r i n g s m
TEACHER’S GUIDE
Everyone knows that math and science go hand in hand. But, in my opinion, this already-happy marriage is happiest in the field of engineering. Engineering is the Our First Engineering Issue
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process of creating things to solve problems. The process shares a profound connection with math. Furthermore, engineering is finally getting its due in elementary school classrooms with the Next Generation Science Standards and emphasis on STEM. We felt like there’s no better time to dedicate an issue of DynaMath to engineering. We crafted this issue to show the breadth of the field, from creating probes that explore the reaches of our solar system to writing computer code and renovating roller coasters. We hope the issue inspires your students to think of all the ways they can #makeitwithmath. Join the conversation and share your engineering lesson ideas on Facebook, Twitter, and Pinterest with the #makeitwithmath hashtag.
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All the best, Elizabeth Carney, Editor ECarney@Scholastic.com
DYNAMATH TEACHER’S GUIDE T1
SKILLS AND STANDARDS
SKILL & ARTICLE TITLE
COMMON CORE STATE STANDARD
FEATURE
PAGE
Numbers & Operations — Fractions: Convert fractions to their equivalent decimal notation.
FRACTIONS TO DECIMALS Recycling Roller Coasters
4
Measurement & Data: Relate area and perimeter to the operations of multiplication and addition.
AREA AND PERIMETER Kids Who Code
8
Numbers & Operations in Base 10: Divide multi-digit whole numbers and decimals, using the inverse relationship with multiplication.
DIVISION RELATIONSHIPS Pluto: King of Dwarfs
10
Operations & Algebraic Thinking: Solve problems with whole numbers using the four operations.
SOLVING MULTISTEP PROBLEMS Bot Builders
14
Teaching Support in a Snap on DynaMath Online www.scholastic.com/dynamath Videos
Games
Math News
Video Lessons
Searchable Magazine Archive
Skills Sheets
T2 DYNAMATH TEACHER’S GUIDE
LESSON PLANS
Recycling Roller Coasters FRACTIONS TO DECIMALS PAGE 4
2 4
3 4
4 4
0
1 4
• Which decimals are the same as the fractions on your number line? Discuss, and then write the decimals above the corresponding fractions.
0
0.25
0.50
0.75
1
0 4 4 • Compare with a partner to see which decimals you have represented. Revise your number lines if necessary. • Explain how you know that 2 4 is equal to five tenths. ( 2 4 can be reduced to 1 2 , which is the same as 5 10 , 0.5, or five tenths.) • Discuss with your partner why 3 4 is equal to 0.75, or 2 4 1 4 3 4 seventy-five hundredths. What numbers or values from the real world does this connect to? (possible responses include: money, measurement, my fraction kits, percent) • Project the number line below. Let’s fill in the missing decimal values.
CONTENT STANDARD Number & Operations—Fractions:
OBJECTIVE Students use number lines and benchmark fractions to convert fractions to decimals. OPTIONAL MATERIALS whiteboards and markers LESSON Engagement 1. After students read the article, facilitate a discussion either as a whole group, in small groups, or with a partner: • How do you feel about roller coasters? • If you have ridden one, do you recall if it was wooden, steel, or both? • What type of mathematical concepts do you think go into roller coaster design? Concept Development 2. Facilitate a discussion focusing on equivalent fractions: • Discuss what math tools you can use to show how fractions and decimals are related. (possible responses include: base 10 blocks, number lines, fraction/decimal kits) • Project the number line at the top of the next column. Ask students to discuss the fractions they see and how those fractions relate to each other. Then ask them to draw the number line themselves. Convert fractions to their equivalent decimal notation. MATHEMATICAL PRACTICES STANDARDS 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.
0
0.25
0.50
0.75
1
1 8
2 8
3 8
4 8
5 8
6 8
7 8
8 8
0
• For those of you who got 1 8 = 0.125, explain how you got this value. (Answer could include: Half of 0.25 equals 0.125, just like half of 25 whole units equals 12.5 whole units.) • Now that you know the decimal value for 1 8 , what does this mean for the rest of your number line? (that each segment is 0.125)
0
0.125
0.25 0.375
0.50
0.625
0.75
0.875
1
1 8
2 8
3 8
4 8
5 8
6 8
7 8
8 8
0
• Plot the fractions and corresponding decimals for a number line with increments of 1 10 . Application 3. Have students work on the article’s problems, encouraging them to use number lines and share ideas with each other. Closure 4. Watch the “Record Roller Coasters!” video. Have students write a paragraph about their favorite roller coaster, using the numbers mentioned in the video to explain why.
DYNAMATH TEACHER’S GUIDE T3
LESSON PLANS LE SO LANS
• Project the image below. Let’s say the rectangular figure is a rug. Does it fit in a 20-square-foot bedroom? Explain your answer.
Kids Who Code AREA AND PERIMETER PAGE 8
6 ft
4 ft
4 ft
6 ft • How can area models help you solve for both area and perimeter of the rug? Use a visual model to represent what you mean. • Show me how writing an addition equation helps you solve for area and perimeter. • The examples below are for your reference. Always use student examples when possible: 4 ft + 4 ft + 6 ft + 6 ft = 20 ft (Perimeter) 6 ft + 6 ft + 6 ft + 6 ft = 24 square ft (Area) • Discuss whether or not you agree with the equations. • Now discuss how you can connect your equations to the formulas for perimeter and area. Encourage students to look for patterns and connect them to the formulas. Don’t worry if they don’t all see the connection right away. The more they work with these methods, the more they will develop a solid understanding. P = 2L + 2W (2 x 4 ft) + (2 x 6 ft) = 20 ft A = L x W 6 ft x 4 ft = 24 square ft • What did you write for your answer? The rug’s total area is 24 square feet. It is too large for a 20-square-foot bedroom. Application 3. Have students use area models to solve the problems from the article. Encourage them to compare their models with each other. Closure 4. Pair students together and assign one partner to draw a rectangle. Then have the other partner solve for the area and perimeter using an area model.
OBJECTIVE Students will use equations, formulas, and area models to solve word problems involving area and perimeter. OPTIONAL MATERIALS whiteboards and markers LESSON Engagement 1. After reading the article, ask questions and facilitate discussion either as a whole group, in small groups, or with a partner: • Do any of you know what computer coding is? Can you describe it in one sentence? • Would you be interested in learning how to code? • If you could create a website of your own, what would it be about? What would it look like? Draw a picture. Concept Development 2. Watch the video lesson “Area and Perimeter” as a class. Then facilitate a discussion with the following questions: • What is the difference between area and perimeter? Discuss what you learned from the video, drawing an example if you need to. • If I wanted to find out how much carpet I need to cover my whole bedroom floor, what would I be solving for? If I wanted to find the distance around the edge of the carpet, what would I be solving for? Discuss. CONTENT STANDARD Measurement & Data: Relate area and perimeter to the operations of multiplication and addition. MATHEMATICAL PRACTICES STANDARDS 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 7. Look for and make use of structure.
T4 DYNAMATH TEACHER’S GUIDE
LESSON PLANS
• Project the problem in the box below. Ask: How you would explain the division steps you see? Think to yourself, then explain to a partner.
Pluto: King of Dwarfs DIVISION RELATIONSHIPS PAGE 10
Estimates with Unit Form 176.9 ÷ 29 =
Solution 29 176.9 6.1 29 176.9
180 ones ÷ 30 = 6 ones 30 tenths ÷ 30 = 1 tenth
−174 ones 29
−29 tenths 0
• Now, discuss what you observe about the left side of this solving strategy compared with the right side. (The left side uses estimation and unit form while the right side uses precise calculations and the division algorithm.) • Why did the student take 176.9 and estimate it to 180 ones? Where do you see the ones in the algorithm? ( 180 ones ÷ 30 = 6 ones. Using this answer, the student plugged 6 ones into the algorithm. 6 ones × 29 = 174 ones. This leaves 2 ones and 9 tenths, or 29 tenths. ) • In the second estimate, why is the student is estimating by tenths? Where do you see the tenths in the algorithm? ( 30 tenths ÷ 30 = 1 tenth. The student then placed 1 tenth in the algorithm. 29 tenths × 1 = 29 tenths, leaving a difference of 0. ) • Does our answer make sense? Where did the student decompose the numbers to make solving simpler? (176.9 was decomposed into 180 ones, so that you could evenly divide without regrouping. 29 groups was also rounded to 30 groups.) • Discuss what the standard form of 29 tenths looks like. (2.9) Do you see this in the algorithm? (No, you see 29 tenths.) Application 3. Have students solve the article’s problems. Continually question them so they make connections between estimation/unit form and the standard algorithm. These connections take time and practice. Closure 4. Give students further practice with the “Division Relationships” skills sheet.
CONTENT STANDARD Numbers & Operations in Base 10:
OBJECTIVE Students will deepen their understanding of division using estimation, precise place value vocabulary, and the standard algorithm. OPTIONAL MATERIALS whiteboards and markers LESSON Engagement 1. Before reading the article, ask questions and facilitate discussion either as a whole group, in small groups, or with a partner: • How many of you have heard of a dwarf planet? Discuss what you think one is. • There are dwarf planets throughout our solar system. What do you think scientists use to study them? Concept Development 2. After reading, ask questions and facilitate discussion with the following questions: • When solving division problems, what other operations or strategies could you use to help you find your answer? (possible responses include: multiplication, repeated addition, skip counting) • Have you ever solved division problems using unit form vocabulary? Discuss how using precise unit form vocabulary can help you solve these problems. Perform division with multi-digit whole numbers and decimals, using the inverse relationship with multiplication. MATHEMATICAL PRACTICES STANDARDS 2. Reason abstractly and quantitatively. 6. Attend to precision. 8. Look for and express regularity in repeated reasoning.
DYNAMATH TEACHER’S GUIDE T5
LESSON PLANS LE SO LANS
plan for solving multistep problems. Read over the Homework Helper and compare the class’s ideas with the steps the Homework Helper suggests. Discuss the similarities and differences. • Divide students into groups and ask them to come up with four to five steps that will help them solve multistep word problems. They can format their list as a poster on chart paper. Remind them to use the class suggestions and the Homework Helper as guides. • Hang up the posters from each group. Have students walk around and read each other’s work. • Now, using each group’s poster, ask the class to come up with a consensus for a class strategy for solving multistep problems. Discuss with students what similarities they saw in the charts and what steps they feel are essential for the class list. By having your class create this tool, they will have ownership and a better understanding of the process. • Finalize your class strategy and post it in the room. Later, type up the list and hand out a copy to students for use in class or at home when doing homework. An example is below:
Bot Builders SOLVING MULTISTEP PROBLEMS PAGE 14
CONTENT STANDARD Operations & Algebraic Thinking: Solve problems with whole numbers using the four operations. MATHEMATICAL PRACTICES STANDARDS 1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively. 7. Look for and make use of structure.
OBJECTIVE Students solve multistep problems, assessing how reasonable their answers are using estimation and rounding. MATERIALS chart paper and markers LESSON Engagement 1. First have students read the article. Then, facilitate discussion either as a whole group, in small groups, or with a partner: • How many of you enjoy playing with LEGO ® bricks? Explain what you like to create with LEGO bricks. • Have any of you ever designed or built a LEGO robot? • This article is about kids like you who participated in a LEGO robot competition. What kinds of math and problem-solving skills are needed for this event? Concept Development 2. Begin exploring the math concept by asking the following questions and facilitating a discussion: • When solving a challenging problem with multiple steps, what solving strategies do you use? • Write students’ ideas for everyone to see. (Possible responses include: Read the whole problem, underline, break it apart, circle important information, read it more than once, discuss it with a partner.) • Explain to students that you’re going to create a class
Multistep Problem-Solving Steps
1. Read the problem twice.
2. Underline all essential information.
3. Circle what the problem is asking you to solve for. 4. Highlight or number what you must solve first, second, etc. 5. Solve using model drawings and equations to persevere to the final step.
Application 3. Have students work with a partner to solve problems 1 through 5 from the article using their class strategy. Closure 4. Assign the “Multistep Problems” skills sheet. Encourage students to continue using their class list, but follow up asking if they’re finding it useful. Would they make any changes to it now that they’ve been using it for several problems?
T6 DYNAMATH TEACHER’S GUIDE
DYNAMATH TEACHER’S GUIDE T7
NAME
r i n g s m
DynaDash: Grade 3 Calculating Area
Are you ready to take the DynaDash? See how many area questions you can answer in 60 seconds. You’re not expected to finish them all—just complete as many as you can. Ready? Go!
Express the area of each shape in square units. Each small square ( ☐ ) has an area of 1 square unit. Fill in the missing lines if you need to.
Complete the models or calculate the missing value.
1.
6.
11. Draw a rectangle that has a length of 7 units and an area of 14 square units.
16.
2 in
A = ? 7 in
Area =
square units
square units
Area =
Area =
8 in
2.
7.
12. Draw a square that has an area of 25 square units.
17.
A = ?
9 in
square units
Area =
Area =
square units
Area =
8.
13. Draw a rectangle with a length of 8 units and a width of 6 units.
18.
3.
9 m
A = ? 20 m
Area =
square units
Area =
square units
Area =
4.
9.
14. Draw a rectangle with a width of 5 units and an area of 35 square units.
19.
9 in
A = ?
3 in
Area =
square units
Area =
square units
Area =
10.
15.
20.
5.
5 in
12 cm
8 cm
A = ?
A = ? 18 in
Area =
Area =
square units
Area =
square units
Area =
T8 DYNAMATH TEACHER’S GUIDE
NAME
r i n g s m
DynaDash: Grade 4 Calculating Area and Perimeter
Are you ready to take the DynaDash? See how many area and perimeter questions you can answer in 60 seconds. You’re not expected to finish them all—just complete as many as you can. Ready? Go!
Calculate the perimeter and area of the rectangles below.
Find the missing value for each shape.
5 units
A = 21 sq units
3 units
7 units
1. P = 2. A =
units
11. w =
units
square units
6 units
A = 128 sq units
8 units
4 units
3. P = 4. A =
units
12. l =
units
square units
3 units
P = 60 units
15 units
13 units
5. P = 6. A =
units
13. l =
units
square units
11 units
P = 72 units
7 units
27 units
7. P = 8. A =
units
14. w =
units
square units
9 units
P = 60 units
6 units
7 units
9. P = 10. A =
units
15. Area =
square units
square units
DYNAMATH TEACHER’S GUIDE T9
NAME
r i n g s m
DynaDash: Grade 5 Calculating Volume
Are you ready to take the DynaDash? See how many volume questions you can answer in 60 seconds. You’re not expected to finish them all—just complete as many as you can. Ready? Go!
Calculate the volume of each rectangular prism. For questions 1 to 4, each small cube equals 1 cubic unit.
Calculate the volume of each rectangular prism.
A = 25 units
5 units
2 units
3 units
6 units
1. V =
6. V =
cubic units
cubic units
7 units
12 units
2 units
8 units
4 units
4 units
2. V =
7. V =
cubic units
cubic units
4 units
A = 42 units
4 units
6 units
4 units
3. V =
8. V =
cubic units
cubic units
11 units
7 units
5 units
2 units
10 units
6 units
4. V =
9. V =
cubic units
cubic units
A = 27 units
6 units
3 units
2 units
10 units
3 units
5. V =
10. V =
cubic units
cubic units
T10 DYNAMATH TEACHER’S GUIDE
ANSWER KEY
T7: PROBLEM OF THE DAY 1. $3.54 2. 0.6
11.
12.
13.
3. 270 grams 4. 4,223 5. P = l + l + w + w A = l x w 6. 3 4 = 6 8 Regular octagon 7.
14.
15. 96 square centimeters 16. 14 square inches 17. 72 square inches 18. 180 square meters 19. 27 square inches 20. 90 square inches
8. 309 × 2 9. l = 192 units 10. 214 + 2(65) = 344 feet 11. 7 × 4 = t 12. 0 1 13. 3.13 inches longer 14. Subtraction Division 1 2
1 1
2
PAGE T9: DYNADASH: GRADE 4 1. P = 16 units 2. A = 15 square units 3. P = 20 units 4. A = 24 square units 5. P = 36 units 6. A = 45 square units 7. P = 36 units 8. A = 77 square units 9. P = 32 units 10. A = 63 square units
Multiplication Addition
15. 7 8 16. Obtuse angles are > 90° 875 1,000 =
Acute angles are < 90° Right angles are = 90°
17. 30.1 tiddlywinks 18. About $0.30 19. 1 2 20. The perimeter of a square is greater.
11. w = 3 units 12. l = 16 units 13. l = 17 units 14. w = 9 units 15. A = 144 square units
PAGE T8: DYNADASH: GRADE 3 1. 2 square units
2. 6 square units 3. 4 square units 4. 10 square units 5. 5 square units 6. 12 square units 7. 12 square units 8. 16 square units 9. 20 square units 10. 20 square units
PAGE T10: DYNADASH: GRADE 5 1. 90 cubic units
2. 56 cubic units 3. 64 cubic units 4. 350 cubic units 5. 36 cubic units 6. 50 cubic units 7. 384 cubic units 8. 252 cubic units 9. 132 cubic units 10. 270 cubic units
DYNAMATH TEACHER’S GUIDE T11
ANSWER KEY
PAGES 2-3: NUMBERS IN THE NEWS
PAGES 10-13: PLUTO: KING OF DWARFS 1a. D 1b. 2 states of Kansas 2a. A and C 2b. 113 islands of Manhattan 3a. B 3b. 24.5 times larger
I Heart My Robo-Pet: $570 Motorcycle Surfer: 1.5 hours Record-breaking Feet: Answers will vary.
PAGES 4-7: RECYCLING ROLLER COASTERS
1a. 2 1 1b. 0.5 2. 0.2 3. 0.25 4. 0.4 5. 0.1
4a. B and D 4b. 17 days 5. 310 Earth years
PAGES 14-15: BOT BUILDERS 1a. 2 x 30 points = 60 points 1b. 20 points 1c. 60 + 20 = 80 points 2a. 3 x 7 = 21 points 2b. 2 x 3 = 6 points 2c. 21 + 6 = 27 points 3. 2(60) + 40 + 4(7) = 188 points 4. 10(3) – 8 = 22 points 5. 3(20) + 6(7) – 2(8) = 86 points
PAGES 8-9: KIDS WHO CODE 1a. l = 1 inch w = 4 inches 1b. P = 2(1) + 2(4) = 10 inches 2. A = 1.5 inches x 1.5 inches = 2.25 square inches 3a. w = 1.25 x 72 pixels = 90 pixels l = 0.25 x 72 pixels = 18 pixels 3b. P = 2(90) + 2(18) = 216 pixels 3c. A = 90 pixels x 18 pixels = 1,620 sq. pixels 4a. Area of box A = 1.5 inches x 0.5 inches = 0.75 square inches = 0.8 square inches 4b. Area of box B = 1 inch x 1.25 inches = 1.25 square inches 5. l = 1.25 x 72 pixels = 90 pixels
PAGE 16: THE PUZZLE PAGE MATH MASTER 1. A 2. B 3. D 4. B 5. C MATH BLOOPER: Floppy did not multiply 7 x 2 when calculating perimeter. The correct equation is: P = 2(15) + 2(7) = 44 tulips BRAINTEASER: 20 students
w = 1 x 72 pixels = 72 pixels P = 2(90) + 2(72) = 324 pixels
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SZDYMTCOCN
T12 DYNAMATH TEACHER’S GUIDE
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