Mathematics: Number and Operations in Base Ten
Mathematics: Number and Operations in Base Ten
Cluster Heading: Understand place value
1.NBT.2
Content Standard: Understand that the two digits of a two-digit number represent amounts of tens and ones.
Practice Standard: MP6 Attend to precision, MP7 Look for and make use of structure
Problem/Task Suggestions Formative Assessment Suggestions
Numbers, Numbers
Each student receives 3 number cards.
Choose 2 cards to make the largest 2-digit number.
4 9 7
Now, choose 2 cards to make the smallest 2-digit number.
Record your answer and explain to a partner how you found your answer.
Differentiation: Supports:
• Provide a recording sheet with 2 blank rectangles for the largest and smallest 2-digit number.
• Provide a number grid.
Extensions:
• Create a word problem that uses the two 2-digit numbers.
• How would have a card that says zero change your answer? Could you get a bigger number or smaller number with a zero and using 2 cards?
Solutions:
• Largest number: 97, smallest number: 7.
• In order to make the largest number, the largest numeral has to go in the tens place and the next largest in the ones place.
• In order to make the smallest number, the smallest numeral has to go in the tens place and the next smallest in the ones place.
Observation of Students
• Do they recognize place value?
• Can they explain their strategy?
• Do they understand what the problem is asking them to do?
• Can they read the number out loud?
Questions to Guide Student Thinking:
• What is a wrong answer? How do you know it is wrong?
• Can you read the number out loud?
• What part of the number can you change to make it greater/smaller?
Misconceptions
Students may:
• Reverse the tens/ones.
• Use numbers not given.
• Not use all 3 numbers (use only the 2 numbers from the largest number to make the smallest number).
Vocabulary Considerations
Greater, greatest, smaller, smallest, place value, tens, ones
Larger/largest
Created by: Sharon Holt, Laura Parat, Laura Rhoney for Aurora University’s OEDC 5204
2.MD.8
Cluster Heading: Work with time and money.
Content Standard: Solve word problems involving dollar bills, quarters, dimes, nickels and pennies, using $ and ¢ symbols appropriately.
Practice Standard: MP1 Make sense of a problem and persevere in solving it.
Problem/Task Suggestions Formative Assessment Suggestions
How Can You Make 25 Cents?
What are all the different ways you can make 25 cents?
(Give students plenty of play/ real money and a blank sheet of paper for
recording. You may want students to work with a partner.)
Differentiation
Supports
• Read a book such as “A Quarter from the Tooth Fairy” that lists four combinations as part of the story. This gives students examples of solutions and a place to start.
• Ask a few questions of struggling students to determine if the student knows the value of each coin. If he/she does not, give the child experiences with coin games and activities, such as matching games (coins to values), race for a dollar (rolling dice, taking designated pennies and trading), playing “fish” or “concentration” where some cards have pictures of coins and other cards have values.
• Ask students to think of a way to make 25 cents with only pennies and only nickels, and ask them if they could trade any of the coins in their combination for a coin that would have the same value. This would generate another combination, which they could write down on their list. Ask, “Do you think trading will give you other combinations?”
• Provide a list of values of each of the coins to use as a reference or have the student create his/her own reference list for coins.
Extensions
• Pick up some coins and hide them. Have your partner ask you yes/no questions about your coins to figure out how many and the value.
• Ask about combinations of coins to make 50 cents or a dollar.
Solution
13 ways: 1 quarter, 25 pennies, 2 dimes, 1 nickel, 1 nickel and 20 pennies, 2
dimes, 5 pennies, 2 nickels and 15 pennies, 1 dime, 3 nickels, 3 nickels and
10 pennies, 1 dime, 2 nickels and 5 pennies, 4 nickels and 5 pennies, 1
dime, 1 nickel and 10 pennies, 5 nickels,1 dime and 15 pennies
Created by: Illinois State Board of Education Content Area Specialists
Observation of the Students
• Is the student able to recognize what the problem is asking? MP1
• Does the student know the value of each of the coins?
• Is the student able to use mental math to do simple computations or do they need to use another tool? MP5
• Does the student recognize that multiple answers are possible?
• Can the student explain his/her process to other students? MP6
• Is the student able to find all 13 ways to make 25 cents?
Questions to Guide Student Thinking
• Is there another way?
• How are you certain that there are no other possibilities?
• When processing the problem with the whole group, ask the students with fewer methods to present first and then have the other students add on so that everyone can participate in the conversation.
Misconceptions
Students may
• Stop after finding only one or two ways.
• Think that larger sized coins are worth more in value.
Vocabulary
• Cents, quarters, dimes, nickels, pennies, equal, value
2.OA.1
Cluster Heading: Represent and solve problems involving addition and subtraction
Content Standard: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions
Practice Standard: MP1 Make sense of problems and persevere in solving them.
Problem/Task Suggestions Formative Assessment Suggestions
All Roads Lead to 100.
Write two math word problems where both answers would be 100. Write the equation that goes with each of your stories and show your computation. Put an “E” by the easy problem and an “H” by the hard problem. Explain what makes the easy problem easy and the hard problem hard.
Differentiation
Supports
• Change the sum in the task to 20 instead of 100
• Ask students who are struggling to write a word problem.
• Scribe for the student as he/she dictates verbally.
• Provide a prompt to get the student started such as “A student was filling boxes with blocks and …” or “I am thinking of some numbers that…”.
• Ask the student to draw a picture or find a picture in a magazine, and then write a story asking about the number of items in the picture.
• Underline words in the prose of the story and show the student how to represent the phrase using a number and/or symbol for the operation.
Extensions
• Increase the number of story problems the student should write.
• Have students write each of their story problems on a separate index card with their name but no “E” and “H” indicated. Students exchange cards with another student, solve the problems on the cards and then indicate which problem they thought the author would have marked “H” and why. Students would then check with the author to see if they agreed on the most difficult problem.
Solutions will vary depending on the story written. All should have the same answer of 100. Some problems may have only 1 operation and only two numbers but others could have multiple numbers and both addition and subtraction included. For instance, “The teacher put 85 colored candies in the class candy jar. 15 students each took a piece from the bowl as an award for good work. A parent brought in another bag of 30 pieces of candy and added it to the jar. How many pieces of candy are in
the jar now?” (85 – 15 + 30 = 100)
Created by: Illinois State Board of Education Content Area Specialist
Observation of Students
• Are the solutions to each of the word problems 100?
• If there are computation errors, are the answers close or are there major errors?
• Are the equations correct for each of the stories?
• Does the problem have only two numbers or multiple numbers?
• Does the student use only addition? Just subtraction? Both
• Does the explanation for selection of the hardest and easiest problem make sense?
• Does the computation work indicate the student used multiple strategies to find their solutions?
Questions to Guide Student Thinking
• Can you think of a list of numbers that add up to 100?
• Do you know a subtraction problem where the answer is 100?
• Have you considered just using numbers that end in zero?
• Can you think of coins that add up to a dollar? Could you use that information to write a word problem?
• If I gave you an equation could you write a word problem to go with it?
Misconceptions
Students may
• Think they can only use two numbers in their story (and equation)
• Think that they can only use addition and don’t consider subtraction problems that result in 100.
Vocabulary Considerations
- Solution, variable, equation, computation
Mathematics: Operations and Algebraic Thinking
Cluster Heading: Represent and solve problems involving addition and subtraction.
Content Standard: Use addition and subtraction within 100 to solve one and two-step word problems involving situations of adding to, taking
2.OA.1
from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Practice Standards: MP1 Make sense of problems and persevere in solving them, MP2 Reason abstractly and quantitatively.
Problem/Task Suggestions Formative Assessment Suggestions
Be the Mathematics Author
Write a word problem for 34 – 16 =
Give each student an 8.5” x 11” recording sheet that has the math equation at the top. Leave half the page for him/her to write the word problem.
Solve the problem using pictures, words, numbers, and mathematical tools that make sense to you.
Leave the bottom half of the page blank for each student to represent or draw his/her solutions.
Differentiation
Support
• Ask the students to describe the word problem and decide if they agree that it matches the original math sentence.
Extensions
• Write your own math sentence and word problem and ask a classmate to solve it.
• Ask the students to solve a problem such as 34 - = 29
Solutions
Any reasonable story and pictorial representation leading to 34 – 16 = 18.
Observation of Students
• Does each student understand the task?
• How does each student represent his/her thinking (counters/physical model, pictures, words, hundreds chart, open-number line)?
• Is each student able to communicate his/her ideas to others?
• Is each student willing to share- individually, small group, or whole group?
Examine student’s written work
Does the student:
• Have a story that makes sense with the numbers in the equation?
• Include accurate calculations?
• Use correct notation?
• Is there a correct label to go with the problem?
Questions to Guide Student Thinking
• What does the 3 in 34 mean? The other digits?
• Could you solve or represent the problem in another way?
Misconceptions
• Student’s story may indicate an addition problem instead of a subtraction problem.
• Students may add 34 + 16.
• Students may not use place value understanding in their calculation.
• Students may calculate an answer of 28 or 22.
Adapted from: How to Assess While You Teach Math: Formative Assessment Practice and Lessons, Grades K-2: A Multimedia Professional Learning Resource, by Dana Islas, Math Solutions, ISBN: 978-1-935099-17-8
Mathematics: Counting and Cardinality
Cluster Headings: Count to tell the number of objects. Compare numbers
.4a
.6
Content Standards: Understand the relationship between numbers and quantities; connect counting to cardinality. a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
Practice Standard: MP6 Attend to precision, MP 1 Make sense of the problem and persevere in solving them.
Problem/Task Suggestion Formative Assessment Suggestions
Rolling a Number Cube
Give pairs of students several unifix cubes and a number cube with numerals
on each side. Adjust the numerals on the number cubes to address the current needs of the students. Students take turns rolling a number cube and counting out the number of unifix cubes indicated. The child then snaps the counted cubes into a train and compares his/her train to a partner to see who has
more.
After each child has had 5 turns, have him/her look at the five individual trains and try to see who would have the most if they snap the trains together. After estimating who may have more they each snap their 5 trains together and lay them beside each other to see which is longer. Finally they may count to see how many are in each long train.
Differentiation
Support
• Use a large foam rubber number cube with dots rather than numerals.
When the student rolls the number cube, he/she places a unifix cube on
each dot, then snaps them together. Next step is to use a number cube with dots and numeral
Extensions
• Roll two number cubes and find the sum to determine the number of unifix cubes to snap together and compare.
• Determine not only who has more but how many more.
Observation of Students
Does the student
• Demonstrate the verbal counting sequence?
• Recognize the rolled numeral and take that many cubes?
• Use 1-to-1 correspondence skills?
• Have a way of keeping track of his/her count?
• Realize the last number said indicates how many she/he has?
• Know which train is bigger by prior knowledge, estimating or by matching?
Questions to Guide Student Thinking
• Do you know which ones you have counted?
• Where could you put the counted cubes?
• Tell me how you know which train has more cubes.
Misconceptions
Students may not understand that the last number they say indicates how many items are in the set.
Vocabulary Considerations
Greater than, less than, equal to
Task adapted from: Developing Number Concepts: Counting, Comparing, and Patterns Book 1 by Kathy Richardson, Apr 16, 1998
.5
Cluster Heading: Count to tell the number of objects.
Content Standard: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
Practice Standard: MP2 Reason abstractly and quantitatively.
Problem/Task Suggestions Formative Assessment Suggestions
Letters in Your Name
Show students a short name and a long name. Ask about differences.
Students will find out how many letters are in their own names by cutting apart
the letters in their names and counting. Students glue their letters on a piece of construction paper. They then glue on a 2" × 2" paper with the number of letters in their name. Have students look at their neighbor's name. Ask, "Who has the shorter name, you or your neighbor?" They share responses with the class, comparing each others’ names. Give each student a paper strip, which reads, “I have letters in my name.” Students will complete the sentence by writing the number of letters in their name, and gluing the sentence on their page. Students can finish their page for the class book by gluing on a snapshot of themselves or drawing a picture of themselves.
Differentiation
Support
• Write their name on 1-inch graph paper with a letter in each square. Don’t cut the letters apart but hold their name up to others to compare lengths. Put a mark on each letter as you say the number word.
Extensions
• Students’ names, written on individual strips, are placed in a box. Have students draw names & count the number of letters.
• Students, in pairs, take turns rolling a die. They look for classmates’ names containing the same number of letters as indicated on the die.
• Students write their names on 1-inch grids/graph paper, one letter per square. They cut their strips and compare with classmates’ name strips.
• Read the story of Chrysanthemum, by Kevin Henkes. Have students count the number of letters in Chrysanthemum’s name & compare the number of letters in her name to the number of letters in their names.
Observation of Students
• Listen for accuracy as each student counts the letters in his or her name.
• Use completed name pages to determine if students were are able to write the number of letters in their names.
Questions to Guide Student Thinking
• Look at names of students in our class. What is different about
everyone's name?
• How did you find out what number you needed to write your sentence?
• Is your name longer or shorter than your neighbor's name?
• If we arrange our class book from shortest name to the longest name, whose name would be first?
• Show students the pages of the book. “If this is the first page what would the next page be?”
Misconceptions
Students may not understand that when they are counting the last number they say indicates how many items (in this case letters) are in the group
Task created by: Deeanna D. Golden Adapted from:
Mathematics: Operations and Algebraic Thinking
Cluster Heading: Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.3
Snap It
Content Standard: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
Practice Standards: MP 1 Make sense of problems and persevere in solving them, MP5 Use appropriate tools strategically.
Problem/Task Suggestion Formative Assessment Suggestions
Observations of Students
Ask each student to count out 10 unifix cubes and snap them together in a train. While standing in a circle, students hold their unifix cubes behind their backs. When the teacher/designated student says “Snap It”, students break their trains into two parts. Students bring one part in each hand around to the front of their bodies for all to see. Each child reports on the combination of
ten by saying, e.g., 4 plus 6 = 10.
Differentiation
Support
• Have the students start with 2-9 unifix cubes instead of 10.
Extension
• Have children reveal what is in one of their hands but keep the other behind their back. Other students try to figure out what is in the hand behind the back.
• Write equations based on the combinations they make.
• Play Tens Go Fish card game. Have students look for pairs of numbers that equal 10. If a child has a 7 in their hand they ask a partner “Do you have a
3?” This will work like a traditional “Go Fish” card game.
Solution
• Each child reports on the combination of ten.
Is the student able to:
• Make combination pairs for a given number?
• Find the missing number from the pair?
• Represent the pair in materials?
Questions to Guide Student Work
• What happens to the ten cubes when you snap them?
• Is there more than one way to snap your train?
• What must the two pieces of the train total?
Misconceptions
• Believe that there is only one to break up ten.
• Believe that they must break it in half.
• Believe that after the break there is no longer a total of ten.
Vocabulary
Sum, equal, addition, adding, subtraction
Adapted by: Joan Barrett from the following resources: “Snap-It” activity from Kathy Richardson’s Developing Number Concepts Book One: Counting, Comparing by Kathy Richardson ; Dale Seymour Publications Compiled Apr 1, 1998; Tens Go Fish –Investigations in Number Data and Space grade 1, TERC 2012
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