1 st Grade Mathematics

Transcription

1 1 st Grade Mathematics Unit # 3: Measurement Pacing: 3 Weeks Unit Overview 1 In this unit, students explore length comparisons both directly and indirectly. Students build and expand on the direct comparison they did in Kindergarten (one object is longer or taller than another) to compare and order three objects directly. Students will have multiple opportunities to practice using cubes, paper clips, pennies, etc. to measure the length of an object from end to end. Through numerous experiences and careful questioning by the teacher, students will recognize the importance of careful measuring to ensure that there are not any gaps or overlaps for accurate measurement (MP. 6). Students will then extend this understanding to do indirect comparison through the use of a third object. Indirect comparison involves using a third object to compare the lengths of two different objects: My string is longer than your book. Your book is longer than my pencil. That means my string is longer than my pencil. This concrete experience with length comparisons supports students understanding of number comparisons and comparison problem solving. The first three lessons focus on indirect comparison. Lessons 4-6 take longer than and shorter than to a new level of precision by introducing the idea of a length unit. Centimeter cubes are laid alongside the length of an object as students learn that the total number of cubes laid end to end with no gaps or overlaps represents the length of that object (1.MD.2). Lessons 7-9 focus on measurement using non-standard units to explore the usefulness of measuring with similar units and the concept of comparing measurements. Lessons close out the unit by revisiting data that was used in the beginning of the year to help students get to know each other. Students focus on interpreting information and then asking and answering questions about various data sets. Prerequisite Skills Vocabulary Mathematical Practices K.CC.5 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. K.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, K.CC.7 Compare two numbers between 1 and 10 presented as written numerals. K.MD.1 Describe measureable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. K.MD.2 Directly compare two objects with a measurable attribute in common, to see which object has more of / less of the attribute, and describe the difference. about aligned bigger compare first gap height higher length less longer longest measure more order overlap second shorter shortest taller third smaller units unifix cubes MP.1: Make sense of problems and persevere in solving them MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others MP.4: Model with mathematics MP.5: Use appropriate tools strategically MP.6: Attend to precision MP.7: Look for and make use of structure MP.8: Look for and express regularity in repeated reasoning 1 Module 3 Unit Overview

2 Common Core State Standards Progression of Skills Additional Standards (10%) Supporting Standards (20%) Major Standards (70%) 1.MD.4: Data and graphs 1.OA.1: Word problems 1.MD.1 Order three objects by length 1.MD.2 Express the length of an object According to the PARCC Model Content Framework, opportunities for connections among standards are found in the following: Measurement standards 1.MD.A.1 and 1.MD.A.2 together support and provide a context for the 1.OA.A.1 goal of solving subtraction problems that involve comparing. To meet standard 1.MD.A.1, students compare the lengths of two objects by means of a third object. When students cannot find the exact difference because of the magnitude of the numbers that arise from measurement as may occur in comparing two students heights they may still compare the measurements to know which is greater (1.NBT.B.3). 2 Units are a connection between place value (1.NBT) and measurement (1.MD). Working with place value and measurement depends on having a sense of the sizes of the base ten and measurement units and being able to see a larger unit as composed of smaller units within the system. In later grades, unit thinking will become important throughout arithmetic, including in the development of multi-digit multiplication and division algorithms and the development of fraction concepts and operations. Kindergarten 1 st Grade 2 nd Grade K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10 K.MD.1 Describe measureable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. K.MD.2 Directly compare two objects with a measurable attribute in common, to see which object has more of / less of the attribute, and describe the difference. N/A K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count 1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions 1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object. 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; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. 1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. 2.OA.1 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 2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. 2.MD.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit 2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. 2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple puttogether, take-apart, and compare problems 1 using information presented in a bar graph 2 See Units, a Unifying Idea in Measurement, Fractions, and Base Ten, unifying-idea/. 2 P a g e

3 Big Ideas Students Will Objects can be compared and ordered based on length. A measurement is a comparison of the size of one object with the size of another. A third object can be used to indirectly compare the lengths of two others. The units used to measure an object determine how the length of an object is expressed. Information can be collected and represented in different ways Know/Understand The length of an object is how long it is from one point to another. The same object can be described using different measurements. The numerical value attached to a measurement depends on the unit of measurement. Information can be collected and recorded using numbers and tallies Information can be represented in different ways using bar graphs and picture graphs Be Able To Comparing objects to find taller, shorter, longer, higher, bigger, or smaller. Indirectly measuring objects by comparing the length of two objects by using a third object as a measuring tool (e.g. using a pencil to measure the height of a bookshelf and the height of a desk to identify which one is longer). Determining which of two objects is longer, by physically aligning the objects. Ordering objects by length, from longest to shortest or shortest to longest (e.g., order pencils, crayons, or markers; build towers with cubes and order them from shortest to tallest). Collect and record information in a data table Represent collected data in a picture or bar graph Ask and answer questions related to collected data and graphs NOTE: The instructional progression for teaching measurement begins by ensuring that students can perform direct comparisons. Then, children should engage in experiences that allow them to connect number to length, using manipulative units that have a standard unit of length, such as centimeter cubes. These can be labeled length-units with the students. Students learn to lay such physical units end-to-end and count them to measure a length. They compare the results of measuring to direct and indirect comparisons 3 3 North Carolina Unpacked Standards: Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June P a g e

4 Student Friendly Objective SWBAT 1 SWBAT directly compare lengths using two or more objects. Unit Sequence Key Points/ Teaching Tips 4 Exit Ticket Instructional Resources As the first lesson in measurement, it Given a visual of three objects, students serves to review direct comparison that will put them in order from longest to Module 3 Lesson 1 students experienced in Kindergarten. shortest. SW identify the longest object Focus on aligning the end-point of and the shortest object. objects to ensure accurate measurement and provide sentence frames using longer than and shorter than to support student expression around comparing the length of various objects. 2 SWBAT compare length using indirect comparison. In this lesson students begin to use indirect comparison (or transitivity) as they compare each item to one consistent item. Items are then compared to each other through the indirect comparison. For instance, if the crayon is shorter than the paper strip and the pencil is longer than the paper strip, we can say that the crayon is also shorter than the pencil. As a way to prove their conclusions from indirect comparisons, students use direct comparison to verify their claims. Students will use drawings to represent the following and use indirect comparison to solve and circle the correct comparison (taller than or shorter than): Tanya s doll is shorter than Aline s doll. Mira s doll is taller than Aline s doll. Tanya s doll is (taller than/shorter than) Mira s doll. Module 3 Lesson 2 My Math Chapter 8 Lesson 1 Printable Paper Strips 4 Module 3 Topic A, B, C, D Overviews 4 P a g e

5 3 SWBAT order three lengths using indirect comparison. This lesson extends the use of indirect comparison to compare distances between objects that cannot be moved next to each other for direct comparison. Students use the same transitive process to compare short distances within the classroom in order to see what the shortest path to their classroom door is, which is helpful to know for lining up and emergencies. Given a map that shows a path between two points, students will use grid paper to construct a shorter path. SW use the grid paper to measure each path. SW compare the paths using shorter than or longer than. Module 3 Lesson 3 My Math Chapter 8 Lesson 3 4 SWBAT express the length of an object using centimeter cubes. In this, centimeter cubes are laid alongside the length of objects as students learn that the total number of cubes laid end to end with no gaps or overlaps is the length measure of that object. The centimeter is introduced before non-standard units because it will help students learn to measure accurately and understand the concept of measurement in a less demanding, real world applicable context. Early use of many non-standard units may actually interfere with students development of basic measurement concepts required to understand the need for standard units. 5 Given two objects, students will use centimeter cubes to measure the length. SW express the length of the objects as a quantity of cubes. Module 3 Lesson 4 5 See 5 P a g e

6 5 SWBAT express the length of an object using centimeters. In this lesson, students lay the same centimeter cubes from the previous lesson alongside a ruler, recognizing the meaning of the numbers on the ruler as describing the number of centimeter length units up to that number. The progressions document suggests that students engage in standard unit measurement in order to develop a solid understanding of why and how to measure, rather than measuring using a plethora of nonstandard measurement units. 6 This lesson allows students to measure and compare sets of three items with centimeter cubes. Students use their measurements and comparisons to put objects in order by length from shortest to longest or vice versa. Students finally solve compare with difference unknown word problems, determining how much longer a given object is than another. Given four objects, students will use centimeter cubes to measure the length. SW express the length of the objects in centimeters. Module 3 Lesson 5 6 SWBAT order, measure, and compare the length of objects using centimeters to solve word problems. Given a visual and the length of three tools, students will order the tools from shortest to longest. Given a new tool, students will use indirect comparison to find the length. SW draw the new tool and express the length in centimeters. Module 3 Lesson 6 Formative Assessment: North Carolina MD Tasks 1a, 1b, 1c 7 SWBAT measure objects using non-standard units. In this lesson, students begin to use nonstandard units to measure familiar objects. Students start with two different units and then move to using the same unit, but different sizes (such as big and small paper clips). Each time they measure one object using both units and receive inconsistent measurement results. Students then begin to ask the question, Why do we measure with same-sized Given three objects, students measure each, once using small paper clips and again using big paper clips. SW record their measurements in a table. Module 3 Lesson 7 My Math Chapter 8 Lesson 3 6 Ibid. 6 P a g e

7 length units? As they explore why it is so important to use the same-sized length unit, they realize that doing so yields consistent measurement results. In this lesson students explore what happens when they use a different unit of measurement from that of their classmates. As students measure the same objects with different non-standard length units, they realize that in order to have discussions about the lengths of objects together, they must measure with the same units. In this lesson students solving compare with difference unknown problems using centimeters. Students explore and solve problems such as, How much longer is the pencil than the marker? (1.OA.1). Revisiting the centimeter here helps students recognize the value of having a consistent way to communicate about various measurements. 8 SWBAT measure and compare the length of objects using the same units. Students will choose a non-standard unit of measurement (small paperclips, big paperclips, toothpicks, cubes) and find two classroom objects to measure. SW use the same unit to measure both objects and record their measurements in a table. Module 3 Lesson 8 9 SWBAT solve word problems by measuring and comparing lengths. Given the following word problem, students will use centimeter cubes to model and draw their solution: Mona s hair grew 7 centimeters. Claire s hair grew 15 centimeters. How much less did Mona s hair grow than Claire s? Moduel 3 Lesson 9 Formative Assessment: North Carolina MD Task 2a, 2b 10 SWBAT organize, represent, and interpret data. Students review data by organizing, representing, and interpreting personally relevant data (1.MD.4). As students work as a class to collect, sort, and organize data into a graph, they find great purpose and excitement in data. They begin to answer, and then ask questions about, the number of data points in a given category, and in two categories. Given a data set, students will use the table to answer 4 questions about the data. Module 3 Lesson 10 My Math Chapter 7 Lessons P a g e

8 11 SWBAT collect, organize, represent, and interpret data. 12 SWBAT ask and answer questions to interpret data and solve word problems. 13 SWBAT ask and answer questions to interpret data and solve word problems. In this lesson, students take a more independent role in the collecting, sorting, organizing, and representing phases involved in graphing. They work on their own to ask and answer questions about the data set. In this lesson students interpret information presented in the graphs by exploring compare with difference unknown problems. They begin with visualizing these problems in their easily accessible equalizing contexts, by answering questions such as, How many more students would Category A need to have the same amount as Category B? Students compare data in three categories. In this lesson students continue working with data in the context of word problems, with students again interpreting data sets to ask and answer varied word problems including, How many students were polled in all? and, How many more students are in Category C than in Category A? (1.OA.1). Given a data set, students will use the table to answer 3 questions about the data. Given a picture, students will use the visual to collect and organize data in a bar graph. SW write a number sentence to answer two questions about the data. 3 Flex Days (Instruction Based on Data) Recommended Resources: Module 3 Assessment Summative Performance Task Designing D.C. Appendix B Given a data set, students will use the table to answer 3 questions about the data. SW write a corresponding number sentence for each question. First Grade Module 3 Lesson 11 My Math Chapter 7 Lesson 3-4 Module 3 Lesson 12 My Math Chapter 7 Lesson 5-6 Module 3 Lesson 13 8 P a g e

9 Standard 1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 1 1 See Glossary, Table 1 Appendix A: Unpacked Standards Guide Source: Public Schools of North Carolina NCDPI Collaborative Workspace Unpacking What do these standards mean a child will know and be able to do? First grade students extend their experiences in Kindergarten by working with numbers to 20 to solve a new type of problem situation: Compare (See Table 1 at end of document for examples of all problem types). In a Compare situation, two amounts are compared to find How many more or How many less. Difference Unknown: How many more? version. Lucy has 7 apples. Julie has 9 apples. How many more apples does Julie have than Lucy? How many fewer? version Lucy as 7 apples. Julie has 9 apples. How many fewer apples does Lucy have than Julie? 7 + = 9 Problem Type: Compare Bigger Unknown: More version suggests operation. Julie has 2 more apples than Lucy. Lucy has 7 apples. How many apples does Julie have? Bigger Unknown: Version with fewer Mastery expected in Second Grade Smaller Unknown: Version with more Mastery expected in Second Grade Smaller Unknown: Fewer version suggests operation. Lucy has 2 fewer apples than Julie. Julie has 9 apples. How many apples does Lucy have? 9 7 = Compare problems are more complex than those introduced in Kindergarten. In order to solve compare problem types, First Graders must think about a quantity that is not physically present and must conceptualize that amount. In addition, the language of how many more often becomes lost or not heard with the language of who has more. With rich experiences that encourage students to match problems with objects and drawings can help students master these challenges. NOTE: Although First Grade students should have experiences solving and discussing all 12 problem types located in Table 1, they are not expected to master all types by the end of First Grade due to the high language and conceptual demands of some of the problem types. Please see Table 1 at the end of this document for problem types that First Grade Students are expected to master by the end of First Grade. (Note: this Table is different than the Table 1 in the original glossary found on the CCSS website.) First Graders also extend the sophistication of the methods they used in Kindergarten (counting) to add and subtract within this larger range. Now, First Grade students use the methods of counting on, making ten, and doubles +/- 1 or +/- 2 to solve problems. 9 P a g e

10 Example: Nine bunnies were sitting on the grass. Some more bunnies hopped there. Now, there are 13 bunnies on the grass. How many bunnies hopped over there? Counting On Method Student: Niiinnneee. holding a finger for each next number counted 10, 11, 12, 13. Holding up her four fingers, 4! 4 bunnies hopped over there. Example: 8 red apples and 6 green apples are on the tree. How many apples are on the tree? Making Tens Method Student: I broke up 6 into 2 and 4. Then, I took the 2 and added it to the 8. That s 10. Then I add the 4 to the 10. That s 14. So there are 14 apples on the tree. Example: 13 apples are on the table. 6 of them are red and the rest are green. How many apples are green? Doubles +/- 1 or 2 Student: I know that 6 and 6 is 12. So, 6 and 7 is 13. There are 7 green apples. In order for students to read and use equations to represent their thinking, they need extensive experiences with addition and subtraction situations in order to connect the experiences with symbols (+, -, =) and equations (5 = 3 + 2). In Kindergarten, students demonstrated the understanding of how objects can be joined (addition) and separated (subtraction) by representing addition and subtraction situations using objects, pictures and words. In First Grade, students extend this understanding of addition and subtraction situations to use the addition symbol (+) to represent joining situations, the subtraction symbol (-) to represent separating situations, and the equal sign (=) to represent a relationship regarding quantity between one side of the equation and the other. 10 P a g e

11 1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object. First Grade students continue to use direct comparison to compare lengths. Direct comparison means that students compare the amount of an attribute in two objects without measurement. Example: Who is taller? Student: Let s stand back to back and compare our heights. Look! I m taller! Example: Find at least 3 objects in the classroom that are the same length as, longer than, and shorter than your forearm. Sometimes, a third object can be used as an intermediary, allowing indirect comparison. For example, if we know that Aleisha is taller than Barbara and that Barbara is taller than Callie, then we know (due to the transitivity of taller than ) that Aleisha is taller than Callie, even if Aleisha and Callie never stand back to back. This concept is referred to as the transitivity principle for indirect measurement. Example: The snake handler is trying to put the snakes in order- from shortest to longest. She knows that the red snake is longer than the green snake. She also knows that the green snake is longer than the blue snake. What order should she put the snakes? Student: Ok. I know that the red snake is longer than the green snake and the blue since it s longer than the green, that means that it s also longer than the blue snake. snake is the red snake. I also know that the green snake and red snake are both blue snake. So, the blue snake is the shortest snake. That means that the green medium sized snake. snake because, So the longest longer than the snake is the NOTE: The Transitivity Principle ( transitivity ) 1 : If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of other quantities as well. Example: Which is longer: the height of the bookshelf or the height of a desk? Student A: I used a pencil to measure the height of the bookshelf and it was 6 pencils long. I used the same pencil to measure the height of the desk and the desk was 4 pencils long. Therefore, the bookshelf is taller than the desk. Student B: I used a book to measure the bookshelf and it was 3 books long. I used the same book to measure the height of the desk and it was a little less than 2 books long. Therefore, the bookshelf is taller than the desk. Another important set of skills and understandings is ordering a set of objects by length. Such sequencing requires multiple comparisons (no more than 6 objects). Students need to understand that each object in a seriation is larger than those that come before it, and shorter than those that come after. Example: The snake handler is trying to put the snakes in order- from shortest to longest. Here are the three snakes (3 strings of different length and color). What order should she put the snakes? Student: Ok. I will lay the snakes next to each other. I need to make sure to be careful and line them up so they all start at the same place. So, the blue snake is the shortest. The green snake is the longest. And the red snake is medium-sized. So, I ll put them in order from shortest to longest: blue, red, green. (Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June 2012.) 11 P a g e

12 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; understand that the length measurement of an object is the number of samesize length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. First Graders use objects to measure items to help students focus on the attribute being measured. Objects also lends itself to future discussions regarding the need for a standard unit. First Grade students use multiple copies of one object to measure the length larger object. They learn to lay physical units such as centimeter or inch manipulatives end-to-end and count them to measure a length. Through numerous experiences and careful questioning by the teacher, students will recognize the importance of careful measuring so that there are not any gaps or overlaps in order to get an accurate measurement. This concept is a foundational building block for the concept of area in 3 rd Grade. Example: How long is the pencil, using paper clips to measure? Student: I carefully placed paper clips end to end. The pencil is 5 paper clips long. I thought it would take about 6 paperclips. When students use different sized units to measure the same object, they learn that the sizes of the units must be considered, rather than relying solely on the amount of objects counted. Example: Which row is longer? Student Incorrect Response: The row with 6 sticks is longer. Row B is longer. Student Correct Response: They are both the same length. See, they match up end to end. In addition, understanding that the results of measurement and direct comparison have the same results encourages children to use measurement strategies. Example: Which string is longer? Justify your reasoning. Student: I placed the two strings side by side. The red string is longer than the blue string. But, to make sure, I used color tiles to measure both strings. The red string measured 8 color tiles. The blue string measure 6 color tiles. So, I was right. The red string is longer. NOTE: The instructional progression for teaching measurement begins by ensuring that students can perform direct comparisons. Then, children should engage in experiences that allow them to connect number to length, using manipulative units that have a standard unit of length, such as centimeter cubes. These can be labeled length-units with the students. Students learn to lay such physical units end-to-end and count them to measure a length. They compare the results of measuring to direct and indirect comparisons. (Progressions for CCSSM: Geometric Measurement, The CCSS Writing Team, June 2012.) 12 P a g e

13 1.MD.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. First Grade students collect and use categorical data (e.g., eye color, shoe size, age) to answer a question. The data collected are often organized in a chart or table. Once the data are collected, First Graders interpret the data to determine the answer to the question posed. They also describe the data noting particular aspects such as the total number of answers, which category had the most/least responses, and interesting differences/similarities between the categories. As the teacher provides numerous opportunities for students to create questions, determine up to 3 categories of possible responses, collect data, organize data, and interpret the results, First Graders build a solid foundation for future data representations (picture and bar graphs) in Second Grade. Example: Survey Station During Literacy Block, a group of students work at the Survey Station. question, creates up to 3 possible answers, and walks around the room classmates. Each student then interprets the data and writes 2-4 sentences When all of the students in the Survey Station have completed their own share with one another what they discovered. They ask clarifying questions the data, and make revisions as needed. They later share their results with Each student writes a collecting data from describing the results. data collection, they each of one another regarding the whole class. Student: The question, What is your favorite flavor of ice cream? is posed and recorded. The categories chocolate, vanilla and strawberry are determined as anticipated responses and written down on the recording sheet. When asking each classmate about his/her favorite flavor, the student s name is written in the appropriate category. Once the data are collected, the student counts up the amounts for each category and records the amount. The student then analyzes the data by carefully looking at the data and writes 4 sentences about the data. 13 P a g e

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