Sunday, October 20, 2013
common core 1st grade module
COMMON CORE MODULE:
Adding and Subtracting Single-Digit Numbers in Grade 1
MODULE SUMMARY
Content area focus: Adding and subtracting single-digit numbers
Priority standards: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). (1.OA.6)
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). (1.OA.5)
Supporting Standard: 1.MD.2 Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end.
Domain: 1.OA Operations and Algebraic Thinking
Instructional time: 13 days.
There are only 10 digits in our number system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These digits are often called the Hindu-Arabic numerals because they originated in India and were brought to Europe by Arabic civilizations during the Middle Ages.
In this base-10 number system, there are exactly 55 unique ways to add single-digit numbers (see chart in Appendix I). Students who master these 55 fact families will build the foundation that is required for all future mathematical endeavors, including multiplication, operations involving fractions, algebra, and geometry.
This module has been developed to guide first-grade students and instructors through the crucial skill of adding and subtracting all of the single-digit numbers by asking students to create representational drawings, measure distances, and become fluent with numerical symbols used to represent these quantities. Ultimately, the goal for the end of the school year is for students to develop a deep understanding between the concept of quantity and numerical representation and to become fluent with all 55 fact families.
Our base-10 number system requires students to develop a deep conceptual understanding of the value of each of the first ten counting numbers and become fluent in the operation of addition as it relates to these ten numbers. The intention is for this module to take place early in the school year, providing students with a strong foundation adding and subtracting single-digit numbers before moving on to double-digit numbers.
The module assessment will seek to determine if a student has a deep understanding of the first ten counting numbers as well as possessing the fluency to quickly and accurately add and subtract all single-digit numbers.
CONTENTS
I. Module overview
a. Content area focus and priority standard
b. Instructional time
c. Assessment goals
d. Assessment tools
II. Integrated tasks
a. Representational/Geometric Tasks
b. Measurement Tasks
c. Computational Tasks
d. Writing Tasks
III. Module outline
a. Section1
i. Section 1 summary
ii. Section 1 pre-assessment
b. Section2
i. Section 2 summary
ii. Activities that emphasize conceptual understanding of section 2 content
iii. Activities that emphasize fluency of section 2 content
iv. Section 2 assessment
c. Section3
i. Section 3 summary
ii. Activities that emphasize conceptual understanding of section 3 content
iii. Activities that emphasize fluency of section 3 content
iv. Section 3 assessment
IV. Day-to-day description of each section
a. Section 1
b. Section 2
c. Section 3
V. Appendices
a. 55 Single-digit addition problems
b. Pre-Assessment
c. Formative Assessment, Section 2
d. Final Module Assessment
MODULE OVERVIEW
Content area focus: Adding and subtracting single-digit numbers
Priority standards: 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). (1.OA.6)
(1.OA.5) Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
Supporting Standard: 1.MD.2 Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end.
Domain: 1.OA Operations and Algebraic Thinking
Instructional time: 13 days.
The timeframe can be adjusted depending on student needs based on mastery of prior knowledge concepts.
Prior knowledge skills required: The most important prior knowledge students must possess is counting from zero to twenty forwards and backwards. If students have not mastered this skill extra time should be built into the beginning of the unit.
Students with Disabilities (SWD) will benefit from structured/extended work with manipulatives and visual models before relying on abstract representations of fact families.
Pacing for English Language Learners (ELL) should plan for extra time when working with text-based questions and word problems. ELL students should be given every opportunity to express fluency with numerical symbols. Mathematics is known as a universal language and ELL students should acquire this mathematical language as rapidly as their general education peers. The visual and numerical representations emphasized within this unit should help facilitate that process.
Assessment goals: Fluency, conceptual understanding, and extension options
Fluency: By the end of this module, students will be able to quickly and accurately determine the sums and differences of all 30 single-digit fact families that result in single-digit solutions, as well as the 5 fact families that result in sums of 10.
Conceptual understanding: Students will be able to model with single-digit numbers and easily translate between visual representations of quantities and numerical symbols.
Extension options: This module can easily be expanded by including single-digit fact families that result in double-digit sums. These additional 20 fact families are included in the chart in the appendix. These decisions will also affect instructional time and pacing.
Assessment tools: This module contains a pre-assessment, several formative assessments, and a summative assessment.
Pre-assessment – The intention is for the instructor to determine the knowledge gaps and misconceptions that students have from the previous year of instruction. It is crucial that these gaps and misconceptions are identified and addressed throughout instruction of this module. Failure to do so will result in frustration for both students and instructors, as math is a supremely cumulative subject.
Results of this pre-assessment may also lead to identifying the specific needs of students with disabilities and English language learners.
The pre-assessment also contains material that will be covered in this module. If a student has already mastered some or all of the material that is contained within this unit, accommodations should be made for the advanced student as well allowing this student to progress with more to new material. Differentiation is not easy, but we all know that students who are bored, either because they are behind or because they are waiting for challenging material. In this case, differentiation is very manageable – offer the advanced student double-digit numbers to add and subtract, starting with problems where there is no borrowing and then including borrowing.
Formative assessments – These are concise assessments, intended to measure the learning targets for this module in small intervals and inform lesson planning for subsequent sessions. All through the module there are descriptions of in-class, formative assessments that the teacher can do, both formal and informal. In the appendix there is an assessment for section 2 that comprises most of the module. This assessment can be divided up and used at intervals throughout the module.
Summative assessment – By the end of the module students are expected to demonstrate a mastery of the basic material presented on this assessment. Student performance on the final assessment should continue to inform the teacher regarding the gaps and misconceptions that students may still possess relative to this material.
Extension material may be included in this assessment for students who have gone beyond the basic expectations.
Integrated tasks
The following tasks will integrate skills that overlap with some of the other core standards from this domain as well as the entirety of the grade 1 standards. A tight focus will be maintained on the priority standard.
Representational/Geometric Tasks:
Students will represent each of the ten digits in our number system through a variety of drawings, ranging from simple tick marks to basic objects and geometric figures. This process will allow students to internalize the meaning and value of each number so that when the student draws 10 circles, for example, 10 is not simply an abstract symbol, rather it is a quantity that can be visualized, counted and named. Further, when those 10 circles are colored in to represent the 5 fact families for 10, namely: {1+9=10, 2+8=10, 3+7=10, 4+6=10, and 5+5=10}, students will have created a visual representation of these fact families.
Students will also experience the value of numbers through tactile experiences like working with building blocks. A teacher may prompt, “If we combine a 7-block row of Lego pieces and a 3-block row, does that create a 10-block row? And if we take 3 blocks away from a 10-block row, how many blocks are left?” Students will experience these fact families in as many different concrete scenarios as possible so that the concept of number eventually transcends the objects at hand and students become truly fluent with the idea of number the same way they become fluent with the words they read in a story.
Measurement Tasks:
Using rulers, students will be asked to be more precise with number and construct line segments with specific whole number lengths using both inches and centimeters. A number now represents a dimension, length. Students will compare the different line segments, further contextualizing the first ten whole numbers in concrete terms that students can see and touch.
Students will measure their hands and objects like books and get a sense of what a length of 8 centimeters signifies, looks like, and feels like. They will be asked to estimate the lengths of other objects in the classroom, using the line segments they constructed as a frame of reference. All of the measurements will be in terms of nearest whole number, which the students can see on their ruler. This task brings in a supporting standard from the geometry domain. This supporting standard will complement the priority standard by allowing students to “see” a number as a very clear quantity.
Computational Tasks:
Students will be asked to perform addition and subtraction with the single-digit fact families, particularly the fact families that add to 10. These five fact families (1+9=10, 2+8=10, 3+7=10, 4+6=10, and 5+5=10) are the key fact families in our base-10 system and, if mastered, they will unlock all of the mysteries of addition and subtraction, including, but not limited to, place value, borrowing, and mental math.
Numerical fluency, as measured through speed and accuracy, can only be achieved through the mastery of these 5 fact families. Working with fact families will also prompt students to consider the basic properties of mathematics, such as the commutative property, i.e.: is the following number sentence true: 3 + 7 = 7 + 3 ?
Writing Tasks:
Students will be asked to reflect upon the discoveries they make and to write about them in a way that summarizes their conceptual understanding of numbers and computation, particularly as their observations relate to the different ways to create ten using pairs of digits and the discovery of the addition properties of equality. Students may be asked to incorporate visual representations to accompany their written work. Ideally, the student will create a drawing and provide the simple text and a basic number sentence that all agree. There are examples below in the day-to-day plans that illustrate this activity.
In their writings, students will be asked to consider even and odd numbers and make observations about adding and subtracting such numbers.
Module Outline: This unit can be broken down into three sections of instruction, followed by additional activities, some designed for enrichment and applications of the learning that happens here.
SECTION 1 What are numbers and how can we represent them?
Section 1 summary: This section serves as a brief introduction to the unit and focuses on assessing whether or not students have developed a deep conceptual understanding of the numbers 0 through 9.
The cumulative nature of mathematics demands that the teacher thoroughly assesses whether or not students have mastered concepts from the prior year, kindergarten. The goal of this section is to remind students of the concepts from the previous year and then to see if students have mastered these concepts. If not, it is imperative that the teacher identifies any gaps or misconceptions that may exist in order for the teacher to plan some strategic re-teaching of particular topics to specific students. If this kind of necessary, targeted instruction does not take place students will be standing on loose foundations.
Section 1 Pre-Assessment: Students will be asked to perform four tasks:
1. Visualization: given a picture of number of objects between zero and ten, quickly and accurately identify the number of objects depicted.
2. Visual representation: accurately draw a picture that represents two groups of objects, for example: 5 fish and 8 birds.
3. Written description: write a simple sentence or two that accurately describes the relative number of objects depicted in the drawing, using phrases such as, “more than” and “less than”.
4. Verbal: count from zero to twenty forwards and backwards. Count up to twenty when prompted from any number along the continuum.
Additionally, there will be material on the pre-assessment that will be covered in this module. The expectation is that students will leave these answers blank. This material appears here for two reasons:
First, if students don’t know this material, they will simply leave it blank and the teacher will use the data as a baseline to compare against the final module assessment. This comparison will allow the teacher and student to clearly see what has been learned during the course of instruction of this module.
The second reason concerns the case of students who are able to correctly fill in these questions. If there are students in the class who already know the material that is about to be taught, they should not be simply sit through the lessons and go through the motions. Instead, the teacher would be well advised to spend time with these students to determine if they have simply memorized facts or if they truly have a deep conceptual understanding of the content. If there are students who have mastered more advanced material, then they should be given every opportunity to progress through the learning standards. Differentiating instructions for students in this situation is as important as differentiating for ELL students and students with disabilities.
SECTION 2 How do numbers combine with each other through addition and subtraction?
Section 2 summary: Students will use their mastery of the concept of numbers as representation of quantities to combine various quantities through addition and to take away portions of quantities through subtraction.
Activities that emphasize conceptual understanding of section 2 content:
• Students will be asked to create drawings that visually represent addition and subtraction sentences.
• Students will be asked to describe in writing the numerical sentences that accompany their drawings. For example, if a student draws five fish swimming towards three fish, the student will add text to the drawing by writing, “5 + 3 = 8” and something along the lines of, “Five fish plus three fish equals eight fish.”
• Students will be introduced to the concept of addition/subtraction fact families. For example, 3, 7 and 10 are a fact family because 7 + 3 = 10, 3 + 7 = 10, 10 – 3 = 7, and 10 – 7 = 3.
• The fact families will be explored and mastered through a series of structured activities. These activities will employ a variety of quantifiable objects including, but not limited to, illustrations provided by the teacher, drawings created by students, drawings of line segments, the number line, geometric shapes, and building blocks, such as Lego pieces.
• Students will discuss, write about and record the observations they make after each of the activities listed above. These observations should include any discoveries that students may make related to one of the addition properties of equality (for example: 2 + 8 = 8 + 2), as well other important observations, such as the following: using the numbers 0 to 9, there are only 5 distinct pairs that have a sum of 10.
Activities that emphasize fluency of section 2 content:
• “Counting up” will be an important skill emphasized in this section. As an example, students will be asked to, “Count up three places from seven.” The student will be taught how to start at 7 and count up three places to 8, 9, and 10. Students will then write, “7 + 3 = 10.” Students who are advancing quickly can then be taught how to count down, (i.e., “Count down two places from 8.”)
• Students will begin with the most rudimentary fact families and catalogue each fact family. Initially, the fact families will be accompanied by visual representations until students feel comfortable with the numerical representations alone.
• Students will begin to master all of the fact families that add up to single-digit sums. There are 30 of these sums. There are 5 fact families that add up to ten. These 35 fact families form the core of this module and the goal is for students to master all of them. (See Appendix I.)
• Advanced students can move on to the single-digit fact families that add up to the double-digit sums other than ten. There are 20 of these fact families.
• Initially, students will be asked to complete fact family sentences in the following format:
3 + 7 = ____. Once students master this format the blank space will then appear anywhere in the sentence, for example: _____ + 7 = 10. True fluency will allow students to fill in randomly placed blanks. Mastery of this skill provides the foundation for subtraction.
Section 2 Assessment: Students will be asked to perform four tasks:
1. Visualization/verbal: solve addition problems by counting up.
2. Visual representation: draw a picture that represents a given addition or subtraction problem.
3. Written description: write a numerical sentence and a word sentence that describes the addition problem.
4. Written work: write out the addition and subtraction sentences for a fact family given the three basic elements. Also, fill in random blanks for the first 35 fact families, with special emphasis on the 10s families.
Note: Depending on what has been covered throughout the unit, all of the assessment tasks for section 2 can be extended by including the 20 single-digit fact families that add up to double-digit sums.
SECTION 3 Why is “10” such an important number?
Section 3 summary: This section will be a brief summary of the importance of the number 10 and will lead into discussions around place value in multi-digit numbers, setting the stage for the next unit, including the concept of carrying over in double-digit addition. This section will emphasize exploration of the advantages of using groups of ten as much as possible. At this point students will focus, almost exclusively, on each of the 5 fact families that add up to 10.
Activities that emphasize conceptual understanding of section 3 content:
• Students will be given large numbers of objects (357 pennies, for example) and be asked to count them in the most efficient way possible. This discovery will lead to students creating groups of tens and then counting the number of tens they have; this will lead students to see that in our system the number “ten” is an incredibly important number because of its convenience and usefulness. Is it a coincidence that we have ten fingers? And is it also a coincidence that fingers are called digits? Are our hands really just built-in calculators?
Activities that emphasize fluency of section 3 content:
• Counting by tens, which is really no different than counting by ones, will be an important emphasis of this section.
• Complete fluency of “tens partners” (single-digit fact families that add to ten) will be a key topic of understanding for this unit.
Section 3 Assessment: Students will be asked to perform four tasks:
1. Verbal: Students will count up by tens to one hundred.
2. Visual representation: Students will represent tens groups through a diagram that includes some kind of a key that represents the number ten as a collective of a set of “ones”.
3. Written: The assessment for this section will ask students to write about the project they were asked to complete where they had to account for a large number of object (for example, the 357 pennies) and how using tens helped them complete this task faster and more accurately.
4. Written work: Students will complete a chart that helped them organize their work for the big counting project.
Appendices
I. 55 Single-digit addition problems
II. Pre-Assessment
III. Formative Assessment, Section 2
IV. Final Module Assessment
DAY-TO-DAY DESCRIPTIONS OF EACH SECTION
A typical day consists of learning experiences that are teacher-directed and student-centered, along with some form of assessment. The sequence of these activities varies. Often, activities play two roles, particularly regarding assessments, which happen throughout the lesson as teachers observe the work students are doing in groups and the work that students do individually.
The day-to-day outline described below is a suggested approach to the material that is covered in this module. The structure of the math lessons as they are described below often function in ways that are similar to balanced literacy lessons where the teacher does the equivalent of a read aloud (or think aloud) with the entire class for a few minutes followed by the class breaking into small groups. During the group work teachers sit with one group at a time to assess individual skills.
These day-to-day outlines are merely a suggested approach. The daily lessons are presented in this format in order to allow teachers to visualize what the instruction of this material might look like in an actual classroom setting. However, the lesson plan activities that take place in any particular classroom should be based on decisions that are made on the district, school and classroom level and are not being prescribed here.
SECTION 1 What are numbers and how can we represent them?
Day 1:
Learning experience:
In a large group, students are shown an example of a story that is centered on a particular number, for example, The Seven Chinese Brothers, by Margaret Mahy.
During the reading the teacher prompts the students to relate number to visual quantities, for example: “Show me how many brothers there are using your fingers to represent the brothers.”
Learning experience:
In small groups, students are asked to illustrate a story that centers on a series of objects where the same number of objects repeats throughout the story. Students are asked to describe what it would be like to illustrate a story about a series of objects that totaled 2 each. Would this be an easier story to illustrate or more difficult? Why? What exactly is the difference? How do they know? How about another story with 3s? Or a story with 10s?
The Pre-Assessment:
The pre-assessment has two parts. The teacher administers the first part to each student individually and records the results. In this part of the assessment, students are asked to identify the number of objects they see on a slide. Students are also asked to count from 1 to 20 forwards and backwards and to count up to 20 from a given number other than 1.
Part 2 of the assessment asks students to work independently. The student is asked to draw a visual representation of two groups of objects and then to write about the picture they have drawn making comparisons between the relative sizes of each group.
While students are working on their illustrations, the teacher assesses individual students based on their ability to translate numbers into pictures in their drawings. The teacher also sits with individual students and asks them to count forwards and backwards to and from 20. This is considered part of the pre-assessment, as it is a skill that comes under the kindergarten standards.
Rationale:
On the first day of any module, a pre-assessment may be administered. It is common in mathematics courses all the way through calculus for students to spend the first part of the school year, or the first day(s) of each unit, reviewing prerequisite skills and sub-skills from prior years.
Students take a pre-assessment based on the relevant skills they were expected to master at the end of kindergarten. In terms of the priority standards in this module, which focus on adding and subtracting single-digit numbers, the prerequisite knowledge includes the ability to count fluently up to 20 and backwards from 20, as well as recognizing and naming the numbers of objects in a group and comparing the relative size of objects in different groups.
The teacher administers the assessment and then uses this data to inform the instruction that will take place on subsequent days. The expectations for the final assessment for this module remain the same, however the pacing of the module may be modified to accommodate strategic re-teaching that addresses the specific gaps and misconceptions in student knowledge as evidenced by the pre-assessment data.
The goal of the pre-assessment is to determine whether or not each student has a strong conceptual understanding of the quantities represented by the first ten counting numbers as well as the fluency to count through these numbers quickly and accurately in both directions. Making sure that students have developed a deep conceptual understanding of the first ten counting numbers and zero is a time-consuming process, however the teacher must spend time determining whether or not students have a “feel” for each individual number from 0 to 10. The abstract sentence 5 + 3 = 8 means nothing to a student who does not have a feel for what each of these numbers really means. Instead, the student is simply memorizing a statement of meaningless symbols.
The pre-assessment also gives the teacher a baseline to determine what the student already knows and what the student has learned. If the teacher does not administer a pre-assessment and only assesses students at the end of the instructional unit, there is no way to know if the student learned anything. Even if the student achieves a perfect score, it may be because the student already knew everything on the assessment even before there was any instruction. Therefore, there are items on the pre-assessment that address topics the student is about to be taught. There is no expectation that students already know this material and it is perfectly acceptable for these questions to be left blank. However, a student who does answer these questions correctly should be moved ahead so as not to waste the student’s time.
Pacing note:
Based on the observations the teacher makes regarding how well students have mastered the material on the pre-assessment, the first few days of this module may be modified to account for gaps and misconceptions that students possess regarding these prior knowledge areas.
SECTION 2 How do numbers combine with each other?
Day 2:
Learning experience:
The teacher works with the entire group and warms them up by having them count from 1 to 20 backwards and forwards.
The teacher shows the students images of groups of objects, for examples circles, from 1 to 10. The teacher asks how many objects there are in the image in order for the teacher to get a sense of student ability to connect numbers with raw quantity. The teacher then asks students to close their eyes and visualize quantity, for example, “Close your eyes and picture three small circles.”
Next, the teacher demonstrates the strategy of counting up. For example, a teacher explains to students, “I will give you a number, let’s say 5, and I will ask you to count up 3 places from 5. You will start by stating the given number and then counting up the number of places I ask you to count up. In the example I gave you, you would reply, ‘I have 5 and I am counting up to 6, 7, 8.’
The teacher uses a visual representation, such as fingers or a number line, to demonstrate the increase of three. The teacher does several examples, all with single-digit numbers. The teacher asks one or two students to try this exercise while the entire class is together to get a sense of what it will be like for students to accomplish this task.
Learning experience:
The class then breaks up into small groups. Students are given sheets and they are asked to draw simple objects in groups that represent each of the first ten numbers. For example, draw 1 cat, 2 ice cream cones, 3 basketballs, etc.
Assessment:
While students are working in small groups on these exercises, the teacher works with one group at a time focusing on the strategy of counting up. The teacher asks individual students to count up from a given number, as demonstrated at the beginning of class. Before these two days are over the teacher should have concise notes on each student describing each student’s fluency with this skill. If a student is particularly strong in this area, the teacher can ask the student to count down from a given number, for example, “Start with 8 and count down three places.”
Rationale:
The same way we give students leveled reading books that are easy enough for them to read on their own, but challenging enough to cause them to acquire new vocabulary, we apply the same approach with math exercises. In order for this to happen, teachers must get to know their students extremely well, particularly regarding the basic fluency and conceptual skills that are addressed in these first few days. If the time is put in now, the pay-offs will be enormous later. If not, students will constantly bump up against the misconceptions and gaps they possess.
Day 3:
Learning experience:
The teacher demonstrates how to use a ruler to measure line segments of different lengths. During this process the teachers discusses the concept of estimation and talking in terms of phrases like, “about how long is this line segment?” Since students will only be using whole number answers, students need to estimate the nearest whole number. The teacher then demonstrates how to construct line segments of specific lengths.
Learning experience:
Students will be given rulers and asked to construct line segments in whole number increments from 0 to 10 inches, and then repeat this activity using centimeters. Students will then be asked to put together the following pairs of line segments: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. Students will be asked to determine the length of the new line segment. This will lead to working with the fact families that add up to 10.
Assessment:
As students measure and construct line segments, the teacher circulates and checks for accuracy and precision. Teachers will begin to analyze whether or not there is a correlation between the skills of measuring and the other skills that students worked on in section 1 of this module.
Rationale:
Using rulers to construct line segments and number lines will be challenging for many students, but the fine motor skills they will develop over time will help students improve their spatial relations skills, impacting their ability to estimate and determine quantity. This tactile skill is time-consuming, but highly engaging for students and is an excellent example of an opportunity to translate abstract numbers into concrete values. Seven is now not an abstract number, but represents 7 inches or centimeters and can easily be compared to 3 inches or centimeters. These numbers are real quantities that can be seen and measured, and discussed objectively and with certainty.
Day 4:
Learning experience:
The teacher will introduce students to the concept of a fact family. For today’s lesson the teacher will only use the first five fact families, for example:
1 + 4 = 5 4 + 1 = 5 5 – 4 = 1 5 – 1 = 4
The teacher will demonstrate this using a visual image.
Example:
1 purple fish swims to meet up with 4 yellow fish. We represent this as: 1 + 4 = ?
4 purple fish swim to meet up with 1 yellow fish. We represent this as: 4 + 1 = ?
Once the students get the hang of this, the teacher uses an example where the sum from the original fact family is diminished:
5 fish are together and 1 fish swims away. We represent this as: 5 – 1 = ?
5 fish are together and 4 fish swim away. We represent this as: 5 – 4 = ?
The teacher guides students to use their counting up and counting down skills to determine the answers and leads a discussion about why these numbers form a family.
Learning experience:
Students start discovering fact families. They use illustrations like the fish example where groups of fish are swimming towards each other to create sums or swimming away from each other to create differences. Students are asked to use two colors to represent the two different groups in each diagram and record what they discover.
For example, given the following scenario, students choose a color for the four fish swimming towards the left and a different color for the one fish swimming away from the others.
The student is prompted to express the picture using numbers.
How many fish are there in the picture?
How many fish are swimming to the left?
How many fish are swimming away to the right?
Can you write two similar number sentences for this picture?
Assessment:
The teacher circulates and monitors student work. At the end of class, the teacher collects one diagram from each student and assesses how successfully the student is able to express the appropriate fact family for each picture.
Rationale:
At this stage, students work with numbers and diagrams below ten to discover fact families. Students are focused on fact families that add to 3, 4, and 5 and master those fact families. Then they move on to fact families focusing on sums of 6 through 9.
Day 5:
Learning experience:
The teacher starts the lesson by modeling for the students how to create a fact family using numbers without visual aids or diagrams. The teacher creates a fact family for the number 5.
“I want to create a fact family for 5. I need two numbers that are smaller than 5 that add up to 5. The numbers that are smaller than 5 are 1, 2, 3, and 4.”
“I’m going to choose 4 and 1 because I can write the number sentence 4 + 1 = 5.”
“Every fact family has four number sentences: two addition and two subtraction.”
“I can also write the number sentence 1 + 4 = 5.”
“There are also two subtraction sentences: 5 – 1 = 4 and 5 – 4 = 1.”
The teacher presents all of these number sentences to the students.
The teacher reminds the students of the visual examples they translated into number sentences the day before, “Remember when we had 5 fish and one swam away? How many fish were left? This is just like our fact family.”
The teacher then works with the students to come up with the other fact family for 5: 2, 3, and 5. This time it is a more collaborative effort with the students writing their ideas down before the teacher presents the fact family.
Learning experience:
Students are given a visual prompt similar to the fish diagram from the previous day and are asked to work on the fact families for 6, one family at a time. They start by listing the numbers smaller then 6 and coming up with pairs that have sums equaling 6. They do this by having three diagrams of fish for each of the different fact families for 6, namely: 1, 5, and 6; 2, 4, and 6; and 3, 3, and 6.
Assessment:
The teacher monitors student work and guides students through the process. If the teacher notices common mistakes, the teacher brings the students back together for a group mini-lesson. By the end of the lesson the student will write number sentences for each fact family and the teacher will collect their work.
Rationale:
Students this young will typically develop the skill of creating fact families very slowly. There are a number of transferable reasoning skills that students are learning how to use while working through these math scenarios. One key skill is using process of elimination. When students are creating lists of fact families and choosing pairs of numbers that add to a larger number they should begin to realize that there are not many numbers to choose from. If students can begin to see that there is a process to the creation of these families, they may apply those skills to other mathematical reasoning. The value of developing these types of transferable skills cannot be overestimated. They must be developed slowly and carefully, allowing students to make discoveries that will stick with them.
Name ____________________________________ Date ________________________________
Problem Set: Generating Fact Families
Fact Family: ___________________
List all the numbers smaller than six:
Make as pairs as you can that add up to six:
Fill in the missing blanks for the subtraction problems:
6 – ______ = 5 6 – ______ = 3 6 – _______ = 1
6 – ______ = 4 6 – ______ = 2
Write the Fact families for six (one family in each box):
How many fact families are there for 6?
Day 6:
Learning experience:
Today will be a big day. The teacher will introduce students to the most important fact families of all – the fact families that sum up to 10. The teacher will ask students to hold their hands out their hands and to count their fingers. Students have a built-in calculator right in their hands, and they can use this calculator to generate the 10s partners.
The teacher will guide the students through a series of exercises where they will be asked to determine the fact families and to list them by using their hands. For example, if a student turns 1 finger down and has 9 fingers up, then that is a pair: 1 and 9. Next, the student turns two fingers down. If we count the fingers we have up, that’s another pair: 2 and 8. The teacher continues this process until all 5 tens pairs have been identified. The teacher then asks the students to try while they are still in front of the teacher in a big group. The teacher immediately starts to assess who is struggling and who is figuring it out. These students can be paired together and they can discuss the possibilities.
Learning experience:
After the teacher reviews how to find the 10s partners and the students have a chance to practice in a group, the list is recorded for all of the students to see. At their seats students are given the chart below. They are then asked to generate the list of the 5 tens partners. Each pair should then be expanded into each of the 5 fact families that sum up to 10.
10s Pairs
10s Fact Families
When they are finished, students may be given another sheet to rewrite/revise their work and organize it better. For example, it probably makes the most sense to start the list on the left with the pair 1 and 9, and then move to 2 and 8, etc. This is more than just an organizational preference. This level of organization also encourages students to see that numbers have a built-in order (i.e.: 1, 2, 3, …) and that taking advantage of this can actually help a student do their work faster and more accurately without leaving out any of the pairs.
Assessment:
When the student is finished, all 5 columns of this chart are filled in and the teacher makes a note of whether or not the student was able to do this independently or, if not, how much help the student required.
Rationale:
This one sheet of paper is potentially the most important sheet of paper in a young math student’s career. Students who understand this particular catalogue of symbols on a deep level will become fluent with numbers in a way that will impact almost all of their future math experiences. Numbers have a pattern and this pattern is meaningful and knowable. When students move on to higher-level mathematics, the time it takes them to solve problems will be dramatically reduced if they master these 5 fact families, leading them to be more confident and capable math students.
Day 7:
Assessment:
The day starts with the teacher giving out the same chart as the previous day and asking the students to complete the table for the 10s pairs and then for the 10s fact families. The teacher can remind the students that they can use their hands to help them, but aside from this comment there is not much prompting.
Rationale:
This is not about sheer memorization. There is a logical reasoning component to this process. Throughout all of the work that students have done during the course of this module, they should be comfortable with pairs of numbers that sum up to another number. More specifically, in the prior day’s lesson students should have made the realization that 10 is a unique number and we literally have it in our hands. Through their ability to generate the 10s pairs and 10s fact families, students will demonstrate their ability to reason abstractly and quantitatively.
Learning experience:
The teacher brings the class together and briefly impresses upon students how important it is for them to know the tens pairs and the tens factor families. The teacher explains to students that for the rest of the lesson today and the next day they are bringing together everything they have done so far. Half of the class uses rulers to create line segments that add up to 10. The other half of the students are writing and illustrating stories that revolve around tens partners. At the end of class each student who made a number line is paired with a student who was involved with a story and they describe to each other what they did to set the stage for the two groups to switch roles the next day. The students working with stories are working independently, while the students who are creating number lines receive more attention and guidance and the teacher is largely focused in this direction.
Learning experience:
As described above by the teacher, students break into groups and work extensively with the following fact families for two days.
1, 9, 10: 1 + 9 = 10 9 + 1 = 10 10 – 1 = 9 10 – 9 = 1
2, 8, 10: 2 + 8 = 10 8 + 2 = 10 10 – 2 = 8 10 – 8 = 2
3, 7, 10: 3 + 7 = 10 7 + 3 = 10 10 – 3 = 7 10 – 7 = 3
4, 6, 10: 4 + 6 = 10 6 + 4 = 10 10 – 4 = 6 10 – 6 = 4
5, 5, 10: 5 + 5 = 10 5 + 5 = 10 10 – 5 = 5 10 – 5 = 5
In our base-10 system, these five fact families are the most important families and form the backbone of all of the addition and subtraction that a student will do throughout the rest of their math career.
Students explore the fact families that add up to 10 using drawings of objects and line segments. Students choose a fact family, (ex: 3, 7, and 10) and do a series of drawings that depict all four elements of the fact family. Sticking with an earlier example, a student may draw 10 fish in the sea and depict four different scenarios:
1) 3 yellow fish swim to meet up with 7 purple fish to make a group of 10 fish
2) 3 purple fish swim to meet up with 7 yellow fish to make a group of 10 fish
3) 3 fish swim away from a group of 10 fish to leave behind 7 fish
4) 7 fish swim away from a group of 10 fish to leave behind 3 fish
Students will put text to their drawings, including number sentences that represent each picture, such as 10 – 3 = 7. Students write about their drawings and articulate their thoughts related to quantity, specifically in terms of the relative size of different numbers, using words and phrases like, greater than, less than, bigger, smaller, a lot, a few, etc.
Assessment:
The teacher will pay particular attention to the students who are creating number lines and collect this work at the end of class. Again, each student’s ability, so far, in the area of spatial relations and attention to detail in their work with number lines potentially says a good deal about the struggles that may come up in other areas of math.
Rationale:
The goal by the end of the year is for students to become completely fluent with the tens pairs and tens fact families and to develop a deep understanding of the numbers involved and the partnerships that are so ingrained in the base-10 system. For anyone to become expert at any one thing there is no substitute for familiarity; the person simply must spend a significant amount of time with the material. The analogy that has been used before is the way people know their way around the neighborhood where they live. People know their neighborhoods so intimately and fluently because that is where they spend their time. They’ve even made wrong turns, but it is even through those experiences that their deep knowledge of their neighborhood has been cemented and becomes second nature. This is the same familiarity that is the goal for Grade 1 students and the tens fact families. The only way to attain it is for them to spend a lot of time with this material in various contexts: drawing number lines, constructing with blocks, using pictures and writing stories. There is no substitute for being immersed in this world of tens and every student deserves the time it takes to absorb it and internalize it.
Day 8:
Day 8 will largely be a repeat of Day 11: students will switch groups and activities. Yesterday’s storywriters and illustrators will now work with rulers, number lines, and blocks, and vice-versa.
Time spent on engaging, various and contextualized tasks is crucial at this point. As stated above, every child deserves the opportunity to discover these basic math facts through “doing” – drawing, writing, coloring, cataloguing, failing, correcting, revising and discussing. These discoveries then need to be absorbed and internalized. This takes time. They need to learn their way around this neighborhood. In terms of teacher-centered activity, student-centered activities, assessment, and rationale, Day 12 looks a lot like day 11, except with students in reversed roles. This allows students the opportunity to engage in a variety of discovery opportunity, as well as discussing their varied experiences with each other and allowing the teacher to focus attention on the students who are engaged in the activities that need more guidance. In this way, each day half the class is engaged in the more guidance-intensive activities rather than the entire class on any given day.
Day 9:
Assessment:
The first thing students will be asked to do almost every day is to catalogue the tens pairs and tens fact families on the sheet used on Day 10. This activity will be timed and every day the students should be able to list these pairs faster and more accurately. Initially, accuracy is the priority. Completing this sheet quickly, but incorrectly is meaningless and the teacher should make the students aware of which goal is the priority. Fluency will be the goal over the course of the year for students, as this work will be revisited through many of the other modules taught throughout the academic year.
Learning experience:
The teacher will give each student the best version of the number line that the student has created. (If more time needs to be spent revising number lines, so that they are functional and can be used as tools for add and subtraction, then now is the time to do it and today’s activities can be moved to the next day. This revision of student work by students with the guidance of the teacher is crucial and must not be cut short.)
Today is all about visualization. Students will be asked to visualize objects and count them. For example, students will close their eyes and they will be asked to visualize 3 cats. When they are ready they will be asked to visualize 2 new cats sitting near the original 3 cats. They will be asked to count up all the cats in their mental picture. Then students will be asked to check their work using their number lines.
Students may also be asked to close their eyes and to visualize a number on their number line, say 5, and then visualize jumping two spaces forward on their number. They will end on 7 and the teacher will then write the number sentence 5 + 2 = 7 to connect what the students did mentally on the concrete number line with the abstract, symbolic representation. Finally, students will be asked to close their eyes and visualize the number 3. Then they will be asked to visualize the number 2 vertically stacked under the number 3 and determine the total.
The teacher will show them on the board that they should picture the math sentence in this form: 3
+ 2
5
The teacher will then show a visual representation of this image and ask the students to do it again and see if their mental image matches the teacher’s visual image. Initially, the goal will be for students to do this for all 5 of the single-digit addition problems that add up to 5 or less. Eventually (throughout the year), students will work their way up to all 30 single-digit addition problems that add up to single-digits (see next page).
This will set the stage for students to develop the internalization of numbers as abstractions. Students should be able to move away from using manipulatives and using their fingers and go directly to number. This is just the beginning and the mental math process and this process should not be rushed. This form of mental math also sets the stage for adding double-digit numbers later in the year.
Learning experience:
Students will be given traditional flashcards with single-digit numbers stacked vertically. They will work in groups. The teacher will sit with one group at a time and practice visualization techniques with students. Even for a high-caliber veteran teacher, this is not an easy exercise to teach; however, if the activity is structured correctly and the teacher makes good use of going back and forth between visuals and student think-alouds, this process can dramatically move students toward the goals of depth of understanding in terms of numbers abstractions.
Part of this process should include students practicing the crucial skill of “one more than.” If a student is given the open sentence, “One more than 5 is…”, the student should be able to answer six. If a student can not do this skill easily and fluently, it means that the student is probably not ready to move to abstractions and may need to go back to the number line and answer the “one more than” questions using the number line first.
Once students master “One more than” they should move to “Two more than”, and so on. If students master this skill then this is an indication that they have internalized the number line and are ready to make the leap into abstract representations of quantities.
Assessment:
The teacher will treat the group work experience the way a balanced literacy interaction would take place, with the teacher taking concise notes on students and their initial ability to perform and articulate these mental math tasks.
Rationale:
As discussed above, students must make the move from concrete experience with numbers through pictures, number lines, building blocks, use of fingers, etc., to mathematical symbols (numbers) as mental abstractions. The transition from concrete to abstract is a balancing act – too early and students may never internalize the facts in a concrete way; too late and students may become dependent on concrete manipulatives and may have a hard time making the leap to abstractions.
Day 10: Section 2 Assessment:
Students will be asked to perform four tasks:
1. Visualization/verbal: solve addition problems by counting up.
2. Visual representation: draw a picture that represents a given addition problem, including addition of line segments and number lines.
3. Written description: write a numerical sentence and a word sentence that describes the addition problem.
4. Written work: write out the addition portion of a fact family given the three basic elements. Also, fill in random blanks for the first 35 fact families.
Note: Depending on what has been covered throughout the unit, all of the assessment tasks for section 2 can be extended by including the 20 single-digit fact families that add up to double-digit sums.
Day 11 and 12:
Students will work on a big project where they are counting quantities of an object into the hundreds, such as pennies. The example used above describes quantifying 357 in such a way as to encourage the use of the number 10 as the key organizing number in this process. Students will be asked to draw representations of how they accounted for all of the objects and how they used the number 10 to help them in this process. The result of this project should look more like a portfolio with a description of the process in words, pictures and number sentences, heavily relying on the concept of 10 (and by extension, 100) as the organizing principle.
Day 13: Section 3 Assessment:
Students will be asked to perform four tasks:
1. Verbal: count up by tens to one hundred.
2. Visual representation: students will represent tens groups through a diagram that includes some kind of a key that represents the number ten as a collective of a set of “ones”.
3. Written: The assessment for this section will ask students to write about the project they were asked to complete where they had to account for a large number of object, using tens.
4. Written work: fill in all of the possible blanks for the 5 fact families that add up to ten.
Single-Digit Addition Problems
In a base-10 number system, there are only 55 single-digit addition problems. (It is not a coincidence that 55 is the summation of the first 10 counting numbers, and this is a fact that some students may notice in later years.)
9 + 9 = 18 18
9 + 8 = 17 17
9 + 7 = 16 8 + 8 = 16 16
9 + 6 = 15 8 + 7 = 15 15
9 + 5 = 14 8 + 6 = 14 7 + 7 = 14 14
9 + 4 = 13 8 + 5 = 13 7 + 6 = 13 13
9 + 3 = 12 8 + 4 = 12 7 + 5 = 12 6 + 6 = 12 12
9 + 2 = 11 8 + 3 = 11 7 + 4 = 11 6 + 5 = 11 11
9 + 1 = 10 8 + 2 = 10 7 + 3 = 10 6 + 4 = 10 5 + 5 = 10 10
9 + 0 = 9 8 + 1 = 9 7 + 2 = 9 6 + 3 = 9 5 + 4 = 9 9
8 + 0 = 8 7 + 1 = 8 6 + 2 = 8 5 + 3 = 8 4 + 4 = 8 8
7 + 0 = 7 6 + 1 = 7 5 + 2 = 7 4 + 3 = 7 7
6 + 0 = 6 5 + 1 = 6 4 + 2 = 6 3 + 3 = 6 6
5 + 0 = 5 4 + 1 = 5 3 + 2 = 5 5
4 + 0 = 4 3 + 1 = 4 2 + 2 = 4 4
3 + 0 = 3 2 + 1 = 3 3
2 + 0 = 2 1 + 1 = 2 2
1 + 0 = 1 1
0 + 0 = 0
30 of the single-digit addition problems result in single-digit sums.
25 of the single-digit addition problems result in double-digit solutions (including the 5 problems that add up to 10).
Every addition problem that students will see for the rest of their lives will be based on these 55 problems.
For example, adding 2,763 + 8,654 is just a series of small addition problems that can all be found in the chart above.
If the student wants to start by adding the thousands place first, that is simply 2 + 8.
If the student wants to stack the two numbers vertically and add the ones place first that is 3 + 4.
Of course, both of these smaller addition problems are found in the chart above, as is every addition problem a student will ever do.
Name: _____________________________________ Date: ___________________________
Teacher: ___________________________________ First Grade Math
Pre-Assessment
Adding and Subtracting Single-Digit Numbers: Section 1
1. One-one-one assessment component (administered by the teacher to each student individually).
a) Student is shown a series of 5 slides depicting images of a various numbers of objects between 1 and 10 (similar to the way a teacher would show a student flashcards).
Slide 1:
Notes to teachers: The teacher shows a series of slides for a predetermined amount of time and records how accurately a student can see a number of objects and translate this into a number.
Later, the teacher can ask the student if the student is using any strategies to count faster.
For example, does the arrangement at the right remind the student of the number 5 on a playing die or on a dominoe?
Student correctly names the number of objects: _____ correct out of 5 slides.
Teacher notes: _____________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
b) Student counts from 0 to 20 forwards and backwards.
Student counts from 0 to 20 forwards: Accurately Inaccurately
Student counts from 0 to 20 forwards: Accurately Inaccurately
Student counts up from 13 to 20: Accurately Inaccurately
Teacher notes: _______________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. Independent assessment component:
a) In the box below, draw a picture of 5 fish and 8 birds:
b) Write a sentence that describes your picture.
Make sure you write about the number of birds and fish in your picture.
Also, answer these questions:
Are there more fish or more birds?
Which number is larger, 5 or 8?
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
3. Find all 30 single-digit sums:
9 + 0 =
3 + 2 =
4 + 1 =
6 + 3 =
5 + 4 =
5 + 1 =
8 + 0 =
3 + 0 =
2 + 1 =
4 + 0 =
1 + 1 =
4 + 3 =
7 + 0 =
3 + 3 =
6 + 1 =
2 + 0 =
1 + 0 =
5 + 2 =
6 + 0 =
2 + 2 =
4 + 2 =
5 + 3 =
3 + 1 =
7 + 1 =
5 + 0 =
4 + 4 =
7 + 2 =
6 + 2 =
0 + 0 =
8 + 1 =
4. Name all 5 tens partners:
+ = 10
+ = 10
+ = 10
+ = 10
+ = 10
5. If there are 15 fish swimming together and 8 fish swim away, how many are left?
Write these as a number sentence. Make sure you use an equal sign.
________________________________________________
6. What is the difference between 7 and 5? _________________
7. Fill in the missing numbers:
9 – 0 =
3 – 2 =
4 – 1 =
6 – 3 =
5 – 4 =
5 – 1 =
8 – 6 =
9 – 5 =
7 – 4 =
4 – 0 =
10 – 1 =
10 – 2 =
10 – 6 =
10 – 7 =
10 – 5 =
8. Fill in each of the missing numbers:
9 + 8 = _____
9 + ______ = 12
______ + 8 = 16
9 + _____ = 16
6 + 5 = _____
7 + _____ = 12
8 + 5 = _____
7 + ______ = 13
7 + 7 = ______
8 + _____ = 15
_____ + 6 = 14
6 + _____ = 12
9 + 4 = _____
8 + ______ = 12
9 + 9 = ______
9 + 2 = ______
8 + 3 = _____
7 + _____ = 11
9 + 6 = _____
_____ + 5 = 14
Name: _____________________________________ Date: ___________________________
Teacher: ___________________________________ First Grade Math
Formative Assessment
Adding and Subtracting Single-Digit Numbers: Section 2
1. One-one-one assessment component (administered by the teacher verbally to each student). The teacher records the answers).
a) Count up 2 from 6: ________ c) Count down 2 from 5: _______
b) Count up 4 from 7: ________ d) Count down 3 from 9: _______
Notes: ___________________________________________________________________________
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2. Student works independently with a ruler.
a) Use your ruler to draw a line segment that has a length of 3 inches:
b) Use your ruler to draw a line segment that has a length of 2 inches:
c) Use your ruler to draw a line segment that is as long as the other two line segments put together:
d) Write an addition sentence that describes the lengths of the line segments:
e) Measure this line. About how many inches is it? (to the nearest whole number)
_____________________________________________________
3. Draw a picture that represents this sentence:
a) Two fish are swimming toward six other fish.
b) How many fish are there total? _________
c) Write a number sentence that represents this picture: _________________________________
4. Tens pairs
Name all of the 10s Pairs
Choose 2 10s pairs and write out the Fact Families
5. Single-digit sums:
3 + 2 =
4 + 1 =
6 + 3 =
8 + 0 =
5 + 4 =
2 + 1 =
4 + 3 =
7 + 0 =
3 + 3 =
4 + 4 =
5 + 2 =
6 + 0 =
5 + 3 =
3 + 1 =
7 + 1 =
6. Double-digit sums:
9 + 3 =
8 + 8 =
9 + 7 =
8 + 5 =
7 + 6 =
7 + 7 =
6 + 6 =
9 + 4 =
8 + 4 =
8 + 3 =
7 + 4 =
9 + 6 =
7. Subtraction:
9 – 0 =
3 – 2 =
4 – 1 =
5 – 1 =
8 – 6 =
9 – 5 =
10 – 1 =
10 – 2 =
10 – 6 =
8. Answer the question: What is the difference between four and one? ____________________
9. a) If Ryan has seven balloons and three fly away how many balloons does he have left? _______
b) Write a number sentence that represents Ryan’s balloons. Make sure to include an equal sign.
____________________________________________________________________
12. Fill in each of the missing numbers:
6 + 5 = _____
9 + ______ = 10
______ + 8 = 16
7 + _____ = 12
8 + ______ = 11
7 + ______ = 17
_____ + 6 = 9
6 + _____ = 12
9 + 4 = _____
Name: ___________________________________________ Date: ___________________________
Teacher: _________________________________________ First Grade Math
Module Assessment
Adding and Subtracting Single-Digit Numbers
1. Find all 30 single-digit sums:
9 + 0 =
3 + 2 =
4 + 1 =
6 + 3 =
5 + 4 =
5 + 1 =
8 + 0 =
3 + 0 =
2 + 1 =
4 + 0 =
1 + 1 =
4 + 3 =
7 + 0 =
3 + 3 =
6 + 1 =
2 + 0 =
1 + 0 =
5 + 2 =
6 + 0 =
2 + 2 =
4 + 2 =
5 + 3 =
3 + 1 =
7 + 1 =
5 + 0 =
4 + 4 =
7 + 2 =
6 + 2 =
0 + 0 =
8 + 1 =
2. Name all 5 tens partners:
+ = 10
+ = 10
+ = 10
+ = 10
+ = 10
3. Create a Fact Family for each set of numbers:
a) 3, 5, 8
example: 3 + 5 = 8
5 + 3 = 8
8 – 5 = 3
8 – 3 = 5
b) 1, 6, 7
c) 4, 5, 9
d) 4, 2, 6
4. There are four sets of numbers. Circle any set that is NOT a Fact Family:
4, 3, 7 7, 1, 5 3, 5, 9 6, 8, 2
5. In the morning, Charlie had eight pencils.
In the afternoon, he had six pencils.
How many pencils is he missing?
Draw a picture of all of Charlie’s pencils in the morning:
Draw a picture of Charlie’s pencils in the afternoon: Draw a picture of Charlie’s missing pencils:
Write a subtraction sentence using the numbers for Charlie’s pencils and the missing number:
________________________________________
6. For each pair of numbers, CIRCLE the number with the larger value and UNDERLINE the number with
the smaller value:
8 and 6
7 and 9
17 and 16
11 and 8
7. Counting up
a) If you start at 7 and count up 1, you get to:
____________
b) If you start at 4 and count up 3, you get to:
____________
c) If you start at five and count up two, you get to:
___________
d) If you count up three from six you get to:
___________
8. Counting down
a) If you start at 8 and count down 2, you get to:
____________
b) If you start at 5 and count down 1, you get to:
____________
c) If you start at nine and count down three, you get: t
___________
d) If you count down two from seven you get to:
___________
9. Fill in the missing numbers
8 + 3 = 3 + ______
7 + 5 = _______ + 7
_______ + 9 = 9 + 4
8 + 4 + 1 = 4 + 1 + _______
10. If there are 15 fish swimming together and 8 fish swim away, how many are left?
Write these as a number sentence. Make sure you use an equal sign.
________________________________________________
11. What is the difference between 7 and 5? _________________
12. Fill in the missing numbers:
9 – 0 =
3 – 2 =
4 – 1 =
6 – 3 =
5 – 4 =
5 – 1 =
8 – 6 =
9 – 5 =
7 – 4 =
4 – 0 =
10 – 1 =
10 – 2 =
10 – 6 =
10 – 7 =
10 – 5 =
13. Fill in each of the missing numbers:
9 + 8 = _____
9 + ______ = 12
______ + 8 = 16
9 + _____ = 16
6 + 5 = _____
7 + _____ = 12
8 + 5 = _____
7 + ______ = 13
7 + 7 = ______
8 + _____ = 15
_____ + 6 = 14
6 + _____ = 12
9 + 4 = _____
8 + ______ = 12
9 + 9 = ______
9 + 2 = ______
8 + 3 = _____
7 + _____ = 11
9 + 6 = _____
_____ + 5 = 14
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