# Unit 5 Overview: Linear Measurement & Analyzing and Interpreting Data

## Overview

**Unit Overview/Summary - FOCUS: **

This unit focuses on measurement and data. Learning in this unit will enable students to:

- Measure and estimate lengths in standard units;
- Represent and solve addition and subtraction problems involving length; and
- Represent and interpret data.

# Relevant Standards:

**Major work of the grade is in bold. **

The standards to be addressed during this unit of study include:

**2.OA.A.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, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

**2.MD.A.1. **Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

**2.MD.A.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.A.3.** Estimate lengths using units of inches, feet, centimeters, and meters.

**2.MD.A.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.B.5.** Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

**2.MD.B.6. **Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

2.MD.D.9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

2.MD.D.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 put together, take-apart, and compare problems using information presented in a bar graph.

# Examples and Explanations:

**2.MD.A.1. **Students in second grade will build upon what they learned in first grade from measuring length with non-standard units to the new skill of measuring length in metric and U.S. Customary with standard units of measure. They should have many experiences measuring the length of objects with rulers, yardsticks, meter sticks, and tape measures. They will need to be taught how to actually use a ruler appropriately to measure the length of an object especially as to where to begin the measuring. Do you start at the end of the ruler or at the zero?

**2.MD.A.2. ** Students need multiple opportunities to measure using different units of measure. They should not be limited to measuring within the same standard unit. Students should have access to tools, both U.S. Customary and metric. The more students work with a specific unit of measure, the better they become at choosing the appropriate tool when measuring.

Students measure the length of the same object using different tools (ruler with inches, ruler with centimeters, a yardstick, or meter stick). This will help students learn which tool is more appropriate for measuring a given object. They describe the relationship between the size of the measurement unit and the number of units needed to measure something. For instance, a student might say, “The longer the unit, the fewer I need.” Multiple opportunities to explore provide the foundation for relating metric units to customary units, as well as relating within customary (inches to feet to yards) and within metric (centimeters to meters).

**2.MD.A.3. **Estimation helps develop familiarity with the specific unit of measure being used. To measure the length of a shoe, knowledge of an inch or a centimeter is important so that one can approximate the length in inches or centimeters. Students should begin practicing estimation with items which are familiar to them (length of desk, pencil, favorite book, etc.).

Some useful benchmarks for measurement are:

- First joint to the tip of a thumb is about an inch
- Length from your elbow to your wrist is about a foot
If your arm is held out perpendicular to your body, the length from your nose to the tip of your fingers is about a yard

**2.MD.A.4.** Second graders should be familiar enough with inches, feet, yards, centimeters, and meters to be able to compare the differences in lengths of two objects. They can make direct comparisons by measuring the difference in length between two objects by laying them side by side and selecting an appropriate standard length unit of measure. Students should use comparative phrases such as “It is longer by 2 inches” or “It is shorter by 5 centimeters” to describe the difference between two objects. An interactive whiteboard or document camera may be used to help students develop and demonstrate their thinking.

**2.MD.B.5. **Students need experience working with addition and subtraction to solve word problems which include measures of length. It is important that word problems stay within the same unit of measure. Counting on and/ or counting back on a number line will help tie this concept to previous knowledge. Some representations students can use are drawings, rulers, pictures, and/or physical objects. An interactive whiteboard or document camera may be used to help students develop and demonstrate their thinking.

Equations include:

- 20 + 35 = c
- c –20 = 35
- c – 35 = 20
- 20 + b = 55
- 35 + a = 55
- 55 = a + 35
- 55 = 20 + b

Example:

A word problem for 5- n = 2 could be: Mary is making a dress. She has 5 yards of fabric. She uses some of the fabric and has 2 yards left. How many yards did Mary use?

There is a strong connection between this standard and demonstrating fluency of addition and subtraction facts. Addition facts through 10 + 10 and the related subtraction facts should be included.

2.MD.B.6. Students represent their thinking when adding and subtracting within 100 by using a number line. An interactive whiteboard or document camera can be used to help students demonstrate their thinking.

Example: 10-6 = 4

2.MD.D.9. This standard emphasizes representing data using a line plot. Students will use the measurement skills learned in earlier standards to measure objects. Line plots are first introduced in this grade level. A line plot can be thought of as plotting data on a number line. An interactive whiteboard may be used to create and/or model line plots.

2.MD.D.10. Students should draw both picture and bar graphs representing data that can be sorted up to four categories using single unit scales (e.g., scales should count by ones). The data should be used to solve put together, take-apart, and compare problems.

In second grade, picture graphs (pictographs) include symbols that represent single units. Pictographs should include a title, categories, category label, key, and data.

Second graders should draw both horizontal and vertical bar graphs. Bar graphs include a title, scale, scale label, categories, category label, and data.

# Transfer Goal: Aligned to district portrait or vision of the learner

**Aligned to district portrait or vision of the learner **

In this unit students will develop the skills of perseverance, reasoning, and precision. This will be accomplished through focus on the following Standards for Mathematical Practice:

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Model with mathematics.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

**Grade level content is in bold **

## Coherence:

## How does this unit build on and connect to prior knowledge and learning?

Students build on their previous understandings of:

- Measuring lengths indirectly and by iterating length units.
- Representing and interpreting data.
- Representing and solving problems involving addition and subtraction.

## How does this unit prepare students for future learning?

The learning of this unit serves as a foundation for content that will be addressed in future years. Specifically, students will utilize this learning to:

- Understand concepts of area and relate area to multiplication and to addition.
- Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
- Measurement data can be organized and analyzed by plotting vales on a line plot.
- Develop understanding of fractions as numbers.
**Relate addition and subtraction to length.**

# Essential Questions:

**Essential Questions can be approached in multiple ways. There should be no more than 2-3 essential questions and they should align with your topics. Questions can be repeated throughout a course or over years, with different enduring understandings. **

- How do we measure length?
- What is the relationship between the sizes of units to the number of units?
- How can measurement data be organized?
- How do visual representations depict addition and subtraction?

# Enduring Understanding:

**Enduring Understanding: The major ideas you want students to internalize and understand deeply. These understandings should be thematic in nature. They are not the end all, be all of the question. They are focused to align to the focus (unit overview). **

Students understand:

- Length is found by counting intervals rather than counting the marks on a number line,
- There is an inverse relationship between the unit size and number of units. (The smaller the unit, the more needed to measure; the larger the unit, the fewer needed to measure.)
- Data can be organized, represented and analyzed by using a line plot, picture graph or bar graph.
- Addition and subtraction can be represented on various models such as number lines, picture graphs and bar graphs.

# What Students Will Know:

**Measurement: **

- The standard tools for linear measurement.
- The location of the beginning point of the appropriate standard measuring tool.
- Length-units.
- Length is measured by using an appropriate tool.
- The length of an object remains constant regardless of where it is placed on a measurement tool.
- Starting points on a measurement tool may vary.
- Units must be of equal size.
- Measurements can be nonstandard or standard units.
- All measurements include a margin of error.
- Numerals on a measuring tool indicate the number of length units.
- Lengths can be estimated.
- Lengths can be compared.
- Length is measured by using an appropriate tool.
- A number line diagram is similar to a ruler in that whole numbers are 1 unit apart.
- Each number on a number line denotes the distance from the labeled point from 0, not the number itself.
- Addition and subtraction strategies can be used to solve real-world measurement problems.
- A symbol can be used to represent an unknown number.

**Data: **

- Length measurement data can be generated and used to create a line plot in whole number units.
- Categorical data results from sorting objects into as many as four categories.
- Given a graph, the data can be used to solve addition, subtraction, and comparison problems.

# What Students Will Do:

**Measurement: **

- Measure the length of an object by selecting and using appropriate standard tools.
- Measure length of an object twice, using units of different lengths for the two measurements.
- Describe how two measurements using different units relates to the size of the unit chosen.
- Estimate lengths using units of inches, feet, centimeters and meters.
- Check for reasonableness of estimates.
- Compare objects visually, side by side, and measure the difference.
- Express the difference between lengths in terms of a standard length unit.
- Select and use appropriate tools: rulers, meter sticks, yard sticks, and measuring tapes.
- Use physical representations of standard units to measure length.
- Discuss and describe how different units can give different measurements.
- Emphasize the use of approximate language using phrases such as “about how many.”
- Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the number 0, 1, 2.
- Represent whole-number sums and differences within 100 on a number line diagram.
- Use number sentences or drawings to solve measurement word problems within 100.
- Explore and explain the use of addition and subtraction to solve real-world problems involving lengths (given in the same whole number units).
- Use drawings and equations with a symbol for the unknown number to represent the problem.
- Find the unknown length unit in real-world situations.
- Explore the relationship between number lines and whole number measurement tools.
- Write an equation using a symbol for the unknown number to represent the problem.

**Data: **

- Generate measurement data by measuring lengths of several objects to the nearest whole unit or by making repeated measurements of the same object.
- Show measurement data by making a line plot, where the horizontal scale is marked off in whole-number units.
- Draw a picture graph to represent data with up to 4 categories (including title, scale label, categories, category labels, and data).
- Draw a bar graph to represent data with up to 4 categories (including title, scale label, categories, category labels, and data).
- Solve put together, take-apart, and compare problems about information presented in a bar graph.
- Explore and record data using line plots in whole number units.
- Explore and record data using picture graphs (when single unit scales are provided).
- Explore and record data using bar graphs (when single unit scales are provided).
- Organize and represent data with up to four categories.
- Interpret data to solve addition, subtraction, and compare problems.

# Demonstration of Learning:

**Additional Assessment Samples: **

# Unit Specific Vocabulary and Terminology:

The purpose of vocabulary work should be to allow all students to access mathematics. Vocabulary is a way to provide opportunities for students to use mathematical language to communicate about how they solved a problem, describe their reasoning, and demonstrate understanding of mathematical content. Vocabulary is inclusive of key words and phrases.

Often multilingual learners/English learners (MLs/ELs) are perceived as lacking academic language and needing remediation. Research shows that MLs/ELs bring standards-aligned background knowledge and experiences to the task of learning, and they need opportunities to extend their language for academic purposes. When considering the language demand of a lesson (at the word level), you can check for cognates and polysemous words. Pointing out these words to students can help them activate and build background knowledge assumed in lessons. TESOL professionals can assist with the identification of cognates and polysemous words, and they can provide guidance about the background knowledge MLs/ELs bring or may need.

**Academic Vocabulary **

- compare
- distance
- equal
- estimate
- length
- longer
- measure
- quantity
- symbol
- taller
- unknown
- wider

**Content Vocabulary **

- centimeter
- customary system
- difference
- foot
- inch
- linear
- measuring tape
- meter
- meter stick
- metric system
- ruler
- subtraction
- sum
- unit
- yardstick
- zero
- compare
- equal
- longer
- shorter
- taller
- wider
- Adding to
- Difference
- line plot
- number diagram
- number line
- Putting together
- subtraction
- sum
- taking apart
- taking from

**Vocabulary resources: **

- Bilingual Glossaries and Cognates
- Using Cognates to Develop Comprehension in English
- Challenges for EL Students to Overcome
- Cognates and Polysemous Words
**Granite School Vocabulary Cards:**Each card has the word and a picture. They are designed to help all students with math content vocabulary, including ELL, Gifted and Talented, Special Education, and Regular Education students.

# Aligned Unit Materials, Resources and Technology:

High-quality instructional resources are critical for improving student outcomes. The alignment guidance is intended to clarify content and support understanding for clear implementation and coherence.

Materials selection is a local control decision and these documents have been provided from participating publishers to assist districts in implementation. Use of the materials from these publishers is not required. These aligned core programs meet expectations as reported by edReports. Strong alignment of curricula and instructional materials have the potential to support student engagement and teacher growth.

# Opportunities for Interdisciplinary Connections:

**Computer Science **

**1A-AP-09**Model the way programs store and manipulate data by using numbers or other symbols to represent information.**1A-DA-07**Identify and describe patterns in data visualizations, such as charts or graphs, to make predictions.

**ELA **

**RI.2.1**Ask and answer such questions as who, what, where, when, why, and how to demonstrate understanding of key details in a text. RI.2.8 Describe how reasons support specific points the author makes in a text.**W.2.6**With guidance and support from adults, use a variety of digital tools to produce and publish writing, including in collaboration with peers.**W.2.7**Participate in shared research and writing projects (e.g., read a number of books on a single topic to produce a report; record science observations).**W.2.8**Recall information from experiences or gather information from provided sources to answer a question.**SL.2.5**Create audio recordings of stories or poems; add drawings or other visual displays to stories or recounts of experiences when appropriate to clarify ideas, thoughts, and feelings.

**Science **

**2-ETS1-1**Ask questions, make observations, and gather information about a situation people want to change to define a simple problem that can be solved through the development of a new or improved object or tool.**2-ETS1-3**Analyze data from tests of two objects designed to solve the same problem to compare the strengths and weaknesses of how each performs.**2-PS1-1**Plan and conduct an investigation to describe and classify different kinds of materials by their observable properties.

# Opportunities for Application of Learning:

**Defined Learning** provides an open access online library of standards-aligned project-based lessons to help students meet the expectations of the Standards. Each project is based on a situation in a relevant career to help students connect classroom content to career pathways. This supplemental resource is available at no cost to teachers and districts.* Create an account and log in to access this free resource to support your curriculum.*

*The tasks below provide additional opportunities to apply the content of this unit.*

# Critical Consciousness for Diversity and Equity:

Culturally relevant mathematics engages and empowers students. Opportunities for teachers to orchestrate discussions where students share not only connections to prior mathematics learned but also to their lived experiences must be provided. It is important to dig deep to find ways to link students’ home cultures and the mathematics classroom. Build authentic relationships with families through two-way, reciprocal conversations that acknowledge families’ cultures as assets for teaching and learning. As you plan to implement this unit, focus on designing experiences that have students at the center. In addition to keeping students engaged, ensure the learning experiences have a context that reflects lived experiences (mirror) or provide opportunities to view and learn about the broader world (window).* *

One crucial link to students’ home cultures is through their language. Students’ language repertoires –all the languages and language varieties they use everyday– are a valuable resource to be engaged in the mathematics classroom. This approach is referred to as a translanguaging stance. It is based on a dynamic view of bilingualism that understands individuals as having one linguistic repertoire composed of various named languages (such as English and Spanish) and/or language varieties on which they draw to make meaning.

The following questions are intended to assist in promoting diverse voices and perspectives while avoiding bias and stereotyping:

- How will students share their experiences with others while attending to the mathematics in the unit?
- What opportunities are there for students to make connections from their life to the mathematics?
- What do I know or need to learn about my students to create lessons free from bias and stereotypes?
- In what ways can the mathematical thinking already taking place in the classroom and community be honored?
- How is relevant background knowledge developed so that all students can access the mathematics of the unit?
- What opportunities are there for students to use their full language repertoires during mathematical discussions and practice? Where can I create these opportunities?
- What do I know or need to learn about students’ languages and how they use them? How can I learn this?

*** Resources to support diversity and equity in the classroom please visit the **DEI collection

# Multilingual Learners/English Learners (ML/EL):

Mathematical symbols, expressions, and methods are not universal; ways of doing math differ across cultures. When working with diverse students, especially with those from different countries, it is important to be aware that these differences exist.

It is also important to remember that communicating about mathematical content and practices requires complex language. Since conceptual learning and language learning are interconnected and acquired through participation in meaningful activities, the research-based strategies listed below focus on making content comprehensible (accessible) and creating opportunities for student voice, both verbal and written.

*Additional resources for ML/EL*

CELP Standards--Linguistic Supports

ML/EL Support Collection for Math

- An awareness of different measurement vocabulary.
- The use of visual tools such as rulers, number diagrams or number lines, and meter sticks.
- An understanding of the universality of measurement.
- The use of visual tools such as number diagrams, tape diagrams, number lines, picture graphs, and bar graphs to represent and solve problems.

This unit presents opportunity to address the following CELP Standards:

**CELP Standard 1: **Construct meaning from oral presentations and literary and informational text through grade-appropriate listening, reading, and viewing

**CELP Standard 2: **Participate in grade-appropriate oral and written exchanges of information, ideas, and analyses, responding to peer, audience, or reader comments and questions

**CELP Standard 3: **Speak and write about grade-appropriate complex literary and informational texts and topics

**CELP Standard 4: **Construct grade-appropriate oral and written claims and support them with reasoning and evidence

**CELP Standard 5:** Conduct research and evaluate and communicate findings to answer questions or solve problems

**CELP Standard 6: **Analyze and critique the arguments of others orally and in writing

**CELP Standard 7:** Adapt language choices to purpose, task, and audience when speaking and writing

**CELP Standard 8:** Determine the meaning of words and phrases in oral presentations and literary and informational text

**CELP Standard 9:** Create clear and coherent grade-appropriate speech and text

**CELP Standard 10:** Make accurate use of standard English to communicate in grade appropriate speech and writing

The document below provides guidance on these standards that is grade appropriate and broken down by language level descriptors which will assist the teacher in making the content of the unit accessible to all students.