CoreConnects Mathematics Sample Pages - Grades 3-5

Professional Development for College and Career Readiness

Teaching a deep understanding of math content and how to use math in the real world

Teacher Resource

Manual

Grades3-5

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©2014 This information is confidential and proprietary to Catapult Learning™. For internal distribution only.

CoreConnects: Mathematics

Teaching Math Skills to Achieve Common Core Outcomes

Levels 3-5

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This information is confidential and proprietary to Catapult Learning™. It is for internal use and distribution only.

Distribution of this document beyond employees of Catapult Learning™ is strictly prohibited.

External Distribution: In the event that any proprietary or confidential information is disclosed, intentionally or otherwise to a School District/Schools, its employees, agents or assigns, the School District/Schools agrees to hold same in strictest confidence and not to disclose same to any other person for any reasons nor utilize same within the School District or Schools without prior written approval by Catapult Learning.

The School District/Schools further agree to use all efforts at its disposal to assure that its employees, agents or assigns are aware of the confidential and proprietary nature of the subject matter, and do not disclose same to any other person for any reasons nor utilize same without prior written approval by Catapult. The School District/Schools acknowledges that unauthorized disclosure of Catapult’s proprietary and confidential information may cause Catapult irreparable harm and may entitle Catapult to injunctive relief in a court of competent jurisdiction.

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CoreConnects: Mathematics 3 ©2014 This information is confidential and proprietary to Catapult Learning™. For internal distribution only.

Table of Contents

Overview

Common Core General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Building Performance Character Traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

How to use the Teaching Math Skills sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Recommended Manipulatives and Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Skills Sheets at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Standards for Mathematical Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Directed Math Activity Format/DMA Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Sequence of TMS 6-8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Skills Sheets

3.OA.A – Represent and solve problems involving multiplication and division . . . . . . . . . . . . . . . 39

3.OA.B – Understand properties of multiplication and the relationship

between multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.OA.C – Multiply and divide within 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.OA.D – Solve problems involving the four operations,

and identify and explain patterns in arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.NBT.A – Use place value understanding and properties

of operations to perform multi-digit arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.NF.A – Develop understanding of fractions as numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.MD.A – Solve problems involving measurement and estimation

of intervals of time, liquid volumes, and masses of objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.MD.B – Represent and interpret data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.MD.C – Geometric measurement: understand concepts of area

and relate area to multiplication and to addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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CoreConnects: Mathematics – Teaching Math Skills Levels 3-5 – Table of Contents

3.MD.D – Geometric measurement: recognize perimeter as an attribute of

plane figures and distinguish between linear and area measures . . . . . . . . . . . . . . . . . . . . . . 71

3.G.A – Reason with shapes and their attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.OA.A – Use the four operations with whole numbers to solve problems . . . . . . . . . . . . . . . . . . 79

4.OA.B – Gain familiarity with factors and multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.OA.C – Generate and analyze patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.NBT.A – Generalize place value understanding for multi-digit whole numbers . . . . . . . . . . . . . 89

4.NBT.B – Use place value understanding and properties of

operations to perform multi-digit arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.NF.A – Extend understanding of fraction equivalence and ordering . . . . . . . . . . . . . . . . . . . . . . 97

4.NF.B – Build fractions from unit fractions by applying and

extending previous understandings of operations on whole numbers . . . . . . . . . . . . . . . . . . . 101

4.NF.C – Understand decimal notation for fractions, and compare decimal fractions . . . . . . . . . . 105

4.MD.A – Solve problems involving measurement and conversion

of measurements from a larger unit to a smaller unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109


4.MD.C – Geometric measurement: understand concepts of angle and measure angles . . . . . . . 117

4.G.A – Reason with shapes and their attributes:

Figures with perpendicular and parallel lines, symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.OA.A – Write and interpret numerical expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.OA.B – Analyze patterns and relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.NBT.A – Understand the place value system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.NBT.B – Perform operations with multi-digit whole numbers

and with decimals to hundredths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.NF.A – Use equivalent fractions as a strategy to add and subtract fractions . . . . . . . . . . . . . . . 141

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5.NF.B – Apply and extend previous understandings of multiplication

and division to multiply and divide fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.MD.A – Convert like measurement units within a given measurement system . . . . . . . . . . . . . 149


5.MD.C – Geometric measurement: understand concepts of volume

and relate volume to multiplication and to addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.G.A – Graph points on the coordinate plane to solve real-world

and mathematical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.G.B – Classify two-dimensional figures into categories based on their properties . . . . . . . . . . 165

Performance Tasks

3-5 Performance Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Performance Task Test Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Performance Task Rubric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Performance Task Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Level 3 (1) What’s In Your Garden? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Level 3 (2) Picture Perfect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Level 3 (3) The Class Pet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Level 4 (1) Snack Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Level 4 (2) A Penny Found is a Penny Earned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Level 4 (3) Do You Have the Time? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Level 5 (1) The Cost of Lunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Level 5 (2) Class Pride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Level 5 (3) Thirst Quencher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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How to Use the Teaching Math Skills Sheets

The Teaching Math Skills sheets are designed to assist teachers in approaching Common Core Standards in the classroom. Each TMS Skills Sheet addresses a Common Core cluster within each domain of the Common Core State Standards for Math.

TMS Sheets are not intended to be used only once. Many of them cover many standards and can and should be used until the teacher feels that those standards have been addressed to sufficiently meet the needs of the students s/he is teaching. Although the Standards for Mathematical Practice are not outlined within these lesson suggestions, teachers should continue to provide opportunities for these practices for students.

An example for the top of a TMS Skills Sheet for Level 2 follows. Note the following:

• The objective which is the cluster, is indicated.

• The prerequisite standards for the skill indicate what is expected to be known at the end of the level prior to the current level.

• The standards for the current grade-level skills are indicated.

» If the current cluster is considered a Major Cluster, it will be indicated with a * in the objective.

» If the current cluster contains a Required Fluency, that standard will be indicated with a ◊.

• Students that show mastery of the current cluster can be challenged to explore the growth cluster.

Performance Task: Exploring Problem Solving Strategies

The skills sheets will each be set up with the following sections: Review and Pre-assessment, Instruction, Scaffolding, and Evidence of Learning.

Review and Pre-assessment

This section provides a set of review and pre-assessment questions. As noted above, they will align to the pre-requisite skills required for the current cluster. Students should have sufficient knowledge of these review items in order to continue with instruction at this level. If they do not, the teacher will refer to the pre-requisite cluster, and start instruction with that skills sheet.

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CoreConnects: Mathematics – How to Use the Teaching Math Skills Sheets

Instruction

This section will present several instructional activities that pertain to the current level cluster. They are bulleted activities that follow a Directed Math Activity Format. Within the instruction, specific Practice Standards have been indicated. Teachers should be looking for student engagement in these areas. Each instructional activity is designed to cover one or two class sessions, depending on session length.

Scaffolding

It is rare that all students will be working at the exact same pace and at the exact same level. A few options are provided to give either additional support or an additional challenge to those students who may need it.

Evidence of Learning

Once an adequate amount of time has been spent on the standards and cluster, the teacher will need to look for evidence of learning. At the end of each skills sheet, there will be a bulleted list of what is expected of students upon mastery of this cluster.

These skills sheets have been provided as a guide for instruction. Teachers are encouraged to supplement with additional activities that align to the standards indicated.

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Teaching Math Skills 3.MD.D3.M

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Objective: Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures

Pre-Requisite

Relate addition and subtraction to length

(2.MD.B.5, 2.MD.B.6)

Performance Task: The School Olympics

Current

Geometric measurement: recognize perimeter as an

attribute of plane figures and distinguish between linear and

area measures

(3.MD.D.8)

Performance Task: Picture Perfect

Growth

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

to a smaller unit

(4.MD.A.1, 4.MD.A.2, 4.MD.A.3)

Performance Task: A Penny Found is a Penny Earned

Review & Pre-AssessmentHave students find the area of the following rectangles using multiplication, images, or manipulatives:

• (12)

• (20)

• (40)

• (54)

• (8)

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3.MD.D Teaching Math Skills3.

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Ask students to measure the following items with a ruler (answers will vary):

• Their pencil

• Each edge of their text book

• The top of a sheet of paper

• The edge of their desk

InstructionUNDERSTANDING PERIMETER:

• Tell students that you would like to plant a garden. In order to keep the animals out, you need to put up a fence. The fence will go around the outside of the garden. Have students create a garden on a Geoboard or sheet of graph paper. Explain that one box equals one foot. Ask students how much fencing they would need to buy. (Answers will vary.) Explain that we call the measurement around something the perimeter. Ask students if they can think of another situation in which they would need to find the perimeter of a shape. (PS.2, PS.3)

• Have students find the perimeter of several differently shaped images. Challenge students to use a ruler or tape measure to find the perimeter of things in the room (e.g., the room, door, desk, poster, book, etc.). Discuss why you may need to find the perimeter of those objects. (PS.3, PS.5, PS.6)

• Have students create several shapes on a sheet of graph paper and identify the perimeter. Have students exchange images and find the perimeters. Discuss the results. Do they agree on the perimeters? Why or why not? (PS.3, PS.5, PS.6, PS.7)

FINDING THE MISSING LENGTH:

• Distribute base ten blocks (rods and units) to students. Explain that in order to find missing side lengths they are going to use base ten blocks. Each unit is 1 cm in length. Each rod is 10 cm in length. Remind students that 1 unit = 1 cm and 1 rod = 10 cm. Draw a triangle. Provide students with the perimeter and the lengths of two of the sides. Use base ten blocks to represent the known sides. For example, if a side is 15 cm, it would be represented with one rod and five units. If the other known side is 10 cm, it would be represented with one rod. Show students that the total is 25 cm. Ask students how many more centimeters are needed to create the given perimeter. (40 cm.) Allow students time to explore the problem with a partner and base ten blocks. Reconvene and discuss the strategies used. (PS.2, PS.5, PS.6)

• Write the corresponding number sentence on the board (e.g.,15 + 10 + ? = 40). Work through the steps to solve the problem with the students. Repeat the process with another example. (PS.1, PS.2)

• Pair up students and have them continue to use the base ten blocks to identify the missing side when given the perimeter and 2 sides of a triangle for 3-5 additional problems. Challenge students to write the corresponding number sentence for each example. Discuss the results. (PS.3, PS.5, PS.6)

• Challenge students to create 3-5 problems (and answers) to exchange with a classmate. Discuss the solutions. (PS.3, PS.5, PS.6, PS.7)

• Extend the activity to find the missing side of a quadrilateral given 3 sides and the perimeter. (PS.5, PS.6, PS.7)

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Teaching Math Skills 3.MD.D3.M

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EXPLORING RECTANGLES WITH THE SAME PERIMETER AND DIFFERENT AREAS:

• Review the concepts of area and perimeter. Explain to students that you have 12 feet of fencing and would like to create a rectangular garden. Have students create a garden with a perimeter of 12 feet on a sheet of graph paper or Geoboard. Challenge students to find the area of their garden. (Answers will vary.) Discuss the results. Not all of the areas are the same. Ask students: Is this correct? Why or why not? How do you know? Explain that rectangles can have the same perimeter but a different area. (PS.2, PS.5, PS.7)

• Provide students with 1-3 perimeters and challenge them to find all of the areas. Have students organize their information in a chart and draw a picture or model the rectangles with manipulatives. Discuss the results. Did students find all of the possible areas? How do they know? (PS.3, PS.5, PS.6, PS.7)

• Challenge students to work independently to find all of the areas for a given perimeter. (PS.5, PS.6, PS.7)

Scaffolding

Evidence of Learning• Students will be able to solve real-world mathematical problems involving perimeters of polygons.

• Given the perimeter of a polygon and all but one length, students will be able to identify the length of the unknown side.

• Students will be able to create rectangles with the same perimeter and different area.

Additional Support

• Allow struggling learners to work with a partner

• Allow students to use manipulatives such as: counters, tokens, or base ten blocks when solving problems

• Have students focus on finding the perimeter of a shape by counting units

• Have students identify perimeters in the classroom

Additional Challenge or Rigor

• Pair students with a struggling learner for peer tutoring

• Challenge students to complete similar problems with larger perimeters

• Have students create their own problems and answers to trade with a classmate

• Have students identify and explain a real-life application of perimeter and area

• Challenge students to maximize the area for a given perimeter and explain why this may be usefulSAMPLE


3.MD.D Teaching Math Skills3.

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Level 3 – Picture PerfectLevel 3

You want to use a piece of wood to build a rectangular picture frame. You have a piece of wood that is 48 inches long and 2 inches wide.

Part ACreate a picture frame using graph paper (Note: allow each unit to represent one inch). Make sure not to waste any wood!

• Did you use all of the wood to create your picture frame? How do you know?

• What is the perimeter of your frame? How does this compare to your classmates’ frames?

Create another picture frame using the same amount of wood, but with a different length and width.

• How do the perimeters of the two frames compare?

Part BLooking at your first picture frame, what is the area of a picture that you can place in that frame?

Now, look at your second frame. What is the area of the picture that would fit inside this frame? How does this compare to the area of the picture in the first frame?

Part CWhich shape of frame that you created would be a better choice for framing a picture? Explain.

Why do you think it is possible to create different areas and perimeters when you started with the same length of wood in the beginning of each?

Can you create a new frame that has the same inside area as the frame you just chose, but with a different perimeter? (You are not limited to the original piece of wood given.)

• How can rectangles with different perimeters have the same area? Can you think of another time that this may happen?

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CoreConnects: Mathematics – Level 3 – Picture PerfectLe

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Answer Key/Teacher Guide

Common Core Standard Assessed Skill3.MD.8 Solve real world and mathematical problems involving perimeters of polygons,

including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

You want to use a piece of wood to build a rectangular picture frame. You have a piece of wood that is 48 inches long and 2 inches wide.

Part ACreate a picture frame using graph paper (Note: allow each unit to represent one inch). Make sure not to waste any wood!

• Did you use all of the wood to create your picture frame? How do you know?

• What is the perimeter of your frame? How does this compare to your classmates’?

Teacher: Students will need to take the wood that is 48 inches long and 2 inches wide, and create a frame using all of that wood by drawing or cutting paper to show. Here are a few examples of possible frames. Encourage students to be creative and find other possible ways of cutting the wood.

16 in

8 in 8 in

16 in

14 in

10 in 10 in

14 in

30 in

18 in

18 in

30 in

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CoreConnects: Mathematics – Level 3 – Picture PerfectLevel 3

Create another picture frame using the same amount of wood but with a different length and width.

• How do the perimeters of the two frames compare?

Teacher: Perimeters should be the same for all 2 inch wide frames, 56 inch perimeter. Perimeters for 1 inch wide frames should equal 100 inches. The perimeters will depend on how the students cut the wood.

Part BLooking at your first picture frame, what is the area of a picture that you can place in that frame?

Teacher: Students will need to determine that the area of the picture is equal to the inside area of the frame. Using the examples above 1) 12 x 8 = 96 in2 2) 10 x 10 = 100 in2 3) 28 x 18 = 504 in2

Now, look at your second frame. What is the area of the picture that would fit inside this frame? How does this compare to the area of the picture in the first frame?

Teacher: Remind students that they need to use the lengths of the interior sides of the frame to find the area where the picture would fit. Have students compare both areas.

Part CWhich shape of frame that you created would be a better choice for framing a picture? Explain.

Teacher: Guide students to determine what a better choice for a picture size would be. Do they know standard picture sizes? Are there typical shapes for pictures?

Why do you think it is possible to create different areas and perimeters when you started with the same length of wood in the beginning of each?

Teacher: By cutting the wood in different ways, you can create different rectangles with lengths and widths of different sizes.

Can you create a new frame that has the same inside area as the frame you just chose, but with a different perimeter? (You are not limited to the original piece of wood given.)

Teacher: Students will need to use the area they found from their chosen picture frame, and try to create a new frame with that same inside area but with a different perimeter. For example, if they chose the frame with an interior area of 100 in2 , they could create a new frame with interior lengths of 20 and 5. Depending on the width of the wood they are using, they would determine the outside perimeter. The example below shows an interior area of 100 in2 , and a perimeter of 24 +9 + 24 + 9 = 66 in.

• How can rectangles with different perimeters 24 in

5 in 5 in

24 in

have the same area? Can you think of another time that this may happen?

Teacher: Bedrooms in a house may have the same area, but have different lengths and widths. Discuss additional examples.

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CoreConnects: Mathematics – Level 3 – Picture PerfectLe

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CoreConnects Mathematics Sample Pages - Grades 3-5