All the resources on this page come from the GADOE Framework 4th Grade Unit 6 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 7 Overview

In this unit students will:

● investigate what it means to measure length, weight, liquid volume, time, and angles

● understand how to use standardized tools to measure length, weight, liquid volume, time, and angles

● understand how different units within a system (customary and metric) are related to each other

● know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz; L, ml; hr, min, sec.

● solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals.

● make a line plot to display a data set of measurements in fractions of a unit ( 1 2 , 1 4 , 1 8 )

● solve problems involving addition and subtraction of fractions by using information presented in line plots

● apply the area and perimeter formulas for rectangles in real world and mathematical problems.

● decompose rectilinear figures into non-overlapping squares and rectangles to find the total area of the rectilinear figure

● recognize angles as geometric shapes that are formed when two rays share a common endpoint, and understand concepts of angle measurement

● measure angles in whole number degrees using a protractor

● recognize angle measurement as additive and when an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.

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 FOR MATHEMATICAL PRACTICE: make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning, should be addressed constantly as well. The first unit should establish these routines, allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.

These tasks are not intended to be the sole source of instruction. They are representative of the kinds of experiences students will need in order to master the content, as well as mathematical practices that lead to conceptual understanding. Teachers should NOT do every task in the unit; they should choose the tasks that fit their students’ needs. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources. 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 for Grade 4.

STANDARDS FOR MATHEMATICAL PRACTICE

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. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students will solve problems involving measurement and the conversion of measurements from a larger unit to a smaller unit.

2. Reason abstractly and quantitatively. Students will recognize angle measure as additive in relation to the reference of a circle.

3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding the relative size of measurement units and relating them to everyday objects.

4. Model with mathematics. Students use line plots to display data of measurements in fractions of a unit.

5. Use appropriate tools strategically. Students select and use tools such as a ruler, balance, graduated cylinders, angle rulers and protractors to measure.

6. Attend to precision. Students will specify units of measure and state the meaning of the symbols they choose.

7. Look for and make use of structure. Students use the structure of a two-column table to generate a conversion table for measurement equivalents.

8. Look for and express regularity in repeated reasoning. Students notice repetitive actions in computations to make generalizations about conversion of measurements from a larger unit to a smaller unit.

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

STANDARDS FOR MATHEMATICAL CONTENT

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

MGSE4.MD.1 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec.

a. Understand the relationship between gallons, cups, quarts, and pints.

b. Express larger units in terms of smaller units within the same measurement system. c. Record measurement equivalents in a two-column table.

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.

MGSE4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

MGSE4.MD.8 Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Represent and interpret data.

MGSE4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1 2 , 1 4 , 1 8 ). Solve problems involving addition and subtraction of fractions with common denominators by using information presented in line plots. For example, from a line plot, find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Geometric Measurement - understand concepts of angle and measure angles.

MGSE4.MD.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles. b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

MGSE4.MD.6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

MGSE4.MD.7 Recognize angle measure as additive. When an angle is decomposed into no overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol or letter for the unknown angle measure.

BIG IDEAS

● To measure something according to a particular attribute means you compare the object to a unit and determine how many units are needed to have the same amount as the object.

● Measurements are estimates.

● When reporting a measurement, you must always indicate the unit you are using.

● The larger the unit, the smaller the number you obtain as you measure.

● Measurement units within a system of measurement have relative sizes. (km, m, cm; kg, g; lb, oz; L, mL; hr, min, and sec.)

● Finding the area of a rectangle or square can be found using the formula l x w. The area should be expressed using square units.

● Finding the perimeter of a rectangle or square can be found using the formula 2l + 2w or 2(l + w). The perimeter should be expressed using linear units.

● Rectilinear figures can be decomposed into smaller rectangles and squares. The area of the smaller rectangles and squares can be determined using the formula a = l x w. The areas of the smaller rectangles and squares can be added together to find the total area of the rectilinear figure.

● The measure of an angle does not depend on the lengths of its sides.

● Angle measurement can be thought of as a measure of rotation.

● Data can be measured and represented on line plots in units of whole numbers or fractions.

● Data can be collected and used to solve problems involving addition or subtraction of fractions.

● Appropriate units should be used to measure weight or mass of an object. (ounce, pound, gram, kilogram)

● Finding the exact measure of an angle involves using a protractor.

● It is helpful to think about benchmark references for various weight, mass, length and angle units.

● The sum of the angles in any triangle is 180°.

● Half rotations are equivalent to 180° or a straight angle. Full rotations are 360° or a full circle.

● Measurement data can be displayed using a line plot to display a data set of measurements in fractions of a unit to the nearest 1 8 of an inch.

ESSENTIAL QUESTIONS

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

• About how heavy is a kilogram?

● Does liquid volume change when you change the measurement material? Why or why not?

● How are a circle and an angle related?

● How are area and perimeter related?

● How is data collected?

● How are fluid ounces, cups, pints, quarts, and gallons related?

● How are grams and kilograms related?

● How are the angles of a triangle related?

● How are the units used to measure perimeter different from the units used to measure area?

● How are the units used to measure perimeter like the units used to measure area?

● How can I decompose a rectilinear figure to find its area?

● How are units in the same system of measurement related?

● How can angles be combined to create other angles?

● How can we estimate and measure capacity?

● How can we measure angles using wedges of a circle?

● How can we use the relationship of angle measures of a triangle to solve problems?

● How do graphs help explain real-world situations?

● How do we compare customary measures of fluid ounces, cups, pints, quarts, and gallons?

● How do we compare metric measures of milliliters and liters?

● How do we determine the most appropriate graph to use to display the data?

● How do we make a line plot to display a data set?

● How do we measure an angle using a protractor?

● How do we use mass/weight measurement?

● How does a circle help with angle measurement?

● How does a turn relate to an angle?

● How does the area change as the rectangle’s dimensions change (with a fixed perimeter)?

● How heavy does one pound feel?

● What are benchmark angles and how can they be useful in estimating angle measures?

● What around us has a mass of about a gram?

● What around us has a mass of about a kilogram?

● What do we actually measure when we measure an angle?

● What does half rotation and full rotation mean?

● What is an angle?

● What unit is the best to use when measuring capacity?

● What unit is the best to use when measuring volume?

● What units are appropriate to measure weight?

● When do we use conversion of units?

● Why are units important in measurement?

● Why do we need a standard unit with which to measure angles?

● Why do we need to be able to convert between capacity units of measurement?

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.

● To measure an object with respect to a particular attribute (For example: length, area, capacity, elapsed time, etc.), we may select another object with the same attribute as a unit and determine how many units are needed to ‘cover’ the object.

● The use of standard units will make it easier for us to communicate with each other.

● When we use larger units, we do not need as many as when we use smaller units. Therefore, the larger unit will result in a smaller number as the measurement.

● Measure and solve problems using hour, minute, second, pounds, ounces, grams, kilograms, milliliters, liters, centimeters, meters, inches (to halves and fourths), feet, ounces, cups, pints, quarts, and gallons.

● Solve problems involving perimeters of polygons and perimeter and area of rectangles.

● Draw a scaled picture graph and bar graph.

● Generate measurement data using length and display data by making a line plot.

● Relate area to multiplication and addition and find the area of a rectangle using whole number side length.

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/nctm-timedtests.pdf

COMMON MISCONCEPTIONS

4.MD.1 & 2 - Student believe that larger units will give larger measures. Students should be given multiple opportunities to measure the same object with different measuring units. For example, have the students measure the length of a room with one-inch tiles, with one-foot rulers, and with yardsticks. Students should notice that it takes fewer yardsticks to measure the room than the number of rulers of tiles needed.

4.MD.4 - Students use whole-number names when counting fractional parts on a number line. The fraction name should be used instead. For example, if two-fourths is represented on the line plot three times, then there would be six-fourths.

Specific strategies may include:

Create number lines with the same denominator without using the equivalent form of a fraction. For example, on a number line using eighths, use 48 instead of 12. This will help students later when they are adding or subtracting fractions with unlike denominators. When representations have unlike denominators, students ignore the denominators and add the numerators only. Have students create stories to solve addition or subtraction problems with fractions to use with student created fraction bars/strips.

4.MD.5 - Students are confused as to which number to use when determining the measure of an angle using a protractor because most protractors have a double set of numbers. Students should decide first if the angle appears to be an angle that is less than the measure of a right angle (90°) or greater than the measure of a right angle (90°). If the angle appears to be less than 90°, it is an acute angle and its measure ranges from 0° to 89°. If the angle appears to be an angle that is greater than 90°, it is an obtuse angle and its measures range from 91° to 179°. Ask questions about the appearance of the angle to help students in deciding which number to use.

STRATEGIES FOR TEACHING AND LEARNING

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

In order for students to have a better understanding of the relationships between units, they need to use measuring devices in class. The number of units needs to relate to the size of the unit. They need to discover that there are 12 inches in 1 foot and 3 feet in 1 yard. Allow students to use rulers and yardsticks to discover these relationships among these units of measurements. Using 12-inch rulers and a yardstick, students can see that the set of three of the 12-inch rulers is the same as 3 feet since each ruler is 1 foot in length and is equivalent to one yardstick. Have students record the relationships in a two-column table or t-charts. A similar strategy can be used with centimeter rulers and a meter stick to discover the relationships between centimeters and meters.

Present word problems as a source for developing students’ understanding of the relationships among inches, feet, and yards. Students are also 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.

Present problems that involve multiplication of a fraction by a whole number (denominators are 2, 3, 4, 5 6, 8, 10, 12 and 100). Problems involving addition and subtraction of fractions should have the same denominators. Allow students to use strategies learned with these concepts.

Students used models to find area and perimeter in Grade 3. They need to relate discoveries from the use of models to develop an understanding of the area and perimeter formulas to solve real world and mathematical problems. Students should also use their knowledge of squares and rectangles to decompose rectilinear figures into smaller rectangles and squares. Then, using the formula developed through their work in fourth grade with area, students can find the area of each smaller rectangle or square and find the area of the rectilinear figure by finding the sum of the areas calculated in the smaller rectangles or squares.

Represent and interpret data

Data has been measured and represented on line plots in units of whole numbers, halves, or quarters. Students have also represented fractions on number lines. Now students are using line plots to display measurement data in fraction units and using the data to solve problems involving addition or subtraction of fractions.

Have students create line plots with fractions of a unit ( 1/2 , 1/4 , 1/8 ) and plot data showing multiple data points for each fraction.

Pose questions that students may answer, such as:

● “How many one-eighths are shown on the line plot?” Expect “two one-eighths” as the answer. Then ask, “What is the total of these two one-eighths?” Encourage students to count the fractional numbers as they would with whole number counting but using the fraction name.

● “What is the total number of inches for insects measuring 3/8 inches?” Students can use skip counting with fraction names to find the total, such as, “three-eighths, six-eighths, nine-eighths. The last fraction names the total. Students should notice that the denominator did not change when they were saying the fraction name. Have them make a statement about the result of adding fractions with the same denominator.

● “What is the total number of insects measuring 1 8 inch or 5 8 inches?” Have students write number sentences to represent the problem and solution such as, 1/8 + 1/8 + 5/8 = 7/8 inches.

Use visual fraction strips and fraction bars to represent problems to solve problems involving addition and subtraction of fractions.

Geometric measurement - understand concepts of angle and measure angles.

Angles are geometric shapes composed of two rays that are infinite in length. Students can understand this concept by using two rulers held together near the ends. The rulers can represent the rays of an angle. As one ruler is rotated, the size of the angle is seen to get larger. Ask questions about the types of angles created. Responses may be in terms of the relationship to right angles. Introduce angles as acute (less than the measure of a right angle) and obtuse (greater than the measure of a right angle). Have students draw representations of each type of angle. They also need to be able to identify angles in two-dimensional figures.

Students can also create an angle explorer (two strips of cardboard attached with a brass fastener) to learn about angles.

They can use the angle explorer to get a feel of the relative size of angles as they rotate the cardboard strips around.

Students can compare angles to determine whether an angle is acute or obtuse. This will allow them to have a benchmark reference for what an angle measure should be when using a tool such as a protractor or an angle ruler.

Provide students with four pieces of straw, two pieces of the same length to make one angle and another two pieces of the same length to make an angle with longer rays.

Another way to compare angles is to place one angle over the other angle. Provide students with a transparency to compare two angles to help them conceptualize the spread of the rays of an angle. Students can make this comparison by tracing one angle and placing it over another angle. The side lengths of the angles to be compared need to be different.

Students are ready to use a tool to measure angles once they understand the difference between an acute angle and an obtuse angle. Angles are measured in degrees. There is a relationship between the number of degrees in an angle and circle which has a measure of 360 degrees. Students are to use a protractor to measure angles in whole-number degrees. They can determine if the measure of the angle is reasonable based on the relationship of the angle to a right angle. They also make sketches of angles of specified measure.

● 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.

● 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 include self-assessment and reflection.

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, or numbers.

Note – At the elementary level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks. The Standards glossary of mathematical terms: 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. ● centimeter(cm) ● cup (c) ● customary ● foot (ft) ● gallon (gal) ● gram (g) ● kilogram (kg) ● kilometer (km) ● liquid volume ● liter (L) ● mass ● measure ● meter (m) ● metric ● mile (mi) ● milliliter (mL) ● ounce (oz) ● pint (pt) ● pound (lb) ● quart (qt) ● relative size ● ton (T) ● weight ● decompose ● yard (yd) ● data ● line plot ● intersect ● acute angle ● angle ● arc ● circle ● degree ● measure ● obtuse angle ● one-degree angle ● protractor ● rectilinear figure ● right angle ● straight angle