Grade: 3 Unit #2: Place Value and Problem Solving with Units of Measure Time: 25 Days

Unit OverviewModule 2 uses place value to unify measurement, rounding skills, and algorithms for addition and subtraction.  The module begins with plenty of hands-on experience using a variety of tools to build practical measurement skills and conceptual understanding of metric and time units.  Estimation naturally surfaces through application; this transitions students into rounding.  In the module’s final topics students round to assess whether or not their solutions to problems solved using the various algorithms are reasonable.

Connection to Prior Learning

In Grade 3, students use their knowledge and experience with the number line, from 2nd grade, to extend multi-digit addition and subtraction skills to solve problems involving elapsed time. In grade 2 students used strategies based on place value, properties and relationships to add and subtract numbers within 100. Students in 3rd grade will extend their thinking from 2nd grade to numbers within 1000.

Major Cluster Standards

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.3.MD.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

Major Cluster Standards UnpackedSolve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.3.MD.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

This standard calls for students to solve elapsed time, including word problems. Students could use clock models or number lines to solve. On the number line, students should be given the opportunities to determine the intervals and size of jumps on their number line. Students could use pre-determined number lines (intervals every 5 or 15 minutes) or open number lines (intervals determined by students). Students in second grade learned to tell time to the nearest five minutes. In third grade, they extend telling time and measure elapsed time both in and out of context using clocks and number lines. Students may use an interactive whiteboard to demonstrate understanding and justify their thinking.

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3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

This standard asks for students to reason about the units of mass and volume using units g, kg, and L. Students need multiple opportunities weighing classroom objects and filling containers to help them develop a basic understanding of the size and weight of a liter, a gram, and a kilogram. Milliliters may also be used to show amounts that are less than a liter emphasizing the relationship between smaller units to larger units in the same system. Word problems should only be one-step and include the same units.

Students are not expected to do conversions between units, but reason as they estimate, using benchmarks to measure weight and capacity.

Example:Students identify 5 things that weigh about one gram. They record their findings with words and pictures. (Students can repeat this for 5 grams and 10 grams.) This activity helps develop gram benchmarks. One large paperclip weighs about one gram.

Example:A paper clip weighs about a) a gram, b) 10 grams, c) 100 grams? Explain why.

Foundational understandings to help with measure concepts:Understand that larger units can be subdivided into equivalent units (partition).Understand that the same unit can be repeated to determine the measure (iteration).

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Understand the relationship between the size of a unit and the number of units needed (compensatory principal).

Before learning to measure attributes, children need to recognize them, distinguishing them from other attributes. That is, the attribute to be measured has to “stand out” for the student and be discriminated from the undifferentiated sense of amount that young children often have, labeling greater lengths, areas, volumes, and so forth, as “big” or “bigger.”

These standards do not differentiate between weight and mass. Technically, mass is the amount of matter in an object. Weight is the force exerted on the body by gravity. On the earth’s surface, the distinction is not important (on the moon, an object would have the same mass, would weigh less due to the lower gravity).

Much of the work involving measure supports the work that is emphasized in third on multiplication.Example:

Common addition and subtraction situations to be applied to measurement problemsResult Unknown Change Unknown Start Unknown

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Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now?5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown2

Put Together/Take Apart3

Three red apples and two green apples are on the table. How many apples are on the table?3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green?3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?5 = 0 + 5, 5 = 5 + 05 = 1 + 4, 5 = 4 + 15 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare4

(“How many more?” version):Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

(“How many fewer?” version):Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?2 + ? = 5, 5 – 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

(Version with “fewer”):Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?2 + 3 = ?, 3 + 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?

(Version with “fewer”):Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?5 – 3 = ?, ? + 3 = 5

Instructional Strategies (3.MD.1-2)A clock is a common instrument for measuring time. Learning to tell time has much to do with learning to read a dial-type instrument rather than with time measurement.

Students have experience in telling and writing time from analog and digital clocks to the hour and half hour in Grade 1 and to the nearest five minutes, using a.m. and p.m. in Grade 2. Now students will tell and write time to the nearest minute and measure time intervals in minutes.

Provide analog clocks that allow students to move the minute hand. Students need experience representing time from a digital clock to an analog clock and vice versa.

Provide word problems involving addition and subtraction of time intervals in minutes. Have students represent the problem on a number line. Student should relate using the number line with subtraction from Grade 2.

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Provide opportunities for students to use appropriate tools to measure and estimate liquid volumes in liters only and masses of objects in grams and kilograms. Students need practice in reading the scales on measuring tools since the markings may not always be in intervals of one. The scales may be marked in intervals of two, five or ten. Allow students to hold gram and kilogram weights in their hand to use as a benchmark. Use water colored with food coloring so that the water can be seen in a beaker.

Students should estimate volumes and masses before actually finding the measuring. Show students a group containing the same kind of objects. Then, show them one of the objects and tell them its weight. Fill a container with more objects and ask students to estimate the weight of the objects.

Use similar strategies with liquid measures. Be sure that students have opportunities to pour liquids into different size containers to see how much liquid will be in certain whole liters. Show students containers and ask, “How many liters do you think will fill the container?”

If making several estimates, students should make an estimate, then the measurement and continue the process of estimating measure rather than all estimates and then all measures. It is important to provide feedback to students on their estimates by using measurement as a way of gaining feedback on estimates.

Common Misconceptions:Students may read the mark on a scale that is below a designated number on the scale as if it was the next number. For example, a mark that is one mark below 80 grams may be read as 81 grams. Students realize it is one away from 80, but do not think of it as 79 grams.

Avoid the use of paper plate clocks. Students need to see the actual relationship between the hour and the minute hand. This is not adequately represented on student made clocks.

Students forget to label the measurement or choose the incorrect unit. Make sure that the expectation is that students will label all units correctly.

Additional Cluster StandardsUse place value understanding and properties of operations to perform multi-digit arithmetic.3. NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.3. NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or therelationship between addition and subtraction.

Additional Cluster Standards Unpacked

Use place value understanding and properties of operations to perform multi-digit arithmetic.3. NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.This standard refers to place value understanding, which extends beyond an algorithm or memorized procedure for rounding. The expectation is that students have a deep understanding of place value and number sense and can explain and reason about the answers they get when they round. Students should have numerous experiences using a number line and a hundreds chart as tools to support their

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work with rounding.

Mrs. Rutherford drives 158 miles on Saturday and 171 miles on Sunday. When she told her husband she estimated how many miles to the nearest 10 before adding the total. When she told her sister she estimated to the nearest 100 before adding the total. Which method provided a closer estimate?

Estimating using number line thinking helps students understand the process and reason about their solution. For example: To round 168 to the nearest 10, students think about counting by 10s….What would be on each side of 168 on a number line that was counted by 10?160 168 170 would be on either side. Then students can think of 165 would be right in the middle…Is 168 closer to 160 or 170 on a number line? Writing the two options down before rounding helps students reason about the estimate.

Another example: 423 rounded to the nearest 100. Think about a number line that is counting by 100. What would be on either side of 423?400 423 500 Is 423 closer to 400 or 500? Explain why? Talking about the exact middle number helps students build number sense as well. If 450 is in the middle then what is 423 closer to? Why?

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

This standard refers to fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as the distributive property). The word algorithm refers to a procedure or a series of steps. There are other algorithms other than the standard algorithm. Third grade students should have experiences with algorithms and strategies based on place value, properties, and relationships between addition and subtraction. The standard algorithm should not be the focus of this work. If a student is using the standard algorithm, make sure that he/she has a solid understanding of place value by helping him/her make the connection between the standard algorithm and other methods: Decomposing numbers, using a number line, and compensation.

Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and associative properties. Students explain their thinking and show their work by using strategies and algorithms, and verify that their answer is reasonable.

Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly.

Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another.

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Focus Standards for Mathematical Practice

MP.2 Reason abstractly or quantitatively. Students decontextualize metric measurements and time intervals in minutes as they solve problems involving addition, subtraction, and multiplication. They round to estimate and then precisely solve, evaluating solutions withreference to units and with respect to real world contexts.MP.4 Model with mathematics. Students model measurements on the place value chart. They create drawings and diagrams and write equations to model and solve word problems involving metric units and intervals of time in minutes.MP.6 Attend to precision. Students round to estimate sums and differences and then use algorithms for addition and subtraction to calculate. They reason about the precision of their solutions by comparing estimations with calculations, and are attentive to specifying units of measure.MP.7 Look for and make use of structure. Students model measurements on the place value chart. Through modeling they relate different units of measure and analyze the multiplicative relationship of the base ten system.

Understandings-Students will understand… Rounding is a method for approximating an answer That time is a continuous measure that can be represented on a number line.

Elapsed time is the interval of time, given a specific unit, from a starting time to an ending time.Metric units of measurement are related to place value and multiples of 10.

Essential Questions

How can understanding the relationship between addition and subtraction aid us in problem solving? How does elapsed time help us to plan and organize real life responsibilities? When would rounding be appropriate to use? How are metric units related to multiples of 10 and place value?

Prerequisite Skills/Concepts: Students should already be able to…

Advanced Skills/Concepts: Some students may be ready to…

Round to the nearest 1000.

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

Estimate lengths using units of inches, feet, centimeters, and meters.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Covert measurements within the same system using a table. Make the connection between strategies for addition and subtraction

and the standard algorithm.

Knowledge: Students will know…

Standard and metric measurement units. Strategies for fluently adding and subtracting within 1000. How to round numbers to the nearest 10 or 100 When to use rounding in real life situations Addition and subtraction computation and problem solving

strategies. A.M. represents time from midnight to noon. P.M. represents time from noon to midnight. 60 min = 1 hour.

Skills: Students will be able to …

Tell and write time to the nearest minute. (3.MD.1) Solve word problems involving elapsed time. (3.MD.1) Use a number line or clocks to model elapsed time and record

calculations. (3.MD.1) Measure and estimate liquid volumes and masses of objects using

standard units of grams (g), kilograms (kg), and liters (l). (3.MD.2) Add, subtract, multiply, or divide to solve one-step word problems

involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. (3.MD.2)

Use place value understanding to round whole numbers to the nearest 10 or 100. (3. NBT.1)

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. (3. NBT.2)

Transfer of Understanding-Students will apply…

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Understanding of rounding to real-life problems Strategies for adding and subtracting to solve time and measurement problems.

Academic Vocabulary

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About (with reference to rounding and estimation, an answer that is not precise)Addend (the numbers that are added together in an addition equation, e.g., in 4 + 5, the numbers 4 and 5 are the addends)Analog clock (a clock that is not digital)Capacity (the amount of liquid that a particular container can hold)Compose (change 10 smaller units for 1 of the next larger unit on the place value chart)Continuous (with reference to time as a continuous measurement)Endpoint (used with rounding on the number line; the numbers that mark the beginning and end of a given interval)Gram (g)Halfway (with reference to a number line, the midpoint between two numbers, e.g., 5 is halfway between 0 and 10)Interval (time passed or a segment on the number line)Kilogram (kg)Liquid volume (the space a liquid takes up)Liter (L)Milliliter (mL)Plot (locate and label a point on a number line)Point (a specific location on the number line)Reasonable (with reference to how plausible an answer is, e.g., “Is your answer reasonable?”)Round (estimate a number to the nearest 10 or 100 using place value)Second (sec)≈ (Symbol used to show than an answer is approximate)Centimeter (cm)Divide (e.g., 4 ÷ 2 = 2)Estimate (approximation of the value of a quantity or number) Pan balancePlace value chart without headings.Number disksHorizontal (with reference to how an equation is written, e.g., 3 + 4 =7 is written horizontally. Problems can also be solve horizontally)Measure (a quantity representing a weight or liquid volume, or the act of finding the size or amount of something)Mental mathMeter (m)Minute (min)MultiplyNumber lineSimplifying strategy (transitional strategies that move students toward mental math, e.g., “make ten” to add 7 and 6, (7 + 3) + 3 = 13)Vertical (with reference to how problems are sometimes written to be solved.

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Unit Resources

Pinpoint: Grade 3 Unit #2

Connection to Subsequent LearningStudents will apply concepts of place value, recognizing that in a multi-digit whole number, the digit on the left represents ten times the value of the digit to its right. They will be able to write multi-digit numbers in expanded form and compare two-digit numbers based on meaning and placement of the digits. They will convey meaning of their comparisons using the symbols: <(less than), >(greater than), and =(equal). Also, addition and subtraction strategies from 3rd grade will become more procedural in 4th grade.

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