does knowing the identity property help us add numbers easily and quickly?
• How is zero different from any other whole number you might add or subtract?
• How does knowing the associative property help us add numbers easily and quickly?
• How do properties work in subtraction problems?
• How can I learn to quickly calculate sums in my head?
• What strategies will help me add multiple numbers quickly and accurately?
• What strategies are helpful when estimating sums in the hundreds?
• How can I use what I understand about addition and subtraction in word problems?
• What is a number sentence and how can I use it to solve word problems?
• How can I use what I understand about money to solve word problems?
• What is the most efficient way to give change?
• How can I show what I know about addition and subtraction, problem solving, and estimation?
• How do we use addition and subtraction to tell number stories?
• How are addition and subtraction alike?
• How are addition and subtraction different?
• In what type of situations do we subtract?
• In what type of situations do we add?
• Why is place value important?
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.
• Addition Facts
• Subtraction Facts
• Place Value
• Money and Counting Change
SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, teachers should pay particular attention to them and how their students are able to explain and apply them.
The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.
← Addend: A number being added
← Associative Property of Addition: When there are three addends, the sum does not change regardless of which two numbers are grouped together first.
o Example: 3 + 5 + 2 = (3 + 5) + 2 = 3 + (5 + 2) = 10; 8 + 2 = 3 + 7 = 10
← Benchmark Number: A number that is easy to work with, such as numbers that end in 0, 5, or numbers that relate easily to money such as 25, 50, or 75.
← Commutative Property of Addition: The order in which two numbers are added does not change the sum.
o Example: 9 + 7 = 16 and 7 + 9 = 16
← Difference: The answer obtained when two numbers are subtracted
← Doubling: Adding the same amount twice; two times a number
← Estimate: An approximation of the actual value for the size, cost, or quantity of something
← Identity Property of Addition: Zero added to any number equals the number. Zero is called the additive identity
← Inverses: Operations that undo each other, such as addition and subtraction as well as multiplication and division
← Operations: Addition, subtraction, multiplication, and division
← Property: A consistent way in which numbers work together in arithmetic. Knowing properties helps us compute sums and differences with speed and reliability
← Sum: Total amount added; total number of elements in sets that are combined
CLASSROOM ROUTINES
The importance of continuing established classroom routines cannot be overstated. Daily routines must include such obvious activities as estimating, analyzing data, describing patterns and answering daily questions. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, and how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students’ number sense, flexibility, fluency, collaborative skills, and communication. These routines contribute to a rich, hands-on, standards based classroom and will support students’ performances on the tasks in this unit and throughout the school year.
STRATEGIES FOR TEACHING AND LEARNING
• Students should be actively engaged by developing their own understanding.
• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, models, symbols, and words.
• Appropriate manipulatives and technology should be used to enhance student learning.
• Students should be given opportunities to revise their work based on timely teacher feedback, peer feedback, and their own reflection.
• Students need to write in mathematics class to explain their thinking, to talk about how they perceive topics, and to justify their work to others.
EVIDENCE OF LEARNING
By the conclusion of this unit, students should be able to demonstrate the following competencies:
• Accurately represent, read, and verbalize numbers from tenths to ten thousands place.
• Demonstrate understanding of the relative sizes of digits in a number.
• Use mental math to add and subtract.
• Use estimation to determine reasonableness of sums and differences computed.
• Count back change efficiently and fluently.
• Read, interpret, solve and compose simple word problems using addition and subtraction.
• Understand how to use an inverse operation to verify computation accuracy.
• Write and solve expressions using symbols in place of numbers.
TASKS
The following tasks represent the level of depth, rigor, and complexity expected of all third grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).
|Task Name |Task Type |Content Addressed |
| |Grouping Strategy | |
|Building Base Ten Numbers |Learning Task Individual/Partner Task |Using base ten blocks to build large numbers |
|I Spy a Number |Learning Task |Place value with reasoning |
| |Partner/Small Group Game | |
|Shake, Rattle, and Roll |Learning Task |Using estimation and mental math with addition |
| |Partner/Small Group Task | |
|Happy to Eat Healthy |Performance Task Individual/Partner Task |Addition/Subtraction Word Problem |
|Mental Mathematics |Learning Task |Using and sharing mental math strategies |
| |Whole Class Task | |
|The Power of Properties |Learning Task Individual/Partner Task |Commutative, Identity and Associative Properties |
|Take 100 |Learning Task |Mental Math with combinations of 100 |
| |Partner/Small Group Game | |
|Perfect 500! |Learning Task |Mental Math with sums of 100 |
| |Small Group/Partner Game | |
|I Have a Story, You Have a Story |Performance Task Individual/Partner Task |Understanding and writing addition and subtraction word |
| | |problems |
|Car Wash at Carter Elementary |Performance Task Individual/Partner Task |Counting back change, money story problems |
|Culminating Task: |Performance Task Individual/Partner Task |Bookmaking to demonstrate standards learned |
|What’s the Story Here? | | |
LEARNING TASK: Building Base Ten Numbers
STANDARDS ADDRESSED
M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.
a. Identify place values from tenths through ten thousands.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
M3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How do the value of digits change when their position in a number changes?
• How can we tell which numbers are larger or smaller than others?
MATERIALS
• Three 6-sided dice per student/group
• Base ten blocks
• “Building Base Ten Numbers” Recording Sheet
• Place Value Charts (optional)
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will roll dice and make three-digit numbers from the results. Students will then draw or use base ten blocks to build models of the numbers created and explain how they know which numbers are larger or smaller.
Comments
Because there are several steps involved in this task, you may want to model this activity while “thinking aloud.” Include a discussion about assigning a place value to the number rolled on a given die. Ask questions such as, “Will you make the 3 on this die represent 3, 30, or 300? How will each of those numbers look if we use base ten blocks?” Some students may need extra support with this concept.
When using dice, base ten blocks, or any other manipulatives, students need to understand procedures and routines associated with using them. You may want to give students a few minutes to explore with the pieces and make observations before moving into the task.
Background Knowledge
Students need to understand how to order digits from largest to smallest and vice versa. They also need to understand place value concepts and how they relate to the base ten pieces.
Task Directions
Students will follow the directions below from the “Building Base Ten Numbers” Recording Sheet.
Your task is to build large numbers and identify them. Use dice and base ten blocks or models to complete this exercise.
1. Roll all 3 dice at once.
2. Record the number formed when you place the digits on the dice in order from smallest to largest.
3. Record the new number formed when you order them from largest to smallest.
4. Using the smallest number of base ten pieces possible, draw (or build) a model of each number you recorded. Have the flats represent hundreds, the rods represent tens, and the unit cubes represent ones. Use words to write how each of the two numbers is spoken.
5. Repeat the same exercise two more times and record.
6. When you have completed all your rolls, put a star beside the greatest number you rolled. Then put a check beside the smallest number you rolled. Explain how you know what the largest and smallest numbers are possible when using three dice.
Questions/Prompts for Formative Student Assessment
• How did you decide in which order to place your dice?
• What would happen if you changed the order of your dice?
• What would happen to the size of your numbers if you used more or fewer dice?
• Which representation of your numbers makes the most sense to you?
Questions for Teacher Reflection
• Are student representations of numbers accurate?
• Which representation of the numbers are most students using? Which representation might need more modeling?
• How ready are students to move away from using the manipulatives and utilize internalized place value concepts?
DIFFERENTIATION
Extension
• If you have access to calculators such as the TI-10, have students verify their partner’s results with the calculator. Discuss the important idea that when using a calculator, incorrect input will yield incorrect results. Let students practice basic or more advanced facts with the calculator by using the games on calculators such as the TI-30 which allows for three levels of practice on each of the operations. The card inside the cover of the calculator explains how to use the computation practice games.
• Students may use four dice instead of three (or use three dice and always have one digit be zero). If you continue to have them draw/build base ten models, you will need to provide the large cube in the base ten blocks to represent the thousands place. If students have access to base ten stamps, those may be used instead of or in addition to the actual manipulatives.
• Students may also use three dice of one color and a fourth of a different color to represent tenths. You can use a dried black-eyed pea or other small object to represent a decimal point. Have students repeat the above activity for ordering numbers but omit the modeling with base tens. Instead, have students find the difference between the largest and smallest numbers formed and show how they used the inverse operation to verify their results.
Intervention
• Students may need to use a place value chart to align their digits.
• Have students complete the task in small groups with direct instruction or modeling for additional support.
• For students who have difficulty counting the dots on a die, dice with numbers printed on them may be used.
TECHNOLOGY CONNECTION
• Place value chart on which students can build numbers up to three digits from hundreds through hundredths.
• Place value chart on which students can build numbers up to four from thousands through thousandths.
• Printable base ten blocks.
Name __________________________________________ Date __________________________
Building Base Ten Numbers
Your task is to build large numbers and identify them. Use dice and base ten blocks or models to complete this exercise.
1. Roll all 3 dice at once.
2. Record the number formed when you place the digits on the dice in order from smallest to largest.
3. Record the new number formed when you order them from largest to smallest.
4. Using the smallest number of base ten pieces possible, draw (or build) a model of each number you recorded. Have the flats represent hundreds, the rods represent tens, and the unit cubes represent ones. Use words to write how each of the two numbers is spoken. Repeat the same exercise two more times and record.
5. When you have completed all your rolls, put a star beside the greatest number you rolled. Then put a check beside the smallest number you rolled. Explain how you know what the largest and smallest numbers are possible when using three dice.
| |Smallest Number |Model |Largest Number |Model |
|2 |___ ___ ___ | |___ ___ ___ | |
|3 |___ ___ ___ | |___ ___ ___ | |
LEARNING TASK: I Spy a Number
STANDARDS ADDRESSED
M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.
a. Identify place values from tenths through ten thousands.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
ESSENTIAL QUESTIONS
• How can we figure out a number we don’t know using clues about the place value of its digits?
• What can we learn about the value of a number by examining its digits?
MATERIALS
• Overhead or Interactive White Board
• “I Spy a Number” Recording Sheet
GROUPING
Partner/Small Group Game
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students play a game in which they must figure out an unknown number by gathering information about the value of its digits. They will use deductive reasoning and problem solving to determine the identity of the target number.
Comments
This is a whole group game that works well on the overhead projector. Make a simple chart like this:
| |Digits Correct |Places Correct |
|Guess | | |
| | | |
| | | |
| | | |
| | | |
Set a range for student guesses such as 10 to 100. Decide on a target number and write it in a place where you can keep it covered. For example, use the number 75. Have a student guess the target number and record the guess in the first column. Are any digits in the guess correct? If not, put a “0” in both the “Digits Correct” and “Places Correct” columns.
If any digits are correct, record how many digits and how many places are correct in the appropriate columns. Students will use deductive reasoning and knowledge of place value to find the target number.
An example is shown below:
|Guess |Digits Correct |Places Correct |
|12 |0 |0 |
|36 |0 |0 |
|45 |1 |1 |
|73 |1 |1 |
|98 |0 |0 |
|75 |2 |2 |
The chart shows that none of the digits in 12 and 36 will be found in the target number. The next guess of 45 is a wise guess because the digits 4 and 5 had not yet been used. We do not know if the 4 or the 5 is correct at this point, but we do know that one digit and one place is correct. The task now is to find out if the 4 or the 5 is the digit to keep.
Next in the chart, 73 is a crucial guess because we already know that the target number has no three. Therefore, the correct digit must be the 7. Since the place is also correct, the number has to be seventy-something. The table confirms that no digits in 98 are correct. So, the only remaining digits possible in the ones place of the target number are 4 and 5. We know 0 is not a consideration because the target number has only two digits. There is a 7 in the tens place and the chart shows us that either 4 or 5 in 45 is in the correct place. We can eliminate the 4 because it is in the tens place and we already know there is a 7 in the tens place. Therefore, the five in forty-five must be correct because it is in the ones place. So, “seventy-something” is definitely 75.
Even though this task yields one correct answer for the target number, it is important for you to see the thought processes and reasoning that students use in the game. Students should play this game many times before they are given the “I Spy a Number” Recording Sheet to play on their own or with a partner.
Background Knowledge
Students must have numerous experiences with the game before they can complete this task in small cooperative groups or with a partner. A clear understanding of place value names and values is also essential.
Task Directions
Students will follow the directions below from the “I Spy a Number” Recording Sheet.
1. This game should be played with a partner. Player one will choose a target number. The player will write it down secretly on the back of this paper and give player two a range of numbers from which to choose (Examples: between 0 and 100 or between 100 and 1,000).
2. Player two will then give his/her first guess. Player one will write the guess in the chart below and use the correct columns to write how many digits and the number of places that are correct. Use a zero to show that neither the digit nor the place value is correct.
3. After each guess, Player two should explain why each guess was made. Continue playing until the target number has been determined.
4. Then Player two will choose the target number and repeat the game.
5. Both players must explain the strategy for the guesses they make.
Questions/Prompts for Formative Student Assessment
• What are your strategies for determining the target number?
• How does knowing that the digit and/or the place value are correct help you figure out the target number?
• What can you conclude if your partner tells you that you didn’t correctly guess the digit or the place value?
• How would you explain the best way to win this game to another student?
• What strategy do you think works best for finding the target number?
• Why did you choose this number?
Questions for Teacher Reflection
• Which students are guessing haphazardly, demonstrating lack of a reasonable strategy?
• Are students able to record their answers in order to reflect on the implications of them?
• How much automaticity are students demonstrating in their understanding of place value?
DIFFERENTIATION
Extension
• Have students use larger numbers and an expanded chart to play the game.
• Have students write an explanation of their reasoning as they find the target number.
Intervention
• As you model the game, verbalize your own strategy, step by step, so that students who need to build reasoning skills can hear the processes that result in discovering the target number.
• The chart below tells which digit is in the correct place in addition to the number of digits that are correct. This added information may make this game more accessible at first. The example below is for the number 45.
|Guess |Tens Digit |Ones Digit |Digits Correct|
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
|Guess |Tens Digit |Ones Digit |Digits Correct|
|47 |1 |0 |1 |
|53 |0 |0 |1 |
|45 |1 |1 |2 |
| | | | |
| | | | |
TECHNOLOGY CONNECTION
• Variations of this game are given on this web site.
Name ___________________________________________ Date _________________________
I Spy a Number
Directions:
1. This game should be played with a partner. Player One will choose a target number. The player will write it down secretly on the back of this paper and give Player Two a range of numbers from which to choose (Examples: between 0 and 100 or between 100 and 1,000).
2. Player Two will then give his/her first guess. Player One will write the guess in the chart below and use the correct columns to write how many digits and the number of places that are correct. Use a zero to show that neither the digit nor the place value is correct.
3. After each guess, Player Two should explain why each guess was made. Continue playing until the target number has been determined.
4. Then Player Two will choose the target number and repeat the game.
5. Both players must explain the strategy for the guesses they make.
|Guess |Digits Correct |Places Correct |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|Guess |Digits Correct |Places Correct |
| | | |
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| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
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LEARNING TASK: Shake, Rattle, and Roll
STANDARDS ADDRESSED
M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
c. Apply and adapt a variety of appropriate strategies to solve problems.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
ESSENTIAL QUESTIONS
• What is an effective way to estimate numbers?
• How can estimation strategies help us build our addition skills?
• When will estimating be helpful to us?
MATERIALS
• Two six-sided dice
• Calculator
• “Shake, Rattle, and Roll” Recording Sheet
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students play a game with dice that enables them to build mental math concepts as they practice addition skills and strategies and determine to which multiple of ten a given number is nearest.
Comments
This task is designed to provide addition practice and mental math/estimation skills. You may want to use a book like Mental Math in the Primary Grades by Jack Hope, R. Reys, Larry Leutizinger, Barbara Reys, and Robert Reys to practice mental math with the class as a whole group.
Use all available opportunities during the day to incorporate the use of estimation, for example, determining to which multiple of ten a given number is nearest. This skill was originally introduced in Grade 1, supported with the use of a number line 0-99 chart and/or a hundreds chart. Students should have these tools available for this task. Alternatively, students can create a number line to determine the closest multiple of ten. A student sheet with open number lines could be provided. An example of an open number line is shown below.
For the number 36, students can fill in the numbers around 36, including the two closest multiples of ten as shown below. Then looking at the number line, students can determine the multiple of ten that is the closest to 36. In this case 40 is 4 away, but 30 is 6 away, so 40 is the closest multiple of ten.
For the number 63, students can follow the same procedure to determine the multiple of ten that is the closest to 63. In this case 60 is 3 away, but 70 is 7 away, so 60 is the closest multiple of ten.
Estimating skills will help students determine reasonableness of answers, a vital skill for standardized tests, as well as everyday living.
If you incorporate calendar activities into your instruction, many opportunities present themselves for activities with estimation. Also, be sure students make connections between counting by tens, multiplying by ten, and estimating to the nearest ten before adding or multiplying.
Background Knowledge
Students should be proficient in determining to which multiple of ten any given two-digit number is nearest. They should also be comfortable adding two-digit multiples of ten (For example, 20 + 60 = 80).
Task Directions
Students will follow the directions below from the “Shake, Rattle and Roll” Recording Sheet.
This is a two player game that will help you practice your estimation skills. The goal of the game is to be the person with the most points at the end of ten turns.
1. Play with a partner. You will need 2 dice, a recording sheet for each player, and a calculator.
2. Roll two dice. Form the two possible numbers as shown below.
Example:
Using the digits 3 and 6, make the numbers 36 and 63. Find the nearest multiple of 10 for each number, and then using mental math, add to find an estimate.
Estimated sum = 40 + 60 = 100
3. Player one records the estimate on the game recording sheet to end round 1. Your partner must agree with your estimation, using a calculator to check if needed.
4. Player two takes a turn, following steps 2 and 3 above.
5. Players take turns for a total of ten rounds.
6. After ten rounds, each player finds the sum of their estimates. The player with the higher sum wins the game.
Questions/Prompts for Formative Student Assessment
• Explain how you found the closest multiple of ten.
• Do you think your estimated sum is higher or lower than the actual sum? Why? How could you check?
• What kinds of situations in life might be easier if you knew how to estimate and add numbers like this?
Questions for Teacher Reflection
• What estimation strategies are students in my class using most? Least?
• Can students explain the process they used to estimate?
DIFFERENTIATION
Extension
• Use one or more additional dice.
Intervention
• Use number lines, hundreds charts, and models to help students who are having difficulty determining to which multiple of ten their number is nearest. Use counting up/counting back to the nearest multiple of ten and compare the results to determine which multiple of ten a number is closest.
TECHNOLOGY CONNECTION
• . A “Four in a Row” game where players get checkers when they quickly and efficiently estimate a sum to two numbers.
• Students estimate the number indicated on a number line.
Name __________________________________________ Date __________________________
Shake, Rattle, and Roll
Game Directions
This is a two player game that will help you practice your estimation skills. The goal of the game is to be the person with the most points at the end of ten turns.
Directions:
1. Play with a partner. You will need 2 dice, a recording sheet for each player, and a calculator.
2. Player one rolls two dice and forms the two possible numbers as shown below.
Example:
Using the digits 3 and 6, make the numbers 36 and 63. Find the nearest multiple of 10 for each number, and then using mental math, add to find an estimate.
Estimated sum = 40 + 60 = 100
3. Player one records the estimate on the game recording sheet to end round 1. Your partner must agree with your estimation, using a calculator to check if needed.
4. Player two takes a turn, following steps 2 and 3 above.
5. Players take turns for a total of ten rounds.
6. After ten rounds, each player finds the sum of their estimates. The player with the higher sum wins the game.
Shake, Rattle, and Roll Game
Player 1 ________________________________
|Round |Dice Numbers |Smaller Number |Larger Number |Estimate |
| |Die 1 |Die 2 |Actual |Nearest Multiple of 10 |
| |Die 1 |
| | |
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| | |
| | |
| | |
| Total |$3.55 |
Combination 2
|Items |Price |
| | |
| | |
| | |
| | |
| | |
| Total |$3.55 |
Combination 3
|Items |Price |
| | |
| | |
| | |
| | |
| | |
| Total |$3.55 |
Healthy Snack List
| | | |
|Baked Chips 45[pic] |Milk 86[pic] |Hot Chocolate 67[pic] |
|[pic] |[pic] |[pic] |
| | | |
|Apple 52[pic] |Low-fat Hot Dog $1.25 |Veggie Burger $1.44 |
|[pic] |[pic] |[pic] |
| | | |
|Popcorn 49[pic] |Orange 62[pic] |Banana 87[pic] |
|[pic] |[pic] |[pic] |
| | | |
| | | |
|Bottle Water 99[pic] |Frozen Yogurt Bar 39[pic] |Raisins 56[pic] |
|[pic] |[pic] |[pic] |
| | | |
| | | |
| | | |
| | | |
Healthy Snack Shopping Rubric
|Elements |Exemplary |Moderate |Incomplete |
|Completed Lists |Three completed |Two completed |One completed list |
| |lists |lists |(20 points) |
| |(30 points) |(25 points) | |
|Calculations |All lists are |Two lists are |One list is |
| |correctly |correctly |correctly |
| |calculated |calculated |calculated |
| |(60 points) |(50 points) |(45 points) |
|Presentation |Neatly written |Readable |Unclear, difficult |
| |(10 points) |(5 points) |to read |
| | | |(0 points) |
|Total |
Sample Table
|Item |Price |
|Hot Dog |$1.25 |
|Orange |.62 |
|Frozen Yogurt |.39 |
|Frozen Yogurt |.39 |
|Chips |.45 |
|Chips |.45 |
| Total |$3.55 |
Other Solutions
|Item |Price |
|Hot Dog |$1.25 |
|Water |.99 |
|Chips |.45 |
|Milk |.86 |
| Total |$3.55 |
|Item |Price |
|Veggie Burger |$1.44 |
|Veggie Burger |1.44 |
|Hot Chocolate |.67 |
| Total |$3.55 |
|Item |Price |
|Veggie Burger |$1.44 |
|Hot Dog |1.25 |
|Milk |.86 |
| Total |$3.55 |
|Item |Price |
|Water |$.99 |
|Water |.99 |
|Chips |.45 |
|Chips |.45 |
|Hot Chocolate |.67 |
| Total |$3.55 |
|Item |Price |
|Hot Dog |$1.25 |
|Hot Dog |$1.25 |
|Popcorn |.49 |
|Raisins |.56 |
| Total |$3.55 |
|Item |Price |
|Raisins |.56 |
|Water |.99 |
|Water |.99 |
|Apple |.52 |
|Popcorn |.49 |
| Total |$3.55 |
|Item |Price |
|Banana |.87 |
|Banana |.87 |
|Hot Dog |1.25 |
|Raisins |.56 |
| Total |$3.55 |
|Item |Price |
|Veggie Burger |1.44 |
|Banana |.87 |
|Orange |.62 |
|Orange |.62 |
| Total |$3.55 |
|Item |Price |
|Orange |.62 |
|Orange |.62 |
|Orange |.62 |
|Orange |.62 |
|Orange |.62 |
|Chips |.45 |
| Total |$3.55 |
Name _____________________________ Date _______________________
Happy to Eat Healthy
(Version 3)
1. Cut out the snack items from the Healthy Snack List.
2. Find a combination of 3 or more items that totals exactly $3.55.
3. Glue the snack items you chose to the bottom of this page. Snack items can be used more than once.
4. Show all of your work.
5. Show your answer in the table below. Be ready to explain how you arrived at your answer.
|Items |Price |
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|Total |$3.55 |
LEARNING TASK: Mental Mathematics
STANDARDS ADDRESSED
M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
ESSENTIAL QUESTIONS
• What is mental math?
• How does mental math help us calculate more quickly and develop an internal sense of numbers?
• What mental math strategies are available to us?
• How can we select among the most useful mental math strategies for the task we are trying to solve?
MATERIALS
• Chalkboard, overhead projector or Interactive Whiteboard
• “Mental Mathematics” Recording Sheet
GROUPING
Whole Class/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will engage in mental math activities and rich group discussions about various strategies used to find the answers to addition and subtraction problems without paper and pencil.
Comments
Discussions should move beyond whether or not the answers are correct. The goal here is to develop efficient ways to group numbers and/or develop compensation strategies for mental addition and subtraction. The value of group discussions and modeling is evident when students gather insights from their classmates that will reinforce basic number sense and develop strategies that will help them become better at mental computation.
This is a valuable opening activity and should be revisited frequently throughout the year. Students should be encouraged to solve problems in ways that make sense to them. If students have never been encouraged to solve problems mentally and share their own strategies with others, they may be reluctant to share or may feel that their strategy is inappropriate. Establish ground rules in your classroom about sharing ideas and how to appropriately respond to each other.
Background Knowledge
Students should have some prior experiences with basic computation strategies allowing them to calculate quickly and reliably. Examples include counting on, doubling, making tens, making hundreds, and using benchmark numbers.
Task Directions
Begin this activity by placing one problem at a time on the board, preferably horizontally. Be aware that students may initially need individual time to solve these problems mentally, so encourage students to be patient and quiet during this time.
After allowing enough time for students to consider the problem, lead a discussion by asking several students to share their solution and/or strategy. Simply stating an answer is not enough to make this a rich activity. Encourage students to share different strategies, asking them to try to make sense of each solution as it is presented. Remind students that the goal is to become efficient and flexible in their thinking and strategies.
Have students follow the directions below:
Solve the following problems as they are placed on the board using no paper or manipulatives. Use your mental math strategies. Be prepared to share your solutions and strategies.
• 15 + 7
Students may solve this problem in a variety of ways. Examples are:
❖ 15 + 5 is 20 and 2 more is 22.
❖ 5 + 7 is 12 and 10 more is 22.
❖ 10 + 7 is 17, 3 more is 20 and 2 more is 22.
• 24 + 16
Students may solve this problem in a variety of ways. Examples are:
❖ 20 + 10 is 30 and 4 + 6 is 10, so 30 + 10 is 40.
❖ 4 + 6 is 10 and 20 + 10 is 30, so 10 + 30 is 40.
❖ 24 + 6 is 30 and 10 more is 40.
• 99 + 17
Students may solve this problem in a variety of ways. Examples are:
❖ 99 and 1 more is 100, 100 + 17 is 117, but take 1 away that was added to the 99 to get 100, so the answer is 116.
❖ Some may attempt a traditional algorithm, but should notice that this is more cumbersome that examining the numbers and using the ideas above to compute.
• 50 - 12
Students may solve this problem in a variety of ways. Examples are:
❖ 50 – 10 is 40, then 40 – 2 is 38.
❖ 50 – 2 is 48, them 48 – 10 is 38.
❖ You need 8 more to get to 20 from 12, then 30 more to get to 50, so the answer is 8 + 30 or 38. Note: Students who use this method are actually finding the difference between the two numbers and not simply “taking away.” This is a wonderful opportunity to discuss different approaches to subtraction.
Questions/Prompts for Formative Student Assessment
• What is one strategy you could use to solve the problem quickly?
• How can you verify your solution?
• Could this problem be solved another way? How?
• Which problem solving strategy works best for you?
Questions for Teacher Reflection
• What routines do I have in place that will encourage students to verbalize their math thinking?
• Were students able to develop additional mental math strategies?
• How effectively did students explain their thinking?
• How will I be sure to listen to each student’s explanations during a given math lesson?
• What mental math strategies do my students seem to use most? Least?
DIFFERENTIATION
Extension
• When you are presenting problems to students, vary the problems you use. Include various operations and numbers.
• Have students develop their own mental math problems, solve them, and explain their solution strategies.
Intervention
• Have students work with smaller, single-digit numbers initially.
• Have students work with a partner to develop strategies.
• Students who struggle with math reasoning often have difficulty communicating their thinking. Extra sensitivity and encouragement must be shown for these students as they develop and strengthen these sets of process skills. Questioning can scaffold students who are challenged by discussing their math thinking.
TECHNOLOGY CONNECTION
• Teacher background information as well as student practice materials on the topic of elementary mental math strategies
• Mental computation strategies with some fun graphics to demonstrate the strategies.
• A fun race against time using mental addition skills.
Name _____________________________________ Date ____________________
Mental Mathematics
When your teacher gives you an addition problem, solve it using mental mathematics and then record your thinking in the correct box below. During student sharing, if you like a strategy used by another student, record it in the same box.
|Problem #1 |Problem #2 |
|Problem #3 |Problem # 4 |
LEARNING TASK: The Power of Properties
STANDARDS ADDRESSED
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
M3P1. Students will solve problems (using appropriate technology).
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• What are the properties that relate to addition and subtraction?
• How can we verify the results of an addition problem?
• How does knowing the commutative property help us add numbers easily and quickly?
• How does knowing the identity property help us add numbers easily and quickly?
• How is zero different from any other whole number you might add or subtract?
• How does knowing the associative property help us add numbers easily and quickly?
• How do properties work in subtraction problems?
MATERIALS
• Counters (i.e., connecting cubes, cardboard cutouts, or paper clips)
• “The Power of Properties” Student Recording Sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will use counters to demonstrate various addition properties and explore these properties with subtraction.
Comments
“Property” is just one of many words in the English language that has more than one meaning. While mathematical meanings may seem obvious to adults, children may not be able to understand “mathematical properties” and “difference” without some explanation and discussion. Also, the word “sum” and its homophone “some” may require clarification. This is particularly true for students who are not yet proficient in the English language.
Most children find it easy to understand the commutative property of addition and the identity property of addition, especially if they have seen them modeled and tried them themselves many times with manipulatives. Some areas where students may have more difficulty are listed below.
• For subtraction there is no commutative property and no associative property.
• The number “zero” should not be referred to as (the letter) “O” since this will cause confusion when working with variables.
• It is important that students first simplify what is inside the parentheses when using the associative property.
Background Knowledge
Be careful about making an inaccurate statement such as, “You cannot subtract a larger number from a smaller number.” It is possible to subtract a larger number from a smaller number; however, the result is a negative number. You want students to have access to correct mathematical information, even though they will not study positive and negative numbers until middle school. Therefore, you might say, “You cannot take away 12 pennies when you only have 8 pennies.” Or use a similar example with concrete materials.
Questions/Prompts for Formative Student Assessment
• Explain how you represented the property.
• What do you notice about the sum of an addition problem if you switch the order of the digits?
• What do you notice about the difference of a subtraction problem if you switch the order of the digits?
• How is understanding the commutative property helpful?
• What happens to a number when you add zero to it?
• What happens to a number when you subtract zero from it?
• How is understanding the identity property helpful?
• What do you notice about the sum of three addends if you change the pair of numbers you add first?
• What do you notice about the difference of three numbers if you change the pair of numbers you subtract first?
• How is understanding the associative property helpful?
Questions for Teacher Reflection:
• Which of my students are able to complete this task independently?
• What level of support do my struggling students need in order to be successful with this task?
• Are students able to model the properties?
• How clearly can my students articulate their understanding of the properties of addition and subtraction?
• What real life connections can I make that will help my students utilize the skills learned from doing this task?
DIFFERENTIATION
Extension
• Have students create story problems that include use of the properties of addition and subtraction.
• Have students compute addition problems that involve larger numbers of addends and prove in more than one way, using parenthesis, that the sums are the same.
• Have students model the properties with larger numbers.
Intervention
• Have students draw a picture to go along with their number sentences that will also demonstrate what happened. Pay close attention to how students model the problem. Have them explain their thinking.
• Pose a story problem to students and have them use counters or other manipulatives to model the problem.
An example is:
TECHNOLOGY CONNECTION
• Students practice identification and application of arithmetic properties
Name ________________________________________ Date ___________________________
The Power of Properties
Use the boxes below to model and correctly identify the properties of addition. For this task, you may use connecting cubes, paper clips or any other small objects your teacher has provided. For each property, decide on the numbers you will use and the correct symbols to use in each number sentence.
|Commutative Property of Addition |
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|Choose two different numbers for addends. Write a number sentence to show the sum. |
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|Now change the order of the addends and write a new number sentence to show the sum. |
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|Draw a picture to illustrate your two number sentences and explain how they are alike and how they are different. |
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|Explain the commutative property of addition in your own words. |
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|Identity Property of Addition |
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|Write an addition number sentence with zero as one of the addends. |
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|Explain the identity property of addition in your own words. |
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|Associative Property of Addition |
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|Decide on three different numbers to use as addends. Write two number sentences following the given format. Keep the order of the addends the|
|same in both equations. Remember to add what is in the parenthesis first. |
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|( ___ + ___ ) + ___ = ___ |
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|___ + ( ___ + ___ ) = ___ |
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|Draw a picture to illustrate your two number sentences and explain how they are alike and how they are different. |
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|Explain the associative property of addition in your own words. |
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|What about Subtraction? |
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|Use counters to model each property again, this time with subtraction. |
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|Do the properties for addition also work for subtraction? Use words, pictures and numbers to explain what happens for each property. |
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|Commutative Property |
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|Identity Property |
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|Associative Property |
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LEARNING TASK: Take 100
STANDARDS ADDRESSED
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
M3P3. Students will communicate mathematically.
c. Analyze and evaluate the mathematical thinking and strategies of others.
ESSENTIAL QUESTIONS
• How can I learn to quickly calculate sums in my head?
• What strategies will help me add multiple numbers quickly and accurately?
MATERIALS
• A deck of cards containing two of each of the following numbers: 10, 20, 30, 40, 50, 60, 70, 80, 90, 50, 5, 95, 15, 85, 25, 75, 35, 65, 45, 55. (Copy 2 game cards sheets for each deck of cards)
• “Take Ten Game” Student Recording Sheet
GROUPING
Partner/Small Group
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
This is a card game during which students must be the first to spot combinations of one hundred.
Comments
Students should have had practice developing strategies to make combinations of one hundred easily using mental math.
Task Directions
Students play this game in pairs. Players place a shuffled deck of cards (see attached cards) between them. Player 1 turns over the top card and lays it to one side of the deck so that the number shows.
Player 2 turns the next card, laying it to the other side of the overturned deck so that the numbers are showing. Whenever the 2 cards total one hundred, the first student to say “one hundred!” gets all the cards that are face up. Play continues until all the cards have been used.
As students play, ask them to record their pairs of 100 as an addition number sentence. This gives students an opportunity to focus on the pairs that make 100 and provides a record of the game.
This game can be adapted to eliminate the speed aspect to the game. Students can take turns turning over two cards and placing them face up next to the deck of cards. If the sum of the numbers is 100 the student gets to take those cards and any others that have been turned over. If the numbers do not equal 100, then the cards are left face up and the student’s turn ends. Play continues until all of the cards have been turned over. The player with the most cards at the end of the game wins.
Questions/Prompts for Formative Student Assessment
• What do you do to help you remember the number combinations that make up one hundred?
• What can you do to find the answer quicker than your partner?
• Why doesn’t 63 + 47 equal 100?
Questions for Teacher Reflection
• Do I need to review strategies for making 100?
• Are my students evenly matched for this activity according to ability level?
• Which students seem to have a more difficult time remembering the combinations of ten? How will I assist them?
DIFFERENTIATION
Extension
• To determine a winner have each student take all the cards he or she won and add them. Students will trade cards and let their partner add the cards with a calculator. When the amounts agree, the student with the larger total wins the game.
• To determine a winner have each student estimate to the nearest 100 the sum of the cards earned. The student with the larger total wins the game.
Intervention
• If two struggling students are going to play this game together, it may help to model the game during small group instruction first. While modeling the game, use the think-aloud strategy to model ways students can think about pairs to one hundred.
• Play a “Pairs to Twenty Game” using two of each of the following cards: 1, 19, 2, 18, 3, 17, 4, 16, 5, 15, 6, 14, 7, 13, 8, 12, 9, 11, 10, 10.
TECHNOLOGY
• Offers ideas for other games and links to additional math sites.
|5 |10 |15 |20 |
|25 |30 |35 |40 |
|45 |50 |55 |60 |
|65 |70 |75 |80 |
|85 |90 |95 |50 |
Name _________________________________________ Date __________________________
Take 100 Game
Student Directions
Number of Players: 2
Materials: Deck of 40 Cards
Directions:
1. Shuffle the cards well and lay them face down in a pile on the desk.
2. Turn the top card over and set it to the side where both partners can see it. Now turn the next card over and set it to the side of the first overturned card.
3. Your goal in this game is to make sets of one hundred.
4. If the first two overturned cards equal one hundred when added together, try to be the first one to say, “One hundred!” loudly enough for your partner to hear you. If you are first to notice, you may take the cards. If your partner is the first to notice, he or she gets to take the cards.
5. If the first two cards do not make a set of one hundred, keep turning cards over and setting them on top of the first overturned cards. When someone spots a combination of one hundred, they can take all of the cards laying face up on the table. Keep playing this way until all cards have been claimed or until no cards are left and the overturned cards do not make a set of one hundred.
6. The player with the most cards at the end of the game is the winner.
LEARNING TASK: Perfect 500!
STANDARDS ADDRESSED
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
M3P3. Students will communicate mathematically.
c. Analyze and evaluate the mathematical thinking and strategies of others.
ESSENTIAL QUESTIONS
• How can I learn to quickly calculate sums in my head?
• What strategies will help me add numbers quickly and accurately?
• What strategies are helpful when estimating sums in the hundreds?
MATERIALS
• Deck of playing cards, (2 copies of the cards provided for a deck of 40 cards)
• “Perfect 500” Directions Sheet
• “Perfect 500” Student Recording Sheet
GROUPING
Partner/Small Group Game
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
This game allows students to look for combinations of numbers that equal 100.
Background Knowledge
Students should have addition skills clearly in place, and strategies for larger numbers, including counting up, counting back, pairs that make ten, pairs that make 100, and compensation strategies.
Students may find this game challenging, particularly at the beginning of the year. When introducing this game you may choose to use one of the variations of the game from the list below.
• Play just one round, the students with the sum closest to 100 wins.
• Play just one round as a class. Put the digits on the board and let students create the sum that is closet to 100.
• Discuss the relationship between pairs of 10 and pairs of 100. (i.e. 4 + 6 = 10, so 40 + 60 = 100 What about 42 + 68? Why doesn’t that equal 100?
Task Directions
The goal of the game is to have a sum as close to but not over 500 at the end of five rounds. To begin, each student is dealt 5 cards. The player uses four of the cards to make 2, two-digit numbers, saving the unused card for the next round. Each player tries to get as close as possible to 100. Students record their addition problem on the recording sheet, keeping a running total as they play.
For the second round, each player gets four cards to which they add the unused card from the first round. The student, who is closest to 500 without going over, after five rounds, is the winner.
Questions/Prompts for Formative Student Assessment
• What is one way to quickly find the answer? Can you think of another way?
• How do you know you will not go over 500?
• How do you decide which numbers to use? How do you choose which cards to use?
Questions for Teacher Reflection
• What strategies are students using successfully?
• Are there strategies that would be helpful to model for students?
DIFFERENTIATION
Extension
• Students can play “Perfect 5,000” during which each player draws 7 cards and uses 6 to make 2, three-digit numbers whose sum is close to 1,000. After 5 rounds, the player with the sum closest to 5,000 without going over is the winner.
Intervention
• Plan for students with like abilities to play against each.
• Students can play “Perfect 20” during which each player draws 4 cards and adds the numbers on three cards to find a sum as close as possible to 20. After 5 rounds, the player with the sum closest to 20 without going over is the winner.
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Name _____________________________________________ Date ______________________
Perfect 500
Number of Players: 2 or 3
Materials: One deck of 40 cards (4 each of the numbers 0-9)
Directions:
1. The goal of the game is to have a sum as close to but not over 500 at the end of five rounds.
2. To begin, shuffle the deck of cards.
3. Deal 5 cards to each player. Use four of the cards to make 2, two-digit numbers, saving the fifth card for the next round.
4. Try to get as close as possible to 100. Record your addition problem and sum on the recording sheet, keeping a running total as you play.
5. For the second round, each player gets four cards to which they add the unused card from the first round.
6. After five rounds, the winner is the player who is closest to 500 without going over.
Perfect 500!
Player 1 ______________________________________ Date ___________________
| | |
|Here is another story: |Write a story for this number sentence: |
| |18 + □ = 61. |
|I had some quarters in my piggy bank. For my 8th birthday, Jacob gave me 8 | |
|quarters. Now I have 85 quarters. | |
|Here is a number sentence for my story. | |
|□ + 8 = 85 | |
| | |
|What number goes in the box? How do you know? | |
| |What number goes in the box? How do you know? |
PERFORMANCE TASK: Car Wash at Carter Elementary
STANDARDS ADDRESSED
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
d. Model addition and subtraction by counting back change using the fewest number of coins.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How can I use what I understand about money to solve word problems?
• What is the most efficient way to give change?
MATERIALS
• Chalkboard, overhead projector, or interactive white board for whole group instruction
• Student Task Sheets for small group or cooperative learning groups
GROUPING
Whole/Small Group/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students will solve four story problems within the context of students holding a car wash to raise money for their elementary school. Afterwards, students will write and solve two word problems within the context of a school fair.
Task Description
Teachers may want to use the book Sluggers’ Car Wash by Stuart J. Murphy or a similar book about fundraising or having a car wash to launch this lesson.
Comments
When students make up their own number stories, teachers gain insight into the students’ understanding of the problem solving process. Simplify or extend these situations to help students grasp how to solve addition problems with the use of subtraction. This is also intended to give students practice with adding and subtracting money amounts since they form important benchmarks (5s, 10s, 25s, & 50s) that will also be used in multiplication and division.
Background Knowledge
In third grade, students begin to use decimal notation to record money amounts. However, they are not expected to compute with two-digit decimals and should not be required to add or subtract money written in decimal form. When computing with money, students should have access to play money and be encouraged to use math pictures or mental math strategies (including the use of benchmark numbers with money, 5, 10, 25, 1.00).
Before working on this task, students will need to work with coins and bills. In pairs, students choose an item from a menu, catalog, or sale paper. Each student is given an amount of money (i.e., $5, $10, $20, $50, or $100) depending on the cost of the items being purchased. The partner determines the amount of change by counting back (counting up) from the given amount of money. Students should give the fewest number of coins and bills possible (use the most efficient method). Also, each student should tell their partner the total value of the change they receive.
An example follows:
Hannah chose a soccer ball for $8.59 from a sports catalog. She pays her partner with a $10 bill. Her partner counts back Hannah’s change by saying, “one cent makes $8.60, one dime makes $8.70, one nickel makes $8.75, and one quarter makes $9.00. One more dollar makes $10.00. Your change is $1.41.
Students should be laying the money on the table as they are counting back, then count the total amount of change. Their partner should be checking for accuracy. If there are discrepancies another group can be asked to check the work or a calculator can be used.
Partners should keep track of the cost of their items, the amount paid, and the change received in a table. See the attached “What’s the Change?” student recording sheet.
Questions/Prompts for Formative Student Assessment
• What is the next benchmark number? What would you need to add to get to the next benchmark number?
• What coin(s) can you use to get to the next benchmark number? Which is most efficient?
• If you added the value of a penny (nickel, dime, or quarter), what would the total be?
• Does your partner agree with you? How do you know your amount is correct?
• How much money did he/she start with? What was the cost of the item?
Questions for Teacher Reflection
• Is student work accurate?
• Which students can fluently count back change?
• At what point do students have difficulty with this task?
• What type of problem(s) or question(s) would help students become more efficient with counting back change?
DIFFERENTIATION
Extension
• Have students add two or more items before determining change.
Intervention
• Provide additional practice with the “What’s My Change” activity. Student recording sheet is provided on page 66.
TECHNOLOGY CONNECTION
provides teachers with resources for a variety of word problems.
Name __________________________________________ Date _____________________
Car Wash at Carter Elementary School
|José, Sarah, and Mika bought supplies for the car wash. The soap cost |The students paid for the supplies with a $20.00 bill. |
|$2.89 for each bottle. The sponges cost $1.29 each. The students |The students were given their change in the most efficient way. What |
|bought one bottle of soap and one sponge. How much money did they |coins and bills did they receive? |
|spend? How do you know? | |
|What was the value of their change? How do you know? |The students charged $4.25 for each car they washed. One customer paid |
| |for two car washes with a $10.00 bill. The students gave the most |
| |efficient change. What coins and bills did they give? |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| |What was the value of the customer’s change? How do you know? |
Name ______________________________________________ Date _____________________
Partner’s Name ___________________________________________
What’s My Change?
1. Choose an item you wish to buy. Record the item and its cost in the chart below.
2. Pay for the item. Record the amount you paid (use only bills to pay, no coins) in the chart.
3. Ask your partner to count back your change and tell you the total amount of change you will receive. Record the total change in the last column.
4. Switch jobs. Let your partner choose an item to buy.
5. Take turns choosing items to buy until the chart below is complete.
|Item |Cost |Amount Paid |Total Change |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
UNIT ONE CULMINATING TASK
PERFORMANCE TASK: WHAT’S THE STORY HERE?
This culminating task represents the level of depth, rigor, and complexity expected of all third grade students to demonstrate evidence of learning.
STANDARDS ADDRESSED
M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.
a. Identify place values from tenths through ten thousands.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.
M3N2. Students will further develop their skills of addition and subtraction and apply them in problem solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
d. Model addition and subtraction by counting back change using the fewest number of coins.
M3A1. Students will use mathematical expressions to represent relationships between quantities and interpret given expressions.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the unknown in a number sentence.
RELATED STANDARDS
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M3P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How can I show what I know about addition and subtraction, problem solving, and estimation?
MATERIALS
• Large paper (11”x 17”) – one sheet per student
• Scissors
• Markers, crayons, or colored pencils
GROUPING
Independent Performance Assessment
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students make a book following given guidelines that demonstrate the concepts learned in this unit.
Background Knowledge
Satisfactory completion of direct instruction, guided practice, independent practice, and performance of concepts contained in this unit, such as addition and subtraction, place value, estimation, and arithmetic properties is necessary prior to asking students to complete this assessment independently.
Encourage students to write all of their word problems based on one topic or theme. For example, students could choose to write all problems about soccer or a favorite hobby.
Below are the student directions for this task.
Your task is to make a book to demonstrate what you have learned in this unit.
There are many ways to make a simple 8-page book, the directions for one foldable are at the following link:
Your book will need 8 pages. Use the following directions to complete your book.
□ Page 1 – title, author, publishing date
□ Page 2 – addition story showing commutative property
□ Page 3 – addition story showing associative property
□ Page 4 – addition story showing identity property
□ Page 5 – subtraction story showing take-away
□ Page 6 – subtraction story showing comparison
□ Page 7 – addition story with a string of at least 3 numbers
□ Page 8 – your choice of an addition or subtraction story using estimation
Make sure each page contains the following:
• Use at least one two-digit and one three-digit number in each story
• Model each story with an illustration or drawing using base ten blocks,
• Put the correct solution on the back of each page or in a separate answer key
• Show how you checked your work by using the inverse operation.
Make sure your book is clearly written, that your math stories are correctly spelled, capitalized, and punctuated, and that you follow the steps above when making your book. Put page numbers on the bottom right hand corner of your book pages and if desired, decorate the title page.
Comments
While this task is intended to serve as a summative assessment, it also may be used for teaching and learning. If used as an assessment, it is important that all elements of the task be addressed throughout the unit so that students understand what is expected of them. Also, if using a rubric, students should be given a copy of the rubric as part of the teacher introduction of the assessment, so that they are aware of the expected rigor and quality for their work. A sample rubric is provided below.
Questions/Prompts for Formative Student Assessment
• What is your plan for completing this assessment?
• Do you have a draft of your project?
• How will you prove that your answers are correct?
Questions for Teacher Reflection
• Have students shown a clear understanding of all the concepts and skills in this unit?
• Are students able to complete the task independently or do they need prompting?
• Which students need more support to perform this task?
• How can I extend the learning of students who are finding this task easy to complete?
DIFFERENTIATION
Extension
• Rather than having a separate page for each of the properties, ask students to identify the use of each property within other pages of the book. In this way, students can create their own problem and solution for the three open pages.
Intervention
• Provide story frames or other supportive structures to allow students to be successful in completing each page of their book.
• Break the task into related, manageable chunks, eliminating unnecessary steps or combining steps (for example, estimation could replace one of the addition or subtraction pages).
TECHNOLOGY
An alternative to a book would be to use PowerPoint, or a similar program, and have some (or all) students make slides instead of a book. Photographs of the students and their work can be inserted into slides for a presentation for parents for the current year or to show benchmark work to students next year.
Name _________________________________________ Date __________________________
WHAT’S THE STORY HERE?
Your task is to make a book to demonstrate what you have learned in this unit.
There are many ways to make a simple 8-page book, the directions for one foldable book are at the following link:
Your book will need 8 pages. Use the following directions to complete your book.
□ Page 1 – title, author, publishing date
□ Page 2 – addition story showing commutative property
□ Page 3 – addition story showing associative property
□ Page 4 – addition story showing identity property
□ Page 5 – subtraction story showing take-away
□ Page 6 – subtraction story showing comparison
□ Page 7 – addition story with a string of at least 3 numbers
□ Page 8 – your choice of an addition or subtraction story using estimation
Make sure each page contains the following:
• Use at least one two-digit and one three-digit number in each story
• Model each story with an illustration or base ten drawing,
• Put the correct solution on the back of each page or in a separate answer key
• Show how you checked your work by using the inverse operation.
Make sure your book is clearly written, that your math stories are correctly spelled, capitalized, and punctuated, and that you follow the steps above when making your book. Put page numbers on the bottom right hand corner of your book pages and if desired, decorate the title page.
|Standard ↓ |Exceeding |Meeting |Not Yet Meeting |
|M3N2. Students will further develop their skills |--Property number sentences use |--Student work shows correct|--Property number sentences |
|of addition and subtraction and apply them in |two- or three-digit numbers |property number sentences |or omitted, misidentified, or|
|problem solving. |correctly |and correct solutions |done incorrectly. |
|Use the properties of addition and subtraction to | |--Addition and subtraction |--Solutions to addition and |
|compute and verify the results of computation. | |number stories show correct |subtraction problems are |
|Use mental math and estimation strategies to add | |solutions |incorrect or omitted. |
|and subtract. | | | |
|Solve problems requiring addition and subtraction.| | | |
|M3A1. Students will use mathematical expressions |--Algebraic expressions are |--Number sentences are |--Number sentences are |
|to represent relationships between quantities and |written correctly and all steps |written correctly with |omitted or written |
|interpret given expressions. |required for solving for the |symbol to represent the |incorrectly. |
|Use a symbol, such as □ and Δ, to represent an |unknown are listed in all cases,|unknown. |--No symbol is used to |
|unknown and find the value of the unknown in a |including property usage as |--Correct solution is found |identify the unknown. |
|number sentence. |needed. |for the unknown in each |--Incorrect solutions are |
| | |number sentence |given for the unknowns in |
| | | |number sentences. |
|M3P1. Students will solve problems (using |--Problem solving strategies are|--Addition and/or |--Problem solving strategies |
|appropriate technology). |apparent from the work shown as |subtraction stories show |are not apparent from work |
|Apply and adapt a variety of appropriate |well as student description of |work that demonstrates |shown. |
|strategies to solve problems. |strategies utilized. |student strategies. | |
|M3P3. Students will communicate mathematically. |--Math stories are |--Addition and subtraction |--Addition and subtraction |
|Communicate their mathematical thinking coherently|sophisticated, with multi-step |stories are logical, fit the|stories are omitted, |
|and clearly to peers, teachers, and others. |problems accurately described |operation designated for |illogical or difficult to |
|Use the language of mathematics to express |and solved in at least one case.|them, and use math |understand and/or solve. |
|mathematical ideas precisely. | |vocabulary. | |
|M3P4. Students will make connections among |--All inverse operations are |--Student work demonstrates |Student work does not |
|mathematical ideas and to other disciplines. |used as appropriate and student |or describes how inverse |demonstrate any connection |
|Recognize and use connections among mathematical |description accompanies the |operations can be used to |between inverse operations. |
|ideas. |work. |check solution found in | |
|Understand how mathematical ideas interconnect and| |number sentences. | |
|build on one another to produce a coherent whole. | | | |
|M3P5. Students will represent mathematics in |--Student drawings are |--Illustrations and base ten|--Models and illustrations |
|multiple ways. |sophisticated or unique and may |drawings support the number |are missing or do not support|
|Create and use representations to organize, |involve representations of |sentences and number stories|the story problems. |
|record, and communicate mathematical ideas. |decimal numbers or numbers in |described. | |
| |the hundreds or thousands. | | |
3rd Grade Math Unit 1 Performance Assessment RUBRIC
-----------------------
Rashad gave his two sisters some of his chewing gum. He gave Samantha 2 pieces in the morning and 5 pieces after lunch. In the evening, he gave Samantha 8 more pieces of gum.
Rashad gave his other sister, Tina, 8 pieces in the morning and 5 pieces after lunch. Tina said he did not give her as much gum as he gave Samantha because he only gave her 2 pieces that evening.
Is Tina correct? Use your mathematical skills to explain whether or not Rashad gave both sisters the same amount of gum.
60 61 62 63 64 65 66 67 68 69 70
30 31 32 33 34 35 36 37 38 39 40
MATHEMATICS
3
6
3
6
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