Rounding can be expanded by having students identify all the numbers that will round to 30 or round to 200.

Strategies used to add and subtract two-digit numbers are now applied to fluently add and subtract whole numbers within 1000. These strategies should be discussed so that students can make comparisons and move toward efficient methods.

Number sense and computational understanding is built on a firm understanding of place value.

Graphing and Data…

Representation of a data set is extended from picture graphs and bar graphs with single-unit scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate to multiplication and division with 100 (product is 100 or less and numbers used in division are 100 or less). In picture graphs, use values for the icons in which students are having difficulty with multiplication facts. For example, represents 7 people. If there are three, students should use known facts to determine that the three icons represent 21 people. The intervals on the vertical scale in bar graphs should not exceed 100.

Students are to draw picture graphs in which a symbol or picture represents more than one object. Bar graphs are drawn with intervals greater than one. Ask questions that require students to compare quantities and use mathematical concepts and skills. Use symbols on picture graphs that student can easily represent half of, or know how many half of the symbol represents.

Examples of Common Graphing Situations

• Pose a question: Student should come up with a question. What is the typical genre read in our class?

• Collect and organize data: student survey

• Pictographs: Scaled pictographs include symbols that represent multiple units. Below is an example of a pictograph with symbols that represent multiple units. Graphs should include a title, categories, category label, key, and data. How many more books did Juan read than Nancy?

• Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs include a title, scale, scale label, categories, category label, and data.

• Analyze and Interpret data:

• How many more nonfiction books were read than fantasy books?

• Did more people read biography and mystery books or fiction and fantasy books?

• About how many books in all genres were read?

• Using the data from the graphs, what type of book was read more often than a mystery but less often than a fairytale?

• What interval was used for this scale?

• What can we say about types of books read? What is a typical type of book read?

• If you were to purchase a book for the class library which would be the best genre? Why? For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

STANDARDS FOR MATHEMATICAL PRACTICE (SMP)

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students make sense of problems involving rounding, addition and subtraction.

2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning by connecting quantity to the relative magnitude of digits in numbers to 1000.

3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding mental math strategies focusing on addition and subtraction.

4. Model with mathematics. Students are asked to use Base Ten blocks to model various understandings of place value and value of a digit. They record their thinking using words, pictures, and numbers to further explain their reasoning.

5. Use appropriate tools strategically. Students utilize a number line to assist with rounding, addition, and subtraction.

6. Attend to precision. Students attend to the language of real-world situations to determine appropriate ways to organize data.

7. Look for and make use of structure. Students relate the structure of the Base Ten number system to place value and relative size of a digit. They will use this understanding to add, subtract, and estimate.

8. Look for and express regularity in repeated reasoning. Students relate the properties and understanding of addition to subtraction situations.

*Mathematical Practices 1 and 6 should be evident in EVERY lesson!

CONTENT STANDARDS

Content standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

MGSE3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

MGSE3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

MGSE3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units – whole numbers, halves, or quarters.

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

BIG IDEAS

Numbers and Operations in Base Ten Place Value and Rounding…

• Place value is crucial when operating with numbers.

• Estimation helps us see whether or not our answers are reasonable.

Addition and Subtraction…

• Addition and subtraction are inverse operations; one undoes the other.

• Addition means the joining of two or more sets that may or may not be the same size. There are several types of addition problems, see the chart above.

• Subtraction has more than one meaning. It not only means the typical “take away” operation, but also can denote finding the difference between sets. Different subtraction situations are described in the chart above.

Data and Graphing

• Charts, tables, line plot graphs, pictographs, Venn diagrams, and bar graphs may be used to display and compare data.

• The scale increments used when making a bar graph is determined by the scale intervals being graphed.

ESSENTIAL QUESTIONS

• Why is place value important?

• How are addition and subtraction related?

• How can graphs be used to organize and compare data?

• How can we effectively estimate numbers?

CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• place value

• standard and expanded forms of numbers

• addition

• subtraction

• addition and subtraction properties

• conceptual understanding of multiplication

• interpreting pictographs and bar graphs

• organizing and recording data using objects, pictures, pictographs, bar graphs, and simple charts/tables

• data analysis

• graphing

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf and: http://joboaler.com/timed-tests-and-thedevelopment-of-math-anxiety/

STRATEGIES FOR TEACHING AND LEARNING
(Information adapted from North Carolina DPI Instructional Support Tools)

Prior to implementing rules for rounding, students need to have opportunities to investigate place value. A strong understanding of place value is essential for the developed number sense and the subsequent work that involves rounding numbers.

Building on previous understandings of the place value of digits in multi-digit numbers, place value is used to round whole numbers. Dependence on learning rules can be eliminated with strategies such as the use of a number line to determine which multiple of 10, or of 100, a number is nearest. As students’ understanding of place value increases, the strategies for rounding are valuable for estimating, justifying and predicting the reasonableness of solutions in problem-solving.

Strategies used to add and subtract two-digit numbers are now applied to fluently add and subtract whole numbers within 1000. These strategies should be discussed so that students can make comparisons and move toward efficient methods.

For additional assistance with this unit, please watch the unit webinar: https://www.georgiastandards.org/unit-webinar

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, teachers should pay particular attention to them and how their students are able to explain and apply them.

The terms below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real-life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. Mathematics Glossary • add • addend • addition o associative property of addition o commutative property of addition o identity property of addition • bar graph • chart • difference • expanded form• graph • increment • interval • inverses • line plot graph • pictograph • place value • properties • round • scale • standard form • strategies • subtract • subtraction • sum • table