All the resources on this page come from the GADOE Framework 4th Grade Unit 5 Overview. Reviewing this information before or during planning can help ensure your instruction is aligned with the standards and is at the depth expected.

Unit 5 Overview

In this unit, students will:

● express fractions with denominators of 10 and 100 as decimals

● understand the relationship between decimals and the base ten system

● understand decimal notation for fractions

● use fractions with denominators of 10 and 100 interchangeably with decimals

● express a fraction with a denominator 10 as an equivalent fraction with a denominator 100

● add fractions with denominators of 10 and 100 (including adding tenths and hundredths)

● compare decimals to hundredths by reasoning their size

● understand that comparison of decimals is only valid when the two decimals refer to the same whole

● justify decimals comparisons using visual models

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight standards of mathematical practice: making sense of problems and persevering in solving them, reasoning abstractly and quantitatively, constructing viable arguments and critiquing the reasoning of others, modeling mathematics, using appropriate tools strategically, attending to precision, looking for and making use of structure, and looking for and expressing regularity in repeated reasoning, should be addressed continually as well. The first unit used each year should establish these routines, allowing students to gradually enhance their understanding of the concept of numbers and to develop computational proficiency.

The Critical Areas are designed to bring focus to the standards at each grade by describing the big ideas that educators can use to build their curriculum and to guide instruction. Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15 9 = 5 3 ), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

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 PRACTICES

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

1. Make sense of problems and persevere in solving them. Students will create and compare decimals to solve problems in the tasks “Taxi Trouble” and “Cell Phone Plans.”

2. Reason abstractly and quantitatively. Students will order decimal fractions to hundredths on a number line or with visual models and understand that fraction equivalency is only valid when comparing parts of the same whole.

3. Construct viable arguments and critique the reasoning of others. Students will communicate why one decimal or fraction is either greater than, less than or equal to another decimal or fraction and be able to question the interpretations of others when discussing the same decimals and fractions.

4. Model with mathematics. Students will use base ten models (blocks, number lines, etc.) to model relative size of decimals and fractions and use the same models to represent fraction and decimal equivalency.

5. Use appropriate tools strategically. Students will determine which tools (blocks, number lines, etc.) would be best used to represent situations involving decimals and decimal fractions.

6. Attend to precision. Students attend to the language of real-world situations to order decimals and decimal fractions.

7. Look for and make use of structure. Students relate the structure of number lines and base ten models to the ordering of decimals and decimal fractions. Furthermore, students will relate the structure of the models to fractional and decimal equivalency.

8. Look for and express regularity in repeated reasoning. Students will use mathematical reasoning to relate new experiences with similar experiences when dealing with fractional and decimal equivalency and with ordering decimals to hundredths.

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

STANDARDS FOR MATHEMATICAL CONTENT

Understand decimal notation for fractions and compare decimal fractions.

MGSE4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/1001 .

MGSE4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

MGSE4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of the comparisons with the symbols >, =, or <, and justify the conclusions, e.g. by using a visual model.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

MGSE4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

BIG IDEAS

● Fractions can be expressed as decimals.

● Decimals can be represented visually in models like hundredths grids and number lines and in written form.

● Decimals are a part of the base ten system.

● Tenths can be expressed using an equivalent fraction with a denominator of 100.

● Comparisons of two decimals are only valid when the two decimals refer to the same whole.

● The sum of two fractions with the respective denominators 10 and 100 can be determined.

ESSENTIAL QUESTIONS

Choose a few questions based on the needs of your students.

● How are decimal fractions written using decimal notation?

● How are decimal numbers and decimal fractions related?

● How are decimals and fractions related?

● How can I combine the decimal length of objects I measure?

● How can I model decimals fractions using the base-ten and place value system?

● How can I write a decimal to represent a part of a group?

● How does the metric system of measurement show decimals?

● What is a decimal fraction and how can it be represented?

● What models can be used to represent decimals?

● What patterns occur on a number line made up of decimal fractions?

● When adding decimals, how does decimal notation show what I expect? How is it different?

● When is it appropriate to use decimal fractions?

● When you compare two decimals, how can you determine which one has the greater value?

● Why is the number 10 important in our number system?

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.

● Recognize and represent that the denominator determines the number of equal sized pieces that make up a whole.

● Recognize and represent that the numerator determines how many pieces of the whole are being referred to in the fraction.

● Compare fractions with denominators of 2, 3, 4, 6, 10, or 12 using concrete and pictorial models.

● Understand that a decimal represents a part of 10

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: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctmtimed-tests.pdf

STRATEGIES FOR TEACHING AND LEARNING

The place value system developed for whole numbers extends to fractional parts of a whole represented as decimals. This is a connection to the metric system. Decimals are another way to represent fractions. The place-value system developed for whole numbers extends to decimals. The concept of one whole used in fractions is extended to models of decimals.

Students can use base-ten blocks to represent decimals. A 10 x 10 block can be assigned the value of one whole to allow other blocks to represent tenths and hundredths. They can show a decimal representation from the base-ten blocks by shading on a 10 x 10 grid.

Students need to make connections between fractions and decimals. They should be able to write decimals for fractions with denominators of 10 or 100. Have students say the fraction with denominators of 10 and 100 aloud. For example, 4 10 would be “four tenths” or 27 100 would be “twenty-seven hundredths.” Also, have students represent decimals in word form with digits and the decimal place value, such as 4 10 would be 4 tenths.

Students should be able to express decimals to the hundredths as the sum of two decimals or fractions. This is based on understanding of decimal place value. For example, 0.32 would be the sum of 3 tenths and 2 hundredths. Using this understanding, students can write 0.32 as the sum of two fractions 3 10 + 2 100 .

Students’ understanding of decimals to hundredths is important in preparation for performing operations with decimals to hundredths in Grade 5.

In decimal numbers, the value of each place is 10 times the value of the place to its immediate right. Students need an understanding of decimal notations before they try to do conversions in the metric system. Understanding of the decimal place value system is important prior to the generalization of moving the decimal point when performing operations involving decimals.

Students extend fraction equivalence from Grade 3 with denominators of 2, 3 4, 6 and 8 to fractions with a denominator of 10. Provide fraction models of tenths and hundredths so that students can express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: fraction, numerator, denominator, equivalent, reasoning, decimals, tenths, hundreds, multiplication, comparisons/compare, ‹, ›, =.

● Students should be actively engaged by developing their own understanding.

● Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words.

● Interdisciplinary and cross-curricular strategies should be used to reinforce and extend the learning activities.

● Appropriate manipulatives and technology should be used to enhance student learning

● Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition, which includes self-assessment and reflection.

● Students should write about the mathematical ideas and concepts they are learning.

● Books such as Fractions and Decimals Made Easy (2005) by Rebecca Wingard-Nelson, illustrated by Tom LaBaff, are useful resources to have available for students to read during the instruction of these concepts.

● Consideration of all students should be made during the planning and instruction of this unit. Teachers need to consider the following:

● What level of support do my struggling students need in order to be successful with this unit?

● In what way can I deepen the understanding of those students who are competent in this unit?

● What real-life connections can I make that will help my students utilize the skills practiced in this unit?

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, instructors should pay particular attention to them and how their students are able to explain and apply them.

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, and numbers.

The websites below are interactive and include a math glossary suitable for elementary children. It has activities to help students more fully understand and retain new vocabulary. (i.e. The definition for dice actually generates rolls of the dice and gives students an opportunity to add them.)

Note – At the elementary level, different sources use different definitions. Please preview any website for alignment to the GSE. http://www.corestandards.org/Math/Content/mathematics-glossary/glossary

The terms below are for teacher reference only and are not to be memorized by the students. ● decimal ● decimal fraction ● decimal point ● denominator ● equivalent sets ● increment ● numerator ● term ● unit fraction ● whole number