Vacaville USD February 12, 2015

AGENDA Problem Solving – Where are the Cookies? Estimating and Measurement Dividing with Decimals Volume Coordinate Graphing

Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?

Analyze Student Work For each piece of work: Describe the problem solving approach the student used. For example, you might: –Describe the way the student has organized the solution. –Describe what the student did to calculate the number of cookies that started on the tray. Explain what the student needs to do to complete or correct his or her solution.

Analyze Student Work Suggestions for feedback Common issues Suggested questions and prompts

Kentucky Department of Education Mathematics Formative Assessment Lessons –Concept-Focused Formative Assessment Lessons –Problem Solving Formative Assessment Lessons Designed and revised by Kentucky DOE Mathematics Specialists –Field-­ ‐ tested by Kentucky Mathematics Leadership Network Teachers formative-assessment-lessons.html

Where are the Cookies? Grades 4 – 6 Problem Solving Formative Assessment Lesson Lesson Format –Pre-Lesson (about 15 minutes) –Lesson (about 1 hour) –Follow-Up (about 10 minutes) Where are the Cookies?

Estimation

Estimation How many cheeseballs are in the vase? 183

Estimation How many cheeseballs are in the original container? 917

Estimation How many peanut m&m’s are in the vase? Are there more m&m’s than cheeseballs or less? –How do you know? 441

Measurement We are going to do a brain dump. In just a minute I am going to show you a math term and you have 60 seconds to throw out as many ideas and thoughts as you can.

foot

Measurement Is a foot larger or smaller than a yard? So suppose I tell you I have 9 feet and I want my answer in yards. –Will I have more than 9 yards or less than 9 yards? –How do you know?

Measurement So what do we know about feet and yards? 1 yard 3 feet

Measurement 9 feet = ____ yards 9 yards = ____ feet 1 yard 3 feet

Measurement So what do we know about meters and centimeters? 1 meter 100 centimeters

Measurement 9 m = ____ cm 800 cm = ____ m 1 m 100 cm

Measurement 50 m = ____ cm 70 cm = ____ m 1 m 100 cm

Measurement For each of the following problems, Decide if the answer is going to be larger or smaller than the original amount Draw a representation showing the relationship Find the missing measurement

larger or smaller (more or less)? Draw representation Find missing measurement 500 m = ____ cm24 ft = ___ yd 600 sec = ___ min32 lb = ___ oz 16 qt = ____ pt500 g = ___ kg

How many green marshmallows will fit on the skewer?

How many green marshmallows are inside the glass?

How many green mallows are needed to complete the 4-leaf clover?

What's the capacity of the tall vase?

What's the capacity of the wide vase?

Order the glasses from least to greatest in capacity.

How many Red Vines are in my hand?

How many Red Vines are in the container?

Division of Decimals

Standard 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Dividing a Decimal by a Whole Number

Example of Division: A relay race lasts 4.65 miles. The relay team has 3 runners. If each runner goes the same distance, how far does each team member run? Make an estimate, find your actual answer, and then compare them. Estimate: My estimate is that each runner runs between 1 and 2 miles. If each runner went 2 miles, that would be a total of 6 miles which is too high. If each runner ran 1 mile, that would be 3 miles, which is too low.

I used the 5 grids above to represent the 4.65 miles. I am going to use all of the first 4 grids and 65 of the squares in the 5 th grid. I have to divide the 4 whole grids and the 65 squares into 3 equal groups. I labeled each of the first 3 grids for each runner, so I know that each team member ran at least 1 mile. I then have 1 whole grid and 65 squares to divide up. Each column represents one-tenth. If I give 5 columns to each runner, that means that each runner has run 1 whole mile and 5 tenths of a mile. Now, I have 15 squares left to divide up. Each runner gets 5 of those squares. So each runner ran 1 mile, 5 tenths and 5 hundredths of a mile. I can write that as 1.55 miles.

Dividing Decimals Dividing a decimal by a whole number –Fair Share models Dividing a decimal by a decimal –Measurement models –Most common representation was hundredths grids

Dividing a Decimal by a Decimal

Example of Division: Using an area model to show 0.30 ÷ The decimal 0.30 is broken into 6 groups of ÷ 0.05 = 6

Example of Division: A bag of jelly beans weighs 7.8 pounds. If each student gets 1.3 pounds of candy, how many students will get candy? Use the grid below to solve this task.

Example of Division: A bag of jelly beans weighs 7.8 pounds. If each student gets 1.3 pounds of candy, how many students will get candy? c

Volume

5. MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a.A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b.A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5. MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5. MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

5. MD.5a Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole- number products as volumes, e.g., to represent the associative property of multiplication.

5. MD.5b Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. 5. MD.5c Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non- overlapping parts, applying this technique to solve real world problems.

Volume Example:

V = base area x height V = x V = 60 cubic cm 15 4

Volume Example: V = length x width x height V = x x V = 60 cubic cm 53 4

Volume Example: V = base area x height V = x V = 64 cubic cm 16 4

Volume Example: V = length x width x height V = x x V = 64 cubic cm 44 4

Volume Example: Determine the volume of concrete needed to build the steps in the diagram below.

Formative Assessment Lesson This Formative Assessment Lesson is designed to be part of an instructional unit. This task should be implemented approximately two-thirds of the way through the instructional unit. The results of this task should then be used to inform the instruction that will take place for the remainder of your unit.

Mathematical goals This lesson is intended to help you assess how well students are able to model three dimensional figures and find their volume. In particular, this unit aims to identify and help students who have difficulties with: Recognizing volume as an attribute of three-dimensional space. Measuring volume by finding the total number of same- size units of volume required to fill the space without gaps or overlaps. Measuring necessary attributes of shapes, in particular the base area, in order to determine volumes to solve real world and mathematical problems.

Formative Assessment Lesson Pre-Lesson: Assessment Task: How Many Cubes? As you work on the task, consider: What are some common student errors or misunderstandings that you might expect to see? What questions might you use to focus students’ attention on those issues?

Formative Assessment Lesson Lesson Return task and pose questions; allow students about 10 min to revisit the task Collaborative work (partner): card sort 1 –Begin with a Task card from Card Set A. –Model this problem with the blocks first. –Then find a card from Card Set B that matches the model you built.

Formative Assessment Lesson Card sort 2 –Keep the pairs you have already sorted –Now look at Card Set C – Base Area –Match each of your sorted pairs with their correct base area figure

Formative Assessment Lesson Card sort 3 –Keep the sets you have already sorted –Now look at Card Set D – Formula Cards –Match each of your sorted sets pairs with the correct V = l x w x h formula; then fill in the missing number(s) Whole class discussion and debrief Improve individual solutions

Box of Clay A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box has twice the height, three times the width, and the same length as the first box. How many grams of clay can it hold?

Box of Clay A box 2 centimeters high, 3 centimeters wide, and 5 centimeters long can hold 40 grams of clay. A second box has twice the height, three times the width, and the same length as the first box. How many grams of clay can it hold?

Coordinate Graphing

5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

Coordinate Grid Example: Plot these points on a coordinate grid. Point A: (2,6)Point B: (4,6) Point C: (6,3)Point D: (2,3) Connect the points in order. Make sure to connect Point D back to Point A. –What geometric figure is formed? What attributes did you use to identify it? –What line segments in this figure are parallel? –What line segments in this figure are perpendicular?

5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Coordinate Grid Example:

Sara has saved $20. She earns $8 for each hour she works. If Sara saves all of her money, how much will she have after working 3 hours? 5 hours? 10 hours? –Create a graph that shows the relationship between the hours Sara worked and the amount of money she has saved. –What other information do you know from analyzing the graph?

Coordinate Grid Example: Use the graph to determine how much money Jack makes after working exactly 9 hours.

Meerkat Coordinate Plane Task Greetings from the Kalahari Desert in South Africa! In this activity, you will learn a lot about the Kalahari’s most playful residents: meerkats. The following ordered pairs show the height of a typical meerkat at different times during the first 20 months of life. Graph the corresponding points and see what you can discover about meerkats.

Meerkat Coordinate Plane Task Once you have graphed them all, connect the points in the order they are given to form a line graph.

Meerkat Coordinate Plane Task b.What does the point (0 months, 3 inches) mean for a typical meerkat's height? c.How tall do you think a typical meerkat gets? Why? d.At what age do meerkats reach their full height? How do you know from this graph? e.If this graph were about a human instead of a meerkat, at what age do you think the height would stop getting larger?

Attributes of Shapes

What is a Trapezoid? Niko and Carlos are studying parallelograms and trapezoids. They agree that a parallelogram is a quadrilateral with 2 pairs of parallel sides.

Niko says, “A trapezoid has one pair of parallel sides and a parallelogram has two pairs of parallel sides. So a trapezoid is also a parallelogram.” Carlos says, “No - a trapezoid can have only one pair of parallel sides.” Niko says, “That’s not not true. A trapezoid has at least one pair of parallel sides, but it can also have another.”

What is a Trapezoid? At your tables, discuss the difference between Niko’s definition and Carlos’ definition for a trapezoid.

Niko says, “A trapezoid has one pair of parallel sides and a parallelogram has two pairs of parallel sides. So a trapezoid is also a parallelogram.” Carlos says, “No - a trapezoid can have only one pair of parallel sides.” Niko says, “That’s not not true. A trapezoid has at least one pair of parallel sides, but it can also have another.”

Some people use Niko’s definition for a trapezoid, and some people use Carlos’ definition. Niko: A trapezoid has at least one pair of parallel sides and a parallelogram has two pairs of parallel sides. Carlos: A trapezoid can have only one pair of parallel sides.”

Which statements below go with Niko’s definition? Which statements below go with Carlos’ definition? –All parallelograms are trapezoids. –Some parallelograms are trapezoids. –No parallelograms are trapezoids. –All trapezoids are parallelograms. –Some trapezoids are parallelograms. –No trapezoids are parallelograms.

Which picture represents Nico’s definition? Which picture represents Carlos’ definition?