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2. 2. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 2 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.
3. 3. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 3 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
5. 5. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 5 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.
10. 10. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 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 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.
15. 15. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 15 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 = ?
16. 16. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 16 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.
17. 17. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 17 “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.
18. 18. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 18 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?
19. 19. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 19 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.
20. 20. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 20 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.
21. 21. Grade 1 Module 110611 {Add Subtract Single Digits} DRAFT 21 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: