vacaville usd february 19, 2015. agenda problem solving – where are the cookies? estimating and...

SIXTH GRADE

CCSS-Math

Vacaville USD

February 19, 2015

AGENDA• Problem Solving – Where are the

Cookies?• Estimating and Measurement• Geometry

– Area of parallelograms, triangles and trapezoids

– Volume of rectangular prisms

Others???

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

http://teresaemmert.weebly.com/middlehigh-school-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

Ratios and Measurement

RP.3d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Measurement

• Is a foot larger or smaller than a yard?

• So suppose I tell you I have 9 feet but I need to know how many yards that is.– 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

1 yard

3 feet

ydft

13

ydft

__9

Measurement

1 yard

3 feet

• 9 yards = ____ feet

ydft

13

ydft

9__

Measurement

• 9 yards = ____ feet

1 yard

3 feet

ftyd31

ftyd

__9

Measurement

• So what do we know about meters and centimeters?

1 meter

100 centimeters

Measurement

• 50 m = ____ cm

• 70 cm = ____ m

1 meter

100 centimeters

Measurement

• 50 m = ____ cm

• 70 cm = ____ m

1 m

100 cm

www.estimation180.com

http://www.estimation180.com/

How many green marshmallows will fit on the skewer?

How many green marshmallows are inside the glass?

http://www.estimation180.com/day-117.html

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?


A Brief Discussion of Division

Dividing Decimals

• Why do we “move” the decimal before we divide?

• Why are we “allowed” to move the decimal (mathematically speaking)?

59.76.0

Dividing Fractions

• Why do we “invert and multiply” instead of dividing?

• Why are we “allowed” to multiply by the reciprocal (mathematically speaking)?

43

85

G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Unpacking the Standard

Students continue to understand that area is the number of squares needed to cover a plane figure. Students should know the formulas for rectangles and triangles. “Knowing the formula” does not mean memorization of the formula. To “know” means to have an understanding of why the formula works and how the formula relates to the measure (area) and the figure. This understanding should be for all students.

G.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Why a rectangle?

Parallelograms

• Can you find a way to turn your parallelograms into rectangles?– Scissors, tape, grid paper

• How can you use what you just discovered to develop a formula for finding the area of a parallelogram (without having to cut it up)?

Parallelograms

Parallelograms 1

NCTM Illuminations• http

://illuminations.nctm.org/Lesson.aspx?id=1882– Rectangles and Parallelograms– Quadrilateral Area Record Sheet

http://illuminations.nctm.org/Lesson.aspx?id=1882



Rectangle A• Have students cut from the lower left

corner to a point on the top edge that is three units in from the upper left vertex;

• This cut will form a 45-degree angle, which divides each of the squares through which it passes exactly in half.

Rectangles B and C • Have students remove a triangle by cutting

from the lower left corner diagonally to any point along the top edge.

• Encourage students to choose a point along the top where the edge and one of the grid lines meet.

• Students should then place the removed triangle at the other end of the rectangle.

• Encourage students to make a different cut than other members of their group.

Rectangles D and E • Have students remove the right triangle on

either the right or left side, and move it to the other side.

• Students should realize that this modification changes the parallelogram to a rectangle with the same area.

Parallelograms 2

Triangles

• Can you find a way to turn your triangle (or triangles) into a rectangle or a parallelogram?– Scissors, tape, grid paper

• How can you use what you just discovered to develop a formula for finding the area of a triangle (without having to cut it up)?

Triangles

Triangles – Version 1

Triangles 1

NCTM Illuminations• http

://illuminations.nctm.org/Lesson.aspx?id=1874– Squares and Rectangles Activity Sheet




Triangles 1

Shapes A, B, and C• Using rulers, draw one diagonal in each of

shapes • Cut each shape into two parts along the

diagonal• Estimate the area of each triangle formed

from each shape.

Shape D• Pick a point along the top of the rectangle• Using rulers, draw a line connecting the

bottom corners of the rectangle to the selected point on the top

• Cut each shape into 3parts along the lines• Estimate the area of the triangles formed• How does the area of the 2 small triangles

relate to the area of the large triangle?

Triangles 2

Trapezoids

• Can you find a way to turn your trapezoid (or trapezoids) into rectangles, parallelograms, or triangles?– Scissors, tape, grid paper

• How can you use what you just discovered to develop a formula for finding the area of a trapezoid (without having to cut it up)?

Trapezoids – Version 1

Trapezoids 1

NCTM Illuminations• http://

illuminations.nctm.org/Lesson.aspx?id=1893– Centimeter Grid Paper– Trapezoids




• Ask students to suggest methods for finding the area of the trapezoid. –Prompt: "What other shapes could you

use to help you? Are there any shapes for which you already know how to find the area?"

• After students have had some time to discuss suggestions in their groups, ask them to share their ideas.

Trapezoids 1

Trapezoids 2

Volume

G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Volume Example:

V = base area x height V = x

V = 60 cubic cm15 4

Volume Example:

V = length x width x height V = x x

V = 60 cubic cm5 3 4

Formative Assessment Lesson

• This is a 5th Grade Formative Assessment Lesson on developing and understanding 2 formulas for finding the area of a rectangular prism:

V = base area x height

V = length x width x height


Mathematical goals• 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.


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?


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.


• 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


• 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

Volume Example:

V = base area x height V = x

V = 88.2 cubic cm17.64 5

4.2 cm

4.2 cm

5 cm

Volume Example:

4.2 cm

4.2 cm

5 cm

V = length x width x height V = 4.2 x 4.2 x 5

V = 88.2 cubic cm

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?

Optimizing: Packing It In

FAL – The Shell Center

MATHEMATICAL GOALS• To help students to:

– Reason precisely and defend their conclusions.

– Use mathematics to model a scenario concerning volume.

Introduction

The lesson unit is structured in the following way:• Before the lesson, students attempt the

task Packing It In individually. You review their responses and formulate questions that will help them improve their work.

• At the start of the lesson, students read your comments and consider ways to improve their work.

Introduction• In pairs or threes, students work together

to develop a better solution, producing a poster to show their conclusions and their reasoning.

• Then, in the same small groups, students look at some sample student work showing different approaches to the problem. They evaluate the strategies used and seek to improve the arguments given.

Introduction• In a whole-class discussion, students

compare different solution methods.• Finally, students reflect individually on their

learning.

Sample Work

Analyze the sample student work• What common errors or

misunderstandings do you see?• If these responses were typical of your

class, what prompts or questions would you use to move the class forward?

vacaville usd february 19, 2015. agenda problem solving – where are the cookies? estimating and...

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